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Continuous and Discrete Time Signals and Systems (Mandal &amp; Asif) solutions - chap01

# Continuous and Discrete Time Signals and Systems (Mandal & Asif) solutions - chap01

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Chapter 1: Introduction to Signals Problem 1.1: i) z [ m , n , k ] is a three dimensional (3D) DT signal. The independent variables are given by m , n , and k , while z is the dependent variable. Digital video is an example of a 3D DT signal of the form z [ m , n , k ]. The intensity z of the pixels in a frame is a function of the spatial coordinates ( m , n ) and frame number k . ii) I ( x , y , z , t ) is a four dimensional (4D) CT signal. The independent variables are given by x , y , z, and t , while I is the dependent variable. Atmospheric pressure is an example of a 4D DT signal of the form I ( x , y , z , t ) if recorded continuously in time and space. The atmospheric pressure I is a function of the spatial coordinates ( x , y , z ) and time t . Problem 1.2 : The CT signals can be plotted using the following MATLAB code. The CT signals are plotted in Fig. S1.2. The students should also try plotting them by hand. -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 t x1(t) cos(3 π t/4 + π /8) -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 t x 2 (t) sin(-3 π t/8 + π /2) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2 4 6 8 10 12 14 t x 3 (t) 5t + 3exp(-t) -1 -0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 t x 4 (t) (sin(3 π t/4+ π /8)) 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -2 -1 0 1 2 t x 5 (t) cos(3 π t/4) + sin( π t/2) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -150 -100 -50 0 50 t x 6 (t) t exp(-2t) Fig S1.2: CT signals plotted for Problem 1.2.

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2 Chapter 1 % MATLAB code for Problem 1.2 clf % signal defined in part (i) t1 =-1:0.01:2 ; x1 = cos(3*pi*t1/4+pi/8) ; subplot(3,2,1), plot(t1, x1), grid on; xlabel('t'); ylabel('x1(t)'); title('cos(3\pi t/4 + \pi/8)'); % signal defined in part (ii) t2 =-1:0.01:2 ; x2 = sin(-3*pi*t2/8+pi/2) ; subplot(3,2,2), plot(t2, x2), grid on; xlabel('t'); ylabel('x_2(t)'); title('sin(-3\pi t/8 + \pi/2)'); % signal defined in part (iii) t3 =-2:0.01:2 ; x3 = 5*t3+ 3*exp(-t3); subplot(3,2,3), plot(t3, x3), grid on xlabel('t'); ylabel('x_3(t)'); title('5t + 3exp(-t)'); % signal defined in part (iv) t4 =-1:0.01:2; x4 = sin(3*pi*t4/4+pi/8); x4 =x4.*x4; subplot(3,2,4), plot(t4, x4), grid on; xlabel('t'); ylabel('x_4(t)'); title('(sin(3\pi t/4+\pi/8))^2'); % signal defined in part (v) t5 =-2:0.01:3 ; x5 = cos(3*pi*t5/4) + sin(pi*t5/2); subplot(3,2,5), plot(t5, x5), grid on; xlabel('t'); ylabel('x_5(t)'); title('cos(3\pi t/4) + sin(\pi t/2)'); % signal defined in part (vi) t6 =-2:0.01:3 ; x6 = t6.*exp(-2*t6) ; subplot(3,2,6), plot(t6, x6), grid on; xlabel('t'); % clear figure % Label of X-axis % Label of Y-axis % Title % Label of X-axis % Label of Y-axis % Title % Label of X-axis % Label of Y-axis % Title % Label of X-axis % Label of Y-axis % Title % Label of X-axis % Label of Y-axis % Title % Label of X-axis
Solutions 3 ylabel('x_6(t)'); title('t exp(-2t)'); print -dtiff plot.tiff; % Label of Y-axis % Title % Save the figure as a TIFF file Problem 1.3 : (i) The value of x1[k] for 5 5 k is shown in the following table. k 5 4 3 2 1 0 1 2 3 4 5 x 1[ k ] 0.38 0.92 0.92 0.38 0.38 0.92 0.92 0.38 0.38 0.92 0.92 The sketch of x1[k] is shown in the top left figure in Fig. S1.3. The other functions can be plotted in a similar way. However, we use MATLAB to plot the six DT. Fig. S1.3 contains the subplots for these sequences followed by the MATLAB code used to generate them. -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -0.5 0 0.5 1 k x 1 [k] cos(3 π k/4 + π /8) -10 -8 -6 -4 -2 0 2 4 6 8 10 -1 -0.5 0 0.5 1 k x 2 [k] sin(-3 π k/8 + π /2) -5 -4 -3 -2 -1 0 1 2 3 4 5 -50 0 50 100 150 200 250 k x 3 [k] 5k + 3 -k -6 -4 -2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 k x 4 [k] |sin(3 π k/4 + π /8)| -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -1 0 1 2 k x 5 [k] cos(3 π

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