Signal Processing First problems_notes

PSfrag replacements

PROBLEM:A discrete-time system is defined by the input/output relation

y[n] = 2x[n + 2] + 6x[n] + 2x[n − 2]. (1)

(a) Determine whether or not the system defined by Equation (1) is (i) linear; (ii) time-invariant; (iii)causal. Explain your answers.

(b) Obtain an expression for the frequency response of this system.

(c) Make a sketch of the frequency response (magnitude and phase) as a function of frequency. Hint: Usesymmetry to simplify your expression before determining the magnitude and phase.

(d) For the system of Equation (1), determine the output y1[n] when the input is

x1[n] = 10 − 10 cos(0.5π(n − 1))

Hint: Use the frequency response and superposition to solve this problem.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Analyze a Difference Equation and Find the Output Due to a Sum of Cosinesstusol/sp_01/pset07_3.pdf

McClellan, Schafer, and Yoder, DSP First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. © 2003 Pearson Education, Inc.

Chap6: Analyze a Difference Equation and Find the Output Due to a Sum of Cosinesstusol/sp_01/pset07_3.pdf


PSfrag replacements

PROBLEM:The intention of the following MATLAB program is to filter a sinusoid via the conv function. However, thecosine signal has a starting point at n = 0; so we assume that it is zero for n < 0.

omegahat = pi/2;nn = [ 0:4000 ];xn = 6*cos(omegahat*nn - pi/2);bb = ones(1,6)/6;yn = conv( bb, xn );

(a) Determine a formula forH(ω) for this FIR filter.

(b) Make a plot of the magnitude ofH(ω) and label all the frequencies where |H(ω)| is zero.Use freqz(bb, 1, ww) in MATLAB, where ww is a vector of frequencies that defines a densegrid for ω.

(c) Use convolution to determine a formula (or table) for y[n], the signal contained in the vector yn.Give the individual values for n = 0, 1, 2, 3, 4, 5, and then provide a general formula for y[n] that iscorrect for 6 ≤ n ≤ 4000. This formula should give numerical values for the amplitude, phase andfrequency of y[n]. Hint: the formula is a sinusoid for n ≥ 6.

(d) Give at least one different value of omegahat such that the output is guaranteed to be zero, for n ≥ 6.

McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Analyze a Filter Implemented in MATLABstusol/sp_00/pset07_4.pdf






PSfrag replacements

PROBLEM:The frequency response of a linear time-invariant filter is given by the formula

H(ω) = (1 + 0.8e− jω)(1 − e− jπ/2e− jω)(1 − e jπ/2e− jω). (1)

(a) Write the difference equation that gives the relation between the input x[n] and the output y[n]. Hint:Multiply out the factors to obtain a sum of powers of e− jω.

(b) What is the impulse response of this system?

(c) If the input is of the form x[n] = Ae jφe jωn , for what values of −π ≤ ω ≤ π will y[n] = 0 for alln?

(d) Use superposition to determine the output of this system when the input is

x[n] = 3 + δ[n − 3] + e j0.5πn for −∞ < n < ∞

Hint: Divide the input into three parts and find the outputs separately each by the easiest method andthen add the results.


Chap6: Analyze an LTI System Defined by a Factored Frequency Responsestusol/sp_01/pset07_8.pdf


Chap6: Analyze an LTI System Defined by a Factored Frequency Responsestusol/sp_01/pset07_8.pdf


PSfrag replacements

PROBLEM:Consider again the cascade system in Figure 1 with

h1[n] = δ[n] − δ[n − 1] and h2[n] = u[n] − u[n − 5].

(a) Determine H1(ω), the frequency response of the first system.

(b) Determine H2(ω), the frequency response of the second system.

(c) By convolution, show that h[n] = h1[n] ∗ h2[n] = δ[n] − δ[n − 5] (see part part (c) of Problem 7.5with α = 1). From h[n] determine H(ω) the frequency response of the overall system (from x[n] toy[n]).

(d) Show that your result in part (c) is the product of the results in parts (a) and (b); i.e., H1(ω)H2(ω) =

H(ω).McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Cascade Connection of LTI Systems; Frequency Responsestusol/sp_01/pset07_6.pdf


PSfrag replacements

PROBLEM:Suppose that three systems are hooked together in “cascade.” In other words, the output of S 1 is the inputto S2, and the output of S2 is the input to S3. The three systems are specified as follows:

S1 : y1[n] = x1[n] + x1[n − 2]

S2 : y2[n] = 7x2[n − 5] + 7x2[n − 6]

S3 : H3(ω) = e− jω − e− j2ω

NOTE: the output of Si is yi [n] and the input is xi [n].The objective in this problem is to determine the equivalent system that is a single operation from the

input x[n] (into S1) to the output y[n] which is the output of S3. Thus x[n] is x1[n] and y[n] is y3[n].

(a) Determine the difference equation for S3.

(b) Determine the frequency response of the first two systems: Hi (ω) for i = 1, 2.

(c) Determine the frequency response of the overall cascaded system.

(d) Write one difference equation that defines the overall system in terms of x[n] and y[n] only.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Cascade Connection of Three FIR Systemsstusol/sp_00/pset07_5.pdf


PSfrag replacements

PROBLEM:Consider the following cascade system:

-

LTISystem #1

H1(e jω)

LTISystem #2

H2(e jω)

- -

x[n] w[n] y[n]

Both systems are 4-point running average systems, i.e.,

H2(e jω) = H1(e jω) =sin(2ω)

4 sin(ω/2)e− j3ω/2

(a) Determine the frequency response of the overall system from the input x[n] to the output y[n].

(b) Plot the magnitude and phase of the overall frequency response for −π ≤ ω ≤ π .

(c) What is the total time delay (in samples) for the overall system?

(d) If the input is x[n] = e jωn , for which values of ω will y[n] = 0 ?


Chap6: Cascade of 2 FIR Filters; Multiplying Frequency Responsesstusol/su_94/f1_5.pdf


PSfrag replacements

PROBLEM:Consider the following cascade system:

- delay by 2LTI

System #2h2[n]

- -

x[n] w[n] y[n]

(a) Find and plot the magnitude of the frequency response of the first filter |H1(ω)|.

(b) If the overall impulse response of the cascade is

heq[n] = δ[n − 3] + 12δ[n − 4]

determine the impulse response of the second filter h2[n].


Chap6: Cascade of FIR Filtersstusol/f_01/q2a_4.pdf


PSfrag replacements

PROBLEM:The diagram in Fig. 1 depicts a cascade connection of two linear time-invariant systems; i.e., the output ofthe first system is the input to the second system, and the overall output is the output of the second system.

-

LTISystem #1H1(ω)

LTISystem #2H2(ω)

- -

x[n] y1[n] y[n]

Figure 1: Cascade connection of two LTI systems.

(a) Show that if the input is x[n] = Ae jφe jωn , then the corresponding output of the overall system is

y[n] = H2(ω)H1(ω)Ae jφe jωn = H(ω)Ae jφe jωn

whereH1(ω) is the frequency response of the first system andH2(ω) is the frequency response of thesecond system. That is, show that the overall frequency response of a cascade of two LTI system isthe product of the individual frequency responses, and therefore the cascade system is equivalent to asingle system with frequency responseH(ω) = H2(ω)H1(ω).

(b) Use the result of part (a) to show that the order of the systems is not important; i.e., show that for thesame input x[n] into the systems of Figs. 1 and 2, the overall outputs are the same (w[n] = y[n]).

-

LTISystem #2H2(ω)

LTISystem #1H1(ω)

- -

x[n] y2[n] w[n]

Figure 2: Equivalent system to system of Figure 1.

(c) Suppose that System #1 is described by the difference equation y1[n] = x[n] + x[n − 2]. and System#2 is described by the frequency response function H2(ω) = (1 − e− jω2). Determine the frequencyresponse function of the overall cascade system.

(d) Sketch the frequency response (magnitude and phase) of the overall cascade system for −π ≤ ω ≤ π .

(e) Obtain a single difference equation that relates y[n] to x[n] in Fig. 1 and w[n] to x[n] in Fig. 2.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.

PROBLEM:A discrete-time system is known to be LTI, and had been measured at two frequencies, O! D 0 and O! D � .

H.ej 0/ D 12

and H.ej�/ D 2e�j11�

(a) Determine the output when the input is

x1Œn� D

(

8 for n even

4 for n odd

(b) If we also know that H.ej�=2/ D H �.e�j�=2/ D 3e�j11�=2, then determine the output y2Œn� whenthe input is a triangle-shaped signal with a period of four:

x2Œn� D f: : : ; 1; 2; 3; 2; 1; 2; 3; 2; 1; 2; 3; 2; : : :g

i.e., x2Œn C 4� D x2Œn� with x2Œ0� D 1, x2Œ1� D 2, x2Œ2� D 3, x2Œ3� D 2, x2Œ4� D 1, and so on.

McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c 2003 Pearson Education, Inc.


H.ej O!/ D .1 � e�j O!/.1 C e�j�=4e�j O!/.1 C ej�=4e�j O!/ (1)

(a) Write the difference equation that gives the relation between the inputxŒn� and the outputyŒn�.Hint: Multiply out the factors to obtain a sum of powers ofe�j O! .


(c) If the input is a complex exponential of the formxŒn� D Aej�ej O!n, for which values of�� O! � �

will yŒn� D 0 for all n?Hint: In this part, the answer is easy to obtain if you use the factored form of Eq.(1).


xŒn� D 3 C ıŒn � 2� C cos.0:5�n C �=4/ for �1 < n < 1

Hint: Divide the input into three parts and find the outputs separately each by the easiest method andthen add the results. This is what it means to apply the principle ofSuperposition.

McClellan, Schafer and Yoder,Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458.c 2003 Pearson Education, Inc.

PSfrag replacements

PROBLEM:

-Ideal

C-to-DConverter

LTI SystemH(z)

- -x(t) x[n] y[n]

6Ts = 1/ fs

The input to the C-to-D converter in the above system is

x(t) = 100 + 50 cos(1000π t − π/4)

The sampling frequency is fs = 2000 samples/second. The LTI system is an L-point moving averagerdefined by the equation

y[n] =1L

L−1∑

k=0

x[n − k]

(a) Is it possible to find a value of L such that y[n] = A for −∞ < n < ∞, where A is a constant? If so,give a rough outline of your plan for finding L .

(b) Determine the minimum value of L such that the cosine term is removed as specified in part (a). Alsodetermine the value of the constant A for your system in part (a).

Lmin = A =


Chap6: Design FIR Low-Pass Averager to Null Sinusoidal Inputs Sampled by C/Dstusol/f_95/q3_5.pdf



yŒn� D �3GxŒn � 1� C 6GxŒn � 2� � 3GxŒn � 3�

whereG is a constant to be determined.

(a) When the input is the signal,x1Œn� D 1 C .�1/n, the output isy1Œn� D 60.�1/nC1. Determine thevalue ofG, and then determine the output when the input is

x2Œn� D

(

5 for n even

25 for n odd

Use linearity and time invariance to simplify your work.

(b) Obtain an expression for the frequency response of this system, using G from part (a).

(c) Make a sketch of the frequency response (magnitude and phase) as a function of frequency.Hint: Use symmetry to simplify your expression before determining the magnitude and phase.

(d) For the system above, determine the outputy1Œn� when the input is

x1Œn� D 4 C 8 cos.0:5�n C �=2/

Hint: Use the frequency response and superposition to solve this problem.


PSfrag replacements

PROBLEM:

We have shown that an LTI system can be represented in several equivalent ways. In each part below, youare given one representation of an LTI system and you are to provide the other representations requested.(Frequency response formulas can be given in any convenient form. You do NOT have to simplify them.)

(a) Frequency response:

Impulse response:

Difference equation: y[n] = x[n] + 2x[n − 1] + x[n − 2]

(b) Frequency response: H(ω) = e− jω(2 cos(ω))

Impulse response:

Difference equation:

(c) Frequency response:

Impulse response:

MATLAB Implementation: y = conv([0,1,0,-1],x)


Chap6: Determine the Impulse Response and Frequency Response of an FIR Filterstusol/f_01/q2a_5.pdf


PSfrag replacements


H(ω) = (1 + e− jω)(1 − e− jπ/4e− jω)(1 − e jπ/4e− jω). (1)

(a) Write the difference equation that gives the relation between the input x[n] and the output y[n].

(b) What is the output if the input is x[n] = δ[n]?



Chap6: Difference equation, nulling and h[n] from frequency responsestusol/f_00/pset06_7.pdf


PSfrag replacements

PROBLEM:An FIR filter is characterized by the following frequency response:

H(e jω) =sin(3ω)

sin( 12 ω)

e− j3ω

(a) If the input to the filter is a signal with the following spectrum, determine a formula for theinput signal, x[n] for −∞ < n < ∞.

-

0

6

−7 j

−π/3

6

11e jπ/4

−π/5

6

11e− jπ/4

π/5

6

7 j

π/3 ω

(b) Using the input signal from part (a), determine the output, y[n] for −∞ < n < ∞.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.

PROBLEM:Consider the following system for discrete-time filtering of a continuous-time signal:

-Ideal

C-to-DConverter

LTISystem

H.ej O!/

IdealD-to-C

Converter- - -

x.t/ xŒn� yŒn� y.t/

6Ts D 1=fs

6Ts D 1=fs

In this problem, assume that the frequency response of the discrete-time system is

H.ej O!/ D 1 C e�j 2 O!

(a) Make a plot of the frequency response magnitude forH.ej O!/ over the frequency range�� < O! � � .

(b) In this part, assume that the input is

x.t/ D 500 C 400 cos.800�t/ for � 1 < t < 1

For a sampling rate offs D 1000 samples/sec, draw the spectrum ofxŒn�, the discrete-time signalafter the C-to-D converter.

(c) For the samex.t/ as in the previous part, and the same sampling rate, determine a simple formula forthe outputy.t/ for �1 < t < 1.



-Ideal

C-to-DConverter

LTISystem

H.ej O!/

IdealD-to-C

Converter- - -


6Ts D 1=fs

6Ts D 1=fs

In this problem, assume that the impulse response of the discrete-time system is

hŒn� D12ıŒn � 4� C ıŒn � 5� C

12ıŒn � 6�

(a) Determine the frequency response formula,H.ej O!/, for the LTI system.

(b) For a sampling rate offs D 800 samples/sec, determine the frequency of an input sinusoid of the formx.t/ D cos.!t/ such that the resulting output will bey.t/ D cos.!t C �/, i.e., the output amplitudeand frequency are the same as the input.

(c) In this part, assume that the input is

x.t/ D 99 C 88 cos.500�t/ for � 1 < t < 1

For a sampling rate offs D 800 samples/sec, determine the outputy.t/ for �1 < t < 1.


PSfrag replacements

PROBLEM:For the aliased sinc function:

asinc(ω, 11) =sin(5 1

2 ω)

sin( 12 ω)

(a) Make a plot of asinc(ω, 11) over the range −4π ≤ ω ≤ +4π . Label all the zero crossings.

(b) Determine the period of asinc(ω, 11). Is it equal to 2π ; why, or why not?

(c) Find the maximum value of the function.

NOTE: the aliased sinc function is defined via: asinc(ω, L) =sin(Lω/2)

sin( 12 ω)

In MATLAB consult help on diric for more information.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Dirichlet (aliased sinc) Function Plot vs. Frequencystusol/f_94/pset6_6.pdf


PSfrag replacements

PROBLEM:The input to the C-to-D converter in the figure below is

x(t) = 3 + 5 cos(4000π t) + 6 cos(6000π t − π/2)

The frequency response for the digital filter (LTI system) is H(ω) =sin(3ω)

6 sin( 12 ω)

e− j2.5ω

The sampling frequency is fs = 12000 samples/second.

-Ideal

C-to-DConverter

LTISystemH(ω)

IdealD-to-C

Converter- - -

x(t) x[n] y[n] y(t)

6T = 1/ fs

6T = 1/ fs

(a) For the Dirichlet function: D(ω, 6) =sin(3ω)

6 sin( 12 ω)

. make a plot of D(ω, 6) over the range −2π ≤

ω ≤ +2π . Label all the zero crossings.

(b) Determine the period of D(ω, 6). Is it equal to 2π ; why, or why not?

(c) Find the maximum value of the function D(ω, 6).

(d) Determine an expression for y(t), the output of the D-to-C converter (as a sum of sinusoids).

In MATLAB consult help on diric for more information about computing the Dirichlet function; also thereis a function called dirich() on the web site.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Discrete-Time Filtering of a Continuous-Time Signal stusol/sp_00/pset07_3.pdf






PSfrag replacements


x(t) = 10 + 4 cos(4000π t − π/8) + 6 cos(11000π t − π/3)

The system function of the LTI system is

H(z) = (1 + z−2)

If fs = 8000 samples/second, determine an expression for y(t), the output of the D-to-C converter.

-Ideal

C-to-DConverter

LTISystemH(z)

IdealD-to-C

Converter- - -

x(t) x[n] y[n] y(t)

6T = 1/ fs

6T = 1/ fs


Chap6: Discrete-Time Filtering of a Continuous-Time Sinusoidal Signalstusol/sp_01/pset08_4.pdf


Chap6: Discrete-Time Filtering of a Continuous-Time Sinusoidal Signalstusol/sp_01/pset08_4.pdf


PSfrag replacements


x(t) = 3 + 4 cos(3000π t + π/2) + 12 cos(20000π t − 2π/3)

The frequency response for the digital filter (LTI system) is

H(ω) =sin(4.5ω)

sin( 12 ω)

e− j4ω


-Ideal

C-to-DConverter

LTISystemH(ω)

IdealD-to-C

Converter- - -

x(t) x[n] y[n] y(t)

6T = 1/ fs

6T = 1/ fs

McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.


-Ideal

C-to-DConverter

LTISystem

H.ej O!/

IdealD-to-C

Converter- - -


6Ts D 1=fs

6Ts D 1=fs

In this problem, assume that the frequency response of the discrete-time system is

H.ej O!/ D 1 C e�j O!

(a) Make a plot of the frequency response magnitude forH.ej O!/ over the frequency range�� < O! � � .

(b) For a sampling rate offs D 300 samples/sec, determine the frequency of an input sinusoid of theform x.t/ D cos.!t/ such that the resulting output will be zero.

(c) In this part, assume that the input is

x.t/ D 10 C 20 cos.100�t/ for � 1 < t < 1

For a sampling rate offs D 300 samples/sec, determine the outputy.t/ for �1 < t < 1.


PSfrag replacements

PROBLEM:A linear time-invariant discrete-time system is described by the difference equation

y[n] = x[n] + 2x[n − 1] + 3x[n − 2] − x[n − 4].

(a) Draw a block diagram that represents this system in terms of unit-delay elements, coefficient multi-pliers, and adders as in Figure 5.13 in the text.

(b) Determine the impulse response h[n] for this system. Express your answer as a sum of scaled andshifted unit impulse sequences.

(c) Use convolution to determine the output due to the input

x[n] = δ[n] − δ[n − 1] + δ[n − 2]

Plot the output sequence y[n] for −3 ≤ n ≤ 10.

(d) Now consider another LTI system whose impulse response is

hd[n] = δ[n] − δ[n − 1] + δ[n − 2].

Use convolution again to determine yd[n] = xd[n] ∗ hd[n], the output of this system when the input is

xd[n] = δ[n] + 2δ[n − 1] + 3δ[n − 2] − δ[n − 4].

How does your answer compare to the answer in part (c)? This example illustrates the general com-mutative property of convolution; i.e., x[n] ∗ h[n] = h[n] ∗ x[n].


Chap6: Draw Block Diagram and Analyze a Difference Equationstusol/sp_01/pset07_7.pdf


Chap6: Draw Block Diagram and Analyze a Difference Equationstusol/sp_01/pset07_7.pdf


PSfrag replacements


H(ω) = (1 + e− jω)(1 − e jπ/4e− jω)(1 − e− jπ/4e− jω)





Chap6: FIR Difference Equation from Frequency Response; Complex Exponential Inputstusol/w_96/pset6_7.pdf


PSfrag replacements


H(ω) = e− jω cos(ω)

(a) If the input to the filter is a signal with the following spectrum, determine the output, y[n] for −∞ <

n < ∞.

-

6 6

4e jπ/6 4e− jπ/6

0−π/3 π/3 ω

(b) Determine the difference equation that relates the input and output.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: FIR Difference Equation from Frequency Response; Response for Input Spectrumstusol/w_96/q3_3.pdf


PSfrag replacements

PROBLEM:Circle the correct answer to each of these short answer questions:

1. If the impulse response of an FIR filter is defined with a scaling parameter β

h[n] = β(2δ[n] − δ[n − 1] + 2δ[n − 2])

Determine β so that the DC value of the frequency response H(e jω) will be equal to one.

(a) β = 1(b) β = 1/2(c) β = 1/3(d) β = 1/4(e) β = 1/5

2. For the following MATLAB code: yy = firfilt( [1,0,0,0,-5], xx )pick the correct difference equation for the filter being implemented.

(a) y[n] = δ[n] − 5δ[n − 1]

(b) y[n] = δ[n] − 5δ[n − 4]

(c) y[n] = x[n] − 5x[n − 1]

(d) y[n] = x[n − 4]

(e) y[n] = x[n] − 5x[n − 4]

3. The MATLAB statement: xx = [ cos(0.13*pi*(0:2000)), cos(0.17*pi*(0:2000))];,

(a) Defines xx as the sum of two sinusoids played simultaneously.(b) Defines xx as the concatenation of two sinusoids played in succession.(c) Defines xx as a frequency response.(d) Defines xx as a spectrogram.

4. If a filter is defined by the MATLAB operation: yy = firfilt(0.2*ones(1,5),xx), then thefilter is:

(a) a highpass FIR filter.(b) a lowpass FIR filter.(c) a highpass IIR filter.(d) a lowpass IIR filter.(e) an allpass filter, i.e., its frequency response magnitude is constant.


Chap6: FIR Filters And Sinusoids Using MATLABstusol/w_99/fv1_4.pdf



H.ej O!/ D

�

1 C e�j 2 O!� �

1 C e�j 4�=3e�j O!� �

1 C e�j 2�=3e�j O!�

(1)

(a) Write the difference equation that gives the relation between the inputxŒn� and the outputyŒn�.Hint: Multiply out the factors to obtain a sum of powers ofe�j O! .

(b) Determine the impulse response of this system, and make a stem plot. Notice that hŒn� is finite length.

(c) If the input is a complex exponential of the formxŒn� D Aej�ej O!n, for which values of�� O! � �

will yŒn� D 0 for all n?Hint: In this part, the answer is easy to obtain if you use the factored form of Eq.(1).


xŒn� D 3 C 7ıŒn � 1� C 13 cos.0:5�n � �=4/ for �1 < n < 1



PSfrag replacements

PROBLEM:The following figure was created by the following MATLAB statements:

n=0:30;x=A+B*cos(omega*n);y=conv(x,h);plot(n,x(n+1),’-’,n,x(n+1),’x’,n,y(n+1),’*’)...

where A, B, omega, and h are previously defined.

0 5 10 15 20 25 30-20

-15

-10

-5

0

5

10

15

time index n

Input (x) and Output (*) of a Discrete-Time Filter

input

output

(a) What is the length of the vector h?

(b) Are there any zeros on the unit circle? ( yes no )If so, where are they?

(c) What is the “d.c. gain” of the discrete-time filter?


Chap6: Filter Characteristics Derived from MATLAB Plot of Output Signalstusol/su_94/f1_10.pdf


PSfrag replacements

PROBLEM:The intention of the following MATLAB program is to filter a sinusoid via the conv function, but the cosinesignal has a starting point at n = 0; we assume that it is zero for n < 0.

omega = pi/2;nn = [ 0:4000 ];xn = cos(omega*nn - pi/2);bb = [ 1 0 0 0 1 ];yn = conv( bb, xn );

(a) DetermineH(ω) for the FIR filter.

(b) Make a plot of the magnitude ofH(ω) and label all the frequencies where |H(ω)| is zero.

(c) Determine a formula for y[n], the signal contained in the vector yn. Give the individual values forn = 0, 1, 2, 3, and then provide a general formula for y[n] that is correct for 4 ≤ n ≤ 4000. Thisformula should give numerical values for the amplitude, phase and frequency of y[n].

(d) Give at least one value of omega such that the output is guaranteed to be zero, for n ≥ 4.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.

PSfrag replacements

PROBLEM:A signal x(t) has the two-sided spectrum representation shown below.

-

28e jπ/4 28e− jπ/440e− jπ 40e jπ

0−35 −10 10 35 f (in Hz)(a) Write an equation for x(t). Make sure to express x(t) as a real-valued signal.

(b) If the signal is sampled at a rate of fs = 25 Hz, sketch the “digital” spectrum of this signal. Indicatethe complex phasor value at each frequency. Only the range −π < ω ≤ π needs to be shown.

(c) If the length-3 FIR filter (below) has filter coefficients {bk} = {1, b1, 1}, show that b1 = −2 cos(0.8π) =

1.618 will make the output signal y[n] equal to zero.

-Ideal

A-to-DConverter

3-pointFIR

FILTER- -

x(t) x[n] y[n]

6Ts = 1/ fs


PSfrag replacements

PROBLEM:The following FIR filter is specified by the filter coefficients {bk} = {0, 2, 3, 2}

- FIR FILTER{bk}

-x[n] y[n]

(a) Determine the impulse response: give your answer as a plot of h[n] vs. n.

-

6h[n]

n−4 −3 −2 −1 0 1 2 3 4 5 6 7

Plot zero values also

(b) Determine the frequency response,H(ω), and select one of the following as the correct answer:

(A) 2 cos ω + 3e− j (2ω−π) (B) (4 cos ω + 3)e− jω (C) (3 + 4 cos ω)e− j2ω (D) 2 cos ω + 3

(c) Determine the magnitude of H(ω) and present your answer as a a plot of the magnitude vs. fre-quency. Label important features.

-

|H(ω)|

0−π π− 12π

12π ω (in rad)


Chap6: Find Impulse and Frequency Response of FIR Filterstusol/f_98/q2a_3.pdf


PSfrag replacements

PROBLEM:- FIR FILTER

{bk}-

x[n] y[n]

(a) If the filter coefficients of an FIR filter are {bk} = {0, 2, 3, 2}, make a plot of the output when the

input is the unit step signal: x[n] = u[n] =

{

0 for n < 01 for n ≥ 0

-

6y[n]

n−4 −3 −2 −1 0 1 2 3 4 5 6 7


(b) Suppose that the frequency response of a different FIR filter is

H(ω) =sin(9ω/2)

sin( 12 ω)

e− j9ω

If the input signal is x[n] = 3 + 2 cos(0.2πn + 0.3π) for − ∞ < n < ∞,determine a simple mathematical expression for the output signal y[n].

y[n] =


Chap6: Find Output of FIR Filter for Step and Cosine Inputsstusol/f_98/q2a_4.pdf


PSfrag replacements


H(ω) = (1 + e− jω)(1 − e jπ/3e− jω)(1 − e− jπ/3e− jω)

(a) Write the difference equation for the FIR filter that gives the relation between the input x[n] and theoutput y[n]. Give numerical values for the filter coefficients.

(b) What is the output of this FIR filter if the input is x[n] = δ[n]?

(c) Evaluate the frequency responseH(ω) at the frequencies ω = π and ω = π/3.

(d) If the input is of the form x[n] = Ae jφe jωn , for what values of −π ≤ ω ≤ π will y[n] = 0 for alln?


PSfrag replacements

PROBLEM:The following MATLAB code will compute a time response and the frequency response of a digitalfilter:

bb = [ 3, 3 ]; aa = [ 1, -1/3 ];xn = [1, -1, 1, -1, 1];yn = filter( bb, aa, xn );subplot(2,1,1), stem( [0:4], yn ); %--- TIME RESPONSEw = -pi : (pi/100) : pi;H = freqz( bb, aa, w );subplot(2,1,2), plot( w, abs(H) ) %--- FREQUENCY RESPONSE (MAG)

(a) Make the plot of yn that will be done by the MATLAB stem function (in line #4).

(b) Again referring to the MATLAB code above, make an approximate sketch of the magnituderesponse versus ω over the range −π ≤ ω ≤ π . Label the sketch where |H(e jω)| is at itspeak value and where it is zero.


PSfrag replacements

PROBLEM:A linear time-invariant system is described by the difference equation

y[n] = x[n] − x[n − 1] + x[n − 2] − x[n − 3]

(a) Find the frequency response H(ω), and then express it as a mathematical formula, in polar form(magnitude and phase).

(b) Plot the magnitude and phase ofH(ω) as a function of ω for −π < ω < π . Do this by hand, but youcould check your answer by using the MATLAB function freqz.

(c) Find all frequencies, ω, for which the response to the input e jωn is zero.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.

PSfrag replacements

PROBLEM:A linear time-invariant system is described by the FIR difference equation

y[n] = x[n] − 3x[n − 1] + 9x[n − 2] − 3x[n − 3] + x[n − 4]

(a) Write a simple formula for the magnitude of the frequency response |H(e jω)|. Express youranswer in terms of real-valued functions only.

(b) Derive a simple formula for the phase of the frequency response 6 H(e jω).McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Frequency Response from FIR Difference Equation; Magnitude & Phasestusol/f_94/f1_6.pdf


PSfrag replacements


y[n] = 2x[n] + 4x[n − 1] − 3x[n − 2] + x[n − 3] − 3x[n − 4] + 4x[n − 5] + 2x[n − 6]

(a) Write a simple formula for the magnitude of the frequency response |H(e jω)|. Express your answerin terms of real-valued functions only.

(b) Derive a simple formula for the phase of the frequency response 6 H(e jω).

(c) Impulse Response: Determine the response of this system to a unit impulse input; i.e., find the outputy[n] = h[n] when the input is x[n] = δ[n]. Plot h[n] as a function of n.


Chap6: Frequency Response from FIR Difference Equation; Magnitude & Phase; h[n]stusol/f_94/pset6_1.pdf


PSfrag replacements


y[n] = 2x[n] + 4x[n − 1] + 2x[n − 2]

(a) Find the frequency response H(e jω), and then express it as a mathematical formula, in polar form(magnitude and phase).

(b) H(e jω) is a periodic function of ω; determine the period.

(c) Plot the magnitude and phase of H(e jω) as a function of ω for −π < ω < 3π . Do this by hand andthen check with the MATLAB function freqz.

(d) Find all frequencies, ω, for which the output response to the input e jωn is zero.

(e) When the input to the system is x[n] = sin(πn/10) determine the functional form for the outputsignal y[n].


Chap6: Frequency Response from FIR Difference Equation; Sinusoidal Responsestusol/f_94/pset5_3.pdf


Chap6: Frequency Response from FIR Difference Equation; Sinusoidal Responsestusol/f_94/pset5_3.pdf


PSfrag replacements

PROBLEM:Suppose that a LTI system has a frequency response function equal to

H(ω) = 2 + 3e− jω + 3e− j3ω + 2e− j4ω

(a) Determine the difference equation that relates the output y[n] of the system to the input x[n].

(b) Determine and plot the impulse response.

(c) Determine the output when the input is a pulse:

p[n] ={ 1 for 0 ≤ n ≤ 3

0 n < 0

Use convolution for a quick solution.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Frequency Response from FIR Difference Equation; h[n]; Convolutionstusol/sp_96/pset6_3.pdf


PSfrag replacements


y[n] = x[n] − 2x[n − 1] + x[n − 2]


(b) Plot the magnitude and phase of H(ω) as a function of ω for −π < ω < π . Do this by hand and withthe MATLAB function freqz.

(c) Find all frequencies, ω, for which the response to the input e jωn is zero.

(d) When the input to the system is x[n] = sin(πn/100) determine the functional form for the outputsignal y[n].

(e) Impulse Response: Determine the response of this system to a unit impulse input; i.e., find the outputy[n] = h[n] when the input is x[n] = δ[n]. Plot h[n] as a function of n.


Chap6: Frequency Response from FIR Difference Equation; h[n]; Sinusoidal Responsestusol/w_94/pset5_3.pdf


Chap6: Frequency Response from FIR Difference Equation; h[n]; Sinusoidal Responsestusol/w_94/pset5_3.pdf


PSfrag replacements

PROBLEM: - FIR FILTER{bk}

FIR FILTERh2[n]

- -x[n] w[n] y[n]

(a) If the filter coefficients of the first FIR filter are {bk} = {0, 1, −2, 1}, and the impulse response of thesecond FIR filter is h2[n] = δ[n] + 2δ[n − 2] + δ[n − 3], use convolution to determine the impulseresponse of the overall system, h[n]. Give your answer as a plot below.

-

6h[n]

n−4 −3 −2 −1 0 1 2 3 4 5 6 7

Plot zero values alsoLabel Carefully

(b) Suppose that the overall frequency response of the cascade system (using different FIR filters fromthose in part (a)) is

H(ω) = (2 + 2 cos(ω))e− jω

If the input signal is x[n] = 10 + 6 cos(0.5πn + π/3) for − ∞ < n < ∞,determine a simple mathematical expression for the overall output signal y[n].

y[n] =


Chap6: Frequency Response of Cascade of FIR Filtersstusol/sp_01/q2a_4.pdf


PSfrag replacements

PROBLEM:Consider the following system diagram

-FIR FilterH(ω)

-

x[n] y[n]

whereH(ω) = e− jω + e− j3ω + e− j5ω.

(a) Write the frequency responseH(ω) in polar form.

(b) Plot the magnitude vs. frequency ofH(ω). Label important features.

-

6

0−π π− 12π

12π ω (in rad)

|H(ω)|

Use ω for digital freq.

(c) For the input x[n] = 2δ[n] − δ[n − 2], plot the output signal y[n].

-

6y[n]

n−4 −3 −2 −1 0 1 2 3 4 5 6 7



Chap6: Frequency Response of FIR Filterstusol/su_01/q2_4.pdf


PSfrag replacements

PROBLEM:The complex-valued frequency response for an L-point moving average filter is

H(ω) =1L

L−1∑

k−0

e− jωk =1L

1 − e− jωL

1 − e− jω

(a) Derive a formula for the phase of H(ω) and make a plot.

(b) Derive a formula for the magnitude of H(ω) and make a plot.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Frequency Response of L-point Running Average FIR Filterstusol/su_93/pset7_4.pdf


PSfrag replacements


y[n] = x[n] + 3x[n − 1] + 3x[n − 2] + x[n − 3]


(b) Plot the magnitude and phase ofH(ω) as a function of ω for −π < ω < π . Do this by hand, but youcould check your answer by using the MATLAB function freqz.

(c) When the input to the system is x[n] = exp( jπn/2) determine the functional form for the outputsignal y[n]. Find numerical values for the magnitude and phase of y[n].


Chap6: Frequency Response of LTI Systemstusol/sp_99/pset5_5.pdf






PSfrag replacements


{bk}-

x[n] y[n]

(a) If the filter coefficients of an FIR filter are {bk} = {0, 1, −1, 1}, make a plot of the output when the

input is the unit step signal: x[n] = u[n] =

{

0 for n < 01 for n ≥ 0

-

6y[n]

n−4 −3 −2 −1 0 1 2 3 4 5 6 7


(b) Suppose that the frequency response of a different FIR filter is

H(ω) = cos( 12 ω)e− jω

If the input signal is x[n] = 1 + 3 cos(πn + π) for − ∞ < n < ∞,determine a simple mathematical expression for the output signal y[n].

y[n] =


Chap6: Frequency Response of an FIR Filterstusol/f_01/q2a_3.pdf


PROBLEM:

(a) Determine the frequency response of the FIR system:

yŒn� D 10xŒn � 2� � 10xŒn � 5�

Give your answer as a formulain the following form: H.ej O!/ D je�j˛ O! ˇ sin.� O!/

by finding numerical values for, ˇ and�.

˛ D ˇ D � D

(b) For the system in part (a), determine the output signalyŒn� when the input signal is

xŒn� Dp

7 C 100 cos.0:1�n/

(c) Write a few lines of MATLAB code that would compute the specific values of the frequency responseneeded in part (b).


PSfrag replacements


y[n] = −3x[n − 2] + 6x[n − 4] − 3x[n − 6]. (1)





x1[n] = 4 + 8 cos(0.5πn + π/2)

Hint: Use the frequency response and superposition to solve this problem.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Frequency response & sinusoidal response of FIR systemstusol/sp_02/pset07_3.pdf


Chap6: Frequency response & sinusoidal response of FIR systemstusol/sp_02/pset07_3.pdf


PROBLEM:A discrete-time system is defined by the input/output relation (given as a difference equation)

yŒn� D �GxŒn � 2�C 2GxŒn � 3� �GxŒn � 4�

whereG is a real-valued constant to be determined.

(a) Obtain an expression (in terms ofG) for the frequency response of this system. Simplify into the“magnitude-phase” form:H.ej O!/ D ej . O!/M. O!/, whereM. O!/ and . O!/ are real.

(b) When the input is the signal,x1Œn� D .�1/n, the output isy1Œn� D 20.�1/nC1. Determine the valueof G.G D

(c) For the system above, setG D 1 and then determine the outputy2Œn� when the input is

x2Œn� D 2 cos.0:5�n � �=3/

Fill in the boxes below:

y2Œn� D cos. nC /


PSfrag replacements


y[n] = 2x[n] − 5x[n − 1] + 2x[n − 2]. (1)





x2[n] = 4 + 4 cos(0.5π(n − 1))

Hint: use the linearity and time-invariance properties.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Frequency response and sinusoidal output of LTI systemstusol/f_00/pset06_3.pdf






PSfrag replacements

PROBLEM:Let h[n] = δ[n] + 2δ[n − 1] + δ[n − 2] be the impulse response of an LTI system and let

x[n] = 2e j (π/2)n, −∞ < n < ∞

be the input to that system.

(a) Determine the frequency responseH(ω) of h[n].Note: We have also used the notation H(e jω) for the frequency response; i.e. H(ω) = H(e jω).

H(ω) =

(b) If y[n] = h[n] ∗ x[n], the output is a complex exponential of the form Ae j (ωon+φ), where A is a realpositive number. Determine A, φ and ωo.

A =

φ =

ωo =


Chap6: Frequency response and sinusoidal responsestusol/sp_02/q2a_5.pdf


PROBLEM:Consider the linear time-invariant system given by the difference equation

yŒn� D xŒn� C xŒn � 1� C xŒn � 2� C xŒn � 3� C xŒn � 4� C xŒn � 5� C xŒn � 6� C xŒn � 7� D

7X

kD0

xŒn � k�

(a) Find an expression for the frequency responseH.ej O!/ of the system.

(b) Show that your answer in (a) can be expressed in the form

H.ej O!/ Dsin.8 O!=2/

sin. O!=2/e�j 3:5 O!

(c) Sketch the frequency response (magnitude and phase) as a function of frequency from the formulaabove. You might want to check your plot by doing it in MATLAB with freekz( ) or freqz( ).

(d) Suppose that the input is

xŒn� D 3 C 3 cos. O!0n/ for � 1 < n < 1

Find all possible non-zero frequencies0 < O!0 < � for which the outputyŒn� is a constant for alln,i.e.,

yŒn� D c for � 1 < n < 1

and find the value forc. (In other words, the sinusoid is removed by the filter.)


PROBLEM:The diagram in Fig. 1 depicts acascade connection of two linear time-invariant systems; i.e., the output ofthe first system is the input to the second system, and the overall output is theoutput of the second system.

-

LTISystem #1

h1Œn�

LTISystem #2

h2Œn�

- -

xŒn� y1Œn� yŒn�


Suppose that System #1 is a filter described by the difference equation

y1Œn� D �12xŒn� C 3xŒn � 1� �

12xŒn � 2�

and System #2 is described by the impulse response

h2Œn� D ıŒn � 2�;

(a) Determine the frequency response sequence,H1.ej O!/, of the first system.

(b) Determine the frequency response,H.ej O!/, of the overall cascade system.

(c) Plot the magnitude and phase of the frequency response of the overall cascaded system.


PROBLEM:The diagram in Fig. 1 depicts acascade connection of two linear time-invariant systems; i.e., the output ofthe first system is the input to the second system, and the overall output is theoutput of the second system.

-

LTISystem #1

h1Œn�

LTISystem #2

h2Œn�

- -

xŒn� y1Œn� yŒn�


Suppose that System #1 is a filter described by the impulse response

h1Œn� D ˛ıŒn� C ıŒn � 2�

and System #2 is described by the difference equation

y2Œn� D y1Œn � 1� C ˛y1Œn � 3�;

(a) Determine the frequency response,H2.ej O!/, of the second system.

(b) Determine the frequency response,H.ej O!/, of the overall cascade system.

(c) For the case where D12, plot the magnitude of the overall frequency response of the cascaded

system.

(d) When˛ D12

and the input to this system is

xŒn� D 10 cos.12�n C 0:25�/

Use the frequency response to compute the values ofyŒn�, over the range�1 � n � 1.


PSfrag replacements

PROBLEM:Consider again the cascade system in Figure 1 with

h1[n] = δ[n] − αδ[n − 1] and h2[n] = αn(u[n] − u[n − 6]).

(a) DetermineH1(ω), the frequency response of the first system.

(b) Show that the frequency response of the second system is

H2(ω) =1 − α6e− jω6

1 − αe− jω .

(c) It is possible to show that h[n] = h1[n] ∗ h2[n] = δ[n] − α6δ[n − 6]. From h[n] determineH(ω) thefrequency response of the overall system (from x[n] to y[n]).

(d) Show that your result in part (c) is the product of the results in parts (a) and (b); i.e., H1(ω)H2(ω) =

H(ω).McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Frequency response of cascaded systemsstusol/sp_02/pset07_6.pdf


PROBLEM:Consider the linear time-invariant system given by the difference equation

yŒn� D xŒn� � xŒn � 1� C xŒn � 2� � xŒn � 3� C xŒn � 4� � xŒn � 5� D

5X

kD0

.�1/kxŒn � k�

(a) Find an expression for the frequency responseH.ej O!/ of the system.

(b) Show that your answer in (a) can be simplified and expressed in the form

H.ej O!/ Dsin.3. O! � �//

sin.12. O! � �//

e�j 2:5. O!��/

Hint: use the fact that.�1/kD e˙j�k .

(c) Sketch the frequency response (magnitude and phase) versusO! from the formula above. Notice thatyou get aDirichlet shape, but its peak is no longer centered atO! D 0. You might want to check yourplot by doing it in MATLAB with freekz( ) or freqz( ).

(d) Suppose that the input signal is

xŒn� D 7 C 9 cos. O!0n/ for � 1 < n < 1

Find all possible non-zero frequencies0 < O!0 < � for which the outputyŒn� is zero for alln. In otherwords, the sinusoid and DC are removed by the filter.


PSfrag replacements

PROBLEM:For the following TRUE/FALSE questions, give an explanation and circle the correct answer:

(a) TRUE or FALSE: A signal x(t) contains a maximum frequency component at f = 200 Hz. If x(t)is sampled at twice its Nyquist rate, then fs = 800 Hz.

(b) TRUE or FALSE: The frequency responseH(ω) is always periodic with a period of 2π .

(c) TRUE or FALSE: If a filter has an impulse response h[n] that is h[n] = δ[n] − δ[n − 1], the theresponse of this filter to the DC input x[n] = 7 will be y[n] = 7δ[n] − 7δ[n − 1]

(d) TRUE or FALSE: If the impulse response is h[n] = δ[n − 1] then the magnitude of the frequencyresponseH(ω) is equal to one for all frequencies.

(e) TRUE or FALSE: A 2-point running average filter, y[n] = 12 x[n] + 1

2 x[n − 2], has a frequencyresponseH(ω) that is zero at ω = 0.


Chap6: Impulse Response, Frequency Response, Samplingstusol/w_99/q2a_2.pdf


PSfrag replacements


H(ω) = (1 + e− jω)(1 − e− jπ/3e− jω)(1 − e jπ/3e− jω). (1)

(a) Write the difference equation that gives the relation between the input x[n] and the output y[n]. Hint:Multiply out the factors to obtain a sum of powers of e− jω.


(c) If the input is of the form x[n] = Ae jφe jωn , for what values of −π ≤ ω ≤ π will y[n] = 0 for alln? Hint: In this part, the answer is most obvious in the factored form of Eq. (1).


x[n] = 3 + δ[n − 2] + cos(0.5πn + π/4) for −∞ < n < ∞

Hint: Divide the input into three parts and find the outputs separately each by the easiest method andthen add the results.


Chap6: Impulse response from factored frequency responsestusol/sp_02/pset07_8.pdf


PSfrag replacements

PROBLEM:Suppose that S is a linear, time-invariant system whose exact form is unknown. It needs to be tested byrunning some inputs into the system, and then observing the output signals. Suppose that the followinginput/output pairs are the result of the tests:

x[n] = δ[n] −→ y[n] = δ[n] − δ[n − 3]

x[n] = cos(2πn/3) −→ y[n] = 0x[n] = cos(πn/3 + π/2) −→ y[n] = 2 cos(πn/3 + π/2)

(a) Make a plot of the signal: x[n] = 3δ[n] − 2δ[n − 2] + δ[n − 3].

(b) What is the output of the system when the input is x[n] = 3δ[n] − 2δ[n − 2] + δ[n − 3].

(c) Determine the output when the input is x[n] = cos(π(n − 3)/3).

(d) Is the following statement true or false: “H(π/2) = 0.” EXPLAINMcClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Linearity & Time-Invariance Propertiesstusol/f_94/pset6_12.pdf


PSfrag replacements

PROBLEM:Suppose that S is a linear, time-invariant system whose exact form is unknown. It needs to be tested byrunning some inputs into the system, and then observing the output signals. Suppose that the followinginput/output pairs are the result of the tests:

x[n] = δ[n] − δ[n − 1] −→ y[n] = 4δ[n] − 4δ[n − 4]

x[n] = cos(πn/2) −→ y[n] = 0x[n] = cos(πn/3) −→ y[n] = 6.93 cos(πn/3 − π/2)

(a) Make a plot of the signal: x[n] = 4δ[n] − 4δ[n − 4].

(b) Use linearity and time-invariance to find the output of the system when the input is

x[n] = 3δ[n] − 3δ[n − 3]

(c) Determine the output when the input is x[n] = 7 cos(π(n − 2)/3).

(d) Determine the output when the input is x[n] = 9 sin(πn/2)


PSfrag replacements

PROBLEM:For each of the following frequency response, pick one of the

representations below that defines exactly the same LTI system. Write your answer S1, S2, S3, S4, S5, or S6,in the box next to each frequency response. In addition, evaluate the frequency response at ω = 0, ±π andω = ± 1

2π as requested for each case;simplify the answer to polar form and write it in the space provided.

ANS = H(ω) = 1 − e− j2ω

H(12π) =

ANS = H(ω) = e− jω − e− j3ω

H(π) =

ANS = H(ω) = e− jω(2 cos(ω))

H(0) =

ANS = H(ω) = −e− j2ω

H(0) =

POSSIBLE ANSWERS: (impulse response, filter coefficients or difference equation)

S1 : bk = {0, 1, 0, −1}

S2 : y[n] = x[n] + x[n − 2]

S3 : h[n] = δ[n] + δ[n − 1] + δ[n − 2]

S4 : h[n] = −δ[n − 2]

S5 : y[n] = x[n − 1] + x[n − 3]

S6 : bk = {1, 0, −1}


Chap6: Match FIR Frequency Response to other Representationsstusol/sp_00/q2a_1.pdf


PSfrag replacements

PROBLEM:Pick the correct frequency response and enter the number in the answer box:Difference Equation or Impulse Response

(a) h[n] = δ[n − 1]

ANS =

(b) y[n] = x[n − 1] − x[n − 3]

ANS =

(c) h[n] = δ[n] − δ[n − 2]

ANS =

(d) y[n] = x[n] + x[n − 1] + x[n − 2]

ANS =

Frequency Response

1. H(ω) = 1 − e− j2ω

2. H(ω) = 2 je− j2ω sin(ω)

3. H(ω) = 2e− j2ω cos(ω)

4. H(ω) = e− jω(1 + 2 cos(ω))

5. H(ω) =sin ω

sin( 12 ω)

6. H(ω) = e− jω


Chap6: Match Frequency Response to Difference Equation or Impulse Responsestusol/sp_99/q2_1.pdf


PSfrag replacements

PROBLEM:For each of the following frequency responses, pick one of the

representations below that defines exactly the same LTI system. Write your answer S1, S2, S3, S4, S5, or S6,in the box next to each frequency response. In addition, evaluate the frequency response at ω = 0, ±π andω = ± 1

2π as requested for each case;simplify the answer to polar form and write it in the space provided.

ANS = H(ω) = e− jω + e− j3ω

H(−π) =

ANS = H(ω) = e− j2ω(2 j sin(ω))

H(12π) =

ANS = H(ω) = 1 − e− j2ω

H(−12π) =

ANS = H(ω) = e− j3ω/2(2 j sin(3ω/2))

H(0) =

POSSIBLE ANSWERS: (impulse response, filter coefficients or difference equation)

S1 : bk = {1, 0, −1}

S2 : h[n] = δ[n] + δ[n − 3]

S3 : h[n] = δ[n − 1] + δ[n − 3]

S4 : y[n] = x[n] − x[n − 3]

S5 : bk = {0, 1, 0, −1}

S6 : y[n] = x[n] + x[n − 1] + x[n − 2]


Chap6: Matching Frequency Responsesstusol/su_01/q2_1.pdf


PROBLEM:Pick the correct frequency response (from the list on the right) and enter the number in the answer box:Time-Domain Description

(a) yŒn� D xŒn� C xŒn � 1� C xŒn � 2�

ANS =

(b) yŒn� D xŒn� C xŒn � 1�

ANS =

(c) hŒn� D ıŒn � 1� C ıŒn � 3�

ANS =

(d) hŒn� D ıŒn � 1� � ıŒn � 3�

ANS =

(e) fbkg D f 1; 0; �1 g

ANS =

(f) Select all systems (from the list on the right) thatnull out DC. Enter all numbers that apply.

ANS =

Frequency Response

1 H.ej O!/ D 1 � e�j 2 O!

2 H.ej O!/ D 2e�j 2 O! cos. O!/

3 H.ej O!/ D 2je�j 2 O! sin. O!/

4 H.ej O!/ Dsin. O!/

sin.12

O!/e�j O!=2

5 H.ej O!/ D e�j O!.1 C 2 cos. O!//

6 H.ej O!/ Dsin.2 O!/

sin.12

O!/e�j 3 O!=2

7 H.ej O!/ D e�j O!

8 None of the above


PSfrag replacements

PROBLEM:For each of the following frequency responses on the left, pick one of the representations, S1 through S8 onthe right, that defines exactly the same LTI system. Write your answer S1, S2, S3, S4, S5, S6, S7, or S8, in thebox next to each frequency response.

ANS = (a) 1 + e jω S1 bk = {1, 0, 1}

ANS = (b) 2e−3 jω S2 y[n] = 13 {x[n] + x[n − 1] + x[n − 2]}

ANS = (c)sin(2ω)

sin(ω/2)e−3 jω/2 S3 h[n] = 0.5δ[n] + 0.5δ[n − 2]

ANS = (d) e− jω cos(ω) S4 bk = {1, 1, 1, 1}

S5 y[n] = x[n] + x[n − 1]

S6 h[n] = δ[n] − δ[n − 1]

S7 y[n] = x[n] + 2x[n − 3]

S8 h[n] = 2δ[n − 3]McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Matching frequency responses to FIR systemsstusol/f_00/q2a_3.pdf


PROBLEM:A few quick questions:

(a) Consider the following MATLAB program:

nn = 0:16000;

yy = 11*cos(1.2*pi*nn+pi/3);

soundsc(yy,8000)

Neglecting the end effects in the convolution, the frequency in hertz for the output signal producedby the soundsc( ) statement, i.e., the frequency that you hear.

Frequency = Hz

(b) Evaluateˇ

ˇ

ˇ

H.ej O!/ˇ

ˇ

ˇ

2, where H.ej O!/ D 2je�j.3 O!2

C O!/ sin.2 O!/.

(c) Determine the minimum period (in seconds) of the following signal

x.t/ D

1X

kD�1

sin.k=20/

kej 20�kt

Period = secs.

(d) Solve the following relationship for A and �

12 cos.3t/ C A cos.3t C �/ D 16 cos.3t � �=2/

A D

� D


PSfrag replacements


y[n] = x[n + 1] + x[n] + x[n − 1]. (1)

(a) Determine whether or not the system defined by Equation (1) is (i) linear; (ii) time-invariant; (iii)causal.

(b) For the system of Equation (1), determine the output y1[n] when the input is

x1[n] = 2 cos(0.75πn) = e j0.75πn + e− j0.75πn.

(c) For the system of Equation (1), determine the output y2[n] when the input is

x2[n] = 4 + 4 cos(0.75π(n − 1)).

A second discrete-time system is defined by the input/output relation

y[n] = (x[n])2. (2)

Note: in parts (b), (c), (e), and (f), express your answer in terms of cosine functions. Do not leave anysquare powers of cosine functions in your answers.

(d) Determine whether or not the system defined by Equation (2) is (i) linear; (ii) time-invariant; (iii)causal.

(e) For the system of Equation (2), determine the output y1[n] when the input is


(f) For the system of Equation (2), determine the output y2[n] when the input is

x2[n] = 4 + 4 cos(0.75π(n − 1)).

(g) For which system does superposition hold?

(h) For which system does the output contain frequencies that are not present in the input signal?

(i) Which system can cause aliasing of sinusoidal components of the input?


Chap6: Nonlinear and linear systemsstusol/f_99/pset06_3.pdf









yŒn� D

(

1 if jxŒn�j � 0:5

0 if jxŒn�j < 0:5

(a) For the system above, determine the outputy1Œn� when the input is

x1Œn� D cos.0:5�n/

(b) Explain why the result from part (a) proves that the system is not anLTI system.

(c) Is the system linear? or time-invariant? or neither?


PROBLEM:The frequency response of a linear time-invariant filter is given by the formula:

H.ej O!/ D

�

1 � je�j O!� �

1 C je�j O!� �

1 � ej 3�=4e�j O!� �

1 � e�j 3�=4e�j O!�

(a) If the input is a complex exponential of the formxŒn� D Aej�ej O!n, for which values of�� O! � �

will yŒn� D 0 for all n?Hint: In this part, the answer is easy to obtain if you use the factored form above.

(b) Use superposition to determine the output of this system when the input is

xŒn� D 5 C 9ıŒn � 4� C 7 cos.0:5�n � 3�=4/ for �1 < n < 1



PSfrag replacements

PROBLEM:The following MATLAB code will compute a time response and the frequency response of a digital filter:

bb = [ 1 0 -1 ];xn = [ 1, -1, -1, -1, 1, zeros(1,3) ];yn = firfilt( bb, xn );subplot(2,1,1), stem( [0:9], yn(1:10) ); %--- TIME RESPONSEw = -pi : (pi/100) : pi;H = freqz( bb, 1, w );subplot(2,1,2), plot( w, abs(H) ) %--- FREQUENCY RESPONSE

(a) Make the plot of yn that will be done by the MATLAB stem function (in line #4).

(b) Again referring to the MATLAB code above, make the plot of the magnitude response vs. ω overthe range −π ≤ ω ≤ π . Justify by giving a simple formula for the frequency response H(e jω).Remember that the magnitude should never be negative.


Chap6: Output Signal & Frequency Response for FIR Filter Defined by MATLAB Codestusol/f_94/q3_6.pdf


PSfrag replacements


H(e jω) =sin(5ω)

sin( 12 ω)

e− j5ω

(a) If the input to the filter is a signal with the following spectrum, determine a formula for theinput signal, x[n] for −∞ < n < ∞.

-

0

6

−7 j

−π/3

6

11e jπ/4

−π/5

6

11e− jπ/4

π/5

6

7 j

π/3 ω

(b) Using the input signal from part (a), determine the output, y[n] for −∞ < n < ∞.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Output Signal when given Input Spectrum and FIR Frequency Responsestusol/w_96/fv1_4.pdf


PSfrag replacements


{bk}-

x[n] y[n]

(a) If the filter coefficients of an FIR filter are {bk} = {9, −19, 9}, make a plot of the output when theinput is the signal: x[n] = δ[n − 2] + δ[n − 3]

-

6y[n]

n−4 −3 −2 −1 0 1 2 3 4 5 6 7


(b) The magnitude of the frequency response,H(ω), for the filter in part (a) has one of the shapes below.The vertical line in each plot is located at ω = 0. Choose the correct one and then draw the horizontalaxis with correct labels. In addition, label the important features such as the locations of peaksand valleys and the values at those frequencies.

HIGH-PASS LOW-PASS


Chap6: Output and Frequency Response of FIR Filterstusol/sp_99/q2_4.pdf


PSfrag replacements

PROBLEM:For a 4-point moving average filter, find the output of the filter when the input signal is a sinusoid:

x[n] = 3 sin(0.2πn)

Express your answer as a formula that is real-valued.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Output from FIR Filter for Sinusoidal Input Signalstusol/su_93/pset7_5.pdf


PSfrag replacements

PROBLEM:A discrete-time signal x[n] has the two-sided spectrum representation shown below.

-

2e− jπ/2 2e jπ/23e jπ

0−0.3π 0.3π ω (in radians)

9-pointAVERAGING

FILTER- -

x[n] y[n]

(a) Write an equation for x[n]. Make sure to express x[n] as a real-valued signal.

(b) Determine the formula for the output signal y[n].McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.

PSfrag replacements

PROBLEM:A discrete-time signal x[n] has the two-sided spectrum representation shown below.

-

2e− jπ/2 2e jπ/23e jπ

0−0.3π 0.3π ω (in radians)

9-pointAVERAGING

FILTER- -

x[n] y[n]

(a) Write an equation for x[n]. Make sure to express x[n] as a real-valued signal.

(b) Use the MATLAB GUI called ltidemo.m to determine the output y[n] when the FIR filter is a9-point averaging filter. Include a screen shot of the result from ltidemo.

(c) Determine the formula for the output signal y[n]. Do this calculation by hand.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.

PSfrag replacements


x(t) = 3 + 4 cos(2000π t) + 5 cos(4000π t − 2π/3)

The frequency response for the digital filter (LTI system) is

H(ω) =sin(2.5ω)

sin( 12 ω)

e− j2ω


-Ideal

C-to-DConverter

LTISystemH(ω)

IdealD-to-C

Converter- - -

x(t) x[n] y[n] y(t)

6T = 1/ fs

6T = 1/ fs


Chap6: Output of C-to-D, FIR system and D-to-Cstusol/f_99/pset06_5.pdf


Chap6: Output of C-to-D, FIR system and D-to-Cstusol/f_99/pset06_5.pdf


PSfrag replacements

PROBLEM:A linear time-invariant filter is described by the difference equation

y[n] = x[n] − x[n − 1] + x[n − 2] − x[n − 3]

(a) Impulse Response: Determine the impulse response of this system. Plot h[n] as a function of n.

(b) When the input to the system is x[n] = exp( jπn/4) determine the functional form for the outputsignal y[n]. Find numerical values for the mangitude and phase.

(c) What is the output if the input is

x[n] = 4 + cos[0.5π(n − 1)] − 3 cos[0.25πn]


PSfrag replacements


H(ω) =sin(5ω)

sin( 12 ω)

e− j5ω

(a) If the input to the filter is a signal with the following spectrum, determine a formula for the inputsignal, x[n] for −∞ < n < ∞.

-

0

−7 j

−π/3

11e jπ/4

−π/5

11e− jπ/4

π/5

7 j

π/3 ω

(b) Using the input signal from part (a), determine the output, y[n] for −∞ < n < ∞.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.

PSfrag replacements


H(ω) = (1 + 12 e− jω)(1 − e jπ/4e− jω)(1 − e− jπ/4e− jω)

(a) Write the difference equation for the FIR filter that gives the relation between the input x[n] and theoutput y[n]. Give numerical values for the filter coefficients.

(b) What is the output of this FIR filter if the input is x[n] = δ[n]?

(c) Evaluate the frequency response H(ω) at the frequencies ω = 0, ω = π and ω = π/4. Do thecalculations by hand by manipulating the complex number formulas.

(d) Use MATLAB to make plots of the magnitude and phase of the frequency response. Include theseplots in your homework solution. Mark the values that you determined in part (b).


Chap6: Output of LTI Filter Given Frequency Response in Factored Formstusol/w_99/pset6_3.pdf


Chap6: Output of LTI Filter Given Frequency Response in Factored Formstusol/w_99/pset6_3.pdf


PSfrag replacements

PROBLEM:A second discrete-time system is defined by the input/output relation

y[n] = (x[n + 1])2. (1)

(a) Determine whether or not the system defined by (1) is (i) linear; (ii) time-invariant; (iii) causal.



Express your answer in terms of cosine functions. Do not leave any squared powers of cosine func-tions in your answers.

(c) For the system of Equation (1), determine the output y2[n] when the input is

x2[n] = 4 + 4 cos(0.75π(n − 1)).

(d) This system produces output contain frequencies that are not present in the input signal. Explain howthis system might cause “aliasing” of sinusoidal components of the input.


Chap6: Output of a Nonlinear System for Cosine Inputsstusol/sp_00/pset07_2.pdf






PSfrag replacements


H(ω) = (1 − e− jω)(1 − e jπ/3e− jω)(1 − e− jπ/3e− jω). (1)




(d) The frequency response in Equation (1) is written as a product of factors suggesting that it couldbe implemented as a cascade of several systems. By suitably grouping the factors and multiplyingthem together, obtain a representation as the cascade of two systems each of which has only real filtercoefficients. Give the frequency responses and impulse responses of the two systems and draw a blockdiagram of the cascade system.


Chap6: Output signal & difference equation from frequency responsestusol/f_99/pset06_6.pdf


PSfrag replacements

PROBLEM:For the aliased sinc function:

asinc(ω, 11) =sin(5 1

2 ω)

sin( 12 ω)

(a) Make a plot of asinc(ω, 11) over the range −4π ≤ ω ≤ +4π . Label all the zero crossings.

(b) Determine the period of asinc(ω, 11). Is it equal to 2π ; why, or why not?


NOTE: the aliased sinc function is defined via: asinc(ω, L) =sin(Lω/2)

sin( 12 ω)

In MATLAB consult help on diric for more information.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc.

PSfrag replacements

PROBLEM:For the modified Dirichlet function:

D(ω, 5) =sin(2.5ω)

sin( 12 ω)

(a) Make a plot of D(ω, 5) over the range −2π ≤ ω ≤ +2π . Label all the zero crossings.

(b) Determine the period of D(ω, 5). Is it equal to 2π ; why, or why not?


Note: the unmodified Dirichlet function is defined via: D(ω, L) =sin(Lω/2)

L sin( 12 ω)

, so D(ω, 5) = 5D(ω, 5).

In MATLAB consult help on diric for more information.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Plot of Dirichlet functionstusol/f_99/pset06_4.pdf


PSfrag replacements

PROBLEM:A linear time-invariant filter is described by the difference equation

y[n] = −x[n] + 2x[n − 1] − x[n − 2]

(a) Obtain an expression for the frequency response of this system.

(b) Make a sketch of the frequency response (magnitude and phase) as a function of frequency. Hint: Usesymmetry to simplify your expression before determining the magnitude and phase.

(c) What is the output if the input is x[n] = 5 + 5 cos(0.5πn + π/2)?

(d) What is the output if the input is the unit impulse sequence x[n] = δ[n] =

{

1 n = 00 n 6= 0.

(e) What is the output if the input is the unit step sequence x[n] = u[n] =

{

0 n < 01 n ≥ 0.


Chap6: Responses of FIR systemstusol/f_99/pset06_1.pdf


Chap6: Responses of FIR systemstusol/f_99/pset06_1.pdf


PSfrag replacements

PROBLEM:Circle the correct answer to each of these short answer questions, and give a brief explanation:

1. Suppose that the discrete-time signal x[n] is x[n] = 99 cos(0.4πn − 0.8π) determine the frequency(in Hz) of the analog signal y(t) that will be reconstructed by the ideal D-to-C converter operating ata sampling rate of 10,000 samples/second.

(a) f = 8000 Hz(b) f = 4000 Hz(c) f = 2000 Hz(d) f = 1000 Hz(e) f = 0.2 Hz

2. A continuous-time signal x(t) is defined by the Fourier Series sum: x(t) =

10∑

k=−10

jke j16πkt .The Nyquist Rate for sampling x(t) is

(a) 20 Hz(b) 40 Hz(c) 80 Hz(d) 160 Hz(e) 320 Hz

3. A rotating disk with one spot is spinning clockwise at the rate of 10 revolutions per second. If the diskis illuminated with a strobe light that flashes once every 0.2 seconds, determine the movement of thespot that you will see.

(a) The spot appears to stand still.(b) The spot appears to rotate counter-clockwise at a rate of 1 revolutions per second.(c) The spot appears to rotate counter-clockwise at a rate of 2 revolutions per second.(d) The spot appears to rotate clockwise at a rate of 1 revolutions per second.(e) The spot appears to rotate clockwise at a rate of 2 revolutions per second.

4. Suppose that the discrete-time signal x[n] = cos(0.8πn) is the input to an FIR filter whose frequencyresponse is shown on the next page. Determine the output signal, y[n].

(a) y[n] = 3 cos(ω) e− jω cos(0.8πn)

(b) y[n] = 0.62 cos(0.4πn + 0.2π)

(c) y[n] = 0.62 cos(0.8πn − 0.2π)

(d) y[n] = 0.62 cos(0.8πn + 0.2π)

(e) y[n] = 0.5 cos(0.8πn + 0.2π)

(f) y[n] = 0


Chap6: Sampling Theorem & Frequency Responsestusol/sp_99/q2_3.pdf


PSfrag replacements

PROBLEM:Consider the linear time-invariant system described by the difference equation

y[n] = x[n] + x[n − 1] + x[n − 2] + x[n − 3] + x[n − 4] =

4∑

k=0

x[n − k]

(a) Find an expression for the frequency response H(e jω) of the system.


H(e jω) =sin(5ω/2)

sin(ω/2)e− jω2.

(c) Sketch the frequency response (magnitude and phase) as a function of frequency from the formulaabove (or plot it using freqz( )).


x[n] = 10 + 10 cos(ω0n) for −∞ < n < ∞

Find a non-zero frequency 0 < ω0 < π for which the output y[n] is a constant for all n, i.e.,

y[n] = c for −∞ < n < ∞

and find the value for c. (In other words, the sinusoid is removed by the filter.)McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Sinusoidal response and frequency responsestusol/sp_02/pset08_6.pdf


Chap6: Sinusoidal response and frequency responsestusol/sp_02/pset08_6.pdf



yŒn� D .�1/nxŒn�

One characteristic of a LTI system is that a sinusoidal input at frequency O!0 will give a sinusoidal output atthe same frequency—no new frequency components will appear in the output. If frequencies other thanO!0

are contained in the output, then we can conclude that the system is not LTI.

(a) For the system above, determine the outputy1Œn� when the input is

x1Œn� D cos.0:5�n/

Does this input-output pairfx1Œn�; y1Œn�g allow us to conclude that the system is not LTI?Hint: Which frequencies are present in the output signal,y1Œn�?

(b) Exhibit one sinusoidal input signal,A cos. O!0n C �/, for which the output contains frequency compo-nents not contained in the input signal.

(c) Is the system linear? or time-invariant? or neither? Explain.


PSfrag replacements

PROBLEM:

-FIR FILTER

H(ω)-

x[n] y[n]

The frequency response of the filter above is

H(ω) = cos( 12 ω)e− jω

If the input signal is x[n] = 7 + 2 cos(0.5πn + π) for − ∞ < n < ∞,determine a simple mathematical expression for the output signal y[n].

y[n] =McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Sinusoidal response from FIR frequency responsestusol/f_00/q2a_4.pdf


PSfrag replacements

PROBLEM:A linear time-invariant system is defined by the system function

H(z) = −z−1 + 2z−3 − z−5

The magnitude and phase of the frequency response of this system are plotted in the following figure. Notethat the frequency scale is ω/π .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

ω/π

Magnitude of Frequency Response Function

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

ω/π

Phase Angle of Frequency Response Function

radi

ans

(a) This filter is a lowpass bandpass highpass filter. (Circle one.)

(b) Use the above graph to determine (as accurately as you can) the output y[n] of this system when theinput is

x[n] = 10 + 10 cos(0.5πn).

Mark the points on the graph that you used in your solution.

(c) Determine an expression for the frequency response, H(e jω). Write your answer in the form H(e jω) = A(ω)e− jωn0,

where A(ω) is real and n0 is an integer.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Sinusoidal response given frequency responsestusol/f_99/q2a_3.pdf


PSfrag replacements


y[n] = (x[n − 1])2. (1)




Express your answer in terms of cosine functions. Do not leave any squared powers of cosine func-tions in your answers. Note that this system produces output contain frequencies that are not presentin the input signal. Explain how this system might cause “aliasing” of sinusoidal components of theinput.


Chap6: Sinusoidal response of nonlinear systemstusol/f_00/pset06_4.pdf


PSfrag replacements


1. Suppose that the discrete-time signal x[n] is x[n] = 8 cos(0.3πn − π/4) determine the frequency (inHz) of the analog signal y(t) that will be reconstructed by the ideal D-to-C converter operating at asampling rate of 20 samples/second.

(a) f = 3 Hz(b) f = 6 Hz(c) f = 17 Hz(d) f = 34 Hz(e) f = 0.3 Hz(f) f = 0.15 Hz

2. A signal x(t) is defined by: x(t) =

50∑

k=−50

k2e j2πkt . The Nyquist Rate for sampling x(t) is


3. For the following MATLAB code: yy = firfilt( [0,1,2,0,-5], xx )pick the correct difference equation for the filter being implemented.

(a) y[n] = δ[n]

(b) y[n] = x[n] + 2x[n − 1] − 5x[n − 2]

(c) y[n] = x[n] + 2x[n − 1] − 5x[n − 3]

(d) y[n] = x[n − 1] + 2x[n − 2] − 5x[n − 3]

(e) y[n] = x[n − 1] + 2x[n − 2] − 5x[n − 4]

4. If H(ω) is the frequency response of a digital filter, and the input is x[n] = 5 + 7 cos(0.3πn), then aconcise way to define the output is:

(a) y[n] = H(0.3π)(5 + 7 cos(0.3πn))

(b) y[n] = <e{5 + 7H(0.3π)e j0.3πn}

(c) y[n] = <e{7H(0.3π)e j0.3πn}

(d) y[n] = <e{5H(0) + 7H(0.3π)e j0.3πn}

(e) y[n] = 5H(0) + 7H(0.3π) cos(0.3πn)


Chap6: Sinusoids and Samplingstusol/w_99/q2b_1.pdf


PSfrag replacements


1. Suppose that the discrete-time signal x[n] is x[n] = 10 cos(0.2πn − π/3) determine the frequency(in Hz) of the analog signal y(t) that will be reconstructed by the ideal D-to-C converter operating ata sampling rate of 2000 samples/second.

(a) f = 3200 Hz(b) f = 1600 Hz(c) f = 400 Hz(d) f = 200 Hz(e) f = 0.3 Hz(f) f = 0.15 Hz

2. A signal x(t) is defined by: x(t) = cos(2π t) cos(10π t). The Nyquist Rate for sampling x(t) is


3. If H(ω) is the frequency response of a digital filter, and the input is x[n] = 16 + 4 cos(0.2πn), thena concise way to define the output is:

(a) y[n] = 16 + 4H(0.2π) cos(0.2πn)

(b) y[n] = <e{16H(0) + 4H(0.2π)e j0.2πn}

(c) y[n] = 16H(0) + 4H(0.2π) cos(0.2πn)

(d) y[n] = H(0.2π)(16 + 4 cos(0.2πn))

(e) y[n] = <e{16 + 4H(0.2π)e j0.2πn}

4. For the following MATLAB code: yy = firfilt( [1,-5,0,3], [1,0,0,0,0,0] )pick the correct mathematical formula for the output signal.

(a) y[n] = δ[n]

(b) y[n] = δ[n] − 5δ[n − 1] + 3δ[n − 2]

(c) y[n] = δ[n] − 5δ[n − 1] + 3δ[n − 3]

(d) y[n] = δ[n − 1] − 5δ[n − 2] + 3δ[n − 3]

(e) y[n] = δ[n − 1] − 5δ[n − 2] + 3δ[n − 4]


Chap6: Sinusoids, Sampling, and Filteringstusol/w_99/q2a_1.pdf


PSfrag replacements


y[n] = (x[n + 1])3. (1)




Express your answer in terms of cosine functions. Do not leave any powers of cosine functions inyour answers. Note that this system produces output contain frequencies that are not present in theinput signal. Explain how this system might cause “aliasing” of sinusoidal components of the input.

See Problem 6.4 of Problem Set #6, Fall 2000 for a problem like this.McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Test for Linearity and Time-Invariance; Find the Output Due to Sum of Cosinesstusol/sp_01/pset07_4.pdf


PSfrag replacements

PROBLEM:Consider the linear time-invariant system described by the difference equation

y[n] = x[n] + x[n − 1] + x[n − 2] + x[n − 3] =

3∑

k=0

x[n − k]

(a) Find an expression for the frequency response H(ω) of the system.


H(e jω) =sin(2ω)

sin(ω/2)e− j3ω/2.

(c) Sketch the frequency response (magnitude and phase) as a function of frequency from the formulaabove (or plot it using freqz( )).


x[n] = 1 + 2 cos(nω0) for −∞ < n < ∞

Find a non-zero frequency 0 < ω0 < π for which the ouput y[n] is a constant for all n, i.e.,

y[n] = c for −∞ < n < ∞

and find the value for c. (In other words, the sinusoid is removed by the filter.)McClellan, Schafer and Yoder, Signal Processing First, ISBN 0-13-065562-7.Prentice Hall, Upper Saddle River, NJ 07458. c© 2003 Pearson Education, Inc. SOLUTION

Chap6: Use the Frequency Response to Find the Output Due to a Sum of Cosinesstusol/sp_01/pset08_6.pdf






Date post:	08-Dec-2015
Category:	Documents
Upload:	scorpiontr
View:	211 times
Download:	20 times

Signal Processing First problems_notes

Documents