Question

Solution required in MATLAB

1. Convolution and Discrete-Time Fourier Series (DTFS) (a) Generate a periodic signal r2[n] with the fundamental period N ralla-sin(2nn/ İ0) + sin(2m, 2 ) + sin(2nn/30) for 0 < n < N-1 Find the fundamental frequency Ω0-2, N, with the fundamental period N. (b) Generate a periodic signal h2[n] with the fundamental period N haln] = (1/2)", for 0 < n < N-1 (e) Using the com ftuction n Matab, compute the compvolution (d) Using the fft function in Matlab, find the DTFS X2써 of a 2[n] and HIk] of h2[n]. (e) Multiply the DTBs of .r2[n] and haln], that is ýk] = NXp[A]H2[A]. (f) Using the ifft function in Matlab, find the inverse DTFS (IDTFS) of Y2(k) to find g2n]. (g) Compare the signal y2n found from the convolution in time domain and y2[n] found from the multiplication in frequency domain and IDTFS. Activate Wind

Answer 1

Answer:)

a.) We see that the fundamental period is such that it is the least common multiple of 10, 20 and 30, which is 60, and we can also see that the three signals give the same value for $x_2[n]$ at n and n + 60. Then, we have that

60 30

The code for generating the signal $x_2[n]$ is as follows:

N = 60;

n = 0:N-1;

x_2 = sin(0.2*pi*n) + sin(0.1*pi*n) + sin(0.2*pi*n/3);

b.)The code for generating $h_2[n]$ is as follows:

h_2 = (0.5).^n;

c.) We see the convolution as follows:

y_2 = conv(x_2, h_2);

One can see the result using y_2, which gives the full output of y_2.

d.)The FFT $X_2[k]$ of $x_2[n]$ is found as :

X_2 = fft(x_2);

And similarly the FFT $H_2[k]$ is found as:

H_2 = fft(h_2);

e.) The product of these two, i.e the DTFS of $x_2[n] \text{ and } h_2[n]$ is given as :

Y_2 = N * X_2 .* H_2;

f.) Using ifft(), we get that the inverse is :

y_inv = ifft(Y_2)

g.) We see that the convolution in time domain, is similar to using fft() and then ifft() on the DTFS. This corroborates the fact that FFT converts convolution in time domain to multiplication of the respective DTFS in frequency domain. However, the magnitudes are different (which is due to our multiplication of the frequency domain components by N), and also, the output of y_inv is repetitive and matches y_2 exactly within $0\leq n \leq N-1$ , and repeats outside that interval.

The plots are attached for reference:

Plot of ifft[fftsx) - fft(h))) 200 100 200 10 20 30 40 50 50

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