Question

please matlab code result is important

5. Consider a system with a cascade connection of two causal LTI systems: • Frequency response of the first system is H, (e) 1-2 and The impulse response of the second system is h, [n] = 5()'u[n] The input to the system is x[n], the output of the first system is w[n) and the output of the overall (complete) system is yn). a. Find the difference equation relating i. The input x[n) to the output w[n] of the first system ii. The input x[n) to the output yín) of the overall system b. Determine the impulse response, h[n], of the overall system e. Find the output, yin), of the overall system if the input of the cascaded system is [n] (a)' [n]. d. Plot the output y[n] (from 0 Sns 20) using MATLAB and give the results of your plot. e. Determine the values of y[n) at 0, and -1.

Answer 1

5 Given; Hilelw= 2 st lion] = 5(+)" ur] HilelW) = 2 gin an] (1-2 eiw wing H(Z) = H, (z) Hz(2) [z=ew] o, o Or, H(Z) = 10 or Y(Z) = 10 X(2) (1 - 1 2 + 1 2 2 2 01 Y.- *******- 10X(a) Abbly Inverse 3-transform m equºs): ( 910) - Hy[n-i] + + y (n-2] = Jox[n] Ans ®) This is the input a n ] related to the aulbeet ylo]. Consider Hilelw) = 2 [2=eiW] or, Hi (2) = 2 2 2 or W(Z) = X(2) 2

en Or, (z)- 2 x W(7) = 2Xlz) Abbly Invesse 7-transtoom on equi) a). 2) Win-2 win-i] = 2x10] Ans This is oled beet whor]. the infect xing related to the b) To find overall system response., hin]. from equ" (A): H(Z) = 10 To find thor], use partial fraction method of Inverse a transfron - A + B ii) H(A) = 10 (1-32)(1-42-) (1-47-) (-2) 01, 10 = A (1-47 1) + 6(1-32-) — Put tl=4; B( 1- 8) = 10 of P żn equne) Or, B = 30 = -6 Put I = 3 ; A (1 - 0) = 10 in egen in Por, A = 80 = 16. Substitute the value of A S B in equn (ii); HZ) = 16 - 6 (1 - 2 27-) (-+ 2-1) or hin] = 16 ( 2 jor urn - 6H)" um]. Ans

5. (d) Plot the output y[n]for N= 21 using Matlab code: From Equation (B); y[n]-(11/12)y[n-1]+(1/6)y[n-2)=10x[n] where, x[n]=(1/4)^n u[n] Matlab Program: cic; clear all; close all; n=0:1:20; b= 10* (0.25. n); a=11 -11/12, 1/6]; N=21; y=impz (b,a,N) steminy) xlabel('n'); vlabel('Magnitude'); title('Step Repsponse');

Results: 10 11.6666666666667 3.32666216563786 0.665649245838285 0.131537321057331 0.0259829493633754 9.65277777777778 2.23486462084191 0.443871068066994 0.0876921061650990 0.0173219690071142 7.06018518518518 4.90210262345679 1.49479255972794 0.997901664167583 0.295942655607488 0.197302852342146 0.0584615531216780 0.0389744083289947 0.01154798006567190.00769865356810848 Waveform: Step Repsponse 12 Magnitude 0 2 4 6 8 10 12 14 16 18 20