The signal at the input of an AM receiver is u(t) = m₁(t) cos(20t) + m₂(t) cos(100t) where the messages m_i(t), i = 1, 2 are the outputs of a low-pass Butterworth filter with inputs

x₁(t) = r(t) ? 2r(t ? 1) + r(t ?2), and x₂(t) = u(t) ? u(t ??4)

respectively. Suppose we are interested in recovering the message m₁(t). The following is a discretized version of the required filtering and demodulation.

(a) Consider x_i (t) and x₂(t) pulses of duration 5 seconds and ?generate sampled signals x₁ (nT_s) and x₂(nTs) with 512 samples. Determine T_s.

(b) Design a 10th order low-pass Butterworth filter with half- power frequency of 10 (rad/sec). Find the frequency response H(j?) of this filter using the function freqs for the frequencies?

obtained for the discretized Fourier transforms obtained in the previous item. From the convolution property M_i

M_i(?) = H(j?) X_i(?), i = 1,2

?(c) To recover the sampled version of the desired message m₁(t), first use a band-pass filter to keep the desired signal containing m₁(t) and to suppress the other. Design a band-pass Butterworth filter of order 20 with half-power frequencies ?_? = 15 and ?_u = 25 rad/sec so that it will pass the signal m₁(t) cos(20t) and reject the other signal.

(d)?Multiply the output of the band-pass filter by a sinusoid cos(20 nT_s)(exactly the carrier in the transmitter used for the modulation of m₁(t)), and low-pass filter the output of the mixer (the system that multiplies by the carrier frequency cosine). Finally, use the previously designed low-pass Butterworth filter, of bandwidth 10 (rad/sec) and order 10, to filter the output of the mixer. The output of the low-pass filter is found in the same way we did before.

(e)?Plot the pulses x_i(nT_s), i = 1, 2, their corresponding spectra and superpose the magnitude frequency response of the low-pass filter on the spectra. Plot the messages m_i(nT_s), i = 1, 2, and their spectra. Plot together the spectrum of the received signal and of magnitude response of the band-pass filter. Plot the output of the band-pass filter and the recovered message signal. Use the function plot in these plots.Use the following functions to compute the Fourier transform and its inverse.

Transcribed Image Text:

function [X,W] AFT(x,Ts) N=length(x); X=fft(x)*Ts: W=[0:N-1]*2*pi/N-pi:W=W/Ts: X=fftshift(X); function [x ,t] AIFT(X,Ts) N=length(X); X=fftshift(X); X=real(ifft(X))/Ts;t=[0:N-1]*Ts;