Question

Please can you solve it using MATLAB.

(1) Generate random signals [n] and hn, each of length N, and measure the time it takes to compute the linear convolution of r[n using the linear convolution definition and using the FFT method . Plot a graph of the results for N 104 to 10 in steps of 10. (2) Consider the signal x[n]-cos(0.3n),。£11S 100. Generate a plot of: ·The magnitude and phase of the DTFT of x[n] for 0 2π The magnitude and phase of the DFT of r[n] for 0 SksK-1 ·The magnitude and phase of the CTFT of x(t) for 0 Hz < f < 10 Hz assuming x[n] was generated by sampling x(t) at a rate = 20 Hz. (3) Consider the signal x(t) = cos( 1200π) + cos(2400π1+ π/4) + cos(3600t + π/2). The signal n consists of 256 samples of r(t) acquired at a rate of f, 4 KHz. Generate a plot over the range 0 KHz 〈 f 〈 2 KHz of: The magnitude of the CTFT of r(t) by applying a Rectangular window to xn .The magnitude of the CTFT of r(t) by applying a Triangular window to r[n] The magnitude of the CTFT of r(t) by applying a Hanning window to r[n]

Answer 1

1. Generate random signals first:

x = rand(N,1);

h = rand(N,1);

% Start timer while computing convolution

tic;

clin = conv(x,h); % Linear convolution

toc; % Time taken for Linear convolution

tic;

  x1 = [x zeros(1,N-1)];

h1 = [y zeros(1,N-1)];

conv_fft = ifft(fft(x1).*fft(h1)); % Convolution using FFT

toc;

% Plot required graphs.

plot(clin(10000:10000:100000)) % start, step-size, end

plot(conv_fft (10000:10000:100000))

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