1、1Name:Section:Laboratory Exercise 2DISCRETE-TIME SYSTEMS: TIME-DOMAIN REPRESENTATION2.1 SIMULATION OF DISCRETE-TIME SYSTEMSProject 2.1 The Moving Average System A copy of Program P2_1 is given below:% Program P2_1% Simulation of an M-point Moving Average Filter% Generate the input signaln = 0:100;s1
2、 = cos(2*pi*0.05*n); % A low-frequency sinusoids2 = cos(2*pi*0.47*n); % A high frequency sinusoidx = s1+s2;% Implementation of the moving average filterM = input(Desired length of the filter = );num = ones(1,M);y = filter(num,1,x)/M;% Display the input and output signalsclf;subplot(2,2,1);plot(n, s1
3、);axis(0, 100, -2, 2);xlabel(Time index n); ylabel(Amplitude);title(Signal #1);subplot(2,2,2);plot(n, s2);axis(0, 100, -2, 2);xlabel(Time index n); ylabel(Amplitude);title(Signal #2);subplot(2,2,3);plot(n, x);axis(0, 100, -2, 2);xlabel(Time index n); ylabel(Amplitude);title(Input Signal);subplot(2,2
4、,4);plot(n, y);axis(0, 100, -2, 2);xlabel(Time index n); ylabel(Amplitude);title(Output Signal); axis;Answers:Q2.1 The output sequence generated by running the above program for M = 2 with xn = s1n+s2n as the input is shown below. 2The component of the input xn suppressed by the discrete-time system
5、 simulated by this program is s2Q2.2 Program P2_1 is modified to simulate the LTI system yn = 0.5(xnxn1) and process the input xn = s1n+s2n resulting in the output sequence shown below:s3=cos(2*pi*0.05*(n-1);s4= cos(2*pi*0.47*(n-1);z=s3+s4;y = 0.5*(x-z);3The effect of changing the LTI system on the
6、input is - Project 2.2 (Optional) A Simple Nonlinear Discrete-Time SystemA copy of Program P2_2 is given below:% Program P2_2% Generate a sinusoidal input signalclf;n = 0:200;x = cos(2*pi*0.05*n);% Compute the output signalx1 = x 0 0; % x1n = xn+1 x2 = 0 x 0; % x2n = xnx3 = 0 0 x; % x3n = xn-1y = x2
7、.*x2-x1.*x3;y = y(2:202);% Plot the input and output signalssubplot(2,1,1)plot(n, x)xlabel(Time index n);ylabel(Amplitude);title(Input Signal)subplot(2,1,2)plot(n,y)xlabel(Time index n);ylabel(Amplitude);title(Output signal);4Answers:Q2.5 The sinusoidal signals with the following frequencies as the
8、input signals were used to generate the output signals:The output signals generated for each of the above input signals are displayed below: The output signals depend on the frequencies of the input signal according to the following rules: This observation can be explained mathematically as follows:
9、Project 2.3 Linear and Nonlinear SystemsA copy of Program P2_3 is given below:% Program P2_3% Generate the input sequencesclf;n = 0:40;a = 2;b = -3;x1 = cos(2*pi*0.1*n);5x2 = cos(2*pi*0.4*n);x = a*x1 + b*x2;num = 2.2403 2.4908 2.2403;den = 1 -0.4 0.75;ic = 0 0; % Set zero initial conditionsy1 = filt
10、er(num,den,x1,ic); % Compute the output y1ny2 = filter(num,den,x2,ic); % Compute the output y2ny = filter(num,den,x,ic); % Compute the output ynyt = a*y1 + b*y2; d = y - yt; % Compute the difference output dn% Plot the outputs and the difference signalsubplot(3,1,1)stem(n,y);ylabel(Amplitude);title(
11、Output Due to Weighted Input: a cdot x_1n + b cdot x_2n);subplot(3,1,2)stem(n,yt);ylabel(Amplitude);title(Weighted Output: a cdot y_1n + b cdot y_2n);subplot(3,1,3)stem(n,d);xlabel(Time index n);ylabel(Amplitude);title(Difference Signal);Answers:Q2.7 The outputs yn, obtained with weighted input, and
12、 ytn, obtained by combining the two outputs y1n and y2n with the same weights, are shown below along with the difference between the two signals:6The two sequences are same ;we can regard 10(-15) as 0The system is a liner systemQ2.9 Program 2_3 was run with the following non-zero initial conditions
13、- ic = 2 2;The plots generated are shown below - 7Based on these plots we can conclude that the system with nonzero initial conditions is as same as the zero initial condition with the time goneProject 2.4 Time-invariant and Time-varying SystemsA copy of Program P2_4 is given below:% Program P2_4% G
14、enerate the input sequencesclf;n = 0:40; D = 10;a = 3.0;b = -2;x = a*cos(2*pi*0.1*n) + b*cos(2*pi*0.4*n);xd = zeros(1,D) x;num = 2.2403 2.4908 2.2403;den = 1 -0.4 0.75;ic = 0 0; % Set initial conditions% Compute the output yny = filter(num,den,x,ic);% Compute the output ydnyd = filter(num,den,xd,ic)
15、;% Compute the difference output dnd = y - yd(1+D:41+D);% Plot the outputssubplot(3,1,1)stem(n,y);ylabel(Amplitude); title(Output yn); grid;8subplot(3,1,2)stem(n,yd(1:41);ylabel(Amplitude);title(Output due to Delayed Input xn ?, num2str(D),); grid;subplot(3,1,3)stem(n,d);xlabel(Time index n); ylabel
16、(Amplitude);title(Difference Signal); grid;Answers:Q2.12 The output sequences yn and ydn-10 generated by running Program P2_4 are shown below - These two sequences are related as follows same, the output dont change with the timeThe system is - Time invariant systemQ2.15 The output sequences yn and
17、ydn-10 generated by running Program P2_4 for non-zero initial conditions are shown below - ic = 5 2;9These two sequences are related as follows just as the sequences aboveThe system is not related to the initial conditions2.2 LINEAR TIME-INVARIANT DISCRETE-TIME SYSTEMSProject 2.5 Computation of Impu
18、lse Responses of LTI SystemsA copy of Program P2_5 is shown below:% Program P2_5% Compute the impulse response yclf;N = 40;num = 2.2403 2.4908 2.2403;den = 1 -0.4 0.75;y = impz(num,den,N);% Plot the impulse responsestem(y);xlabel(Time index n); ylabel(Amplitude);title(Impulse Response); grid;Answers
19、:10Q2.19 The first 41 samples of the impulse response of the discrete-time system of Project 2.3 generated by running Program P2_5 is given below:Project 2.6 Cascade of LTI SystemsA copy of Program P2_6 is given below:% Program P2_6% Cascade Realizationclf;x = 1 zeros(1,40); % Generate the inputn =
20、0:40;% Coefficients of 4th order systemden = 1 1.6 2.28 1.325 0.68;num = 0.06 -0.19 0.27 -0.26 0.12;% Compute the output of 4th order systemy = filter(num,den,x);% Coefficients of the two 2nd order systemsnum1 = 0.3 -0.2 0.4;den1 = 1 0.9 0.8;num2 = 0.2 -0.5 0.3;den2 = 1 0.7 0.85;% Output y1n of the
21、first stage in the cascadey1 = filter(num1,den1,x);% Output y2n of the second stage in the cascadey2 = filter(num2,den2,y1);% Difference between yn and y2nd = y - y2;% Plot output and difference signalssubplot(3,1,1);stem(n,y);11ylabel(Amplitude);title(Output of 4th order Realization); grid;subplot(
22、3,1,2);stem(n,y2)ylabel(Amplitude);title(Output of Cascade Realization); grid;subplot(3,1,3);stem(n,d)xlabel(Time index n);ylabel(Amplitude);title(Difference Signal); grid;Answers:Q2.23 The output sequences yn, y2n, and the difference signal dn generated by running Program P2_6 are indicated below:T
23、he relation between yn and y2n is yn is the Convolution of y2n and y1nThe 4th order system can do the same job as the cascade systemQ2.24 The sequences generated by running Program P2_6 with the input changed to a sinusoidal sequence are as follows: x = sin(2*pi*0.05*n);12The relation between yn and
24、 y2n in this case is same as the relation aboveProject 2.7 ConvolutionA copy of Program P2_7 is reproduced below:% Program P2_7clf;h = 3 2 1 -2 1 0 -4 0 3; % impulse responsex = 1 -2 3 -4 3 2 1; % input sequencey = conv(h,x);n = 0:14;subplot(2,1,1);stem(n,y);xlabel(Time index n); ylabel(Amplitude);t
25、itle(Output Obtained by Convolution); grid;x1 = x zeros(1,8);y1 = filter(h,1,x1);subplot(2,1,2);13stem(n,y1);xlabel(Time index n); ylabel(Amplitude);title(Output Generated by Filtering); grid;Answers:Q2.28 The sequences yn and y1n generated by running Program P2_7 are shown below:The difference betw
26、een yn and y1n is - sameThe reason for using x1n as the input, obtained by zero-padding xn, for generating y1n is the length of x is 7,but the length of the Convolution is 14,and n=14,we need the length of filter to be 14Project 2.8 Stability of LTI SystemsA copy of Program P2_8 is given below:% Pro
27、gram P2_8% Stability test based on the sum of the absolute % values of the impulse response samplesclf;num = 1 -0.8; den = 1 1.5 0.9;N = 200;14h = impz(num,den,N+1);parsum = 0;for k = 1:N+1;parsum = parsum + abs(h(k);if abs(h(k) 10(-6), break, endend% Plot the impulse responsen = 0:N;stem(n,h)xlabel
28、(Time index n); ylabel(Amplitude);% Print the value of abs(h(k) disp(Value =);disp(abs(h(k);Answers:Q2.32 The discrete-time system of Program P2_8 is - h,t = impz(hd) computes the instantaneous impulse response of the discrete-time filter hd choosing the number of samples for you, and returns the re
29、sponse in column vector h and a vector of times or sample intervals in t where (t = 0 1 2.). impz returns a matrix h if hd is a vector. Each column of the matrix corresponds to one filter in the vector. When hd is a vector of discrete-time filters, impz returns the matrix h. Each column of h corresp
30、onds to one filter in the vector hd. impz(hd) uses FVTool to plot the impulse response of the discrete-time filter hd. If hd is a vector of filters, impz plots the response and for each filter in the vector.The impulse response generated by running Program P2_8 is shown below:15The value of |h(K)| h
31、ere is - 1.6761e-005From this value and the shape of the impulse response we can conclude that the system is - ?By running Program P2_8 with a larger value of N the new value of |h(K)| is - 9.1752e-007N = 400;16From this value we can conclude that the system is - ?Project 2.9 Illustration of the Fil
32、tering ConceptA copy of Program P2_9 is given below:% Program P2_9% Generate the input sequenceclf;n = 0:299;x1 = cos(2*pi*10*n/256);x2 = cos(2*pi*100*n/256);x = x1+x2;% Compute the output sequencesnum1 = 0.5 0.27 0.77;y1 = filter(num1,1,x); % Output of System #1den2 = 1 -0.53 0.46;num2 = 0.45 0.5 0
33、.45;y2 = filter(num2,den2,x); % Output of System #2% Plot the output sequencessubplot(2,1,1);plot(n,y1);axis(0 300 -2 2);ylabel(Amplitude);title(Output of System #1); grid;subplot(2,1,2);plot(n,y2);axis(0 300 -2 2);xlabel(Time index n); ylabel(Amplitude);title(Output of System #2); grid;Answers:Q2.34 The output sequences generated by this program are shown below:17The filter with better characteristics for the suppression of the high frequency component of the input signal xn is system#2Date: Signature:
