COURSE FILEkgr.ac.in/wp-content/uploads/2019/12/DSP.pdf · 2019-12-02 · 2. To familiarize the ......

COURSE FILE Subject (Name) : Digital signal processing

Name (of the Faculty Member) : P.USHA

Designation : Asst. Prof.

Regulation /Course Code : R 16/ EE721PE

Year / Semester : IV/ I

Department : ECE

Academic Year : 2019-20

DEPARTMENT OF ELECTRICAL AND ELECTRONICS

ENGINEERING

R-16

DIGITAL SIGNAL PROCESSING

IV B. Tech

2019-20

Course File Prepared by

P.USHA

Assistant Professor

DIGITAL SIGNAL PROCESSING(EE721PE)

P.USHA, Assistant. Professor,dept of ECE,KGRCET

3) Course Objectives, Course Outcomes and Topic Outcomes

a) Course Objectives

1. To provide background and fundamental material for the analysis and processing of digital

signals.

2. To familiarize the relationships between continuous time and discrete time signals and systems.

3. To study fundamentals of time, frequency and Z-plane analysis and to discuss the inter-

relationships of these analytic method.

4. To study the designs and structures of digital (IIR and FIR) filters from analysis to synthesis for a

given specifications.

5. The impetus is to introduce a few real-world signal processing applications.

6. To acquaint in FFT algorithms, Multi-rate signal processing technique and finite word length

effects.

b) Course Outcomes

1. Construct time, frequency and Z -transform analysis on signals and systems.

2. Compare the inter-relationship between DFT and various transforms.

3. Describe the significance of various filter structures.

4. Design a digital filter for a given specification.

5. Identify the tradeoffs between normal and multi rate DSP techniques and finite length word effects.



c) Topic outcomes

S.No Topic Topic outcome At the end of the topic the

student will be able to

1 UNIT-1 Introduction to DSP Explain a digital processing system.

2 Discrete time signals and sequences Compare discrete time signal and sequence.

3 Linear shift invariant systems Describe a LTI system.

4 problems

5 Stability and Casuality Describe stability and causality of

system.

6 Linear constant coefficient difference

equation

Solve the difference equations.

7 Frequency domain representation of DTS Demonstrate the frequency domain

representation of DTS.

8 Application of Z transform List the applications of Z transform.

9 Solution of difference equation of digital filters

Solve the difference equation.

10 System function, Explain stability criterion and its

frequency response.

11 Stability criterion Explain stability criterion and its frequency response.

12 Frequency response of stable systems Explain stability criterion and its

frequency response.

13 Realization of digital filters-direct and canonic form

Design a digital filter using different

methods.

14 Cascade and parallel form Design a digital filter using different

methods.

15 problems Summarize the important concepts of

this unit. 16 Revision, Discussion of previous question

papers

17 UNIT II-Discrete Fourier Series: DFS

representation

Describe the representation of DFS.

18 Properties of DFS list the properties of DFS

19 DFT and Properties of DFT list the properties of DFT

20 Linear convolution using DFT Solve linear convolution using

different methods.

21 Overlap Add method Solve linear convolution using different

methods.

22 Overlap Save method Solve linear convolution using different

methods.

23 Relation between DTFT and DFS Analyze the relation between different

transforms.



24 Relation between DFT and Z transform Analyze the relation between different

transforms.

25 FFT Explain FFT.

26 Radix-2 decimation in time FFT algorithm Select an appropriate algorithm to

perform FFT.

27 Radix-2 decimation in frequency Select an appropriate algorithm to

perform FFT.

28 Inverse FFT describe Inverse FFT

29 FFT with general radix N describe FFT with general radix N.

30 Problems on overlap add method and

save method.

Solve the problems on overla add and

save method

31 Problems on DIT algorithm Solve the DIT algorithm

32 Problems on DIF algorithm Solve the DIF algorithm

33 Revision, Discussion of previous question

papers

revision

34 UNIT III- IIR Digital Filters Explain IIR digital filters.

35 Analog filter approximations List the steps to design analog filter.

36 Butterworth filter Compare Butterworth and Chebyshev filters. 37 Chebyshev filters

38 Design of IIR filters from Analog filters Design IIR filters using different

techniques.

39 Step Technique Design IIR filters using different techniques.

40 Impulse invariant technique Design IIR filters using different

techniques.

41 Bilinear transformation method Design IIR filters using different techniques.

42 Spectral transformation Design IIR filters using different

techniques.

43 Conversion of low pass to other filters Convert IIR filters from low pass designing.

44 Problems on approximations of different

filters.

Solve the problems and summarize the

important concepts of this unit.

45 Revision

46 Discussion of previous question papers

47 UNIT IV-FIR Digital filters Explain characteristics of FIR filters

and its frequency response. 48 Frequency response

49 Design of FIR filters using fourier method Design FIR filters using various

techniques.

50 Digital filters using window technique:

Frequency sampling technique

Deisgn FIR filter using various

techniques.

51 Comparison of FIR and IIR filters Compare FIR and IIR filters.

52 Problems on Design of FIR filter Summarize the important concepts of

this unit. 53 Revision

54 Discussion of previous question papers

55 UNIT V-Multirate digital signal processing Explain multirate DSP and its

applications.



56 Downsampling Compare up sampling and down

sampling.

57 Decimation Compare up sampling and down sampling.

58 Upsampling Compare up sampling and down

sampling.

59 Interpolation Compare up sampling and down sampling.

60 Sampling rate conversion Explain sampling rate conversion.

61 Finite word length effects Discuss different finite word length

effects.

62 Limit cycle Discuss different finite word length

effects.

63 Overflow oscillations Discuss different finite word length

effects.

64 Round off noise Describe round off noise.

65 Computational output round off noise

66 Methods to prevent overflow List different methods to prevent

overflow.

67 Trade off between round off and overflow

noise

Discuss different finite word length

effects.

68 Dead band effects Explain dead band effects.

69 Revision Summarize the important concepts of this unit. 70 Discussion of previous question papers



4) COURSE PRE–REQUISITES

1. Basics of mathematics

2. Different types of signals and systems

3. Difference between continuous and discrete signals



5) CO’s, PO’s mapping

CO PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12

CO1 3 3 1 3 3

CO2 3 3 1 3 3

CO3 3 3 3

CO4 3 3 3

CO5 2 2 1 2 1

Legends: 1 – Low

2 – Medium

3 – High



5.COURSE INFORMATION SHEET

5.a). COURSE DESCRIPTION:

PROGRAMME: B. Tech. (Electrical and Electronics

Engineering.)

DEGREE: BTECH

COURSE: DIGITAL SIGNAL PROCESSING YEAR: IV SEM: I CREDITS: 4

COURSE CODE: EE721PE

REGULATION: R16

COURSE TYPE: ELECTIVE

COURSE AREA/DOMAIN: Signal processing CONTACT HOURS: 4+1 (L+T)) hours/Week.

CORRESPONDING LAB COURSE CODE (IF ANY):NA LAB COURSE NAME: NA

5.b). SYLLABUS:

Unit Details Hours

I

Introduction: Introduction to Digital Signal Processing: Discrete Time signals &

Sequences, Linear Shift Invariant Systems, Stability, and Causality, Linear Constant

Coefficient Difference Equations, Frequency Domain Representation of Discrete Time Signals

and Systems

Realization of Digital Filters: Applications of Z -Transforms, Solution Difference Equations

of Digital Filters, System Function, Stability Criterion, frequency Response of Stable Systems,

Realisation of Digital Filters-Direct,Canonic,Cascade and Parallel Forms.

13

II

Discrete Fourier series: DFS Representation of Periodic Sequences, properties of Discrete Fourier

Series, Discrete Fourier Transforms: Properties of DFT, Linear Convolution of Sequences using

DFT, Computation of DFT: Over-Lap Add Method, Over-Lap Save Method, Relation between DTFT, DFS, DFT and Z-Transform.

Fast Fourier Transforms: Fast Fourier Transforms (FFT) – Radix-2 Decimation-in-Time and

Decimation-in-Frequency FFT Algorithms, Inverse FFT, and FFT with General Radix-N.

17

III

IIR Digital Filters: Analog filter approximations – Butter worth and Chebyshev, Design of IIR

Digital Filters from Analog Filters, Step and Impulse Invariant Techniques, Bilinear

Transformation Method, Spectral Transformations.

10

IV

FIR Digital Filters: Characteristics of FIR Digital Filters, Frequency Response, Design of FIR

Filters: Fourier Method, Digital Filters using Window Techniques, Frequency Sampling Technique,

Comparison of IIR & FIR filters

08

V

Multi rate Digital Signal Processing: Introduction, Down Sampling Decimation, Up sampling,

Interpolation, Sampling Rate Conversion.Finite Word Length Effects: Limit cycles, Overflow

Oscillations, Round-oft Noise in IIR Digital Filters, Computational Output Round Off Noise,

Methods to Prevent Overflow, Trade Off Between Round Off and Overflow Noise, Dead Band

Effects.

16

Contact classes for syllabus coverage 64

Lectures beyond syllabus 02

Tutorial classes 00

Classes for gaps&Add-on classes 02

Total No. of classes 68



5.c). GAPS IN THE SYLLABUS - TO MEET INDUSTRY/PROFESSION REQUIREMENTS:

S.NO. DESCRIPTION No. Of Classes

1 Discrete time fourier transform 1

2 Applications of digital filters 1

5.d). TOPICS BEYOND SYLLABUS/ADVANCED TOPICS:

S.NO. DESCRIPTION No. Of Classes

1 Architecture of TMS320C5X processor 1

2 Wavelet Transforms 1

5. e). WEB SOURCE REFERENCES:

Sl. No. Name of book/ website

a. http://nptel.ac.in/courses/117104070/

b. http://nptel.ac.in/courses/117104070/9

c. http://nptel.ac.in/courses/117104070/6

d. http://nptel.ac.in/courses/117102060/39

5. f). DELIVERY/INSTRUCTIONAL METHODOLOGIES:

CHALK & TALK STUD. ASSIGNMENT WEB RESOURCES

LCD/SMART BOARDS STUD. SEMINARS ADD-ON COURSES

5.g). ASSESSMENT METHODOLOGIES-DIRECT



ASSIGNMENTS STUD.

SEMINARS

TESTS/MODEL

EXAMS

UNIV.

EXAMINATION

STUD. LAB

PRACTICES

STUD. VIVA MINI/MAJOR

PROJECTS

CERTIFICATIONS

ADD-ON

COURSES

OTHERS

5.h). ASSESSMENT METHODOLOGIES-INDIRECT

ASSESSMENT OF COURSE OUTCOMES

(BY FEEDBACK, ONCE)

STUDENT FEEDBACK ON

FACULTY (TWICE)

ASSESSMENT OF MINI/MAJOR PROJECTS

BY EXT. EXPERTS

OTHERS

5.i). TEXT/REFERENCE BOOKS:

T/R BOOK TITLE/AUTHORS/PUBLICATION

Text Book Digital Signal Processing, Principles, Algorithms, and Applications John G.

Proakis, Dimitris G. Manolakis, Pearson Education / PHI, 2007.

Text Book Discrete Time Signal Processing — A. V. Oppenheim and R.W Schaffer,

PHI, 2009 Fundamentals of Digital Signal Processing — Loney Ludeman,

John Wiley, 2009

ReferenceBo

ok

Digital Signal Processing — Fundamentals and Applications — Li

Tan, Elsevier, 2008

Reference

Book Digital Signal Processing — S.Salivahanan, A.Vallavaraj and C.Gnanapriya,

TMH, 2009

Reference

Book Discrete Systems and Digital Signal Processing with MATLAB —

Taan EIAIi. CRC press, 2009.



Reference

Book Digital Signal Processing – Nagoor Khani, TMG, 2012

6) Micro Lesson Plan

Topic wise Coverage [Micro Lesson Plan]

S.No. Topic Scheduled date Planned date

Unit-I

1 UNIT-1 Introduction to DSP

2 Discrete time signals and sequences

3 Linear shift invariant systems

4 Stability and Causality

5 Linear constant coefficient difference

equation

6 Frequency domain representation of DTS

7 Application of Z transform

8 Solution of difference equation of digital

filters

System function,

9 Stability criterion

10 Frequency response of stable systems

11 Realization of digital filters-direct and

canonic form

12 Cascade and parallel form

13 Revision of 1st unit

14 Discrete time fourier transform



15 UNIT II- Discrete Fourier Series: DFS

representation

16 Properties of DFS

17 DFT and Properties of DFT

18 Linear convolution using DFT

19 Overlap save method

20 Overlap add method

21 Relation between DTFT and DFS

22 Relation between DFT and Z transform

23 FFT

24 Radix-2 decimation in time FFT algorithm

25 Radix-2 decimation in frequency

26 Inverse FFT

27 FFT with general radix N

28 Problems on overlap add method and save

method.

29 Problems on DIT algorithm

30 Problems on DIF algorithm

31 Revision

32 UNIT III- IIR Digital Filters

33 Analog filter approximations

Butterworth filter

34 Chebyshev filters

35 Design of IIR filters from Analog filters

36 Impulse invariant technique



37 Bilinear transformation method

38 Spectral transformation

39 Relation between analog and digital

frequencies

40 Problems on approximations of different

filters.

41 Revision

42 UNIT IV-FIR Digital filters

43 Frequency response

44 Design of FIR filters using Fourier method

45 Design of FIR filters using window

technique

46 Window technique problems

47 Problems on windowing techniques

48 Problems on Digital filters

49 Frequency sampling technique

50 Comparison of FIR and IIR filters

51 Applications of digital filters

52 Revision

53 UNIT V-Multi rate digital signal processing

54 Decimation

55 Interpolation

56 Sampling rate conversion

57 Finite word length effects



58 Limit cycle

59 Overflow oscillations

60 Round off noise

61 Computational output round off noise

62 Methods to prevent overflow

63 Trade off between round off and overflow

noise

64 Dead band effects

65 Problems

66 Revision

67 Architecture of TMS320C5X processor

68 Wavelet Transforms



7. Teaching Schedule

Subject Digital Signal Processing

Text Books (to be purchased by the Students)

Book 1 Digital Signal Processing – Principles, Algorithms, and Applications: John G. Proakis, Dimitris

G.Manolakis, Pearson Education / PHI, 2007.

Book 2 Discrete Time Signal Processing – A. V. Oppenheim and R.W.Schaffer, PHI, 2009.

Reference Books

Book 3 Digital Signal Processing – Fundamentals and Applications – Li Tan, Elsevier, 2008.

Book 4 Digital Signal Processing – S.Salivahanam, A.Vallavaraj and C.Gnanapriya, TMH, 2009.

Unit

Topic

Chapters Nos No of

classes Book 1 Book 2 Book 3 Book 4

I

Introduction to DSP 1 4

Realization of Digital Filters 1 1 1 9

II

Discrete Fourier series 2 2 2 3 2

Discrete Fourier Transforms 3 3 4 6

Fast Fourier Transforms 6 2 4 9

III

IIR Digital Filters 7 6 4 2

Analog Filter approximations 7 6 1 4

Design of IIR Digital Filters from

Analog Filters. 7 6 1 4

IV

FIR Digital Filters 8 7 9 4

Design of FIR Filters 8 7 9 6

Multirate Digital Signal Processing 9 10 10 4



V

Finite Word Length effects 10 8

Dead Band Effects 9 11 11 2

Contact classes for syllabus coverage 64

Lectures beyond syllabus 02

Tutorial classes 00

Classes for gaps&Add-on classes 02

Total No. of classes 68



11. MID EXAM DESCRIPTIVE QUESTION PAPERS

K. G. Reddy College of Engineering &Technology

(Approved by AICTE, Affiliated to JNTUH)

Chilkur (Vil), Moinabad (Mdl), RR District

College Code

QM

Name of the Exam: I Mid Examinations FEB-2018 Marks: 10

Year-Sem & Branch: IV Year I Sem & EEE Duration: 60 Min

Subject: DSP Date &

Session

Answer ANY TWO of the following Questions 2X5=10

1. Find the total response of the system described by difference equation y(n)-4y(n- 1) +4y(n-2)

= x(n)-x(n- 1) when the input is x(n)= (-1)n u(n) with the initial condition y (-1) =y (-2) =1.

2. A) Find x(k) of the given sequence x(n) = 1,2,3,4,4,3,2,1 using DIT-FFT algorithm.

B) Determine the number of additions & multiplications required to implement 16-point DFT&

16 point FFT

3.A) Find the IDFT of the given sequence X (k) = 2, 2-3j, 2+3j,-2

B) Obtain the relation between DFT & Z-transform

4.Derive the transfer function and order for the analog low pass butterworth filter



K. G. Reddy College of Engineering &Technology

(Approved by AICTE, Affiliated to JNTUH)

Chilkur (Vil), Moinabad (Mdl), RR District

_____________________________________________________________________________

Name of the Exam: II mid Examinations April– 2018

Year-Sem & Branch: IV-I EEE Duration: 60 Min

Subject: DIGITAL SIGNAL PROCESSING Date & Session:

Answer ANY TWO of the following Questions 2X5=10

Q.NO QUESTION Bloom’s

level Course

outcome

1

Design a Chebyshev filter for the following specifications using impulse invariance method

0.8 ≤ |H(ejw) | ≤ 1 , 0 ≤ w ≤ 0.2π |H(ejw) | ≤ 0.2 , 0.6π≤ w ≤ π

L6,L3 CO5

2

For the desired frequency response given by Hd (ejw) = e-j3w , |w| ≤ 3π/4 = 0 , 3π/4 < |w| <π

Find the filter coefficients for N=7 using hanning window. Plot the magnitude response.

L3 CO6

3

a) Discuss in detail down sampling process with a neat diagram b) Consider a signal x(n)=u(n) i)obtain a signal with a decimation factor 3 ii)obtain a signal with a interpolation factor 3

L2,L3 CO7

4

a) What are limit cycles and discuss various types of limit cycles in brief b) Compare IIR and FIR digital filters

L1,L4 CO8,CO6



12. MID EXAM OBJECTIVE QUESTION PAPERS

Code No: 126VK Set No. 1

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

III B.Tech. II Sem., I Mid-Term Examinations, February-2018 DIGITAL SIGNAL PROCESSENIG

Objective Exam

Name: ______________________________ Hall Ticket No. A

Answer All Questions. All Questions Carry Equal Marks. Time: 20 Min. Marks: 10.

I. Choose the correct alternative:

1. A discrete time signal has [ ]

a) Continuous time continuous amplitude b) Continuous time discrete amplitude

c) Discrete time continuous amplitude d) Discrete time discrete amplitude

2. The necessary and sufficient condition for causality of an LTI system is, [ ]

its unit sample response

a) h(n) = 0 for negative values of n b) h(n) = 0 for positive values of n

c) h(n) = 0 for integer values of n d) h(n) = 0 for complex values of n

3. FFT algorithm calculates [ ]

a) DTFT b) DCT c) DFT d) DST

4. The no. Of complex multiplications for computing DIF-FFT algorithm are [ ]

a) (N/2)log2N. a) (N/4)log4

N. a) (N/8)log8N. d) (N/16)log16

N

5. The advantage of cascade realization is [ ]

a) Quantization error can be minimized b) Random error can be minimized

c) System error can be minimized d) None of the above

6. Linear convolution of two real sequences with P and Q points, result for zero appended real sequence

after executing circular convolution using FFT will go wrong if FFT of length N is used such that

[ ]

a) N < P+Q-1 b) N=P+Q-1 c) N>P+Q-1 d) N<P+Q

7 he fundamental period of

x(n)=cos(𝜋𝑛2𝑓0

8) is



a) N=8 b) N=4 c) N=16 d)none

8. Z-transform reduces to fourier transform when it is evaluated on [ ]

a) A half circle b) Z-circle c) Unit circle d) Imaginary circle

9. An LTI system having system function H(z) stable if and only if all the poles of H(z) are ____________

the unit circle [ ]

a) Outside b) On c) Inside d) None of the above

10. The zero padding of sequence x(n) to find DFT results in [ ]

a) Better display of the frequency spectrum b) Can be used for non-linear filtering

c) Better display of the phase spectrum d) None of the above

Code No: 126VK :2: Set No. 1

II Fill in the Blanks

11. The fourier transform of a discrete and a periodic sequence is ______________

12. A ___________ signal exhibits no un certainity of value at any given instant of time

13. A linear system is one which satisfies the principle of ___________

14. The _____________ theorem states that the fourier transform of autocorrelation sequence is the

energy density spectrum

15. Average power of energy signals is__________

16. When a sequence is circularly shifted in time by 5 units, the magnitude response __________

17. DTFT can be obtained by performing _________ in time domain

18. In DIF the complex multiplication takes place after the ___________ operation

19. The transition band is ________ in butter worth filter compared to Chebyshev filter

20. The ___________ is discrete time counterpart of the laplace transform



Code No:126VK Set No. 1


KEY

I Choose the correct alternative:

1. c

2. a

3. c

4. a

5. a

6. b

7. b

8. c

9. c

10. a

II Fill in the Blanks 11. Continuous and periodic

12. Deterministic

13. Superposition

14. Wiener Khintchine

15. Zero



16. Remains unchanged

17. Sampling

18. Add-subtract

19. Less

20. Z-transform

Code No: 126VK Set No. 1

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

B.Tech. III Year, II Sem., II Mid-Term Examinations, April-2018


Objective Exam

Name: ______________________________ Hall Ticket No. A

Answer All Questions. All Questions Carry Equal Marks. Time: 20 Min. Marks: 10.

JJJ Choose the correct alternative:

1. Bilinear transformation provides [ ]

a) One-to-many mapping b) Many-to-one mapping

c) Many-to-many mapping d) One-to-one mapping

2. For recursive realization the present output y(n) is a function of [ ]

a) Past outputs, past and present inputs b) Future outputs, present and future inputs

c) Past outputs, past and future inputs d) None of the above

3. The direct form I realization of IIR digital filters require [ ]

a) M+N+1 multiplications b) M+N-1 multiplications c) M+N multiplications d) None of the above

4.The impulse response, which is symmetric having odd number of samples can be used to design

[ ]

a) low pass b) High pass c) Band pass d) All of the above

5. The advantage of cascade realization is [ ]

a) Quantization error can be minimized b) Random error can be minimized

c) System error can be minimizedd) None of the above

6. The magnitude of the side lobe level in Hanning window is [ ]

a) 31dB b) -31 dB c) 30dB d) -30dB

7. Frequency sampling realization of FIR filter introduces [ ]

a) More number of poles on unit circle b) More number of zeros on unit circle

c) Equal number of poles and zeros on the unit circle d) None of the above



8. Limit cycle is [ ]

a) Zero input limit cycle b) Overflow limit cycle

c) Both zero input and overflow limit cycle d) None of the above

9. Decimation is a process in which the sampling rate is [ ]

Reducedb) Enhanced c) Stable d) Unpredictable

Code No: 126VK :2: Set No. 1 II Fill in the Blanks

21. The anti- symmetrical impulse response can be used to design ______________

22. For ___________ realization the current output y(n) is a function of only past and present inputs

23. A ____ filter can have perfectly linear phase

14 The direct form FIR filter needs ................ between the adders to reduce the delay of the

adder tree and to achieve high throughput.

15. The minimum stop band attenuation of rectangular window is ____________

16. Kaiser window is based on __________functions

17. __________ filter is used specifically after upsampling process for removal of unwanted images

18. A __________is formed by an interconnection of the up-sampler, the down-sampler, and the components of an LTI digital filter

19. The process of quantization introduces ___________

20. The frequency response of the system with input h(n) and window length M is given by________



Code No:126VK Set No. 1


KEY

I Choose the correct alternative: 1. D

2. A

3. A

4. D

5. A

6. B

7. C

8. C

9. A

10. C

II Fill in the Blanks 11. Hilbert transformers and differentiators

12. Non-recursive

13. FIR

14. Extra pipeline registers

15. -21 dB

16. Prolate spheroidal

17. Anti –imaging

18. Complex multi rate system

19. Error

20. H(w)=∑ h(n) e-jwn



13. ASSIGNMENT TOPICS WITH MATERIALS

UNIT-I

1. Classification of systems

A discrete-time system can be thought of as a transformation or operator that maps an input sequence

x[n] to an output sequence

By placing various conditions on T(·) we can define different classes of systems. We give some properties

of systems. Basic System Properties are as given below

1) Systems with or without memory: A system is said to be memoryless if the out put for each value of the

independent variable at a given time n depends only on the input value at time n. For example system specified

by the relationship y[n] = cos(x[n]) + z is memoryless. A particularly simple memoryless system is the identity

system defined by y[n] = x[n]

In general we can write input-output relationship for memoryless system as

y[n] = g(x[n])

Not all systems are memoryless. A simple example of system with memory is a delay defined by

y[n] = x[n − 1]

A system with memory retains or stores information about input values at times other than the current input

value.

2) Invertibility: A system is said to be invertible if the input signal x[n] can be recovered from the output

signal y[n]. For this to be true two different input signals should produce two different outputs. If some

different input signal produce same output signal then by processing output we cannot say which input

produced the output.

Example of an invertible system is

then x[n] = y[n] − y[n − 1] Example if a non-invertible system is y[n]=0. That is the system produces an all zero

sequence for any input sequence. Since every input sequence gives all zero sequence, we can not find out which

input produced the output. The system which produces the sequence x[n] from sequence y[n] is called the

inverse system. In communication system, decoder is an inverse of the encoder.

3) Causality: A system is causal if the output at any time depends only on values of the input at the present

time and in the past.

All memoryless systems are causal. An accumulator system defined by

is also causal. The system defined by



is noncausal.

2. Linear constant coefficient difference equations

An important subclass of difference equations is the set of linear constant coefficient difference

equations. These equations are of the form

Cy(n)=f(n)

where C is a difference operator of the form given

C=cNDN+cN−1DN−1+...+c1D+c0

in which D is the first difference operator

D(y(n))=y(n)−y(n−1).

Note that operators of this type satisfy the linearity conditions, and c0,...,cn are real constants.

However, Equation 2 can easily be written as a linear constant coefficient recurrence equation without

difference operators. Conversely, linear constant coefficient recurrence equations can also be written in

the form of a difference equation, so the two types of equations are different representations of the same

relationship. Although we will still call them linear constant coefficient difference equations in this

course, we typically will not write them using difference operators. Instead, we will write them in the

simpler recurrence relation form

∑k=0Naky(n−k)=∑k=0Mbkx(n−k)

where x is the input to the system and y is the output. This can be rearranged to find y(n) as

y(n)=1a0(−∑k=1Naky(n−k)+∑k=0Mbkx(n−k))

The forms provided by equations will be used in the remainder of this course.

A similar concept for continuous time setting, differential equations, is discussed in the chapter on time

domain analysis of continuous time systems. There are many parallels between the discussion of linear

constant coefficient ordinary differential equations and linear constant coefficient differece equations.

3. Frequency domain representation of discrete time signals

Let us assume we have an LTI system

If nfj denx

2][ then

h[.] y[n] x[n]

http://pilot.cnxproject.org/content/collection/col10064/latest/module/m12325/latest#uid2



)(

][

][][][][

2

22

)(2

jwnfj

k

kfjnfj

k

knfj

k

eHe

ekhe

ekhknxkhny

d

dd

d

Eigenfunction eigenvalue

Example:

Let fnje

fnje

AfnAnx

22

22cos][

fnje

jweH

fnje

jweH

A

k k

knfjekh

knfjekh

A

k

knxkhny

22

2

2][

2][

2

][][][

A special case of this problem exist when h[n] is real

jwjw eHeH

In this case

jwjw eangHwherefneHAny 2cos][

In other words a sinusoidal input to a discrete time LTI system provides a sinusoidal output.

Frequency Domain Representation of Discrete-Time Signals and Systems



Discrete time Fourier Transform is a tool by which a time-domain sequence is mapped into a continuous

function of a frequency variable. Because the DTFT is periodic the parent discrete-time sequence can be simply

obtained by computing its Fourier Series representation.

Definition of the Forward Transform

Discrete-time Fourier transform jweX of a sequence x[n] is defined as:

n

jwnjw enxeX ][ (1)

In general jweX is a complex function of the real variable w and can be written as:

jw

im

jw

re

jw ejXeXeX

jweX can alternatively be expressed in polar form as:

jwwjjwjw eXwwhereeeXeX arg)(,)( (2)

In many applications the Fourier transform is called the Fourier Spectrum and likewise jweX and )(w are

referred to as the “magnitude spectrum” and “phase spectrum” respectively.

Note from eq.(2) that if we replace )(w with )(w + k2 , where k is an integer, jweX remains unchanged

implying that the phase function cannot be uniquely specified for any Fourier Transform.

4. Solution of difference equations of digital filters

The difference equation is a formula for computing an output sample at time based on past and present

input samples and past output samples in the time domain.6.1We may write the

general, causal, LTI difference equation as follows:

(6.1)

https://www.dsprelated.com/freebooks/filters/footnode.html#foot9508

http://www.dsprelated.com/dspbooks/filters/Causal_Recursive_Filters.html

http://www.dsprelated.com/dspbooks/filters/Linear_Time_Invariant_Digital_Filters.html



where is the input signal, is the output signal, and the constants

, are called the coefficients

As a specific example, the difference equation

specifies a digital filtering operation, and the coefficient sets and fully characterize

the filter. In this example, we have .

When the coefficients are real numbers, as in the above example, the filter is said to be real. Otherwise, it

may be complex.

Notice that a filter of the form of Eq. (5.1) can use ``past'' output samples (such as ) in the

calculation of the ``present'' output . This use of past output samples is calledfeedback. Any filter

having one or more feedback paths ( ) is called recursive. (By the way, the minus signs for the

feedback in Eq. (5.1) will be explained when we get to transfer functions in §6.1.)

More specifically, the coefficients are called the feedforward coefficients and the coefficients are

called the feedback coefficients.

A filter is said to be recursive if and only if for some . Recursive filters are also called infinite-

impulse-response (IIR) filters. When there is no feedback ( ), the filter is said to be

a nonrecursive or finite-impulse-response (FIR) digital filter.

When used for discrete-time physical modeling, the difference equation may be referred to as

an explicit finite difference scheme.6.2

Showing that a recursive filter is LTI (Chapter 4) is easy by considering its impulse-response

representation (discussed in §5.6). For example, the recursive filter

has impulse response , . It is now straightforward to apply the analysis of the

previous chapter to find that time-invariance, superposition, and the scaling property hold.

http://www.dsprelated.com/dspbooks/filters/Definition_Signal.html

http://www.dsprelated.com/dspbooks/filters/

http://www.dsprelated.com/dspbooks/filters/What_Filter.html

http://mathworld.wolfram.com/RealNumber.html

https://www.dsprelated.com/freebooks/filters/Difference_Equation_I.html#eq:tpnine

https://www.dsprelated.com/freebooks/filters/Difference_Equation_I.html#eq:tpnine

http://www.dsprelated.com/dspbooks/filters/Transfer_Function_Analysis.html

https://www.dsprelated.com/freebooks/filters/Z_Transform.html#sec:zt

http://www.dsprelated.com/dspbooks/filters/Impulse_Response_Representation.html

http://www.dsprelated.com/dspbooks/filters/Impulse_Response_Representation.html

http://en.wikipedia.org/wiki/Model_(physical)

http://www.dsprelated.com/dspbooks/pasp/Finite_Difference_Schemes.html


https://www.dsprelated.com/freebooks/filters/Linear_Time_Invariant_Digital_Filters.html#chap:theory

https://www.dsprelated.com/freebooks/filters/Impulse_Response_Representation.html#sec:imprep



5. Realization of digital filters using direct and cascade form

UNIT-II

1. Properties of DFT

As a special case of general Fourier transform, the discrete time transform shares all properties (and their

proofs) of the Fourier transform discussed above, except now some of these properties may take different

forms. In the following, we always assume and .

Linearity

Time Shifting



Proof:

If we let , the above becomes

Time Reversal

Frequency Shifting

Differencing

Differencing is the discrete-time counterpart of differentiation.

Proof:

Differentiation in frequency

proof: Differentiating the definition of discrete Fourier transform with respect to , we get



Convolution Theorems

The convolution theorem states that convolution in time domain corresponds to multiplication in frequency

domain and vice versa:

Recall that the convolution of periodic signals and is

Here the convolution of periodic spectra and is similarly defined as

Proof of (a):

Proof of (b):



Parseval's Relation

2. Computation of DFT using Over-Lap Add Method and Over-Lap Save Method

There are many DSP applications where a long signal must be filtered insegments. For instance,

high fidelity digital audio requires a data rate of about 5 Mbytes/min, while digital video requires

about 500 Mbytes/min. With data rates this high, it is common for computers to have insufficient

memory to simultaneously hold the entire signal to be processed. There are also systems that process

segment-by-segment because they operate in real time. For example, telephone signals cannot be

delayed by more than a few hundred milliseconds, limiting the amount of data that are available for

processing at any one instant. In still other applications, the processing may require that the signal be

segmented. An example is FFT convolution, the main topic of this chapter.

The overlap-add method is based on the fundamental technique in DSP:

(1) Decompose the signal into simple components,

(2) Process each of the components in some useful way, and

(3) Recombine the processed components into the final signal.

When an N sample signal is convolved with an M sample filter kernel, the output signal is N + M - 1

sample long. For instance, the input signal,

(a), is 300 samples (running from 0 to 299), the filter kernel,

(b), is 101 samples (running from 0 to 100), and the output signal, (i), is 400 samples.

When an N sample signal is filtered, it will be expanded by M - 1 point to the right. (This is

assuming that the filter kernel runs from index 0 to M. If negative indexes are used in the filter kernel,

the expansion will also be to the left). In (a), zeros have been added to the signal between sample 300

and 399 to illustrate where this expansion will occur. Don't be confused by the small values at the ends



of the output signal, (i). This is simply a result of the windowed-sinc filter kernel having small values

near its ends. All 400 samples in (i) are nonzero, even though some of them are too small to be seen in

the graph.

Below Figures show the decomposition used in the overlap-add method. The signal is broken into

segments, with each segment having 100 samples from the original signal. In addition, 100 zeros are

added to the right of each segment. In the next step, each segment is individually filtered by convolving it

with the filter kernel. This produces the output segments shown in figures. Since each input segment is

100 samples long, and the filter kernel is 101 samples long, each output segment will be 200 samples long.

The important point to understand is that the 100 zeros were added to each input segment to allow for the

expansion during the convolution.

Notice that the expansion results in the output segments overlapping each other. These overlapping output

segments are added to give the output signal, (i). For instance, samples 200 to 299 in (i) are found by

adding the corresponding samples in (g) and (h). The overlap-add method produces exactly the same

output signal as direct convolution. The disadvantage is a much greater program complexity to keep track

of the overlapping samples.

3. Relation between DTFT, DFS, DFT and Z-Transform

4. Decimation-in-Time and Decimation-in-Frequency FFT Algorithms

The FFT is a complicated algorithm, and its details are usually left to those that specialize in such things.

This section describes the general operation of the FFT, but skirts a key issue: the use of complex numbers.

In complex notation, the time and frequency domains each contain one signal made up of N complex

points. Each of these complex points is composed of two numbers, the real part and the imaginary part.

For example, when we talk about complex sample X[42], it refers to the combination of ReX[42]

and ImX[42]. In other words, each complex variable holds two numbers. When two complex variables are

multiplied, the four individual components must be combined to form the two components of the product

The following discussion on "How the FFT works" uses this jargon of complex notation. That is, the

singular terms: signal, point, sample, and value, refer to the combination of the real part and the imaginary



part.

The FFT operates by decomposing an N point time domain signal into N time domain signals each

composed of a single point. The second step is to calculate the N frequency spectra corresponding to

these N time domain signals. Lastly, the N spectra are synthesized into a single frequency spectrum.

Below figure shows an example of the time domain decomposition used in the FFT. In this example, a 16

point signal is decomposed through four

Separate stages. The first stage breaks the 16 point signal into two signals each consisting of 8 points. The

second stage decomposes the data into four signals of 4 points. This pattern continues until there

are N signals composed of a single point. An interlaced decomposition is used each time a signal is broken

in two, that is, the signal is separated into its even and odd numbered samples.

There are Log2N stages required in this decomposition, i.e., a 16 point signal (24) requires 4 stages, a 512

point signal (27) requires 7 stages, a 4096 point signal (212) requires 12 stages, etc. Remember this



value, Log2N.

This simple flow diagram is called a butterfly due to its winged

appearance. The butterfly is the basic computational element of the FFT,

transforming two complex points into two other complex points.

Figure shows the structure of the entire FFT. The time domain

decomposition is accomplished with a bit reversal sorting algorithm.

Transforming the decomposed data into the frequency domain

involves nothing and therefore does not appear in the figure.

The frequency domain synthesis requires three loops. The outer loop runs

through the Log2N stages (i.e., each level in Fig., starting from the bottom

and moving to the top). The middle loop moves through each of the

individual frequency spectra in the stage being worked on (i.e., each of the

boxes on any one level in Fig.). The innermost loop uses the butterfly to

calculate the points in each frequency spectra (i.e., looping through the

samples inside any one box in Fig.). The overhead boxes in Fig. determine the beginning and ending

indexes for the loops, as well as calculating the sinusoids needed in the butterflies. Now we come to the

heart of this chapter, the actual FFT programs.

5. Inverse FFT

If you need to compute inverse fast Fourier transforms (inverse FFTs) but you only have forward FFT

software (or forward FFT FPGA cores) available to you, below are four ways to solve your problem.

Preliminaries

To define what we're thinking about here, an N-point forward FFT and an N-point inverse FFT are

described by:

Forward FFT→X(m)=N−1∑n=0x(n)e−j2πnm/N(1)(1)Forward FFT→X(m)=∑n=0N−1x(n)e−j2πnm/NInv

erse FFT→x(n)=1NN−1∑m=0X(m)ej2πmn/NInverse FFT→x(n)=1N∑m=0N−1X(m)ej2πmn/N

=1NN−1∑m=0[Xreal(m)+jXimag(m)]ej2πmn/N(2)(2)=1N∑m=0N−1[Xreal(m)+jXimag(m)]ej2πmn/N

Inverse FFT Method# 1

The first method of computing inverse FFTs using the forward FFT was proposed as a "novel" technique

in 1988 [1]. That method is shown in Figure 1.

Figure 1: Method# 1 for computing the inverse FFT

using forward FFT software.




The second method of computing inverse FFTs using the forward FFT, similar to Method#1, is shown in

Figure 2(a). This Method# 2 has an advantage over Method# 1 when the input X(m)X(m) spectral samples

are conjugate symmetric. In that case, shown in Figure 2(b), only one data flipping operation is needed

because the output of the forward FFT will be real-only.

UNIT-III

1. Analog filter approximations -Butter worth

The classical method of analog filters design is Butterworth approximation. The Butterworth filters are also

known as maximally flat filters. Squared magnitude response of a Butterworth low-pass filter is defined as

follows

where - radian frequency, - constant scaling frequency, - order of the filter.

Some properties of the Butterworth filters are:

gain at DC is equal to 1;

has a maximum at

The first derivatives of (3.1) are equal to zero at . This is why Butterworth filters are

known as maximally flat filters.

2. Impulse Invariant Techniques

the impulse-invariant method converts analog filter transfer functions to digital filter transfer functions in

such a way that the impulse response is the same (invariant) at the sampling instants [346], [365, pp. 216-

219]. Thus, if denotes the impulse-response of an analog (continuous-time) filter, then the digital

(discrete-time) filter given by the impulse-invariant method will have impulse response ,

where denotes the sampling interval in seconds. Moreover, the order of the filter is preserved, and

https://ccrma.stanford.edu/~jos/filters/What_Filter.html

https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

https://ccrma.stanford.edu/~jos/filters/

https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

https://ccrma.stanford.edu/~jos/pasp/Bibliography.html#ParksAndBurrus

https://ccrma.stanford.edu/~jos/pasp/Bibliography.html#RabinerAndGold

https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html



IIR analog filters map to IIR digital filters. However, the digital filter's frequency response is

an aliasedversion of the analog filter's frequency response.9.3

To derive the impulse-invariant method, we begin with the analog transfer function

(9.1)

3. and perform a partial fraction expansion (PFE) down to first-order terms [452]:9.4

where is the th pole of the analog system, and is its residue [452]. Assume that the system is at

least marginally stable [452] so that there are no poles in the right-half plane ( ). Such a PFE is

always possible when is a strictly proper transfer function (more poles than zeros

[452]).9.5 Performing the inverse Laplace transform on the partial fraction expansion we obtain the impulse

response in terms of the system poles and residues:

We now sample at intervals of seconds to obtain the digital impulse response

Taking the z transform gives the digital filter transfer function designed by the impulse-invariant method:

We see that the -plane poles have mapped to the -plane poles

(9.2)

https://ccrma.stanford.edu/~jos/filters/Analog_Filters.html

https://ccrma.stanford.edu/~jos/filters/Difference_Equation.html

https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

https://ccrma.stanford.edu/~jos/mdft/Alias_Operator.html

https://ccrma.stanford.edu/~jos/pasp/footnode.html#foot26676

https://ccrma.stanford.edu/~jos/filters/Partial_Fraction_Expansion.html


https://ccrma.stanford.edu/~jos/pasp/Bibliography.html#JOSFP


https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html






https://ccrma.stanford.edu/~jos/filters/Laplace_Transform_Analysis.html



and the residues have remained unchanged. Clearly we must have , i.e., the analog

poles must lie within the bandwidth spanned by the digital sampling rate . Otherwise, the pole

angle will be aliased into the interval . Stability is preserved

since

3.Analog filter approximations - Chebyshev

The Chebyshev polynomial of degree n is denoted Tn(x), and is given by the explicit formula Tn(x) = cos(n arccos x) This

may look trigonometric at first glance (and there is in fact a close relation between the Chebyshev polynomials and the

discrete Fourier transform); however ( can be combined with trigonometric identities to yield explicit expressions for

Tn(x) ,

T0(x)=1

T1(x) = x

T2(x)=2x2 − 1

T3(x)=4x3 − 3x

T4(x)=8x4 − 8x2 + 1 ···

Tn+1(x)=2xTn(x) − Tn−1(x) n ≥ 1.

4. Bilinear Transformation Method

The bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often

used to convert a transfer function of a linear, time-invariant (LTI) filter in the continuous-time

domain (often called an analog filter) to a transfer function of a linear, shift-invariant filter in

the discrete-time domain (often called a digital filteralthough there are analog filters constructed

http://en.wikipedia.org/wiki/Bandpass

https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

https://ccrma.stanford.edu/~jos/filters/Stability_Revisited.html

https://en.wikipedia.org/wiki/Conformal_map

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation

https://en.wikipedia.org/wiki/Transfer_function

https://en.wikipedia.org/wiki/Linear

https://en.wikipedia.org/wiki/Time-invariant

https://en.wikipedia.org/wiki/LTI_system_theory

https://en.wikipedia.org/wiki/Continuous_function

https://en.wikipedia.org/wiki/Analog_filter

https://en.wikipedia.org/wiki/Discrete_signal

https://en.wikipedia.org/wiki/Digital_filter



with switched capacitors that are discrete-time filters). It maps positions on the axis, , in the s-

plane to the unit circle, , in the z-plane. Other bilinear transforms can be used to warp the frequency

response of any discrete-time linear system (for example to approximate the non-linear frequency

resolution of the human auditory system) and are implementable in the discrete domain by replacing a

system's unit delays with first order all-pass filters.

The transform preserves stability and maps every point of the frequency response of the continuous-time

filter, to a corresponding point in the frequency response of the discrete-time filter, although

to a somewhat different frequency, as shown in the Frequency warping section below. This means that

for every feature that one sees in the frequency response of the analog filter, there is a corresponding

feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a

somewhat different frequency. This is barely noticeable at low frequencies but is quite evident at

frequencies close to the Nyquist frequency.

5. Spectral Transformations.

When designing a particular filter, it is common practice to first design a Low-pass filter, and then using

a spectral transform to convert that lowpass filter equation into the equation of a different type of filter.

This is done because many common values for butterworth, cheybyshev and elliptical low-pass filters are

already extensively tabulated, and designing a filter in this manner rarely requires more effort then

looking values up in a table, and applying the correct transformations. This page will enumerate some of

the common transformations for filter design.

It is important to note that spectral transformations are not exclusively used for analog filters. There are

digital variants of these transformations that can be applied directly to digital filters to transform them into a

different type. This page will only talk about the analog filter transforms, and the digital filter transforms will

be discussed elsewhere.

UNIT-IV

1. Characteristics of FIR Digital Filters

Linear phase is a property of a filter, where the phase response of the filter is a linear

function of frequency. The result is that all frequency components of the input signal are shifted in time

(usually delayed) by the same constant amount, which is referred to as the phase delay. And

consequently, there is no phase distortion due to the time delay of frequencies relative to one another.

https://en.wikipedia.org/wiki/Switched_capacitor

https://en.wikipedia.org/wiki/S-plane

https://en.wikipedia.org/wiki/S-plane

https://en.wikipedia.org/wiki/Unit_circle

https://en.wikipedia.org/wiki/Complex_plane

https://en.wikipedia.org/wiki/Frequency_response


https://en.wikipedia.org/wiki/All-pass_filter

https://en.wikipedia.org/wiki/BIBO_stability


https://en.wikipedia.org/wiki/Bilinear_transform#Frequency_warping

https://en.wikipedia.org/wiki/Nyquist_frequency

https://en.wikipedia.org/wiki/Filter_(signal_processing)

https://en.wikipedia.org/wiki/Phase_response

https://en.wikipedia.org/wiki/Linear_function_(calculus)


https://en.wikipedia.org/wiki/Frequency

https://en.wikipedia.org/wiki/Phase_delay

https://en.wikipedia.org/wiki/Phase_distortion



For discrete-time signals, perfect linear phase is easily achieved with a finite impulse response (FIR)

filter. Approximations can be achieved with infinite impulse response (IIR) designs, which are more

computationally efficient.

Linear-phase FIR filter can be divided into four basic types.

Type impulse response

I symmetric length is odd

II symmetric length is even

III anti-symmetric length is odd

IV anti-symmetric length is even

When h(n) is nonzero for 0 ≤ n ≤ N −1 (the length of the impulse response h(n) is N), then the symmetry of

the impulse response can be written as

h(n) = h(N − 1 − n)

and the anti-symmetry can be written as

h(n) = −h(N − 1 − n).

2. Digital Filters using Window Techniques

(a) Rectangular

(b) Bartlett (or triangle)

(c) Hanning

https://en.wikipedia.org/wiki/Discrete-time

https://en.wikipedia.org/wiki/Finite_impulse_response

https://en.wikipedia.org/wiki/Infinite_impulse_response



(d) Harming

(e) Blackman

(f) Kaiser

where is modified zero-order Bessel function of the first kind given by

The main lobe width and first side lobe attenuation increase as we proceed down the window listed above.

An ideal lowpass filter with linear phase and cut off is characterized by

The corresponding impulse response is



Since this is symmetric about , if we change and use one of the windows listed above the will get linear

phase FIR filter. Transition width and minimum stopped attenuation are listed in the Table 9.3.

Window Transition Width Minimum stopband attenuation

Rectangular

-21db

Bartlett

-25dB

Hanning

-44dB

Hamming

-53dB

Blackman

-74dB

Kaiser variable variable

3. Design of FIR Filters using Fourier Method

In designing FIR filter using Fourier series method the infinite duration impulse response

is truncated at n= ± (N-1/2).Direct truncation of the series will lead to fixed percentage

overshoots and undershoots before and after an approximated discontinuity in the

frequency response .

4. Comparison of IIR & FIR filters

FIR filter IIR filter

These filters can be easily designed to have perfectly linear phase.

These filters do not have linear phase.

FIR filters can be realized recursively and non- recursively.

IIR filters are easily realized recursively

Greater flexibility to control the shape of their magnitude response.

Less flexibility, usually limited to specific kind of filters.

Errors due to round off noise are less severe in FIR filters, mainly because feedback is not used.

The round off noise in IIR filters is more.

5. Frequency Response of FIR digital filters



The frequency response of an LTI filter may be defined as the spectrum of the output signaldivided by

the spectrum of the input signal. In this section, we show that the frequency response of any LTI filter is

given by its transfer function evaluated on the unit circle, i.e., . We then show that this is the same result we

got using sine-wave analysis in Chapter 1.

Beginning with Eq. (6.4), we have

where X(z) is the z transform of the filter input signal , is the z transform of the output signal , and is the

filter transfer function.

A basic property of the z transform is that, over the unit circle , we find the spectrum[84].8.1To

show this, we set in the definition of the z transform, Eq. (6.1), to obtain

which may be recognized as the definition of the bilateral discrete time Fourier

transform(DTFT) when is normalized to 1 When is causal, this definition reduces to the usual

(unilateral) DTFT definition:

DTFT

(8.1)

Applying this relation to gives

UNIT-V

1. Decimation

Decimation can be regarded as the discrete-time counterpart of sampling. Whereas in sampling we start with a

continuous-time signal x(t) and convert it into a sequence of samples x[n], in decimation we start with a discrete-

time signal x[n] and convert it into another discrete-time signal y[n], which consists of sub-samples of x[n]. Thus,

the formal definition of M-fold decimation, or down-sampling, is defined by Equation. In decimation, the sampling

rate is reduced from Fs to Fs/M by discarding M – 1 samples for every M samples in the original sequence

∞

y[ n]= v[nM ]=∑ h[k] x[nM -k] k=-∞


http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html

http://www.dsprelated.com/dspbooks/filters/Definition_Signal.html



http://www.dsprelated.com/dspbooks/filters/What_Filter.html

http://www.dsprelated.com/dspbooks/filters/Transfer_Function_Analysis.html

http://www.dsprelated.com/dspbooks/mdft/Sinusoids.html

https://www.dsprelated.com/freebooks/filters/Simplest_Lowpass_Filter.html#chap:partOne

https://www.dsprelated.com/freebooks/filters/Z_Transform_Convolution.html#eq:tpsixteen


https://www.dsprelated.com/freebooks/filters/Bibliography.html#MDFT


https://www.dsprelated.com/freebooks/filters/Z_Transform.html#eq:zt

http://www.dsprelated.com/dspbooks/mdft/Discrete_Time_Fourier_Transform.html



http://www.dsprelated.com/dspbooks/filters/Causal_Recursive_Filters.html



2. Interpolation

Interpolation is the exact opposite of decimation. It is an information preserving operation, in that all samples of

x[n] are present in the expanded signal y[n]. The mathematical definition of L-fold interpolation is defined by

Equation and the block diagram notation is depicted in below Figure Interpolation works by inserting (L–1) zero-

valued samples for each input sample. The sampling rate therefore increases from Fs to LFs. With reference to

Figure, the expansion process is followed by a unique digital low-pass filter called an anti-imaging filter. Although

the expansion process does not cause aliasing in the interpolated signal, it does however yield undesirable replicas

in the signal’s frequency spectrum.

y[n] =L h[k]w[n-k]

3. Round-off Noise in IIR Digital Filters

the roundoff noise for infinite-impulse-response (IIR) digital filters realized in some canonical forms.

The minimum roundoff noise realization of this class is achieved by letting the finite-impulse-response

(FIR) coefficients be identical to the corresponding partial impulse response of the desired filter. When

the order of the numerator of the transfer function of the desired filter is greater than or equal to that of

the denominator, there is an optimal decomposition which is shown to reduce the roundoff noise without

adding additional multipliers or adders. The proposed structure is applied to parallel and cascade forms,

with real as well as complex arithmetic and shown to result in lower roundoff noise. By increasing the

order of the FIR filter, the optimal synthesis becomes a noise reduction scheme at the cost of additional

computational complexity

4. Limit cycles

When a stable IIR filter digital filter is excited by a finite sequence, that is constant, the output will

ideally decay to zero. However, the non-linearity due to finite precision arithmetic operations often

causes periodic oscillations to occur in the output. Such oscillations occur in the recursive systems are

called Zero input Limit Cycle Oscillation. Normally oscillations in the absence of output u (k) =0 by

equation given below is called Limit cycle oscillations



Y[k] = -0.625.y [k-1] +u[k]

5. Dead Band Effects

A deadband (sometimes called a neutral zone or dead zone) is a band of input values in the domain of

a transfer function in a control system or signal processing system where the output is zero (the output is

'dead' - no action occurs). Deadband regions can be used in control systems such as servoamplifiers to

prevent oscillation or repeated activation-deactivation cycles (called 'hunting' in proportional

control systems). A form of deadband that occurs in mechanical systems, compound machines such

as gear trains is backlash.

https://en.wikipedia.org/wiki/Domain_of_a_function

https://en.wikipedia.org/wiki/Transfer_function

https://en.wikipedia.org/wiki/Control_system

https://en.wikipedia.org/wiki/Servoamplifier

https://en.wikipedia.org/wiki/Oscillation

https://en.wikipedia.org/wiki/Proportional_control

https://en.wikipedia.org/wiki/Proportional_control

https://en.wikipedia.org/wiki/Simple_machine#compound_machines

https://en.wikipedia.org/wiki/Gear_train

https://en.wikipedia.org/wiki/Backlash_(engineering)



15) UNIT WISE-QUESTION BANK

UNIT-I

2 MARKS QUESTIONS WITH ANSWERS

1) What is a continuous and discrete time signal?

Continuous time signal: A signal x(t) is said to be continuous if it is defined for all

time t. Continuous time signal arises naturally when a physical waveform such as acoustics wave or light

wave is converted into an electrical signal.

Discrete time signal: A discrete time signal is defined only at discrete instants of time.

The independent variable has discrete values only, which are uniformly spaced. A discrete time signal is

often derived from the continuous time signal by sampling it at a uniform rate.

2) Give the classification of signals?

There are 5 types of classifications of signals

1. Continuous-time and discrete time signals

2. Even and odd signals

3. Periodic signals and non-periodic signals

4. Deterministic signal and Random signal

5. Energy and Power signal

3) What are the types of systems?

Continuous time and discrete time systems

1. Linear and Non-linear systems

2. Causal and Non-causal systems

3. Static and Dynamic systems

4. Time varying and time in-varying systems

5. Distributive parameters and Lumped parameters systems

6. Stable and Un-stable systems.

4) What are even and odd signals?

Even signal: continuous time signal x(t) is said to be even if it satisfies the condition x(t)=x(-t) for

all values of t.

Odd signal: he signal x(t) is said to be odd if it satisfies the condition x(-t)=-x(t) for all t. In other

words even signal is symmetric about the time origin or the vertical axis, but odd signals are anti-

symmetric about the vertical axis.

5) What are deterministic and random signals?



Deterministic Signal: deterministic signal is a signal about which there is no certainty with respect

to its value at any time. Accordingly, we find that deterministic signals may be modeled as

completely specified functions of time.

Random signal: random signal is a signal about which there is uncertainty before its actual

occurrence. (eg.) The noise developed in a television or radio amplifier is an example for random

signal.


1) What are elementary signals and name them?

The elementary signals serve as a building block for the construction of more complex signals. They

are also important in their own right, in that they may be used to model many physical signals that

occur in nature.

There are five elementary signals. They are as follows

1) Unit step function

2) Unit impulse function

3) Ramp function

4) Exponential function

5) Sinusoidal function

2) What are time invariant systems?

A system is said to be time invariant system if a time delay or advance of the input signal leads to

an identical shift in the output signal. This implies that a time invariant system responds identically no matter

when the input signal is applied. It also satisfies the condition

Rx (n-k) = y (n-k).

3) What do you mean by periodic and non-periodic signals?

A signal is said to be periodic if x (n + N) = x (n), Where N is the time period. A signal is said to be

non-periodic if x (n + N) = -x (n).

4) Define linear and non-linear system.

Linear system is one which satisfies superposition principle. Supeosition principle:

The response of a system to a weighted sum of signals be equal to the corresponding weighted sum of

responses of system to each of individual input signal.

i.e., T [a1x1=(n)+a2x2(n)]=a1T[x1(n)]+a2T[x2(n)]

A system, which does not satisfy superposition principle, is known as non-linear system.

5) Define causal and non-causal system.

The system is said to be causal if the output of the system at any time ‘n’ depends only on present

and past inputs but does not depend on the future inputs.



e.g.:- y (n) =x (n)-x (n-1)

A system is said to be non-causal if a system does not satisfy the above definition.


1) Explain in detail about discrete time systems

A discrete-time system can be thought of as a transformation or operator that maps an input sequence

x[n] to an output sequence

By placing various conditions on T(·) we can define different classes of systems. We give some properties

of systems. Basic System Properties are as given below

1) Systems with or without memory: A system is said to be memoryless if the out put for each value of the

independent variable at a given time n depends only on the input value at time n. For example system specified

by the relationship y[n] = cos(x[n]) + z is memoryless. A particularly simple memoryless system is the identity

system defined by y[n] = x[n]

In general we can write input-output relationship for memoryless system as

y[n] = g(x[n])

Not all systems are memoryless. A simple example of system with memory is a delay defined by

y[n] = x[n − 1]

A system with memory retains or stores information about input values at times other than the current input

value.

2) Invertibility: A system is said to be invertible if the input signal x[n] can be recovered from the output

signal y[n]. For this to be true two different input signals should produce two different outputs. If some

different input signal produce same output signal then by processing output we cannot say which input

produced the output.

Example of an invertible system is

then x[n] = y[n] − y[n − 1] Example if a non-invertible system is y[n]=0. That is the system produces an all zero

sequence for any input sequence. Since every input sequence gives all zero sequence, we can not find out which

input produced the output. The system which produces the sequence x[n] from sequence y[n] is called the

inverse system. In communication system, decoder is an inverse of the encoder.

3) Causality: A system is causal if the output at any time depends only on values of the input at the present

time and in the past.

All memoryless systems are causal. An accumulator system defined by

is also causal. The system defined by



is noncausal.

4) Stability

There are several definitions for stability. Here we will consider bounded input bonded output(BIBO) stability. A

system is said to be BIBO stable if every bounded input produces a bounded output. We say that a signal x[n]

is bounded if |x[n]|<M < ∞ for all n

The moving average system

is stable as y[n] is sum of finite numbers and so it is bounded. The accumulator system defined by

is unstable. If we take x[n] = u[n], the unit step then y[0] = 1, y[1] =2, y[2] = 3, are y[n] = n +1, n ≥ 0 so y[n]

grows without bound.

5) Time invariance

A system is said to be time invariant if the behavior and characteristics of the system do not change with time.

Thus a system is said to be time invariant if a time delay or time advance in the input signal leads to identical

delay or advance in the output signal. Mathematically if y[n] = T (x[n]), Then y[n – no] = T(x[n – no]) for

any no

Let us consider the accumulator system

If the input is now x1[n] = x [n – no] then the corresponding output is



The shifted output signal is given by

The two expression look different, but infact they are equal. Let us change the index of summation by l = k – no

in the first sum then we see that

Hence, y[n] = y[n – no] and the system is time-invariant. As a second example consider the system defined

by y[n] = nx[n]

if

x1[n] = x[n – no]

y1[n] = nx1[n] = n x[n − n0]

while y[n – no] = (n – no)x[n – no] and so the system is not time-invariant. It is time varying. We can also see this

by giving a counter example. Suppose input is x[n] = δ[n] then output is all zero sequence. If the input is δ[n

−1] then output is δ[n−1] which is definitely not a shifted version version of all zero sequence.

6) Linearity

This is an important property of the system. We will see later that if we have system which is linear and time

invariant then it has a very compact representation. A linear system possesses the important property of

superposition: If an input consists of weighted sum of several signals, the and the output is also weighted sum

of the responses of the system to each of those input signals. Mathematically let y1[n] be the response of the



system to the input x1[n] and let y2[n] be the response of the system to the input x2[n]. Then the system is

linear if:

2) Prove that the impulse response of an LTI system is absolutely summable for stability of the system

Due to its convolution property, the z-transform is a powerful tool to analyze LTI systems

When the input is the eigenfunction of all LTI system, i.e., , the operation on this input by

the system can be found by multiplying the system's eigenvalue H(z) to the input.

Causal LTI systems

An LTI system is causal if its output y[n] depends only on the current and past input x[n] (but not the future).

Assuming the system is initially at rest with zero output , then its response to an

impulse at is at rest for , i.e., . Its response to a general

input is:

Due to the properties of the ROC, we know that If an LTI system is causal (with a right sided impulse response

function for ), then the ROC of its transfer function is the exterior of a circle

including infinity. In particular, when is rational, then the system is causal if and only if its ROC is the

exterior of a circle outside the out-most pole, and the order of numerator is no greater than the order of the

denominator.

Note the requirement for the orders of the numerator and denominator guarantees the existence of even

when .

Stable LTI systems



An LTI system is stable if its response to any bounded input is also bounded for all :

As the output and input of an LTI is related by convolution, we have:

which obviously requires:

In other words, if the impulse response function of an LTI system is absolutely integrable, then the

system is stable. We can show that this condition is also necessary, i.e., all stable LTI systems' impulse

response functions are absolutely integrable. Now we have:

An LTI system is stable if and only if its impulse response is absolutely summable, i.e., the frequency response

function exits, i.e. the ROC of its transfer function includes the unit circle

.

2) Explain the principle of operation of analog to digital conversion with a neat diagram

Digital signal processing (DSP) is the use of digital processing, such as by computers, to perform a wide variety

of signal processing operations. The signals processed in this manner are a sequence of numbers that

represent samples of a continuous variable in a domain such as time, space, or frequency. Digital signal

processing and analog signal processing are subfields of signal processing. DSP applications

include audio and speech signal processing, sonar, radar and other sensor array processing, spectral

estimation, statistical signal processing, digital image processing, signal processing

for telecommunications, control of systems, biomedical engineering, seismic data processing, among others.

Digital signal processing can involve linear or nonlinear operations. Nonlinear signal processing is closely

related to nonlinear system identification and can be implemented in the time, frequency, and spatial-temporal

domains.

Signal sampling

The increasing use of computers has resulted in the increased use of, and need for, digital signal processing. To

digitally analyze and manipulate an analog signal, it must be digitized with an analog-to-digital converter.

Sampling is usually carried out in two stages, discretization and quantization.

Discretization means that the signal is divided into equal intervals of time, and each interval is represented by a

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https://en.wikipedia.org/wiki/Nonlinear_system_identification

https://en.wikipedia.org/wiki/Discretization

https://en.wikipedia.org/wiki/Quantization_(signal_processing)



single measurement of amplitude.

Quantization means each amplitude measurement is approximated by a value from a finite set. Rounding real

numbers to integers is an example.

The Nyquist–Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the

sampling frequency is greater than twice the highest frequency of the signal. In practice, the sampling

frequency is often significantly higher than twice that required by the signal's limited bandwidth.

Theoretical DSP analyses and derivations are typically performed on discrete-time signal models with no

amplitude inaccuracies (quantization error), "created" by the abstract process of sampling. Numerical methods

require a quantized signal, such as those produced by an analog-to-digital converter (ADC). The processed

result might be a frequency spectrum or a set of statistics. But often it is another quantized signal that is

converted back to analog form by a digital-to-analog converter (DAC).

Domains:

In DSP, engineers usually study digital signals in one of the following domains: time domain (one-dimensional

signals), spatial domain (multidimensional signals), frequency domain, and wavelet domains. They choose the

domain in which to process a signal by making an informed assumption (or by trying different possibilities) as

to which domain best represents the essential characteristics of the signal. A sequence of samples from a

measuring device produces a temporal or spatial domain representation, whereas a discrete Fourier

transform produces the frequency domain information, that is, the frequency spectrum.

Time and space domains

The most common processing approach in the time or space domain is enhancement of the input signal through

a method called filtering. Digital filtering generally consists of some linear transformation of a number of

surrounding samples around the current sample of the input or output signal. There are various ways to

characterize filters; for example:

A "linear" filter is a linear transformation of input samples; other filters are "non-linear". Linear filters

satisfy the superposition condition, i.e. if an input is a weighted linear combination of different signals,

the output is a similarly weighted linear combination of the corresponding output signals.

A "causal" filter uses only previous samples of the input or output signals; while a "non-causal" filter

uses future input samples. A non-causal filter can usually be changed into a causal filter by adding a

delay to it.

A "time-invariant" filter has constant properties over time; other filters such as adaptive filters change in

time.

A "stable" filter produces an output that converges to a constant value with time, or remains bounded

within a finite interval. An "unstable" filter can produce an output that grows without bounds, with

bounded or even zero input.

A "finite impulse response" (FIR) filter uses only the input signals, while an "infinite impulse response"

filter (IIR) uses both the input signal and previous samples of the output signal. FIR filters are always

stable, while IIR filters may be unstable.

A filter can be represented by a block diagram, which can then be used to derive a sample

processing algorithm to implement the filter with hardware instructions. A filter may also be

described as a difference equation, a collection of zeroes and poles or, if it is an FIR filter,

an impulse response or step response.

The output of a linear digital filter to any given input may be calculated by convolving the

input signal with the impulse response.

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https://en.wikipedia.org/wiki/Wavelet

https://en.wikipedia.org/wiki/Discrete_Fourier_transform

https://en.wikipedia.org/wiki/Discrete_Fourier_transform

https://en.wikipedia.org/wiki/Frequency_spectrum

https://en.wikipedia.org/wiki/Digital_filter

https://en.wikipedia.org/wiki/Linear_transformation

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https://en.wikipedia.org/wiki/Block_diagram

https://en.wikipedia.org/wiki/Algorithm

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https://en.wikipedia.org/wiki/Zero_(complex_analysis)

https://en.wikipedia.org/wiki/Pole_(complex_analysis)

https://en.wikipedia.org/wiki/Impulse_response

https://en.wikipedia.org/wiki/Step_response

https://en.wikipedia.org/wiki/Convolution

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Block Diagram:

3) What are the advantages of DSP over ASP?

The main advantage of digital signals over analog signals is that the precise signal level of the digital

signal is not vital. This means that digital signals are fairly immune to the imperfections of real

electronic systems which tend to spoil analog signals. As a result, digital CD's are much more robust

than analog LP's.

Codes are often used in the transmission of information. These codes can be used either as a means of

keeping the information secret or as a means of breaking the information into pieces that are

manageable by the technology used to transmit the code, e.g. The letters and numbers to be sent by a

Morse code are coded into dots and dashes.

Digital signals can convey information with greater noise immunity, because each information

component (byte etc) is determined by the presence or absence of a data bit (0 or one). Analog signals

vary continuously and their value is affected by all levels of noise.

Digital signals can be processed by digital circuit components, which are cheap and easily produced in

many components on a single chip. Again, noise propagation through the demodulation system is

minimized with digital techniques.

Digital signals do not get corrupted by noise etc. You are sending a series of numbers that represent the

signal of interest (i.e. audio, video etc.)

Digital signals typically use less bandwidth. This is just another way to say you can cram more

information (audio, video) into the same space.

Digital can be encrypted so that only the intended receiver can decode it (like pay per view video,

secure telephone etc.)

Enables transmission of signals over a long distance.

Transmission is at a higher rate and with a wider broadband width.

It is more secure.

It is also easier to translate human audio and video signals and other messages into machine language.

There is minimal electromagnetic interference in digital technology.



It enables multi-directional transmission simultaneously.

4) Discuss the steps involved in calculating convolution sum by taking simple practical example

The steps involved in calculating sum are

1) Folding

2) Shifting

3) Multiplication

4) Summation

.Let's say a chef decides to offer private cooking lessons. A starting pupil receives a 1-hour lesson to explore

interests and background. Next day, and all the days following, the private lessons are 3 hours long. If the

function h is the hours per day for one pupil, the graph would look like:

Hours per day in training one student.

We have 1 hour of instruction for the start day d = 0. Afterwards, for d = 1, 2, …, we have 3 hours.On the first

day, there are 2 pupils. On the second day, 4 new pupils arrive and on the third day, there are 6. No additional

pupils enroll after the third day. If p is the number of pupils who start on a given day, then a graph of the

function p is:

Number of new pupils who enroll on each day.

How many hours of instruction are given on the first day? Two pupils times 1 hour each gives 1*2 = 2. If T is

the total hours per day, then on the zeroth day, T(0) = 2. Now it starts to get interesting.On the following day,

the 2 pupils from the previous day are each getting 3 hours of instruction, and the 4 new pupils receive their 1

hour of orientation. This means T(1) = 3*2 + 1*4 = 10.On the following day, T(2) = 3*2 + 3*4 + 1*6 = 24.

Seems like everybody wants to learn to make croissants! Time to hire more help.But what is this type of

computation? What we have here is called a convolution. It's a type of sum of multiplications that keeps track

of what happened before. Here's how to set up the convolution. First, we fold one of the functions. Here is

the p function folded:



Folding the p function.

Then we line up the folded p with h:

Lining up the functions.

Multiplying point-by-point gives 1*2 = 2. This is the T(0) result.Next day, the pupils shift over to the right by

1.

Shifting p by one day.

We multiply and add to get T(1) = 3*2 + 1*4 = 10. Let's use a new symbol λ to keep track of the day. Our

function h is written as h(λ). To fold h we write h(-λ). To shift to the right, h(-λ) becomes h(d - λ). We're almost

there! The Σ means to add. Now, we can write the convolution as:

The λ = 0 at the bottom of the Σ is our start value for λ. We will have d multiplies to add: one for λ = 0, another

for λ = 1, and we keep going until λ = d. See the 2 above the Σ in the next equation? The number 2 tells us the

end value for λ. To clarify, let's work out the number of hours of instruction for d = 2.



.

5) What are the Characteristics of a DSP?

A unit that can handle floating numbers is present directly in the data flow path.

The accumulators or multipliers that are present are highly parallel in nature.

Special hardware is included in order to carry looping at a very low cost.

Their architecture is specially designed so that fetching multiple data at the same time is possible.

The calculations are usually carried out by fixed point arithmetic process in order to speed them up.

Most of the registers present in computers today move the data to the lower-most bit if an overflow

occurs. However, in case of digital signal processors, the overflow is retained at the maximum point

itself.

Specialised instructions are present for modulo and reversed bit addressing.

FILL IN THE BLANKS

1) _______ is a System which carries out mathematical operations on a sequences of samples related to a

signal.

2) _______ is defined as any physical quantity that varies with time, space, or any other independent

variable

3) A discrete time signal having a set of discrete values is ____________

4) The Process of converting a continuous-valued signal into discrete-valued signal is called

__________________.

5) Any signal that can be uniquely described by an explicit mathematical expression, a table of data, or a

well-defined rule is ______________

6) A system is said to be _________ if the output of the system at any time n depends only on present and

past inputs

7) ____________ is a constant function that describes the magnitude and phase shift of a filter over a range

of frequencies

8) ___________ is the property that states the Z-transforms of a sum of signals is the sum of individual Z-

transforms.

9) __________ of time signals corresponds to the multiplication of Z-transform.

10) If the ROC extends outward from the outermost pole, then the system is _________.



1) DSP

2) Signal

3) Digital Signal

4) Quantization

5) Deterministic

6) Causal

7) Frequency Response

8) Linearity

9) Convolution

10) Causal

MULTIPLE CHOICE QUESTIONS

1) The interface between an analog signal and a digital processor is

a. D/A converter

b. A/D converter

c. Modulator

d. Demodulator

ANSWER: (b) A/D converter

2) The speech signal is obtained after

a. Analog to digital conversion

b. Digital to analog conversion

c. Modulation

d. Quantization

ANSWER: (b) Digital to analog conversion

3) Telegraph signals are examples of

a. Digital signals

b. Analog signals

c. Impulse signals

d. Pulse train

ANSWER: (a) Digital signals

4) As compared to the analog systems, the digital processing of signals allow

1) Programmable operations

2) Flexibility in the system design

3) Cheaper systems

4) More reliability

a. 1, 2 and 3 are correct

b. 1 and 2 are correct



c. 1, 2 and 4 are correct

d. All the four are correct

ANSWER: (d) All the four are correct

5) The Nyquist theorem for sampling

1) Relates the conditions in time domain and frequency domain

2) Helps in quantization

3) Limits the bandwidth requirement

4) Gives the spectrum of the signal



c. 1 and 3 are correct


ANSWER: (c) 1 and 3 are correct

6) Roll-off factor is

a. The bandwidth occupied beyond the Nyquist Bandwidth of the filter

b. The performance of the filter or device

c. Aliasing effect

d. None of the above

ANSWER: (a) The bandwidth occupied beyond the Nyquist Bandwidth of the filter

7) A discrete time signal may be

1) Samples of a continuous signal

2) A time series which is a domain of integers

3) Time series of sequence of quantities

4) Amplitude modulated wave





ANSWER: (a) 1, 2 and 3 are correct

8) The discrete impulse function is defined by

a. δ(n) = 1, n ≥ 0

= 0, n ≠ 1

b. δ(n) = 1, n = 0

= 0, n ≠ 1

c. δ(n) = 1, n ≤ 0

= 0, n ≠ 1

d. δ(n) = 1, n ≤ 0

= 0, n ≥ 1

ANSWER: (b) δ(n) = 1, n = 0

= 0, n ≠ 1

9) DTFT is the representation of

a. Periodic Discrete time signals



b. Aperiodic Discrete time signals

c. Aperiodic continuous signals

d. Periodic continuous signals

ANSWER :(b) Aperiodic Discrete time signals

10) The transforming relations performed by DTFT are

1) Linearity

2) Modulation

3) Shifting

4) Convolution





ANSWER: (d) All the four are correct

UNIT 2 2 MARKS QUESTIONS WITH ANSWERS

1. State the properties of DFT

Periodicity Linearity and symmetry Multiplication of two DFTs Circular convolution Time reversal Circular time shift and frequency shift Complex conjugate Circular correlation

2. Define circular convolution

Let x1(n) and x2(n) are finite duration sequences both of length N with DFTs X1 (k) and X2 (k). If

X3(k) = X1(k) X2(k) then the sequence x3(n) can be obtained by circular.

3. Define sectional convolution

If the data sequence x (n) is of long duration it is very difficult to obtain the output sequence y(n) due

to limited memory of a digital computer. Therefore, the data sequence is divided up into smaller sections.



These sections are processed separately one at a time and controlled later to get the output.

4. What is FFT?

The Fast Fourier Transform is an algorithm used to compute the DFT. It makes use of the symmetry and

periodicity properties of twiddle factor to effectively reduce the DFT computation time. It is based on the

fundamental principle of decomposing the computation of DFT of a sequence of length N into successively

smaller DFTs.

5. How many multiplications and additions are required to compute N point DFT using radix- 2FFT?

The number of multiplications and additions required to compute N point DFT using radix-2 FFT are N log2 N

and N/2 log2 N respectively


1. What is overlap-add method?

In this method the size of the input data block xi (n) is L. To each data block we append M-1 zeros

and perform N point cicular convolution of xi (n) and h(n). Since each data block is terminated with M-1

zeros the last M-1 points from each output block must be overlapped and added to first M- 1 points of the

succeeding blocks. This method is called overlap-add method.

2. Why FFT is needed?

The direct evaluation DFT requires N2 complex multiplications and N2 –N complex additions. Thus

for large values of N direct evaluation of the DFT is difficult. By using FFT algorithm, the number of

complex computations can be reduced. Therefore, we use FFT.

.

3. What are the applications of FFTalgorithm?

The applications of FFT algorithm includes

Linear filtering

Correlation Spectrum

analysis

4. Distinguish between linear convolution and circular convolution of two sequences.

Linear convolution Circular convolution



If x(n) is a sequence of L number of samples

and h(n) with M number of samples, after

convolution y(n) will have N=L+M-1 samples.

If x(n) is a sequence of L number of samples

and h(n) with M samples, after convolution

y(n) will have N=max(L,M)samples.

It can be used to find the response of a linear

filter

It cannot be used to find the response of a filter

Zero padding is not necessary to find the

response of a linear filter.

Zero padding is necessary to find the response

5. What are the differences and similarities between DIF and DITalgorithms?

Differences:

1) The input is bit reversed while the output is in natural order for DIT, whereas for DIF the output is bit reversed while the input is in naturalorder.

2) The DIF butterfly is slightly different from the DIT butterfly, the difference being that the complex multiplication takes place after the add-subtract operation inDIF.

Similarities:

Both algorithms require same number of operations to compute the DFT. Both algorithms can be done in

place and both need to perform bit reversal at some place during the computation.


1. Explain properties of Discrete Fourier Transform.

As a special case of general Fourier transform, the discrete time transform shares all properties (and their

proofs) of the Fourier transform discussed above, except now some of these properties may take different

forms. In the following, we always assume and .

Linearity

Time Shifting

Proof:

If we let , the above becomes



Time Reversal

Frequency Shifting

Differencing

Differencing is the discrete-time counterpart of differentiation.

Proof:

Differentiation in frequency

proof: Differentiating the definition of discrete Fourier transform with respect to , we get

Convolution Theorems

The convolution theorem states that convolution in time domain corresponds to multiplication in frequency

domain and vice versa:



Recall that the convolution of periodic signals and is

Here the convolution of periodic spectra and is similarly defined as

Proof of (a):

Proof of (b):



Parseval's Relation

2) Describe the decimation in time [DIT] radix-2 FFT algorithm to determineN-point

DFT.

This algorithm is very similar in concept to the Decimation in Frequency (DIF) Algorithm discussed earlier,

so the presentation will be a little less detailed. Have a look at the section describing the DIF algorithm first.

We defined the FFT as:

If N is even, the above sum can be split into 'even' (n=2n') and 'odd' (n=2n'+1) halves, where n'=0..N/2-1,

and re-arranged as follows:

This process of splitting the 'time domain' sequence into even an odd samples is what gives the algorithm its

name, 'Decimation In Time'. As with the DIF algorithm, we have succeeded in expressing an N point

transform as 2 (N/2) point sub-transforms. The principal difference here is that the order we do things has

changed. In the DIF algorithm the time domain data was 'twiddled' before the two sub-transforms were

performed. Here the two sub-transforms are performed first. The final result is obtained by 'twiddling' the

resulting frequency domain data. There is a slight problem here, because the two sub-transforms only give

values for k=0..N/2-1. We also need values for k=N/2..N-1. But from the periodicity of the DFT we know:



Also..

So, for k=0..N/2-1:

and..

where:

This all we need to produce a simple recursive DIT FFT routine for any N which is a regular power of 2

(N=2p).

3) Explain overlap add method

There are many DSP applications where a long signal must be filtered insegments. For instance, high

fidelity digital audio requires a data rate of about 5 Mbytes/min, while digital video requires about 500

Mbytes/min. With data rates this high, it is common for computers to have insufficient memory to

simultaneously hold the entire signal to be processed. There are also systems that process segment-by-

segment because they operate in real time. For example, telephone signals cannot be delayed by more than a

few hundred milliseconds, limiting the amount of data that are available for processing at any one instant. In

still other applications, the processing may require that the signal be segmented. An example is FFT

convolution, the main topic of this chapter.

The overlap-add method is based on the fundamental technique in DSP:

(1) Decompose the signal into simple components,

(2) Process each of the components in some useful way, and

(3) Recombine the processed components into the final signal.

When an N sample signal is convolved with an M sample filter kernel, the output signal is N + M - 1 sample

long. For instance, the input signal,

(a), is 300 samples (running from 0 to 299), the filter kernel,

(b), is 101 samples (running from 0 to 100), and the output signal, (i), is 400 samples.

When an N sample signal is filtered, it will be expanded by M - 1 point to the right. (This is assuming

that the filter kernel runs from index 0 to M. If negative indexes are used in the filter kernel, the expansion

will also be to the left). In (a), zeros have been added to the signal between sample 300 and 399 to illustrate

where this expansion will occur. Don't be confused by the small values at the ends of the output signal, (i).

This is simply a result of the windowed-sinc filter kernel having small values near its ends. All 400 samples

in (i) are nonzero, even though some of them are too small to be seen in the graph.

Below Figures show the decomposition used in the overlap-add method. The signal is broken into

segments, with each segment having 100 samples from the original signal. In addition, 100 zeros are added

to the right of each segment. In the next step, each segment is individually filtered by convolving it with the

filter kernel. This produces the output segments shown in figures. Since each input segment is 100 samples

long, and the filter kernel is 101 samples long, each output segment will be 200 samples long. The important



point to understand is that the 100 zeros were added to each input segment to allow for the expansion during

the convolution.

Notice that the expansion results in the output segments overlapping each other. These overlapping output

segments are added to give the output signal, (i). For instance, samples 200 to 299 in (i) are found by adding

the corresponding samples in (g) and (h). The overlap-add method produces exactly the same output signal

as direct convolution. The disadvantage is a much greater program complexity to keep track of the

overlapping samples.

4) Explain FFT

The FFT is a complicated algorithm, and its details are usually left to those that specialize in such things.

This section describes the general operation of the FFT, but skirts a key issue: the use of complex numbers.

In complex notation, the time and frequency domains each contain one signal made up of N complex points.

Each of these complex points is composed of two numbers, the real part and the imaginary part. For

example, when we talk about complex sample X[42], it refers to the combination of ReX[42] and ImX[42]. In

other words, each complex variable holds two numbers. When two complex variables are multiplied, the

four individual components must be combined to form the two components of the product

The following discussion on "How the FFT works" uses this jargon of complex notation. That is, the

singular terms: signal, point, sample, and value, refer to the combination of the real part and the imaginary

part.

The FFT operates by decomposing an N point time domain signal into N time domain signals each

composed of a single point. The second step is to calculate the N frequency spectra corresponding to

these N time domain signals. Lastly, the N spectra are synthesized into a single frequency spectrum.

Below figure shows an example of the time domain decomposition used in the FFT. In this example, a 16

point signal is decomposed through four



Separate stages. The first stage breaks the 16 point signal into two signals each consisting of 8 points. The

second stage decomposes the data into four signals of 4 points. This pattern continues until there

are N signals composed of a single point. An interlaced decomposition is used each time a signal is broken in

two, that is, the signal is separated into its even and odd numbered samples.

There are Log2N stages required in this decomposition, i.e., a 16 point signal (24) requires 4 stages, a 512

point signal (27) requires 7 stages, a 4096 point signal (212) requires 12 stages, etc. Remember this

value, Log2N.

This simple flow diagram is called a butterfly due to its winged

appearance. The butterfly is the basic computational element of the FFT,

transforming two complex points into two other complex points.

Figure shows the structure of the entire FFT. The time domain

decomposition is accomplished with a bit reversal sorting algorithm.

Transforming the decomposed data into the frequency domain

involves nothing and therefore does not appear in the figure.

The frequency domain synthesis requires three loops. The outer loop runs

through the Log2N stages (i.e., each level in Fig., starting from the bottom

and moving to the top). The middle loop moves through each of the

individual frequency spectra in the stage being worked on (i.e., each of the

boxes on any one level in Fig.). The innermost loop uses the butterfly to

calculate the points in each frequency spectra (i.e., looping through the

samples inside any one box in Fig.). The overhead boxes in Fig. determine the beginning and ending indexes

for the loops, as well as calculating the sinusoids needed in the butterflies. Now we come to the heart of this

chapter, the actual FFT programs.

5) Explain about inverse FFT



If you need to compute inverse fast Fourier transforms (inverse FFTs) but you only have forward FFT

software (or forward FFT FPGA cores) available to you, below are four ways to solve your problem.

Preliminaries

To define what we're thinking about here, an N-point forward FFT and an N-point inverse FFT are described

by:

Forward FFT→X(m)=N−1∑n=0x(n)e−j2πnm/N(1)(1)Forward FFT→X(m)=∑n=0N−1x(n)e−j2πnm/NInvers

e FFT→x(n)=1NN−1∑m=0X(m)ej2πmn/NInverse FFT→x(n)=1N∑m=0N−1X(m)ej2πmn/N

=1NN−1∑m=0[Xreal(m)+jXimag(m)]ej2πmn/N(2)(2)=1N∑m=0N−1[Xreal(m)+jXimag(m)]ej2πmn/N


The first method of computing inverse FFTs using the forward FFT was proposed as a "novel" technique in

1988 [1]. That method is shown in Figure 1.




The second method of computing inverse FFTs using the forward FFT, similar to Method#1, is shown in

Figure 2(a). This Method# 2 has an advantage over Method# 1 when the input X(m)X(m) spectral samples

are conjugate symmetric. In that case, shown in Figure 2(b), only one data flipping operation is needed

because the output of the forward FFT will be real-only.



Figure 2: Method# 2 Processing flow: (a) standard Method# 2;

(b) Method# 2 when X(m) samples are conjugate symmetric.

The next two inverse FFT methods are of interest because they avoid the data reversals necessary in

Method# 1 and Method# 2.


The third method of computing inverse FFTs using the forward FFT, by way of data swapping, is shown in

Figure 3.




The fourth method of computing inverse FFTs using the forward FFT, by way of complex conjugation, is

shown in Figure 4.



FILL IN THE BLANKS

1) The important tools used in frequency analysis of signals are _____________and __________ ___

2) __________ can be used to find the response of a linear filter.

3) _________ is not necessary to find the response of a linear filter in linear convolution.

4) The linear convolution of 2 finite duration sequences x(n) and h(n) of lengths L samples and M samples

will result in a output sequence of duration ___________ samples

5) By using with Zero padding, DFT can be used in ____________



6) The two methods to compute DFT are ____________ and __________________

7) In overlap save method the size of the input data block is __________

8) In overlap add method the size of the input data block is ____________

9) ___________ is defined as an algorithm that efficiently computes the DFT.

10) ___________multiplications are required to compute DFT

1 DFT , IDFT

2 Linear

Convolution

3 Zero Padding

4 L+M-1

5 Linear Filtering

6 Overlap Add and

Overlap Save

7 N=L+M-1

8 L

9 FFT

10 N2

Multiple Choice Questions

1. Given that , W = 𝑒−𝑖(2𝜋

𝑁) ,where N = 3. Then F = 𝑊𝑁can be computed as F =

(A) 0 (B) 1

(C)- 1 (D) 5

Solution

The correct answer is (B).



2) Given that W = 𝑒−𝑖(2𝜋

𝑁), where N = 3 and F = 𝑊𝑁/2 can be computed as F =

(A) 0 (B) 1

(C) -1 (D) e

Solution

The correct answer is (C).

3.Given that N=2, f = 4 − 6𝑖

−2 + 4𝑖. The valuesfor vector shown in

Can be computed as

(A) −2−6

(B)−26

(C) 2

−6 (D)

26

Solution

The correct answer is(D).



4. Given that N=2, f = 4 − 6𝑖

−2 + 4𝑖. The values for vector shown in below equation

can be computed as

(A) −2

−10

(B) −1

−10

(C) −2−5

(D) −1 −5

Solution

The correct answer is(A).

5. If the forcing function F(t) is given as F(t) = ∑ 10 sin(2𝜋𝑛𝑡).7𝑛=0 Then, to avoid aliasing phenomenon, the

minimum number of sample data points Nminshould be

(A)8 (B)16

(C)24 (D) 32

Solution

The correct answer is (B).

6. Based on the figure below, aliasing phenomena will not occur because there were



(A) 2 sample data points per cycle.

(B) 4 sample data points per cycle.

(C) 4 sample data points per 2 cycles.

(D) 6 sample data points per 2 cycles.

Solution

The correct answer is (D)

7. Using the definition E= 𝑒−𝑖2𝜋/𝑁, and the Euler identity𝑒±𝑖𝜃 = 𝑐𝑜𝑠𝜃 ± 𝑠𝑖𝑛𝜃, the value E N/6 can be computed

as

(A) 0.866 − 0.5i

(B) − 0.866 + 0.5i

(C) − 0.5 − 0.866i

(D) 0.5 − 0.866i

Solution



The correct answer is (D),

8. Using the definition E = 𝑒−𝑖2𝜋/𝑁, and the Euler identity 𝑒±𝑖𝜃 = 𝑐𝑜𝑠𝜃 ± 𝑠𝑖𝑛𝜃, the value E 6N can be computed

as

(A) i+1

(B) i−1

(C) 1

(D) -1

Solution

The correct answer is (C).



9)

Solution

The correct answer is(A).

(10) For N=24=16, levelL=2and referring to the figure shown below, the only terms of vector f2(−)which only

need to compute are:



(A) f2(4−7,12−15) (B )f2(0−3,8−11)

(C) f2(0−7) (D) f2(8−15)

Solution

The correct answer is(B).

UNIT III 2 MARKS QUESTIONS WITH ANSWERS

1. What are the different types of filters based on impulseresponse?

Based on impulse response the filters are of two types

1. IIRfilter 2. FIRfilter

The IIR filters are of recursive type, whereby the present output sample depends on the present input,

past input samples and output samples.

The FIR filters are of non-recursive type, whereby the present output sample depends on the present

input sample and previous input samples.

2. What are the different types of filters based on frequencyresponse? Based on frequency response the filters can be classified as

1. Low passfilter 2. High passfilter 3. Bandpassfilter 4. Band rejectfilter

3. What are the advantages and disadvantages of FIRfilters? Advantages:

1. FIR filters have exact linearphase. 2. FIR filters are alwaysstable. 3. FIR filters can be realized in both recursive and non-recursivestructure. 4. Filters with any arbitrary magnitude response can be tackled using FIRsequence.

Disadvantages:

1. For the same filter specifications the order of FIR filter design can be as high as 5 to 10 times that in an IIRdesign.

2. Large storage requirement isrequirement



3. Powerful computational facilities required for theimplementation.

4. How one can design digital filters from analogfilters? · Map the desired digital filter specifications into those for an equivalent analogfilter. · Derive the analog transfer function for the analogprototype. · Transform the transfer function of the analog prototype into an equivalent digital filter transferfunction.

5. Mention the procedures for digitizing the transfer function of an analogfilter. The two important procedures for digitizing the transfer function of an analog filter are

· Impulse invariancemethod. · Bilinear transformationmethod.

. Approximation of derivatives


1. Distinguish between FIR filters and IIRfilters.

FIR filter IIR filter

These filters can be easily designed to have perfectly linear phase.

These filters do not have linear phase.

FIR filters can be realized recursively and non- recursively.

IIR filters are easily realized recursively

Greater flexibility to control the shape of their magnitude response.

Less flexibility, usually limited to specific kind of filters.

Errors due to round off noise are less severe in FIR filters, mainly because feedback is not used.

The round off noise in IIR filters is more.

2. What is the mapping procedure between S-plane & Z-plane in the method of mapping differentials? What

are its characteristics?

The mapping procedure between S-plane & Z-plane in the method of mapping of differentials is

given by

H(Z) =H(S)|S=(1-Z-1)/T

The above mapping has the following characteristics

a. The left half of S-plane maps inside a circle of radius ½ centered at Z= ½ in the Zplane. b. The right half of S-plane maps into the region outside the circle of radius ½ in the Z-plane.



c. The j .-axis maps onto the perimeter of the circle of radius ½ in theZ-plane.

3. Write a short note on pre-warping.

The effect of the non-linear compression at high frequencies can be compensated. When the desired

magnitude response is piece-wise constant over frequency, this compression can be compensated by

introducing a suitable pre-scaling, or pre-warping the critical frequencies by using the formula.

4. What are the advantages & disadvantages of bilineartransformation? Advantages:

· The bilinear transformation provides one-to-onemapping. · Stable continuous systems can be mapped into realizable, stable digitalsystems. · There is noaliasing.

Disadvantage:

· The mapping is highly non-linear producing frequency, compression at highfrequencies. · Neither the impulse response nor the phase response of the analog filter is preserved in a digital filter obtained by bilineartransformation.

5. Distinguish analog and digitalfilters

Analog Filter Digital Filter

Constructed using active or passive components and it is described by a differential equation

Consists of elements like adder, subtractor and delay units and it is described by a difference equation

Frequency response can be changed by changing the components

Frequency response can be changed by changing the filter coefficients

It processes and generates analog output Processes and generates digital output

Output varies due to external conditions Not influenced by external conditions


1) Explain about the Butterworth filter approximations

The classical method of analog filters design is Butterworth approximation. The Butterworth filters are also

known as maximally flat filters. Squared magnitude response of a Butterworth low-pass filter is defined as

follows



where - radian frequency, - constant scaling frequency, - order of the filter.

Some properties of the Butterworth filters are:

gain at DC is equal to 1;

has a maximum at

The first derivatives of (3.1) are equal to zero at . This is why Butterworth filters are known as maximally flat filters.

Using above properties the expression can be rearranged to the form

Given this expression can be written as follows

Function has poles and doesn't have any finite zeros. It is easy to see that if is a pole of

(3.2), then is also a pole of (3.2). In order to find the poles of transfer function that satisfy (3.2), we

have to select one pole from each pair of the poles of expression (3.2). As it was mentioned

before, the poles of a valid filter have to have negative real parts.



The poles of (3.2) can be found as roots of equation

Observing that , where stands for the odd number, roots of can be obtained as

solutions to the equation

The solution of (3.4) can be presented in the form

So, roots (3.5) are the poles of .

All poles lie on a circle of radius in the complex s-plane. Since the difference has the

same value for all roots, it can be concluded that the poles are equally spaced on the circumference.

Fig 3.1 Pole locations of the squared

magnitude response for the Butterworth

low-pass filters with orders N=6 and N=5.

Scaling frequency .

It is easy to see that for

the . Therefore, N roots of (3.5)

have the negative real parts; these are the

poles of . The remaining roots are

the poles of . Real parts in (3.5) are also never zero, so poles never fall to the imaginary axis. For the

Butterworth low-pass filters with odd orders, two of the poles have zero imaginary parts, so they fall on the

real axis in the s-plane. For the filters with even orders, all imaginary and real parts are nonzero.

The poles of Butterworth filter lie on left half of the s-plane, and they can be given as follows



2) Write down the advantages and disadvantages of Digital filters

Advantages

It has linear phase response. Thermal and environmental variation cannot change the performance. It is so-called adaptive filter because the frequency response can be possible to adjust automatically

with implementation of programmable processor. It is possible to filter several input sequences without any hardware replication. All data can be stored It has repeatable performance unit to unit. Due to its operating level is at low frequency, it is used where the use of analog system is impractical.

Disadvantages

In reality, the signal bandwidth of the digital sequence is much lower than the analog sequence. Signal processing speed is one of the key factors of calculating the total device performance. Actually the speed operation totally depends on the number of the arithmetic operation in the processor.

Finite word-length effect, which results quantizing noise and round-off noise, is another major drawback during computation.

It needs much longer time to design and develop the digital sequences though it can be used on other tasks or applications once developed. Ordinarily good support of computer aided design can convert them into a enjoyable tasks.

3) Discuss magnitude characteristics of an analog Butterworth filter and give its pole locations

Ideal Frequency Response for a Butterworth Filter



Note that the higher the Butterworth filter order, the higher the number of cascaded stages there are within

the filter design, and the closer the filter becomes to the ideal “brick wall” response.

In practice however, Butterworth’s ideal frequency response is unattainable as it produces excessive passband

ripple.

Where the generalised equation representing a “nth” Order Butterworth filter, the frequency response is given

as:

Where: n represents the filter order, Omega ω is equal to 2πƒ and Epsilon ε is the maximum pass band gain,

(Amax). If Amax is defined at a frequency equal to the cut-off -3dB corner point (ƒc), ε will then be equal to one

and therefore ε2 will also be one. However, if you now wish to define Amax at a different voltage gain value, for

example 1dB, or 1.1220 (1dB = 20logAmax) then the new value of epsilon, ε is found by:

Where: H0 = the Maximum Pass band Gain, Amax. H1 = the Minimum Pass band Gain.



Transpose the equation to give:

The Frequency Response of a filter can be defined mathematically by its Transfer Function with the standard

Voltage Transfer Function H(jω) written as:

Where: Vout = the output signal voltage. Vin = the input signal voltage. j = to the square root of -1 (√-1) ω = the radian frequency (2πƒ)

Note: ( jω ) can also be written as ( s ) to denote the S-domain. and the resultant transfer function for a second-

order low pass filter is given as:

Normalized Low Pass Butterworth Filter Polynomials

To help in the design of his low pass filters, Butterworth produced standard tables of normalized second-order

low pass polynomials given the values of coefficient that correspond to a cut-off corner frequency of 1

radian/sec.

1 (1+s)

2 (1+1.414s+s2)



3 (1+s)(1+s+s2)

4 (1+0.765s+s2)(1+1.848s+s2)

5 (1+s)(1+0.618s+s2)(1+1.618s+s2)

6 (1+0.518s+s2)(1+1.414s+s2)(1+1.932s+s2)

7 (1+s)(1+0.445s+s2)(1+1.247s+s2)(1+1.802s+s2)

8 (1+0.390s+s2)(1+1.111s+s2)(1+1.663s+s2)(1+1.962s+s2)

9 (1+s)(1+0.347s+s2)(1+s+s2)(1+1.532s+s2)(1+1.879s+s2)

10 (1+0.313s+s2)(1+0.908s+s2)(1+1.414s+s2)(1+1.782s+s2)(1+1.975s+s2)

Filter Design – Butterworth Low Pass

Find the order of an active low pass Butterworth filter whose specifications are given as: Amax = 0.5dB at a pass

band frequency (ωp) of 200 radian/sec (31.8Hz), and Amin = 20dBat a stop band frequency (ωs) of 800

radian/sec. Also design a suitable Butterworth filter circuit to match these requirements.

Firstly, the maximum pass band gain Amax = 0.5dB which is equal to a gain of 1.0593(0.5dB = 20log A) at a

frequency (ωp) of 200 rads/s, so the value of epsilon ε is found by:

Secondly, the minimum stop band gain Amin = 20dB which is equal to a gain of 10 (20dB = 20log A) at a stop

band frequency (ωs) of 800 rads/s or 127.3Hz.



Substituting the values into the general equation for a Butterworth filters frequency response gives us the

following:

Since n must always be an integer ( whole number ) then the next highest value to 2.42 is n = 3, therefore a “a

third-order filter is required” and to produce a third-order Butterworth filter, a second-order filter stage

cascaded together with a first-order filter stage is required.

From the normalised low pass Butterworth Polynomials table above, the coefficient for a third-order filter is

given as (1+s)(1+s+s2) and this gives us a gain of 3-A = 1, or A = 2. As A = 1 + (Rf/R1), choosing a value for both

the feedback resistor Rf and resistor R1 gives us values of 1kΩ and 1kΩ respectively, ( 1kΩ/1kΩ + 1 = 2 ).

We know that the cut-off corner frequency, the -3dB point (ωo) can be found using the formula 1/CR, but we

need to find ωo from the pass band frequency ωp then,



So, the cut-off corner frequency is given as 284 rads/s or 45.2Hz, (284/2π) and using the familiar

formula 1/CR we can find the values of the resistors and capacitors for our third-order circuit.

Note that the nearest preferred value to 0.352uF would be 0.36uF, or 360nF.

Third-order Butterworth Low Pass Filter

and finally our circuit of the third-order low pass Butterworth Filter with a cut-off corner frequency of 284

rads/s or 45.2Hz, a maximum pass band gain of 0.5dB and a minimum stop band gain of 20dB is constructed as

follows.



4.

5.




1. IIR digital filters are of the following nature

a. Recursive

b. Non Recursive

c. Reversive

d. Non Reversive

ans-a

2. In IIR digital filter the present output depends on

a. Present and previous Inputs only

b. Present input and previous outputs only

c. Present input only

d. Present Input, Previous input and output

ans-d



3. The most common technique for the design of I I R Digital filter is

a. Direct Method

b. In direct method

c. Recursive method

d. non recursive method

ans-a

4. In the design a IIR Digital filter for the conversion of analog filter in to Digital domain the desirable property

is

a. The axis in the s - plane should map outside the unit circle in the z - Plane

b. The Left Half Plane(LHP) of the s - plane should map in to the unit circle in the Z -

Plane

c. The Left Half Plane(LHP) of the s-plane should map outside the unit circle in the z-

Plane

d. The Right Half Plane(RHP) of the s-plane should map in to the unit circle in the Z -

Plane

Ans-d

5.The s plane and z plane are related as

a. z = esT

b. z = e2sT

c. z = 2esT

d. z = esT/2

ANSWER: (a) z = esT

6. The IIR filter designing involves



a. Designing of analog filter in analog domain and transforming into digital domain

b. Designing of digital filter in analog domain and transforming into digital domain

c. Designing of analog filter in digital domain and transforming into analog domain

d. Designing of digital filter in digital domain and transforming into analog domain

ANSWER: (b) Designing of digital filter in analog domain and transforming into digital domain

7. For a system function H(s) to be stable a. The zeros lie in left half of the s plane

b. The zeros lie in right half of the s plane

c. The poles lie in left half of the s plane

d. The poles lie in right half of the s plane

ANSWER: (c) The poles lie in left half of the s plane

8.IIR filter design by approximation of derivatives has the limitations 1) Used only for transforming analog high pass filters

2) Used for band pass filters having smaller resonant frequencies

3) Used only for transforming analog low pass filters

4) Used for band pass filters having high resonant frequencies





ANSWER: (c) 2 and 3 are correct

9.The filter that may not be realized by approximation of derivatives techniques are

1) Band pass filters

2) High pass filters

3) Low pass filters

4) Band reject filters





ANSWER: (b) 2 and 4 are correct

10. Which among the following represent/s the characteristic/s of an ideal filter?

a. Constant gain in pass band

b. Zero gain in stop band

c. Linear Phase Response

d. All of the above



ANSWER: (d) All of the above

FILL IN THE BLANKS

1. Filters exhibit their dependency upon the system design for the stability purpose

2. The impulse invariant method is obtained by

3. The transformation technique in which there is one to one mapping from s-domain to z-domain is

4. IIR filters have both

5. The poles of chebyshev filter lies on the

6. LTI IIR systems are stable if all the poles of the system function lie the unit circle

7. Bilinear transformation avoids in digitizing IIR filter

8. Impulse response of IIR filter have number of samples

9. IIR filter can have linear phase , if and only if ,it is both and

10. The advantage of direct form-II over direct form-I realization is needs

ANSWERS

1. IIR

2. Sampling the impulse response of an equivalent analog filter

3. Bilinear transformation method

4. poles and zeros

5. ellipse

6. inside



7. aliasing

8. infinite

9. stable ,physically realizable

10. less memory locations

UNIT IV 1. 2 MARKS QUESTIONS WITH ANSWERS

2. How phase distortion and delay distortion are introduced?

The phase distortion is introduced when the phase characteristics of a filter is nonlinear within the

desired frequency band. The delay distortion is introduced when the delay is not constant within the desired

frequency band.

3. What is mean by FIR filter? The filter designed by selecting finite number of samples of impulse response h (n) obtained from

inverse Fourier transform of desired frequency response H(w) are called FIR filters

4. Write the steps involved in FIR filter design Choose the desired frequency response Hd(w)

Take the inverse Fourier transform and obtain Hd(n) Convert the

infinite duration sequence Hd(n) to h(n) Take Z transform of h(n)

to get H(Z)

5. List the disadvantages of FIR FILTER The duration of impulse response should be large to realize sharp cutoff filters. The non integral

delay can lead to problems in some signal processing applications.

6. Define necessary and sufficient condition for the linear phase characteristic of a FIR filter? The phase function should be a linear function of w, which in turn requires constant group delay

and phase delay.


1. List the well-known design technique for linear phase FIR filter design?

Fourier series method and window method Frequency

sampling method

Optimal filter design method

2.What are the properties of FIR filter?



1. FIR filter is always stable. 2. A realizable filter can always be obtained. 3. FIR filter has a linear phase response.

3.What are the disadvantages of Fourier series method?

In designing FIR filter using Fourier series method the infinite duration impulse

response is truncated at n= ± (N-1/2).Direct truncation of the series will lead to fixed

percentage overshoots and undershoots before and after an approximated discontinuity in the

frequency response .

4. Give the desirable characteristics of the windows?

The desirable characteristics of the window are

1. The central lobe of the frequency response of the window should contain most of the energy and should be narrow.

2. The highest side lobe level of the frequency response should be small. 3. The side lobes of the frequency response should decrease in energy rapidly as w tends to p.

5.State Frequency Warping

Because of the non-linear mapping: the amplitude response of digital IIR filter is expand

at lower frequencies and compressed at higher frequencies in comparison to the

analog filter.


1. Compare FIR and IIR filter

S.No FIR Filter IIR Filter

1. The impulse response of this filter is restricted

to finite number of samples

The impulse response extends to infinite

duration

2. FIR Filters have linear phase IIR filter don’t have linear phase

3. Always stable Not always stable

4. Greater flexibility Less flexibility



5. Errors due to roundoff noise are less severe IIR filters are more susceptible to errors due to

roundoff noise

2. What do you understand by Linear Phase Response in filters?

In linear phase filter ( ) α, the linear phase filter does not alter the shape of the

original signal. If phase response of the filter is nonlinear the output signal is

distorted one. In many cases Linear Phase filter is required throughout the pass

band of the filter to preserve the shape of the given signal within the pass band.

An IIR filter cannot produce a linear phase. The FIR filter can give linear phase,

when the impulse response of the filter is symmetric about its midpoint.

3. Explain various window techniques

(a) Rectangular

(b) Bartlett (or triangle)

(c) Hanning

(d) Harming

(e) Blackman

(f) Kaiser

where is modified zero-order Bessel function of the first kind given by

The main lobe width and first side lobe attenuation increase as we proceed down the window listed above.

An ideal lowpass filter with linear phase and cut off is characterized by

The corresponding impulse response is

Since this is symmetric about , if we change and use one of the windows listed above the will get linear

phase FIR filter. Transition width and minimum stopped attenuation are listed in the Table 9.3.

Window Transition Width Minimum stopband attenuation

Rectangular

-21db

Bartlett

-25dB

Hanning

-44dB

Hamming

-53dB

Blackman

-74dB

Kaiser variable variable

4. Explain Linear Phase characteristics of FIR filters

Linear phase is a property of a filter, where the phase response of the filter is a linear

function of frequency. The result is that all frequency components of the input signal are shifted in

time (usually delayed) by the same constant amount, which is referred to as the phase delay. And

consequently, there is no phase distortion due to the time delay of frequencies relative to one another.

For discrete-time signals, perfect linear phase is easily achieved with a finite impulse response (FIR)

filter. Approximations can be achieved with infinite impulse response (IIR) designs, which are more

computationally efficient.

Linear-phase FIR filter can be divided into four basic types.

Type impulse response

I symmetric length is odd

II symmetric length is even

III anti-symmetric length is odd

IV anti-symmetric length is even

When h(n) is nonzero for 0 ≤ n ≤ N −1 (the length of the impulse response h(n) is N), then the

symmetry of the impulse response can be written as

h(n) = h(N − 1 − n)

and the anti-symmetry can be written as

h(n) = −h(N − 1 − n).

5.Why are FIR filters generally preferred over IIR filters in multi rate (decimating and interpolating)

systems?

Because only a fraction of the calculations that would be required to implement a decimating or interpolating

FIR in a literal way actually needs to be done.

Since FIR filters do not use feedback, only those outputs which are actually going to be used have to be

calculated. Therefore, in the case of decimating FIRs (in which only 1 of N outputs will be used), the other

https://en.wikipedia.org/wiki/Filter_(signal_processing)

https://en.wikipedia.org/wiki/Phase_response



https://en.wikipedia.org/wiki/Frequency

https://en.wikipedia.org/wiki/Phase_delay

https://en.wikipedia.org/wiki/Phase_distortion




N-1 outputs don’t have to be calculated. Similarly, for interpolating filters (in which zeroes are inserted

between the input samples to raise the sampling rate) you don’t actually have to multiply the inserted zeroes

with their corresponding FIR coefficients and sum the result; you just omit the multiplication-additions that

are associated with the zeroes (because they don’t change the result anyway.)

In contrast, since IIR filters use feedback, every input must be used, and every input must be calculated

because all inputs and outputs contribute to the feedback in the filter.


1. FIR filters ________ A. are non-recursive

B. do not adopt any feedback

C. are recursive

D. use feedback

a. A & B

b. C & D

c. A & D d. B & C

ANSWER:(a) A & B

2. How is the sensitivity of filter coefficient quantization for FIR filters? a. Low

b. Moderate

c. High

d. Unpredictable

ANSWER: (a) Low

3.The Chebyshev filters have

1) Flat pass band

2) Flat stop band

3) Equiripple pass band

4) Tapering stop band

a. 1 and 2 are correct b. 2 and 4 are correct



ANSWER: 2 and 3 are correct

4. Consider the assertions given below. Which among them is an advantage of FIR Filter?

a. Necessity of computational techniques for filter implementation

b. Requirement of large storage

c. Incapability of simulating prototype analog filters

d. Presence of linear phase response

ANSWER: Presence of linear phase response

5. In FIR filter design, which among the following parameters is/are separately controlled by using Kaiser

window?

a. Order of filter (M)

b. Transition width of main lobe

c. Both a and b


ANSWER: Both a and b

6. Which window function is also regarded as 'Raised-cosine window'?

a. Hamming window

b. Hanning window

c. Barlett window

d. Blackman window

ANSWER: Hanning window

7. In Barlett window, the triangular function resembles the tapering of rectangular window sequence

_______ from the middle to the ends.

a. linearly

b. elliptically

c. hyperbolically

d. parabolically

ANSWER: linearly

8. In FIR filters, which among the following parameters remains unaffected by the quantization effect?

a. Magnitude Response

b. Phase Characteristics

c. Both a and b


ANSWER: Phase Characteristics

9. In the frequency response characteristics of FIR filter, the number of bits per coefficient should be

_________in order to maintain the same error.

a. Increased

b. Constant

c. Decreased


ANSWER: Increased

10. A filter is said to be linear phase filter if the phase delay and group delay are _______

a. High

b. Moderate

c. Low

d. Constant

ANSWER: Constant

FILL IN THE BLANKS

1. In linear phase FIR filters filter is most versatile

2. FIR filters are always stable because its poles lies at

3. FIR filters have exactly phase

4. Width of the main lobe is wider than the in window design method

5. FIR filter is of length M then the order of FIR filter N is given as

6. Filters also known as feed forward filters

7. Transfer function of the FIR filter is given as

8. Width of main lobe of rectangular window is

9. Width of the transition region between pass band and stop band in H(W) increases with the

of the main lobe

10. Width of main lobe of blackmann window is

Answers

1. Type-I

2. Origin

3. Linear

4. Transition bandwidth

5. N=M-1

6. FIR

m N

7. ∑bk z-k/1+∑ak z

-k K=0 k=1

8. 4π/M

9. Width

10. 12π/M

UNIT V


1.Give the different quantization errors occur due to finite word length registers in digital filters?

Input quantization errors

Coefficient quantization errors

Product quantization errors

2. Mention the different quantization methods available for Finite Word Length Effects? 1. Truncation 2. Rounding

3. State truncation?

Truncation is a process of discarding all bits less significant than LSB that is retained

4. Define Rounding?

Rounding a number to b bits is accomplished by choosing a rounded result as the b

bit number closest number being unrounded.

5. List the two types of limit cycle behavior of DSP?

1. Zero limit cycle behavior 2. Over flow limit cycle behavior


MA.Sohana Parveen, Assistant. Professor


1. Mention the methods to prevent overflow?

Saturation arithmetic and Scaling

2. Define overflow oscillations

The overflow caused by adder makes the filter output to oscillate between

maximum amplitude limits and such oscillations is referred as overflow

oscillations

3. What do you understand by input quantization error?

.In digital signal processing, the continuous time input signals are converted into

digital by using b bit ADC. The representation of continuous signal amplitude by

a fixed digit produces an error, which is known as input quantization error.

4. Determine Dead Band of the Filter.

The Limit cycle occurs as a result of quantization effect in multiplication. The

amplitude of output during a limit cycle are confined to a range of values called

the dead band of the filter

5. Why Rounding is preferred to truncation in realizing digital filter?

• The quantization error due to rounding is independent of type arithmetic • The mean of rounding error is zero • The variance of rounding error is low


1. Distinguish between fixed point and floating point arithmetic

S.No Fixed Point Arithmetic Floating Point Arithmetic

1. Fast Operation Slow Operation

2. Relatively Economical More expensive because of

costlier hardware

3. Overflow occurs in addition Overflow does not arise

4. Used in small computers Used in large general purpose

computers

5. Small Dynamic Range Increased dynamic range



2. Draw the quantization noise model for a I order System

3. What do you understand by (Zero input) Limit cycle oscillations?

When a stable IIR filter digital filter is excited by a finite sequence, that is constant, the

output will ideally decay to zero. However, the non-linearity due to finite precision

arithmetic operations often causes periodic oscillations to occur in the output. Such

oscillations occur in the recursive systems are called Zero input Limit Cycle Oscillation.

Normally oscillations in the absence of output u (k) =0 by equation given below is called

Limit cycle oscillations

Y[k] = -0.625.y [k-1] +u[k]

4. Explain decimation process

Decimation can be regarded as the discrete-time counterpart of sampling. Whereas

in sampling we start with a continuous-time signal x(t) and convert it into a

sequence of samples x[n], in decimation we start with a discrete-time signal x[n]

and convert it into another discrete-time signal y[n], which consists of sub-samples

of x[n]. Thus, the formal definition of M-fold decimation, or down-sampling, is

defined by Equation. In decimation, the sampling rate is reduced from Fs to Fs/M

by discarding M – 1 samples for every M samples in the original sequence



∞

y[ n]= v[nM ]=∑ h[k] x[nM -k] k=-∞

5. write short notes on interpolation

Interpolation is the exact opposite of decimation. It is an information preserving

operation, in that all samples of x[n] are present in the expanded signal y[n]. The

mathematical definition of L-fold interpolation is defined by Equation and the

block diagram notation is depicted in below Figure Interpolation works by

inserting (L–1) zero-valued samples for each input sample. The sampling rate

therefore increases from Fs to LFs. With reference to Figure, the expansion process

is followed by a unique digital low-pass filter called an anti-imaging filter.

Although the expansion process does not cause aliasing in the interpolated signal,

it does however yield undesirable replicas in the signal’s frequency spectrum.

y[n] =L h[k]w[n-k]




1. Decimation is a process in which the sampling rate is __________.

a. enhanced b. stable

c. reduced

d. unpredictable

ANSWER:(c) reduced

2. Anti-imaging filter with cut-off frequency ωc = π/ I is specifically used _______ upsampling process for the removal of unwanted images.

a. Before

b. At the time of

c. After d. All of the above

ANSWER: (c) After



3. The effects caused due to finite word lengths are

1) Coefficient quantization error

2) Adder overflow limit cycle

3) Round off noise

4) Limit cycles



c. 1 and 4 are correct d. All the four are correct

ANSWER: All the four are correct

4. The error in the filter output that results from rounding or truncating calculations within the filter

is called

a. Coefficient quantization error b. Adder overflow limit cycle

c. Round off noise

d. Limit cycles

ANSWER: Round off noise

5. To change the sampling rate for better efficiency in two or multiple stages, The decimation and

interpolation factors must be _________unity.

a. Less than

b. Equal to

c. Greater than


ANSWER: Greater than

6. How is/are the roundoff errors reduced in the digital FIR filter?

a. By representation of all products with double-length registers

b. By rounding the results after acquiring the final sum

c. Both a and b





7. Consider the assertions (steps) given below. Which among the following is a correct sequence of

designing steps for the sampling rate converters?

A. Computation of decimation/interpolation factor for each stage.

B. Clarification of anti-aliasing / anti-imaging filter requirements.

C. Designing of filter at each stage.

D. Calculation of optimum stages of decimation/ interpolation yielding maximum efficient

implementation.

a. A, B, C, D

b. C, A, D, B

c. D, A, B, C d. B, D, A, C

ANSWER: B, D, A, C

8. How is the sampling rate conversion achieved by factor I/D?

a. By increase in the sampling rate with (I)

b. By filtering the sequence to remove unwanted images of spectra of original signal

c. By decimation of filtered signal with factor D

d. All of the above

ANSWER: All of the above

9. Which is/are the correct way/s for the result quantization of an arithmetic operation?

a. Result Truncation

b. Result Rounding c. Both a and b



10.the quantization noise can be reduced

a. by increasing step size

b. by reducing step size

c. by reducing no.of bits used for quantization



d. none

ANSWER: b

FILL IN THE BLANKS

1. occurs as a result of the quantization effects in multiplication 2. The conversion of continuous time input signal into digital value produces an error, which is

known as 3. Does not arise in floating point arithmetic 4. Is a process used to increase the sampling rate 5. The amplitudes of output during a limit cycle are confined to the range of values that is

called the of the filter 6. Filter is used in decimation process 7. In fixed point arithmetic round off error occurs only for 8. Arise at the output of multiplier 9. Is a process of discarding all bits less significant than LSB that is retained 10. Decimation is a process which is used to the sampling rate

Answers

1. Limit cycle 2. Input quantization error 3. Overflow 4. Interpolation 5. Dead band 6. Anti-aliasing 7. Addition 8. Product quantization error 9. Truncation 10. decrease



COURSE FILE CONTENTS

S.No. Topics Page No.

1 Vision, Mission, PEO’s, PO’s & PSO’S

2 Syllabus (University Copy)

3 Course Objectives, Course Outcomes And Topic Outcomes

4 Course Prerequisites

5 Course Information Sheet (CIS)

a). Course Description

b). Syllabus

c). Gaps in Syllabus

d). Topics beyond syllabus

e). Web Sources-References

f). Delivery / Instructional Methodologies

g). Assessment Methodologies-Direct

h). Assessment Methodologies –Indirect

i). Text books & Reference books

6 Micro Lesson Plan

7 Teaching Schedule

8 Unit Wise Hand Written notes

9 OHP/LCD SHEETS /CDS/DVDS/PPT (Soft/Hard copies)

10 University Previous Question papers

11 MID exam Descriptive Question Papers

12 MID exam Objective Question papers

13 Assignment topics with materials

14 Tutorial topics and Questions

15 Unit wise-Question bank

1 Two marks question with answers 5 questions

2 Three marks question with answers 5 questions

3 Five marks question with answers 5 questions

4 Objective question with answers 10 questions

5 Fill in the blanks question with answers 10 questions

16 Course Attainment

17 CO-PO Mapping

18 Beyond syllabus Topics with material

19 Result Analysis-Remedial/Corrective Action

20 Record of Tutorial Classes

21 Record of Remedial Classes

22 Record of guest lecturers conducted



Part – 2

S.NO TOPICS

1 Attendance Register/Teacher Log Book

2 Time Table

3 Academic calendar

4 Continuous Evaluation – marks (Test, Assignments etc)

5 Status Report Internal Exams & Syllabus coverage

6 Teaching Dairy/Daily Delivery Record Micro lesson Plan

7 Continuous Evaluation – MID marks

8 Assignment Evaluation-marks/Grades

9 Special Descriptive Tests Marks

10 Sample students descriptive answer sheets

11 Sample students assignment sheets



1. Vision, Mission, Program Educational Objectives (PEOs),

Program Outcomes (POs), Program Specific Outcomes

(PSOs).

VISION

To be recognized as a full-fledged center for learning and research in various fields of

Electronics and Communication Engineering through industrial collaboration and provide

consultancy for solving the real time problems

MISSION

To inculcate a spirit of research and teach the students about contemporary

technologies in Electronics and Communication to meet the growing needs of the

industry.

To enhance the practical knowledge of students by implementing projects based

on real time problems through industrial collaboration



PROGRAMME EDUCATIONAL OBJECTIVES (PEO’s)

After 3-5 years from the year of graduation, our graduates will,

1. With an exposure to the areas of VLSI, Embedded Systems, Signal / Image

Processing, Communications, Wave theory, and Electronic circuits in modern

electronics and communications environment.

2. Demonstrate the impact of Electronics and Communications Engineering on the

society, ethical, social and professional responsibilities/implications of their work.

3. With strong foundations in mathematical, scientific and engineering fundamentals

necessary to formulate, solve and analyze Electronics and communications.

Engineering problems.

4. Have strong communication inter-personal skills, multicultural adoptability and to

work effectively in multidisciplinary teams.

5. Engage life-long learning to become successful in their professional work.

PEO 1: To inculcate the adaptability skills into the students for design and use of

analog and digital circuits and communications and any other allied fields of

Electronics.

PEO 2: Ability to understand and analyze engineering issues in a broader perspective

with ethical responsibility towards sustainable development.

PEO 3: To develop professional skills in students that prepares them for immediate

employment and for lifelong learning in advanced areas of Electronics and

communications and related fields.

PEO 4: To equip with skills for solving complex real-world problems related to VLSI,

Embedded Systems, Signal/Image processing, Communications, and Wave

theory.

PEO 5: Graduates will make valid judgment, synthesize information from a range of

sources and communicate them in sound ways in order to find an

economically viable solution.

PEO 6: To develop overall personality and character with team spirit, professionalism,

integrity, and moral values with the support of humanities, social sciences and

physical educational courses.



PROGRAMME OUTCOMES (PO’s)

A graduate of the Electronics and Communication Engineering Program will demonstrate:

PO1: Engineering knowledge: Apply the knowledge of mathematics, science, engineering

fundamentals, and an engineering specialization to the solution of complex engineering

problems.

PO2: Problem analysis: Identify, formulate, review research literature, and analyze complex

engineering problems reaching substantiated conclusions using first principles of

mathematics, natural sciences, and engineering sciences.

PO3: Design/development of solutions: Design solutions for complex engineering problems

and design system components or processes that meet the specified needs with appropriate

consideration for the public health and safety, and the cultural, societal, and environmental

considerations.

PO4: Conduct investigations of complex problems: Use research-based knowledge and

research methods including design of experiments, analysis and interpretation of data, and

synthesis of the information to provide valid conclusions.

PO5: Modern tool usage: Create, select, and apply appropriate techniques, resources, and

modern engineering and IT tools including prediction and modeling to complex

engineering activities with an understanding of the limitations.

PO6: The engineer and society: Apply reasoning informed by the contextual knowledge to

assess societal, health, safety, legal and cultural issues and the consequent responsibilities

relevant to the professional engineering practice.

PO7: Environment and sustainability: Understand the impact of the professional engineering

solutions in societal and environmental contexts, and demonstrate the knowledge of, and

need for sustainable development.

PO8: Ethics: Apply ethical principles and commit to professional ethics and responsibilities and

norms of the engineering practice.

PO9: Individual and team work: Function effectively as an individual, and as a member or

leader in diverse teams, and in multidisciplinary settings.

PO10: Communication: Communicate effectively on complex engineering activities with the



engineering community and with society at large, such as, being able to comprehend and

write effective reports and design documentation, make effective presentations, and give

and receive clear instructions.

PO11: Project management and finance: Demonstrate knowledge and understanding of the

engineering and management principles and apply these to one’s own work, as a member

and leader in a team, to manage projects and in multidisciplinary environments.

PO12: Life-long learning: Recognize the need for, and have the preparation and ability to engage

in independent and life-long learning in the broadest context of technological change.



PROGRAMME SPECIFIC OUTCOMES (PSO’s)

PSO 1: Problem Solving Skills – Graduate will be able to apply latest electronics

techniques and communications principles for designing of communications

systems.

PSO 2: Professional Skills – Graduate will be able to develop efficient and effective

Communications systems using modern Electronics and Communications

engineering techniques.

PSO 3: Successful Career – To produce graduates with a solid foundation in

Electronics and Communications engineering who will pursue lifelong

learning and professional development including post graduation.

PSO 4: The Engineer and Society– Ability to apply the acquired knowledge for the

advancement of society and self.

(EE721PE) DIGITAL SIGNAL PROCESSING

IV year B.Tech.EEE-I sem L/T/P/D C

3/-/-/- 3

UNIT -I

Introduction: Introduction to Digital Signal Processing: Discrete Tim signals &

Sequences, Linear Shift Invariant Systems, Stability, and Causality, Linear Constant

Coefficient Difference Equations, Frequency Domain Representation of Discrete Time

Signals and Systems

Realization of Digital Filters: Applications of Z — Transforms, Solution Difference

Equations of Digital Filters, System Function, Stability Criterion frequency Response of

Stable Systems, Realization of Digital Filters — Direct, Canonic, Cascade and Parallel

Forms.

UNIT -II Discrete Fourier series: DFS Representation of Periodic Sequence properties of Discrete

Fourier Series, Discrete Fourier Transforms: Properties of DFT, Linear Convolution of

Sequences using DFT, Computation of D Over-Lap Add Method, Over-Lap Save Method,

Relation between DTFT, DFS, DFT and Z-Transform.

Fast Fourier Transforms: Fast Fourier Transforms (FFT) – Radix-2 Decimation-in-Time and

Decimation-in-Frequency FFT Algorithms, Inverse FFT, and FFT with General Radix-N.

UNIT- III IIR Digital Filters: Analog filter approximations – Butter worth and Chebyshev, Design of IIR

Digital Filters from Analog Filters, Step and Impulse Invariant Techniques, Bilinear

Transformation Method, Spectral Transformations.

UNIT-IV FIR Digital Filters: Characteristics of FIR Digital Filters, Frequency Response, Design of FIR

Filters: Fourier Method, Digital Filters using Window Techniques, Frequency Sampling

Technique, Comparison of IIR & FIR filters

UNIT-V Multi rate Digital Signal Processing: Introduction, Down Sampling Decimation, Up sampling,

Interpolation, Sampling Rate Conversion. Finite Word Length Effects: Limit cycles, Overflow

Oscillations, Round-off Noise in IIR Digital Filters, Computational Output Round Off Noise,

Methods to Prevent Overflow, Trade Off Between Round Off and Overflow Noise, Dead

Band Effects.

TEXT BOOKS

Digital Signal Processing, Principles, Algorithms, and Applications John G. Proakis,

Dimitris G. Manolakis, Pearson Education / PHI, 2007.

Discrete Time Signal Processing — A. V. Oppenheim and R.W Schaffer, PHI,

2009 Fundamentals of Digital Signal Processing — Loney Ludeman, John Wiley, 2009

REFERENCE BOOKS

Digital Signal Processing — Fundamentals and Applications — Li Tan, Elsevier, 2008

Fundamentals of Digital Signal Processing using MATLAB — Robert Schilling, Sandra L.

Harris, Thomson, 2007

Digital Signal Processing — S.Salivahanan, A.Vallavaraj and C.Gnanapriya, TMH, 2009

Discrete Systems and Digital Signal Processing with MATLAB — Taan EIAIi. CRC press,

2009.

Digital Signal Processing – A Practical approach, Emmanuel C Ifeachor and Barrie W.

Jervis, 2nd Edition, Pearson Education, 2009

Digital Signal Processing – Nagoor Khani, TMG, 2012

GAPS IN THE SYLLABUS

1.DISCRETE TIME FOURIER TRANSFORM

Discrete Time Fourier Transform of an aperiodic discrete time signal

Given a general aperiodic signal of finite duration, that is; for some integer N,

. From this aperiodic signal we can construct a periodic signal for which

is one period. As we chose period N to be larger than the duration of , is identical

to . As the period , for any finite value of n.

The Fourier series representation of is :

Since over a period that includes the interval , it is convenient to choose the

interval of summation to be this period, so that can be replaced by in the summation.

Therefore,

Another way of representing DTFT of a periodic discrete signal

In continuous time, the fourier transform of is an impulse at .However in discrete

time ,for signal the discrete time fourier transform is periodic in with period

. The DTFT of is a train of impulses at

i.e Fourier Transform can be written as :

Consider a periodic sequence x[n] with period N and with fourier series representation

Then discrete time Fourier Transform of a periodic signal x[n] with period N can be written as :

2.APPLICATIONS OF DIGITAL FILTERS

The applications of digital filters are

1. Noise suppression

(a) imaging devices (medical, etc)

(b) biosignals (heart, brain)

(c) signals stored on analog media (tapes)

2. Enhancement of selected frequency ranges

(a) equalizers for audio systems (increasing the bass)

(b) edge enhancement in images

3. Removal or attenuation of selected frequencies

(a) removing the DC component of a signal

(b) removing interferences at a specific frequency, for example those caused by power supplies

4. Bandwidth limiting

(a) anti-aliasing filters for sampling

(b) ensuring that a transmitted signal occupies only its alloted frequency band.

5. Special operations

(a) differentiation

(b) integration

(c) Hilbert transform

6. Simulation/Modeling

(a) simulating communication channels

(b) modeling human auditory system

TOPICS BEYOND SYLLABUS

1.ARCHITECTURE OF TMS320C5X PROCESSOR

2.WAVELET TRANSFORM

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then

decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by

https://en.wikipedia.org/wiki/Wave

https://en.wikipedia.org/wiki/Oscillation

https://en.wikipedia.org/wiki/Amplitude

a seismograph or heart monitor. Generally, wavelets are intentionally crafted to have specific

properties that make them useful for signal processing. Using a "reverse, shift, multiply and

integrate" technique called convolution, wavelets can be combined with known portions of a

damaged signal to extract information from the unknown portions.

All wavelet transforms may be considered forms of time-frequency representation for continuous-

time (analog) signals and so are related to harmonic analysis. Almost all practically useful discrete

wavelet transforms use discrete-time filterbanks. These filter banks are called the wavelet and

scaling coefficients in wavelets nomenclature. These filterbanks may contain either finite impulse

response (FIR) or infinite impulse response (IIR) filters. The wavelets forming a continuous

wavelet transform (CWT) are subject to the uncertainty principle of Fourier analysis respective

sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact

time and frequency response scale to that event. The product of the uncertainties of time and

frequency response scale has a lower bound. Thus, in the scaleogram of a continuous wavelet

transform of this signal, such an event marks an entire region in the time-scale plane, instead of

just one point. Also, discrete wavelet bases may be considered in the context of other forms of the

uncertainty principle.

Wavelet transforms are broadly divided into three classes: continuous, discrete and

multiresolution-based.

Continuous wavelet transforms (continuous shift and scale parameters)

In continuous wavelet transforms, a given signal of finite energy is projected on a continuous

family of frequency bands (or similar subspaces of the Lp function space L2(R)). For instance, the

signal may be represented on every frequency band of the form [f, 2f] for all positive frequencies

f > 0. Then, the original signal can be reconstructed by a suitable integration over all the resulting

frequency components.

The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at scale 1. This

subspace in turn is in most situations generated by the shifts of one generating function ψ in

L2(R), the mother wavelet.

https://en.wikipedia.org/wiki/Seismograph

https://en.wikipedia.org/wiki/Heart_monitor

https://en.wikipedia.org/wiki/Signal_processing

https://en.wikipedia.org/wiki/Convolution

https://en.wikipedia.org/wiki/Time-frequency_representation

https://en.wikipedia.org/wiki/Continuous-time

https://en.wikipedia.org/wiki/Continuous-time

https://en.wikipedia.org/wiki/Harmonic_analysis


https://en.wikipedia.org/wiki/Filterbank




https://en.wikipedia.org/wiki/Continuous_wavelet_transform


https://en.wikipedia.org/wiki/Fourier_uncertainty_principle

https://en.wikipedia.org/wiki/Scaleogram


https://en.wikipedia.org/wiki/Lp_space

https://en.wikipedia.org/wiki/Function_space

Discrete wavelet transforms (discrete shift and scale parameters)

It is computationally impossible to analyze a signal using all wavelet coefficients, so one may

wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct

a signal from the corresponding wavelet coefficients. One such system is the affine system for

some real parameters a > 1, b > 0. The corresponding discrete subset of the halfplane consists of

all the points (am, namb) with m, n in Z.

Multiresolution based discrete wavelet transforms

Fig- D4 wavelet

In any discretised wavelet transform, there are only a finite number of wavelet coefficients for

each bounded rectangular region in the upper halfplane. Still, each coefficient requires the

evaluation of an integral. In special situations this numerical complexity can be avoided if the

scaled and shifted wavelets form a multiresolution analysis. This means that there has to exist an

auxiliary function, the father wavelet φ in L2(R), and that a is an integer. A typical choice is a = 2

and b = 1. The most famous pair of father and mother wavelets is the Daubechies 4-tap wavelet.

https://en.wiktionary.org/wiki/affine

https://en.wikipedia.org/wiki/Multiresolution_analysis

https://en.wikipedia.org/wiki/Daubechies_wavelets

UNIT-I

Learning Objectives

• Why process signals digitally?

• Definition of a real-time application.

• Why use Digital Signal Processing processors?

• What are the typical DSP algorithms?

• Parameters to consider when choosing a DSP processor.

Why go digital?

• Digital signal processing techniques are now so powerful that sometimes it is extremely difficult, if not impossible, for analogue signal processing to achieve similar performance.

• Examples:

– FIR filter with linear phase.

– Adaptive filters.

Why go digital?

• Analogue signal processing is achieved by using analogue components such as:

– Resistors.

– Capacitors.

– Inductors.

• The inherent tolerances associated with these components, temperature, voltage changes and mechanical vibrations can dramatically affect the effectiveness of the analogue circuitry.

Why go digital?• With DSP it is easy to:

– Change applications.

– Correct applications.

– Update applications.

• Additionally DSP reduces:– Noise susceptibility.

– Chip count.

– Development time.

– Cost.

– Power consumption.

Why NOT go digital?

• High frequency signals cannot be processed digitally because of two reasons:

–Analog to Digital Converters, ADCcannot work fast enough.

–The application can be too complex to be performed in real-time.

Real-time processing• DSP processors have to perform tasks in

real-time, so how do we define real-time?

• The definition of real-time depends on the application.

• Example: a 100-tap FIR filter is performed in real-time if the DSP can perform and complete the following operation between two samples:

Real-time processing

• We can say that we have a real-time application if:

– Waiting Time 0

Processing TimeWaiting Time

Sample Time

n n+1

Why do we need DSP processors?• Why not use a General Purpose Processor

(GPP) such as a Pentium instead of a DSP processor?

– What is the power consumption of a Pentium and a DSP processor?

– What is the cost of a Pentium and a DSP processor?

Why do we need DSP processors?

• Use a DSP processor when the following are required:

– Cost saving.

– Smaller size.

– Low power consumption.

– Processing of many “high” frequency signals in real-time.

Introduction to DSP

Analog system Discrete system

Block diagram of DSP system

Discrete Time Signals & Sequences

• Digital signals are discrete in both time (theindependent variable) and amplitude (thedependent variable). Signals that are discretein time but continuous in amplitude arereferred to as discrete-time signals.

Linear-Time Invariant System

• Special importance for their mathematical tractability

• Most signal processing applications involve LTI systems

• LTI system can be completely characterized by their impulse response

Stability

Causality

Linear Constant-Coefficient Difference Equations

• A linear constant-coefficient difference equation of order N looks like:

• All solutions y[n] can be expressed as a sum yh[n]+yp[n]

Frequency Response

Applications of z-transform

• Inputs and outputs are related by difference equations and z-transform techniques are used to solve those difference equations

• Voice transmission

• Transmission systems

Cascade form

Parallel form

23

UNIT-II

Discrete Fourier Series

• Given a periodic sequence with period N so that

• The Fourier series representation can be written as

• The Fourier series representation of continuous-time periodic signals require infinite many complex exponentials

• Not that for discrete-time periodic signals we have

• Due to the periodicity of the complex exponential we only need N exponentials for discrete time Fourier series

]n[x~]rNn[x~]n[x~

k

knN/2jekX~

N

1]n[x~

knN/2jmn2jknN/2jnmNkN/2j eeee

1

0

/2~1][~

N

k

knNjekXN

nx

Discrete Fourier Series Pair

• A periodic sequence in terms of Fourier series coefficients

• The Fourier series coefficients can be obtained via

• For convenience we sometimes use

• Analysis equation

• Synthesis equation

25

1

0

/2][~~ N

n

knNjenxkX

1N

0k

knN/2jekX~

N

1]n[x~

Nj

N eW /2

1N

0k

knNWkX

~

N

1]n[x~

1N

0n

knNW]n[x

~kX~

Example 1

• DFS of a periodic impulse train

• Since the period the signal is N

• We can represent the signal with the DFS coefficients as

else0

rNn1rNn]n[x~

r

1ee]n[e]n[x~kX~ 0kN/2j

1N

0n

knN/2j1N

0n

knN/2j

1N

0k

knN/2j

r

eN

1rNn]n[x~

Example 2

• DFS of an periodic rectangular

• pulse train

• The DFS coefficients

10/sin

2/sin

1

1~ 10/4

10/2

510/24

0

10/2

k

ke

e

eekX kj

kj

kj

n

knj

Properties of DFS

• Linearity

• Shift of a Sequence

• Duality

kX~bkX

~anx~bnx~a

kX~

nx~kX

~nx~

21DFS

21

2DFS

2

1DFS

1

mkX~

nx~e

kX~

emnx~kX~

nx~

DFSN/nm2j

N/km2jDFS

DFS

kx~NnX~

kX~

nx~

DFS

DFS

Symmetry Properties

Symmetry Properties Cont’d

Periodic Convolution• Take two periodic sequences

• Let’s form the product

• The periodic sequence with given DFS can be written as

• Periodic convolution is commutative

kX

~nx~

kX~

nx~

2DFS

2

1DFS

1

kXkXkX 213

~~~

1N

0m213 mnx~mx~nx~

1

0

123~~~

N

m

mnxmxnx

Periodic Convolution Cont’d

• Substitute periodic convolution into the DFS equation

• Interchange summations

• The inner sum is the DFS of shifted sequence

• Substituting

1N

0m213 mnx~mx~nx~

1N

0n

knN2

1N

0m13 W]mn[x~]m[x~kX

~

1N

0m

knN

1N

0n213 W]mn[x~]m[x~kX

~

kXWWmnx km

N

kn

N

N

n

2

1

0

2

~][~

kXkXkXWmxWmnxmxkXN

m

km

N

N

m

kn

N

N

n

21

1

0

21

1

0

1

0

213

~~~][~][~][~~

Graphical Periodic Convolution

The Fourier Transform of Periodic Signals

• Periodic sequences are not absolute or square summable– Hence they don’t have a Fourier Transform

• We can represent them as sums of complex exponentials: DFS

• We can combine DFS and Fourier transform

• Fourier transform of periodic sequences– Periodic impulse train with values proportional to DFS coefficients

– This is periodic with 2 since DFS is periodic

• The inverse transform can be written as

k

j

N

k2kX~

N

2eX~

1N

0k

nN

k2j

nj2

0k

nj2

0k

nj2

0

j

ekX~

N

1de

N

k2kX~

N

1

deN

k2kX~

N

2

2

1deeX

~

2

1

Relation between Finite-length and Periodic Signals

• Consider finite length signal x[n] spanning from 0 to N-1

• Convolve with periodic impulse train• The Fourier transform of the periodic sequence is

• This implies that

• DFS coefficients of a periodic signal can be thought as equally spaced samples of the Fourier transform of one period

rr

rNnxrNn]n[x]n[p~

]n[x]n[x~

N

keX

NeX

N

k

NeXePeXeX

k

N

kj

j

k

jjjj

22~

22~~

2

N

k2jN

k2j

eXeXkX~

Example• Consider the following sequence

• The Fourier transform

• The DFS coefficients

else0

4n01]n[x

2/sin

2/5sineeX 2jj

10/ksin

2/ksinekX

~ 10/k4j

Discrete Fourier Transform (DFT)

• The DFT provides uniformly spaced samples of the Discrete-Time Fourier Transform (DTFT)

• DFT definition:

• Requires N2 complex multiplies and N(N-1) complex additions

1

0

2

][][N

n

N

nkj

enxkX

1

0

2

][1

][N

n

N

nkj

ekXN

nx

Faster DFT computation?

• Take advantage of the symmetry and periodicity of the complex exponential (let WN=e-j2/N)

– symmetry:

– periodicity:

• Note that two length N/2 DFTs take less computation than one length N DFT: 2(N/2)2<N2

• Algorithms that exploit computational savings are collectively called Fast Fourier Transforms

*][ )( kn

N

kn

N

nNk

N WWW

nNk

N

Nnk

N

kn

N WWW ][][

Decimation-in-Time Algorithm

• Consider expressing DFT with even and odd input samples:

1

0

2/

1

0

2/

1

0

2

1

0

2

1

0

22

22

]12[]2[

)](12[)](2[

][][

][][

NN

NN

r

rk

N

k

N

r

rk

N

r

rk

N

k

N

r

rk

N

oddn

nk

N

evenn

nk

N

N

n

nk

N

WrxWWrx

WrxWWrx

WnxWnx

WnxkX

DIT Algorithm (cont.)• Result is the sum of two N/2 length DFTs

• Then repeat decomposition of N/2 to N/4 DFTs, etc.

samples odd ofDFT N/2

sampleseven ofDFT N/2

][][][ kHWkGkX k

N

X[0…7]

x[0,2,4,6]

x[1,3,5,7]

N/2

DFT

N/2

DFT

7...0

NW

Detail of “Butterfly”

• Cross feed of G[k] and H[k] in flow diagram is called a “butterfly”, due to shape

r

NW

)(

)2(

r

N

Nr

N

W

W

r

NW -1

or simplify:

8-point DIT-FFT Diagram

0

NW

0

NW

0

NW

0

NW

0

NW

0

NW

0

NW

2

NW

2

NW

2

NW

1

NW

3

NW

1

1

1

1

1

1

1

1

1

1

1

1

X[0…7]x[0,4,2,6,1,5,3,7]

8-point DIF-FFT Diagram

UNIT-III

Chebyshev 1 Approximation:

The Chebyshev 1 approximation for an ideal lowpass filter has equal-valued ripples in the passband . It is known as minimax approximation and also known as the equirippleapproximation.

The magnitude squared function of Chebyshevapproximation:

where Cn() is the Chebyshev polynomial of degree n. It is defined by

For n = 2, 3, 4, 5, these polynomials are

Chebyshev II Approximation: (or inverse Chebyshevfilters)

Maximally flat at w=0; decreases monotonically as the frequency increases and has an equiripple response in the stopband.

The magnitude square function of the inverse Chebyshevlow pass filter: |H(j)|2 =

Chebyshev I

Chebyshev II

UNIT-IV

Frequency response of FIR digital filters we consider the FIR filters in which the impulse response h[n] are assumed to be symmetric or antisymmetric.

Since the order of the polynomial in each of these two types can be either odd or even, we have four types of filters with different properties.

Linear phase FIR filters:

Type 1: The coefficients are symmetric, i.e., h[n] = h[Nn], and the order N is even.


Type III: The coefficients are antisymmetric, i.e.,

h[n] = h[Nn], and the order N is even.


Type IV: The coefficients are antisymmetric, i.e.,

h[n] = h[Nn], and the order N is odd.

Frequency Sampling Method

1

0 1

2

.1

1).(

1)(

N

k kN

j

N

ze

zkH

NzH

In this approach we are given H(K) we need to

find H(Z)

This is an interpolation problem and the solution

is given in the DFT part of the course

Comparison of FIR & IIR digital filters

UNIT-IV

Frequency response of FIR digital filters we consider the FIR filters in which the impulse response h[n] are assumed to be symmetric or antisymmetric.

Since the order of the polynomial in each of these two types can be either odd or even, we have four types of filters with different properties.


Type 1: The coefficients are symmetric, i.e., h[n] = h[Nn], and the order N is even.


Type III: The coefficients are antisymmetric, i.e.,

h[n] = h[Nn], and the order N is even.


Type IV: The coefficients are antisymmetric, i.e.,

h[n] = h[Nn], and the order N is odd.

Frequency Sampling Method

1

0 1

2

.1

1).(

1)(

N

k kN

j

N

ze

zkH

NzH

In this approach we are given H(K) we need to

find H(Z)

This is an interpolation problem and the solution

is given in the DFT part of the course

Comparison of FIR & IIR digital filters

Thank you

SET-1

III B.TECH - II SEMESTER EXAMINATIONS, APRIL/MAY, 2011


(COMMON TO EEE, ECE, EIE, ETM, ICE)

Time: 3hours Max. Marks: 80

Answer any FIVE questions

All Questions Carry Equal Marks

1. a) Define an LTI System and show that the output of an LTI system is given by the convolution of

Input sequence and impulse response.

b) Prove that the system defined by the following difference equation is an LTI system y(n) =

x(n+1)- 3x(n)+x(n-1) ; n≥0. [8+8 ]

2. a) Define DFT and IDFT. State any Four properties of DFT.

b) Find 8-Point DFT of the given time domain sequence x(n) = 1, 2, 3, 4. [8+8]

3. a) Derive the expressions for computing the FFT using DIT algorithm and hence draw the standard

butterfly structure.

b) Compare the computational complexity of FFT and DFT. [8+8]

4. Discuss and draw various IIR realization structures like Direct form – I, Direct form-II, Parallel and

cascade forms for the difference equation given y(n) = - 3/8 Y(n-1) + 3/32 y(n-2) + 1/64 y(n-3) + x(n)

+ 3 x(n-1) + 2 x(n-2).

5. a) Compare Butterworth and Chebyshev approximation techniques.

b) Design a Digital Butterworth LPF using Bilinear transformation technique for the following

specifications 0.707 ≤ | H(w) | ≤ 1 ; 0 ≤ w ≤ 0.2π | H(w) | ≤ 0.08 ; 0.4 π ≤ w ≤ [ 8+8]

6. a) Compare FIR and IIR filters

b) Design an FIR Digital High pass filter using Hamming window whose cut off freq is 1.2 rad/s

and length of window N=9. [8+8]

7. a) Define Multirate systems and Sampling rate conversion

b) Discuss the process of n Decimation by a factor D and explain how the aliasing effect can be

eliminated. [8+8]

8. Discuss various Modified Bus structures of Programmable DSP Processors.[16]

SET-2







1. a) Write short notes on classification of systems.

b) Derive BIBO stability criteria to achieve stability of a system. [8+8]

2 .a) Define DFS. State any Four properties of DFS.

b) Find the IDFT of the given sequence x(K) = 2, 2-3j, 2+3j, -2. [8+8]

3.a) Find X(K) of the given sequence x(n) = 1,2,3,4,4,3,2,1using DIT- FFT algorithm

b) Compare the computational complexity of FFT and DFT. [8+8]

4. What are the various basic building blocks in realization of Digital Systems and hence discuss

transposed form realization structures.

5. a) Compare Impulse Invariant and Bilinear transformation techniques.

b) Compute the poles of an Analog Chebyshev filter TF that satisfies the Constraints 0.707 ≤ |

H(jΩ)| ≤ 1 ; 0 ≤ Ω≤ 2 | H(jΩ)| ≤ 0.1 ; Ω ≥ 4 and determine Ha(s) and hence obtain H(z) using Bilinear

transformation. [16]

6. a) Derive the conditions to achieve Linear Phase characteristics of FIR filters

b) Design an FIR Digital Low pass filter using Hanning window whose cut off freq is 2 rad/s and

length of window N=9. [8+8]

7. a) Discuss the implementation of Polyphase filters for Interpolators with an example

b) Discuss the sampling rate conversion by a factor I/D with the help of a Neat block Diagram.

[8+8]

8. Write short notes on:

a) VLIW Architecture of Programmable Digital Signal Processors

b) Multiplier and Multiplier Accumulator

SET-3







1. a) Discuss various discrete time sequences.

b) Give the Basic block diagram of Digital Signal Processor. [8+8]

2. a) Define DFS. State any Four properties of DFS.

b) Find the IDFT of the given sequence x(K) = 2, 2-3j, 2+3j, -2. [8+8]

3.a) Find IFFT of the given X(K) = 1,2,3,4,4,3,2,1using DIF algorithm

b) Bring out the relationship between DFT and Z-transform. [8+8]

4. a) Define Z-Transform and List out its properties.

b) Discuss Direct form, Cascade and Linear phase realization structures of FIR filters. [8+8 ]

5. a) Discuss digital and analog frequency transformation techniques.

b) Discuss IIR filter design using Bilinear transformation and hence discuss frequency warping

effect. [8+8]

6. a) Compare various windowing functions.

b) Design an FIR Digital Low pass filter using rectangular window whose cut off freq is 2 rad/s and

length of window N=9. [8+8]

7. a) Define Interpolation and Decimation. List out the advantages of Sampling rate conversion.

b) Discuss the sampling rate conversion by a factor I with the help of a Neat block Diagram.[8+8]

8. a) Discuss Various Addressing modes of Programmable Digital Signal Processors.

b) Give the Internal Architecture of TMS320C5X 16 bit fixed point processor.[ 8+8]

SET-4







1. a) Define Linearity, Time Invariant, Stability and Causality.

b) The discrete time system is represented by the following difference equations in which x(n) is

input and y(n) is output. Y(n) = 3y 2 (n-1)- nx(n)+4x(n-1)-2x(n-1). [8+8]

2. a) Define Convolution. Compare Linear and Circular Convolution techniques.

b) Find the Linear convolution of the given two sequences x(n)=1,2 and h(n) =1,2,3 using

DFT and IDFT. [8 +8]

3. a) Develop DIT-FFT algorithm and draw signal flow graphs for decomposing the DFT for N=6 by

considering the factors for N = 6 = 2.3.

b) Bring out the relationship between DFT and Z-transform. [8+8]

4. a) Discuss transposed form structures with an example.

b) Discuss Direct form, Cascade realization structures of FIR filters. [8+8]

5. a) Discuss digital and analog frequency transformation techniques.

b) Discuss IIR filter design using Impulse Invariant transformation and list out its advantages and

Limitations. [8+8]

6. a) Compare various windowing functions

b) Design an FIR Digital Band pass filter using rectangular window whose upper and lower cut off

freq.’s are 1 & 2 rad/s and length of window N = 9. [8+8]

7. a) Define Interpolation and Decimation.

b) Discuss the sampling rate conversion by a factor I/D with the help of a Neat block Diagram. [8

+8]

8. a) Write a short notes on On-Chip peripherals of Programmable DSP’s.

b)Give the Internal Architecture of TMS320C5X 16 bit fixed point processor. [8+8]

SET-1







1. (a) Discuss impulse invariance method of deriving IIR digital filter from corresponding analog

filter.

(b) Use the Bilinear transformation to convert the analog filter with system function H(S) = S +

0.1/(S + 0.1)2 + 9 into a digital IIR filters. Select T = 0.1 and compare the location of the zeros in

H(Z) with the locations of the zeros obtained by applying the impulse invariance method in the

conversion of H(S). [8+8]

2. (a) Design a high pass filter using hamming window with a cut-off frequency of 1.2 radians/second

and N=9

(b) Compare FIR and IIR filters. [10+6]

3. (a) For each of the following systems, determine whether or not the system is i. stable ii. causal iii.

linear iv. shift-invariant. A. T[x(n)] = x(n − n0 ) B. T [x(n)] = e x (n) C. T[x(n)] = a x(n) + b. Justify

your answer.

(b) A system is described by the difference equation y(n)-y(n-1)-y(n-2) = x(n1). Assuming that the

system is initially relaxed, determine its unit sample response h(n). [8+8]

4. (a) Implement the decimation in time FFT algorithm for N=16.

(b) In the above Question how many non - trivial multiplications are R e q u i r e d .

5. (a) Discuss the frequency-domain representation of discrete-time systems and signals by deriving

the necessary relation.

(b) Draw the frequency response of LSI system with impulse response h(n) = a n u(−n) (|a| < 1)

6. (a) Describe how targets can be decided using RADAR

(b) Give an expression for the following parameters relative to RADAR i. Beam width ii.

Maximum unambiguous range

(c) Discuss signal processing in a RADAR system. [6 +6+4]

7. (a) An LTI system is described by the equation y(n)=x(n)+0.81x(n-1)- 0.81x(n2)-0.45y(n-2).

Determine the transfer function of the system. Sketch the poles and zeroes on the Z-plane.

(b) Define stable and unstable systems. Test the condition for stability of the first-order IIR filter

governed by the equation y(n)=x(n)+bx(n1). [8+8]

8. (a) Compute Discrete Fourier transform of the following finite length sequence considered to be of

length N. i. x(n) = δ(n + n0 ) where 0 < n0 < N ii. x(n) = a n where 0 < a < 1.

(b) If x(n) denotes a finite length sequence of length N, show that x((−n))N =x((N − n))N . [8+8]

SET-2







1. (a) Design a high pass filter using hamming window with a cut-off frequency of 1.2 radians/second

and N=9

(b) Compare FIR and IIR filters. [10+6]

2. (a) Describe how targets can be decided using RADAR

(b) Give an expression for the following parameters relative to RADAR i. Beam width ii.

Maximum unambiguous range (c) Discuss signal processing in a RADAR system. [6 +6+4]

3. (a) An LTI system is described by the equation y(n)=x(n)+0.81x(n-1)- 0.81x(n2)-0.45y(n-2).

Determine the transfer function of the system. Sketch the poles and zeroes on the Z-plane.

(b) Define stable and unstable systems. Test the condition for stability of the first-order IIR filter

governed by the equation y(n)=x(n)+bx(n1). [8+8]

4. (a) Compute Discrete Fourier transform of the following finite length sequence considered to be of

length N. i. x(n) = δ(n + n0 ) where 0 < n0 < N ii. x(n) = a n where 0 < a < 1.

(b) If x(n) denotes a finite length sequence of length N, show that x((−n))N = x((N − n))N . [8+8]

5. (a) For each of the following systems, determine whether or not the system is i. stable ii. causal iii.

linear iv. shift-invariant. A. T[x(n)] = x(n − n0 ) B. T [x(n)] = e x (n) C. T[x(n)] = a x(n) + b. Justify

your answer

(b) A system is described by the difference equation y(n)-y(n-1)-y(n-2) = x(n1). Assuming that the

system is initially relaxed, determine its unit sample response h(n). [8+8]

6. (a) Discuss the frequency-domain representation of discrete-time systems and signals by deriving

the necessary relation.

(b) Draw the frequency response of LSI system with impulse response h(n) = a n u(−n) (|a| < 1)

[8+8]

7. (a) Implement the decimation in time FFT algorithm for N=16.

(b) In the above Question how many non - trivial multiplications are required.

8. (a) Discuss impulse invariance method of deriving IIR digital filter from corresponding analog

filter. (b) Use the Bilinear transformation to convert the analog filter with system function H(S) =

S + 0.1/(S + 0.1)2 + 9 into a digital IIR filters. Select T = 0.1 and compare the location of the zeros in

H(Z) with the locations of the zeros obtained by applying the impulse invariance method in the

conversion of H(S). [8+8]

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