Question

please write code in MATLAB as well

5. (12 points) Create the following signal in Matlab: x[n] = u(n) - u[n-6] a. Mathematically compute yi[n] = x[n] * x[n] where * means convolution. Now use the conv command to perform the convolution. Graph x[n) and the resulting y(n), both over the interval Osns 20. How many non-zero terms does y(n) have? Does your computational result agree with the Matlab result? b. Repeat a. but this time with yz[n] = x[n]*h[n) where hin) = n(u[n] - u[n-6]). For your mathematical convolution, you will need to use the formula: Sn-N(N+1) 2 6. (8 points) Assume you have a causal LTI system defined by the difference equation y[n] + 0.5y[n-1] = x[n] - 3 x[n-1]. you can use the command filter to obtain the output if you define the coefficients multiplying x as b = [1-3] and the coefficients multiplying y as a = (1 0.5). Then, for any defined input, x, the corresponding output, y, is y = filter(b,a,x); a. Start by generating an impulse vector, d, with a 1 in the first index and 20 zeros after it. Generate h = filter(bad); Plot h vs n. b. Generate a new signal x = [0 1 2 3 400). Generate y using filter, using the same b and a as before. c. Convolve x with h to generate y1. d. Using subplots, stem plot y vs n and y1 vs n. Set both horizontal axes to span ons 20 Discuss the similarities and differences between the two plots.

Answer 1

Solution of 5 by hand

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Here are the Matlab codes

n=0:20; x=[1 1 1 1 1 1 0 00000000000000); y=conv (x,x); subplot (211); stem (n,x), title('x(n)') subplot (212); stem (y), title('y (n)')

x(n) 8 12 ا 14 16 18 | 20 y(n) 5 10 هههههههههههههههههههههههههههههه ا 15 20 ا 30 ا 35 25 40 45

n=0:20; x=[1 1 1 1 1 1 000OOOOOOOOOOOO; h=[0 1 2 3 4 5 000000000000000); y=conv(x,h); subplot (211); stem(n,x), title('x(n)') subplot (212); stem (y), title('y (n)')

x(n) 1 2 4 6 8 10 12 14 16 18 20. y(n) 0 0 0 . 60000000000000000000000000 10. 25 40. | 15 20 30 35

Herr is the solution of 6

n=[0:20]; d=[10 00 00 00 00 00 00 00 00 00 0]; b=[1 -3]; a=(1 0.5]; h=filter (b,a,d); subplot (311); stem (n,h), title('h vs. n'); x=[0 1 2 3 4 0000000000000000); y=filter (b, a,x); yl=conv (x,h); subplot (312); stem (n,y);title('y vs. n'); subplot (313); n=[0:40]; stem (n, y1);title('yl vs. n');

h vs. n 1 246 | 8 10 12 14 16 18 2 0 y vs. n "- 09 . | 0 0 0 T | 2 4 6 8 10 12 14 16 1820 y1 vs. n . 'ਚ 90000000000000000000000000 | 10 | 15 20 25 30 35 40

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