Signals and Systems - Electrical Engineering

358 CHAPTER 5: Frequency Analysis: The Fourier Transform

(b)Suppose you wish to obtain a band-pass filterG(j)fromH(j). If the desired center frequency of

|G(j)|is 5 π, and its desired magnitude is 1 at the center frequency, how would you processh(t)to

get the desired filter? Explain your procedure.

(c)Use symbolic MATLAB to findh(t),g(t), andG(j). Plot|H(j)|,h(t),g(t), and|G(j)|.

5.24. Magnitude response from poles and zeros—MATLAB

Consider the following filters with the given poles and zeros and dc constant:

H 1 (s): K= 1 poles p 1 =−1,p2,3=− 1 ±jπ; zeros z 1 =1,z2,3= 1 ±jπ

H 2 (s): K= 1 poles p 1 =−1,p2,3=− 1 ±jπ; zeros z1,3=±jπ

H 3 (s): K= 1 poles p 1 =−1,p2,3=− 1 ±jπ; zero z 1 = 1

Use MATLAB to plot the magnitude responses of these filters and indicate the type of filters

they are.

5.25. Different types of AM modulations—MATLAB

Let the signal

m(t)=sin( 2 πt)[u(t)−u(t− 1 )]

be the message or input to different types of AM systems with the output the following signals. Carefully

plotm(t)and the following outputs in 0 ≤t≤ 1 and their corresponding spectra using MATLAB. Let the

sampling period beTs=0.001.

(a) y 1 (t)=m(t)cos( 20 πt)

(b)y 2 (t)=[1+m(t)] cos( 20 πt)

5.26. Windows—MATLAB

The signalx(t)in Problem 5.17 is called a raised-cosine window. Notice that it is a very smooth signal and

that it decreases at both ends. The rectangular window is the signaly(t)=u(t+ 1 )−u(t− 1 ).

(a) Use MATLAB to compute the magnitude spectrum ofx(t)andy(t)and indicate which is the smoother

of the two by considering the presence of high frequencies as an indication of roughness.

(b)When computing the Fourier transform of a very long signal it makes sense to break it up into smaller

sections and compute the Fourier transform of each. In such a case, windows are used to smooth out

the transition from one section to the other. Consider a sinusoidz(t)=cos( 2 πt)for 0 ≤t≤1000 sec.

Divide the signal into two sections of duration 500 sec. Multiply the corresponding signal in each of

the sections by a raised-cosinex(t)and rectangulary(t)windows of length 500 and compute using

MATLAB the corresponding Fourier transforms. Compare them to the Fourier transform of the whole

signal and comment on your results. Sample all the signals usingTs= 1 /( 4 π)as the sampling period.

(c)Consider the computation of the Fourier transform of the acoustic signal corresponding to a train

whistle, which MATLAB provides as a sampled signal in “train.mat” using the discrete approximation

of the Fourier transform. The frequency content of the whole signal (hard to find) would not be as

meaningful as the frequency content of a smaller section of it as they change with time. Compute

the Fourier transform of sections of 1000 samples by windowing the signal with the raised-cosine

window (sampled with the same sampling period as the “train.mat” signal orTs= 1 /FswhereFsis the

sampling frequency given for “train.mat”). Plot the spectra of a few of these segments and comment

on the change in the frequency content as time changes.