EE6403 DISCRETE TIME SYSTEMS AND SIGNAL PROCESSING...

EE6403 DISCRETE TIME SYSTEMS AND SIGNAL PROCESSING

1 | Prepared by Mr.S.SANTHOSH M.E., ASST.PROF/ECE/SJCE


QUESTION BANK

UNIT I INTRODUCTION

1. What are energy and power signals? (APR/MAY 17)

Energy signal: A finite energy signal is periodic sequence, which has a finite energy but zero

average power.

Power signal: An Infinite energy signal with finite average power is called a power signal.

2.Describe sampling theorem. (APR/MAY 17)

According to the sampling theorem, a band limited continuous time signal x(t) can

be represented in its samples and can be recovered back when sampling frequency fs is

greater than or equal to the twice the highest frequency component of message signal Also,

the sampling rate of 2fm samples per second is called the nyquist rate and its reciprocal

1/2πfm is called nyquist period.

3. Distinguish between digital signal & discrete-signal representation.(NOV/DEC 16)

Discrete signal is well known as Discrete time signal which holds a specific value at a

specific time instant. It can be obtained by sampling a continuous signal. Here the sequence is

formed by collecting samples at various time instants these instants may be regular or

irregular.

Fig: Discrete Time signal

A digital signal is a signal that represents a sequence of discrete values. In digital

electronics a digital signal is a train of pulses, i.e. a sequence of fixed-width square-wave

Electrical pulses or light pulses, each occupying one of a discrete number of levels of

amplitude.



Fig:Digital Signal

4.IF x(n)=x(n+1)+x(n-2),is the system causal system? (NOV/DEC 16)

ANS: The system is non causal system(present, past& future )

5.Determine the system is causal system y(n)=x(n)+1/(x(n-1)) (MAY/JUNE 16)

ANS: The system is causal system(present, past only)

6. What is anti-aliasing filter. (MAY/JUNE 16)

A filter is used to reject frequency signals before it is sampled to reduce the aliasing is anti-

aliasing filter

7. Given a continuous time signal x(t)= 2cos500πt. What is the Nyquist rate and

fundamental frequency of the signal? (NOV/DEC 15)

ω=500π.

2πf=250π.

f=250Hz.

Nyquist rate Fs=2fm= 2x250= 500Hz.

8. Determine whether x[n]=u[n] is a power signal or an energy signal.

The energy of a discrete time signal x(n) is defined as

The average power of a discrete time signal x(n) is defined as

Here E=infinite and P= Finite. Therefore the given signal is a power signal.

9. . Define static and dynamic systems

When the output of the system depends only upon the present input sample, then it is called

static system, otherwise if the system depends pa t values of input then it is called dynamic

system

10. Define stability of a DT system.



11. Mention few applications of Digital Signal processing

* Speech processing – Speech compression & decompression for voice storage system .

* Communication – Elimination of noise by filtering and echo cancellation.

* Bio-Medical – Spectrum analysis of ECG, EEG etc.

12. Write down the expressions for discrete time unit impulse and unit step functions?

Impulse signal δ(n): The impulse signal is defined as a signal having unit magnitude at n = 0

and zero for other values of n

Unit step signal u(n): The unit step signal is defined as a signal having unit magnitude

for all values of n ≥ 0 . U(n)=1 for n≥1 ,0; n<0

13. What is Discrete Time Systems?

The function of discrete time systems is to process a given input sequence to generate output

sequence. In practical discrete time systems, all signals are digital signals, and operations on

such signals also lead to digital signals. Such discrete time systems are called digital filter.

14. Write the Various classifications of Discrete-Time systems.

Linear & Non linear system

Causal & Non Causal system

Stable & Un stable system

15. Define Linear system.

A system is said to be linear system if it satisfies Super position principle. Let us consider

x1(n) & x2(n) be the two input sequences & y1(n) & y2(n) are the responses respectively,

T[ax1(n) + bx2(n)] = a y1(n) + by2(n).

16. Define Signal& System.

A signal is a function of one or more independent variables which contain some information.

Eg: Radio signal, TV signal, Telephone signal etc.

A system is a set of elements or functional block that is connected together and produces an

output in response to an input signal. Eg: An audio amplifier, attenuator, TV set etc.

17. Is the system Y(n)=ln{x(n)} linear and time invariant?

Sol: Nonlinear, Time Invariant.

18. What is quantization error? (May 2015)

Quantization also forms the core of essentially all lossy compression algorithms. The

difference between an input value and its quantized value (such as round-off error) is

referred to as quantization error.



19. What are the advantages of DSP?

Cheaper, Greater accuracy

Ease of data storage

Implementation of sophisticated algorithms.

Flexibility in configuration

Repeatability

Adaptability

Universal Compatibility

VLF Signal Applications

20. What are the basic elements used to construct the block diagram of discrete time

system?

The basic elements used to construct the block diagram of discrete time Systems are Adder,

constant multiplier &Unit delay element.

UNIT II DISCRETE TIME SYSTEM ANALYSIS

1. What are the properties of (ROC) region of convergence? (APR/MAY 17)

ROC contains strip lines parallel to jω axis in s-plane.

If x(t) is absolutely integral and it is of finite duration, then ROC is entire s-plane.

If x(t) is a right sided sequence then ROC : Re{s} > σo.

If x(t) is a left sided sequence then ROC : Re{s} < σo.

If x(t) is a two sided sequence then ROC is the combination of two regions.

2.Find the convalution of the input signal {1,2,1} and its impulse is {1,1,1} using Z

transform? (APR/MAY 17)

Sol:{1,3,4,3,1}

3. Explain the relationship between laplace transform & fourier transform.

(NOV/DEC 16)

X(jΩ) = X(s) when s=jΩ This states that laplace transform is same as fourier transform when

s=jΩ.

4. What is meant by region of convergence(ROC) of Z transform?state its properties.

The region of conve X(z) is the set of all values of z for which X(z) attains a finite value.

5. State initial value theorem& final value theorem.

If x(n)and X(z) are z-Transform pairs, then the initial value of x(z) ,

provided that the first derivative of x(t) should be Laplace transformable.

If x(n)and X(z) are z-Transform pairs, then the final value of x(z) is given as



have no pole on or outside the unit circle.

6. State the convolution property of fourier transform.

If x1(t) and x1(f) is fourier transform pairs and x2(t) and x2(f) are fourier transform pairs,

then F[x(t)*h(t)] = X(jω).H(jω)

7. Find the Laplace transform of the signal x(t) = e-at u(t). (Apr/May, 2010)

X(s) = 1/s+a

8. What are the properties of convolution?

1. Commutative property x(n) * h(n) = h(n) * x(n)

2. Associative property [x(n) * h1(n)]*h2(n) = x(n)*[h1(n) * h2(n)]

9. Find Z transform of x(n)={1,2,3,4}.

SOL:x(n)={1,2,3,4}

X(z)= ∑x(n)z^-n = 1+2z-1+3z-2+4z-3.

10. List the methods of obtaining inverse Z transform.

Inverse z transform can be obtained by using

Partial fraction expansion.

Contour integration.

Power series expansion.

Convolution.

11. Obtain the inverse z transform of X(z)=1/z-a,|z|>|a|

Given X(z)=z-1/1-az-1 By time shifting property X(n)=an.u(n-1)

12. State parseval’s theorem.

Consider the complex valued sequences x(n) and y(n).If x(n)y*(n)=1/N X(k)Y*(k).

13. Define Z transform.

The Z transform of a discrete time signal x(n) is denoted by X(z) and is given by

14. What z transform of X(n-m)?

Sol: By time shifting property Z[X (n-m)] = z-k X(z)

15. State the time shifting property of discrete time Fourier transform. (Apr/May,

2012)

Sol: If F[x(n)] =X(ejω) Then F[x(n-k)] = e-jωkX(ejω)

16. What is the Z transform of δ(n+k)? May13

Sol: X(Z)=ZkX(Z)

17. State the relation between DTFT and Z-transform. Nov 15

sol: Z= ejω

18. State the convolution property of z – transform. (Nov 2007) (NOV 2013)

The convolution property states that the convolution of two sequences in time domain is

equivalent to multiplication of their Z transform.

19. List the methods of obtaining inverse Z transform.

Inverse z transform can be obtained by using Partial fraction expansion. Contour integration

Power series expansion Convolution.

20. Compare linear convolution and circular convolution.(Nov 2011)



S.No Linear Convolution Circular Convolution

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

samples and h(n) with M number of

samples,after convolution y(n) will

contain N = L + M -1 samples.

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

samples and h(n) with M number of

samples,after convolution y(n) will

contain N = Max(L, M) samples

2 Linear convolution can be used to find

the response of a linear filter.

Circular convolution cannot be used to

find the response of a linear filter

3 Zero padding is not necessary to find

the response of a linear filter

. Zero padding is necessary to find the

response of a linear filter.

UNIT III DISCRETE FOURIER TRANSFORM & COMPUTATION

1. Define twiddle factor. Write its magnitude & phase angle. (APR/MAY 17)

The Twiddle factor is defined as WN=e-j2 /N

The complex number N is called phase factor or twiddle factor.

2. Calculate the number of multiplications & additions for 32 point DFT and FFT.

(APR/MAY 17)

For the computation of N-point DFT, N^2 complex multiplications and N[N-1] Complex

additions are required.

For the computation of N-point FFT ,Complex multiplications = N/2 log2N &

Complex additions = N log2N

For N=32, the number of the complex multiplications is equal to 32/2log232=16*5=80.

3. Why is it required to do zero padding in DFT analysis? (NOV/DEC 16)

Let the sequence x (n) has a length L. If we want to find the N point DFT (N>L) of the

sequence x(n),we have to add (N-L) zeros to the sequence x(n). This is known as zero

padding. The uses of padding a sequence with zeros are (i) Better display of the frequency

spectrum.(ii) With zero padding, the DFT can be used in linear filtering.

4.What is the need for windowing techniques in Fourier transform signals ?

(NOV/DEC 16)

In FIR filter design by Fourier series method the infinite duration impulse response is

truncated to finite duration impulse response at n= . The abrupt truncation of impulse

response introduces oscillations in the passband and stopband. This effect is known as Gibb’s

phenomenon (or Gibb’s Oscillation). These oscillations can be reduced through the use of

less abrupt truncation of the fourier series.This can be achieved by multiplying the infinite

impulse response by a finite weighing sequence w(n),called a window.

5. Draw the basic butterfly diagram for radix -2 DIT -FFT .



6. What are the applications of FFT algorithms?

1. Linear filtering 2. Correlation 3. Spectrum analysis

7. what is the speed of improvement factor in calculating 256-point DFT of a sequence

using direct computation and computation and FFT algorithms? (NOV/DEC 15)

The number of complex multiplications required using direct computation is N^2 = 256^2.

The number of complex multiplications required using FFT is = N/2 LOG N.Speed

improvement factor = N^2/ N/2 LOG N.

8. Define DTFT.

Let us consider the discrete time signal x(n).Its DTFT is denoted as X(w).It is given as X(w)=

x(n)e-jwn.

9. State the condition for existence of DTFT?

The conditions are • If x(n)is absolutely summable then |x(n)|< • If x(n) is not absolutely

summable then it should have finite energy for DTFT to exit.

10. List the properties of DTFT.

Periodicity Linearity Time shift Frequency shift Scaling Differentiation in frequency domain

Time reversal Convolution Multiplication in time domain.

11. What is DFT?

It is a finite duration discrete frequency sequence, which is obtained by sampling one period

of Fourier transform. Sampling is done at N equally spaced points over the period extending

from w=0 to 2л.

12. What is the way to reduce number of arithmetic operations during DFT

computation?

Number of arithmetic operations involved in the computation of DFT is greatly reduse using

different FFT algorithms as follows.

1. Radix-2 FFT algorithms. -Radix-2 Decimation in Time (DIT) algorithm. - Radix-2

Decimation in Frequency (DIF) algorithm.



2. Radix-4 FFT algorithm.

13. What is the computational complexity using FFT algorithm?

1. Complex multiplications = N/2 log2N 2. Complex additions = N log2N.

14. What is zero padding? What are its uses?

This is known as zero padding. The uses of padding a sequence with zeros are (i) We can get

‘better display’ of the frequency spectrum. (ii) With zero padding, the DFT can be used in

linear filtering.

15. How many multiplications and additions are required to compute N-point DFT

using redix-2 FFT?

The number of multiplications and additions required to compute N-point DFT using redix-2

FFT are N log2N and N/2 log 2N respectively.

16. What is meant by radix-2 FFT?

The FFT algorithm is most efficient in calculating N-point DFT. If the number of output

points N can be expressed as a power of 2, that is, N=2M, where M is an integer, Then this

algorithm is known as radix-s FFT algorithm.

17. What are the applications of FFT algorithms?

1. Linear filtering 2. Correlation 3. Spectrum analysis

18. What is a decimation-in-frequency algorithm?

In this the output sequence X (K) is divided into two N/2 point sequences and each N/2 point

sequences are in turn divided into two N/4 point sequences.

19. Distinguish between DFT and DTFT.

20. List any four properties of DFT. (Nov 2005, Nov 2009) (MAY 2014) (May 2015)

(i)Periodicity If X(k) is N-point DFT of a finite duration sequencex(n),then X(n+N)=x(n) for

all n X(k+N)=X(k)

(ii)Linearity If X1(k)=DFT[x1(n)] and X2(k)=DFT[x2(n)], Then

DFT[a1x1(n)+a2x2(n)]=a1X1(k)+a2X2(k)

(iii)Time reversal of a sequence If DFT[x(n)]=X(k),then DFT[x(-n))N]=DFT[x(N-n)]=

x((-k))N=X(N-k)



(iv)Circular time shifting of a sequence If DFT[x(n)]=X(k) thenDFT[x((n-l))n]=X(k)e-

j2πkl/N.

UNIT IV DESIGN OF DIGITAL FILTERS

1. What are the advantages and disadvantages of digital filters? {AM17}

High thermal stability due to absence of resistors, inductors and capacitors.

Increasing the length of the registers can enhance the performance characteristics like

accuracy, dynamic range, stability and tolerance.

The digital filters are programmable.

Multiplexing and adaptive filtering are possible.

Highly immune to noise.

No problems in input or output impedance matching.

Disadvantages of digital filters:

The bandwidth of the discrete signal is limited by the sampling frequency.

The performance of the digital filter depends on the hardware used to implement the

filter.

The quantization error arises due to finite word length of digital samples.

2. What is prewarping?{ND15}{AM17}

The effect of the non-linear compression at high frequencies can be compensated by

Prewarping, i.e the conversion of the digital frequencies to analog frequencies are called

prewarping .

3.Why are digital filters more useful than analog filters? (NOV/DEC 16)

* High thermal stability due to absence of resistors, inductors and capacitors.

*Increasing the length of the registers can enhance the performance characteristics like accuracy,

dynamic range, stability and tolerance.

*The digital filters are programmable.

*Multiplexing and adaptive filtering are possible.

4.Name one method that converts the transfer function of a analog into the digital

filter. (NOV/DEC 16)

*Impulse Invariant Transformation

Bilinear Transformation

5.Mention the advantage of IIR filters over FIR filters(AM 16)



The advantage of IIR filters over FIR filters is that IIR filters usually require fewer

coefficients to execute similar filtering operations, that IIR filters work faster, and require

less memory space.

6. What are the various methods to design IIR filters?

Approximation of derivatives

Impulse invariance

Bilinear transformation.

7. Distinguish between FIR and IIR filters.

8. Compare the Butterworth and Chebyshev Type-1 filters.

Butterworth Chebyshev Type-1



9. What are the techniques of designing FIR filters?

There are three well-known methods for designing FIR filters with linear phase. These are

1) windows method 2) Frequency sampling method 3) Optimal or mini max design.

10. Write the steps involved in FIR filter design.

* Choose the desired (ideal) frequency response Hd(w).

*Take inverse Fourier transform of Hd(w) to get hd(n).

*Convert the infinite duration hd(n) to finite duration h(n).

*Take Z-transform of h(n) to get the transfer function H(z) of the FIR filter

11. Write the expression for transfer function of a first order Butterworth analog filter

having low pass behavior.

12.Compare the rectangular window and hanning window.



13. What is the main drawback of impulse invariant mapping?

In impulse invariant transformation ,the mapping from s-plane to z-plane is many to one to

one i.e all the poles in the s plane between the intervals (2k-1)/T to (2k-1) π/T is mapped into

the entire z-plane. Thus there is an infinite number of poles that map to the same location in

the z-plane, producing aliasing effect. Due to spectrum aliasing the impulse invariance

method is inappropriate for designing high pass filters .That is why the impulse invariance

method is not preferred in the design of IIR filter other than low pass filters.

14. What are the steps for converting a stable analog filter into a stable digital filter?

Map the desired digital filter specifications into those for an equivalent analog filter.

Derive the analog transfer function for the analog prototype.

Transform the transfer function of the analog prototype into an equivalent digital filter

transfer function.

15. What is the importance of poles in filter design? (Nov 2011)

The stability of a filter is related to the location of the poles. For a stable analog filter the

poles should lie on the left half of s-plane. For a stable digital filter the poles should lie inside

the unit circle in the z-plane.

16. What is meant by coefficient quantization error?

Quantization of filter coefficients in digital filters lead to slight changes in their value. This

change in value of filter coefficients modifies the pole-zero locations. This will create

deviations in the frequency response of the system and sometimes the system may drive into

instability. This is called coefficient quantization error.

17. What are the desirable characteristics of windows?

The desirable characteristics of the frequency response of window function are

energy as possible.

18. What is meant by FIR filter and why is it stable?

The specifications of the desired filter will be given in terms of ideal frequency response

Hd(w). The impulse response hd(n) of the desired filter can be obtained by inverse fourier

transform of Hd(w), which consists of infinite samples. The filters designed by selecting

finite number of samples of impulse response are called FIR filters. FIR filter is always stable

because all its poles are at origin.

19. What do you mean by limit cycle oscillations in digital filter?

For an IIR Filter, implemented with finite precision arithmetic, the output may enter into a

fixed value or it may oscillate between finite positive and negative values i.e. periodic

oscillations .this effect is referred to as even when the input is zero. This effect is referred to

as [zero –input] limit cycle oscillations and it is due to the non-linear nature of the arithmetic

quantization.



20. List out different forms of structural realizations available for realizing a FIR

system.

The different forms of realizations a. Transversal b. Linear Phase c. Lattice structure

d. Poly phase

UNIT-V DIGITAL SIGNAL PROCESSOR

1. What is meant by pipeline technique? How to determine depth?(AM17)

Pipelining a processor means breaking down its instruction into a series of discrete pipeline

stages which can be completed in sequence by specialized hardware. The number of pipeline

stages is referred to as the pipeline depth.

2. What is Gibb’s phenomenon (or Gibb’s Oscillation)? (NOV/DEC 16)

In FIR filter design by Fourier series method the infinite duration impulse response is

truncated to finite duration impulse response at n=±(N-1)/2. The abrupt truncation of impulse

response introduces oscillations in the pass band and stop band. This effect is known as

Gibb’s phenomenon (or Gibb’s Osc)

3.what is the advantage of Harvard architecture ?

Advantages

· Efficient Pipelining - Operand Fetch and Instruction Fetch can be overlapped.

· Separate Buses for data and instructions.

· Tailored towards an FPGA implementation.

Disadvantages

· Not widely used.

· More difficult to implement.

· More pins.

4.List some applications of DSP processor?(AM17)

Digital cell phones, automated inspection, voicemail, motor control, video conferencing,

noise cancellation, medical imaging, speech synthesis, satellite communication etc.

5. What are the advantages of VLIW architecture? ?(AM16)

Advantages of VLIW architecture a. Increased performance b. Better compiler targets c.

Potentially easier to program d. Potentially scalable e. Can add more execution units; allow

more instructions to be packed into the VLIW instruction

6. What are the factors that influence selection of DSPs? ?(AM16)

* Architectural features * Execution speed * Type of arithmetic * Word length

7. Write short notes on general purpose DSP processors



General-purpose digital signal processors are basically high speed microprocessors with hard

ware architecture and instruction set optimized for DSP operations. These processors make

extensive use of parallelism, Harvard architecture, pipelining and dedicated hardware

whenever possible to perform time consuming operations

8. Write notes on special purpose DSP processors.

There are two types of special; purpose hardware. (i) Hardware designed for efficient

execution of specific DSP algorithms such as digital filter, FFT. (ii) Hardware designed for

specific applications, for example telecommunication, digital audio.

9.What about of Harvard architecture?

The principal feature of Harvard architecture is that the program and the data memories lie in

t o separate spaces, permitting full overlap of instruction fetch and execution. Typically these

types of instructions would involve their distinct type. 1. Instruction fetch 2. Instruction

decode 3. Instruction execute

10. What are the types of MAC is available?

There are two types MAC’S available 1. Dedicated & integrated 2. Separate multiplier and

integrated unit

11. what is meant by pipeline technique?

The pipeline technique is used to allow overall instruction executions to overlap. That is

where all four phases operate in parallel. By adapting this technique, executi n speed is

increased.

12. What are four phases available in pipeline technique?

The four phases are

(i) Fetch

(ii) Decode

(iii)Read (iv) Execution

13.Write down the name of the addressing modes.

Direct addressing. Indirect addressing. Bit-reversed addressing. Immediate addressing. Short

immediate addressing. Long immediate addressing. Circular addressing

14.What are the instructions used for block transfer in C5X Processors?

The BLDD, BLDP and BLPD instructions use the BMAR to point at the source or destination

space of a block move. The MADD and MADS also use the BMAR to address an operand in

program memory for a multiply accumulator operation

15. What is meant by auxiliary register file?

The auxiliary register file contains eight memory-mapped auxiliary registers (AR0-AR7),

which can be used for indirect addressing of the data memory or for temporary data storage.

16. Write the name of various part of C5X hardware.

1. Central arithmetic logic unit (CALU) 2. Parallel logic unit (PLU) 3. Auxiliary register

arithmetic unit (ARAU) 4. Memory-mapped registers. 5. Program controller.



17. In a non-pipeline machine, the instruction fetch, decode and execute take 30 ns, 45

ns and 25 ns respectively. Determine the increase in throughput if the instruction were

pipelined.

Assume a 5ns pipeline overhead in each stage and ignore other delays. The average

instruction time is = 30 ns+45 ns + 25 ns = 100 ns Each instruction has been completed in

three cycles = 45 ns * 3 = 135ns Throughput of the machine = The average instruction

time/Number of M/C per instruction = 100/135 = 0.7407 But in the case of pipeline machine,

the clock speed is determined by the speed of the slowest stage plus overheads. In our case is

= 45 ns + 5 ns =50 ns The respective throughput is = 100/50 = 2.00 The amount of speed up

the operation is = 135/50 = 2.7 times

18. Assume a memory access time of 150 ns, multiplication time of 100 ns, addition time

of 100 ns and overhead of 10 ns at each pipe stage. Determine the throughput of MAC .

After getting successive addition and multiplications The total time delay is 150 + 100 + 100

+ 5 = 355 ns System throughput is = 1/355 ns.

19. What are multirate systems?

The discrete –time systems that process data at more than one sampling rate are known as

multirate systems.

20. What is adaptive filter? .

An adaptive filter is a system with a linear filter that has a transfer function controlled by

variable parameters and a means to adjust those parameters according to an optimization

algorithm. Because of the complexity of the optimization algorithms, almost all adaptive

filters are digital filters. Adaptive filters are required for some applications because some

parameters of the desired processing operation (for instance, the locations of reflective

surfaces in a reverberant space) are not known in advance or are changing. The closed loop

adaptive filter uses feedback in the form of an error signal to refine its transfer function.



PART B

UNIT I

1. What is meant by energy and power signal? Determine whether the following signals are

energy or power or neither energy nor power signals.

(1) x1(n)=(1/2)nu(n)

(2) x2(n)=sin(πn/6)

(3) x3(n)=ej(πn/3+π/6)

(4) x4(n)=e2n

u(n)

2.Check for following systems are linear, causal, time invariant, stable, static

(i) y(n)=x(1/2n)

(ii) y(n)=sin(x(n))

(iii) y(n) = cos [ x(n) ]

(iv )y(n)=x(-n+2)

(v)y(n) =x(2n)

(vi) y(n)=x(2n)

(vii) y(n)=x(n).cosω0n

3. Consider the analog signal X(t) = 3 cos 2000πt + 5sin 6000πt +10cos 12000πt.

1)What is the nyquist rate for the signal?

2)what is the analog signal y(t) that we can reconstruct from the samples if we use ideal

interpolation ?

3)Assume now that we sample this signal using a sampling rate F= 5000 sample / s .What is

the discrete time signal obtained after sampling ?

4.(i) Check whether the following systems are linear time invariant.

i. y(n)=A+Bx(n)

ii. y(n)=Ax(n)+B[x(n-1)] 2

(ii) Check whether the following are periodic

(1) x(n ) =cos(3πn)

(2) x(n ) =sin(3n) .

5. For each of the discrete time system, determine whether or not the system is

linear, time variant, causal and stable.

i. y(n)=x(n+7)

ii. y(n)=nx(n)

iii. y(n)=x3(n)

6. (i) Check whether the following are periodic (1) x(n ) =cos(3πn) (2) x(n ) =sin(3n)

(ii) Check whether the following are energy or power signals. (1) x(n) = u(n) (2) x(n) = Aejwn

7 (i) What do you mean by Nyquist rate? Give its significance.

(ii) Explain the classification of discrete signal.

8. (i) Given y[n]=x[n2] , Determine whether the system is linear, time invariant, memoryless

and causal.

(ii) Determine whether the following is an energy signal or power signal.

(1) x (n) = 6cos π n



(2) x (n) = 3[0.5]n u(n)

9. State and explain sampling theorem both in time domain and in frequency domain.

10. Explain the digital signal processing system with necessary sketches and give its merits

and demerits.

UNIT II

1. Find the z –transform and ROC of

2. Determine z-transform of the following

i. x(n)=n(-1)nu(n)

ii. x(n)=(-1)n cos(nπ/3)u(n)

3. Find the inverse z-transform of

4. (i)Determine the z transform of the signal and plot the ROC.

(i)

(ii)

(ii) Determine the Z transform of

(1) x(n)=an cosw0n u(n)

(2) x(n)=3 nu(n)

5. (i) Determine the impulse response of the system described by the difference equation

y(n) = y(n − 1) − y(n − 2) + x(n) + x(n − 1) using Z transform and discuss its stability.

(ii) Find the linear convolution of x(n)={2,4,6,8,10} with h(n)={1,3,5,7,9}

6. (i) Explain the properties of Z-transform.

(ii)Obtain x(n) for the following for ROC ;|z|>1, |z|<0.5, 05<|z|<1 .

7. (i) Test the stability of given systems.

y(n)=cos(x(n))

y(n)=x(-n-2)

y(n)=n x(n)

(ii) Find the convolution.

8. (i) Find the z-transform and ROC of x(n)=r

ncos(nθ)u(n)

(ii) Find Inverse z-transform of X(z) = z/[3z2-4z+1], ROC |z|>1.

9. (i) Determine the DTFT of the given sequence x[n]=an (u(n)-u(n-8)), |a|<1.

(ii) Prove the linearity and frequency shifting theorems of the DTFT.

10. Find the inverse z-transform of using

i. Residue method

ii. Convolution method.



UNIT III

1. i) State and prove convolution property of DFT.

(ii) Find the inverse DFT of }

2. (i) Derive the decimation-in time radix-2 FFT algorithm and draw signal flow graph for 8-

point sequence.

(ii) Using FFT algorithm, compute the DFT of x(n)={2,2,2,2,1,1,1,1}.

3. (i) Explain the following properties of DFT.

(1) Convolution. (2) Time shifting (3) Conjugate Symmetry.

(ii) Compute the 4 point DFT of x(n ) ={0,1, 2,3}.

4. (i) Explain the Radix 2 DIF - FFT algorithm for 8 point DFT.

(ii) Obtain the 8 point DFT using DIT - FFT algorithm for x(n)={1,1,1,1,1,1,1,1}

5. An 8-point sequence is given by x(n)={2, 2, 2, 2, 1,1,1,1 }. Compute 8-point DFT of x(n)

by radix DIT-FFT method also sketch the magnitude and phase.

6. Determine the response of LTI system when the input sequence is x(n)={-1,1,2,1,-1} using

radix 2 DIF FFT. The impulse response is h(n)={-1,1,-1,1}.

7. (i) Describe the following properties of DFT.

(1) Time reversal (2) Circular convolution.

(ii) Obtain the circular convolution of x1(n)= {1, 2, 2, 1} x2( n) ={1, 2, 3, 1}

8. Find the output y[n] of a filter whose impulse response is h[n]={1,1,1} and input signal

x[n]={3,-1,0,1,3,2,0,1,2,1} using overlap save method.

9. (i) The first five points of the eight point DFT of a real valued sequence are {0.25, 0.125 –

j0.3018, 0 , 0.125 - j0.0518 , 0 }. Determine the remaining three points. (4)

(ii) Compute the eight point DFT of the sequence x=[1,1,1,1,1,1,1,1], using Decimation-

inFrequency FFT algorithm.

10. (i) Determine 8 point DFT of the sequence x(n)={1,1,1,1,1,1,0,0,0}.

(ii) Find circular convolution of the sequence using concentric circle method x1={1,1,2,1}

and x1={1,2,3,4}.

UNIT IV

1.(i) Obtain cascade and parallel realization for the system having difference equation

y(n)+0.1y(n-1)-0.2y(n-2)=3x(n)+3.6x(n-1)+0.6x(n-2)

(ii) Design a length-5 FIR band reject filter with a lower cut-off frequency of 2KHz, an upper

cut-off frequency of 2.4KHz, and a sampling rate of 8000Hz using Hamming window.

2. (i) Explain impulse invariant method of designing IIR filter

(ii) Design a second order digital low pass Butterworth filter with a cut-off frequency 3.4

KHz at a sampling rate of 8 KHz using bilinear transformation.

3. Design an FIR linear phase, digital filter approximating the ideal frequency response

Determine the coefficients of a 25 tap filter based on the window method with a rectangular

window.

4. (i) Convert the analog filter with system function



into a digital IIR filter by means of the impulse invariance method.

(ii) Draw the direct form I and direct form II structures for the given difference equation

y(n)=y(n-1)-0.5y(n-2)+x(n)-x(n-1)+x(n+2).

5. Design a Chebyshev filter for the following specification using bilinear transformation.

6. For the analog transfer function H(s)=2/(s+1)(s+3) determine H(z) using bilinear

transformation with T=0.1 sec.

7. Design an ideal high pass filter using Hamming window with N=11.

8. (i)Realize the following using cascade and parallel form (12

(ii) Explain how an analog filter maps into a digital filter in Impulse Invariant transformation.

9. Design a butterworth filter using the Impulse invariance method for the following

specifications.

10. Design and realize a digital filter using bilinear transformation for the following

specifications. Monotonic pass band and stop band -3.01 dB cut off at 0.5 π rad magnitude

down atleast 15dB at w=0.75 π rad.

UNIT V

1. (i) Draw the block diagram of Harvard architecture and explain.

(ii) Explain the advantages and disadvantages of VLIW architecture.

2. Write short notes on i. Memory mapped register addressing ii. Circular addressing mode

iii. Auxiliary registers

3. Explain various addressing modes of a digital signal processor.

4. Draw the functional block diagram of a digital signal processor and explain.

5. Explain Von Neumann, Harvard architecture and modified Harvard architecture for the

computer.

6. (i) Explain how convolution is performed using a single MAC unit.

(ii) What is MAC unit? Explain its functions.

7. (i) Explain about pipelining in DSP.

(ii) Discuss the addressing modes used in programmable DSP’s

8. Explain the architecture of TMS320C50 with a neat diagram.

9. Describe the Architectural details and features of a DSP processor.

10. (i) Explain the types of operations performed by L functional mode

(ii) Explain what is meant by bit reversed addressing mode.

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