Signals and Systems - Electrical Engineering

5.6 Spectral Representation 315

Solution

The modulated signals are

• y 1 (t)=x 1 (t)cos( 10 t)=e−|t|cos( 10 t) −∞<t<∞


• y 2 (t)=x 2 (t)cos( 10 t)=0.2[r(t+ 5 )− 2 r(t)+r(t+ 5 )] cos( 10 t)

The signalx 1 (t)is very smooth, although of infinite support, and thus most of its frequency com-

ponents are of low frequency. The signalx 2 (t)is not as smooth and has a finite support, so that its

frequency components are mostly low pass but its spectrum also displays higher frequencies.

The MATLAB scripts used to compute the Fourier transform of the modulated signals and to plot

the signals, their modulated versions, and the magnitude and phase of the Fourier transforms are

very similar. The following script indicates how to generatey 1 (t)and how to find the magnitude

and phase of its Fourier transformY 1 (). Notice the way the phase is computed.

%%%%%%%%%%%%%%%%%%%%%

% Example 5.7---Modulation

%%%%%%%%%%%%%%%%%%%%%

y1 = exp(−abs(t)).∗cos(10∗t);

% magnitude and phase of Y1(Omega)

Y1 = fourier(y1); Ym = abs(Y1); Ya = atan(imag(Y1)/real(Y1));

The signalx 2 (t)is a triangular signal. The following script shows how to generate the signalx 2 (t).

Instead of multiplyingx 2 (t)by the cosine, we multiply it by the cosine-equivalent representa-

tion in complex exponentials, which will give better plots of the Fourier transforms when using

ezplot.

m = heaviside(t + 5)−heaviside(t);

m1 = heaviside(t)−heaviside(t−5);

x2 = (t + 5)∗m + m1∗(−t + 5); x2 = x2/5;

x = x2∗exp(−j∗ 10 ∗t)/2; y = x2∗exp(+j∗ 10 ∗t)/2;

X = fourier(x); Y = fourier(y);

Y2m = abs(X) + abs(Y); % magnitude of Y2(Omega)

X2 = fourier(x2); X2m = abs(X2); % magnitude of X2(Omega)

The results are shown in Figure 5.5. n

Why Modulation?
The use of modulation to change the frequency content of a message from its baseband frequencies
to higher frequencies makes its transmission over the airwaves possible. Let us explore why it is nec-
essary to use modulation to transmit a music or a speech signal. Typically, acoustic signals such as
music are audible up to frequencies of about 22 KHz, while speech signals typically display frequen-
cies from about 100 Hz to about 5 KHz. Thus, music and speech signals are relatively low-frequency