Signals and Systems - Electrical Engineering

314 CHAPTER 5: Frequency Analysis: The Fourier Transform

Applying the frequency shifting to

x(t)cos( 0 t)=0.5x(t)ej^0 t+0.5x(t)e−j^0 t

we obtain the Fourier transform of the modulated signal (Eq. 5.13). In communications, themessage

x(t)(typically of lower frequency content than the frequency of the cosine) modulates thecarrier

cos( 0 t)to obtain themodulated signal x(t)cos( 0 t). Modulation is an important application of the

Fourier transform, as it allows us to change the original frequencies of a message to much higher

frequencies, making it possible to transmit the signal over the airwaves.

Remarks

n As indicated before, amplitude modulation consists in multiplying an incoming signal x(t), or

message, by a sinusoid of frequency higher than the maximum frequency of the incoming signal. The

modulated signal is

x(t)cos( 0 t)=0.5[x(t)ej^0 t+x(t)e−j^0 t]

with a Fourier transform, according to the frequency shifting property, of

F[x(t)cos( 0 t)]=0.5[X(− 0 )+X(+ 0 )]

Thus, modulation shifts the frequencies of x(t)to frequencies around± 0.

n Modulation using a sine, instead of a cosine, changes the phase of the Fourier transform of the incoming

signal besides performing the frequency shift. Indeed,

F[x(t)sin( 0 t)]=F

[

x(t)ej^0 t−x(t)e−j^0 t

2 j

]

=

1

2 j

X(− 0 )−

1

2 j

X(+ 0 )

=

−j

2

X(− 0 )+

j

2

X(+ 0 )

where the−j and j terms add−π/ 2 andπ/ 2 , respectively, radians to the signal phase.

n According to the eigenfunction property of LTI systems, modulation systems are not LTI. Modulation shifts

the frequencies at the input to new frequencies at the output. Nonlinear or time-varying systems are

typically used as amplitude modulation transmitters.

nExample 5.7

Consider modulating a carrier cos( 10 t)with the following signals:

1. x 1 (t)=e−|t|,−∞<t<∞. Use MATLAB to find the Fourier transform ofy 1 (t)=x 1 (t)cos( 10 t)
   and ploty 1 (t)and its magnitude and phase spectra.


2. x 2 (t)=0.2[r(t+ 5 )− 2 r(t)+r(t+ 5 )], wherer(t)is the ramp signal. Use MATLAB to plotx 2 (t)
   andy 2 (t)=x 2 (t)cos( 10 t)and compute and plot the magnitude of their Fourier transforms.