Signals and Systems - Electrical Engineering

6.4 Application to Communications 387

Using the spectrum of a cosine and the modulation theorem, we get

S()≈π[δ(−c)+δ(+c)]−

1

2 j

[2(−c)−2(+c)] (6.26)

where2()is the spectrum of the angle, which is found to be (using the derivative property of the
Fourier transform)

2()=

1

j

M() (6.27)

If the angleθ(t)is not small, we havewideband FMand its spectrum is more difficult to obtain.

nExample 6.6: Simulation of FM modulation with MATLAB

In these simulations we will concern ourselves with the results and leave the discussion of issues

related to the code for the next chapter since the signals are approximated by discrete-time signals.

For the narrowband FM we consider a sinusoidal message

m(t)=80 sin( 20 πt)u(t),

and a sinusoidal carrier of frequencyfc=100 Hz, so that for1=0.1πthe FM signal is

x(t)=cos( 2 πfct+0.1π

∫t

−∞

m(τ)dτ)

Figure 6.19 shows on the top left the message and the narrowband FM signalx(t)right below it, and

on the top right their corresponding magnitude spectra|M()|and below|X()|. The narrowband

FM has only shifted the frequency of the message. The instantaneous frequency (the derivative of

the argument of the cosine) is

IF(t)= 2 πfc+0.1πm(t)= 200 π+ 8 πsin( 20 πt)≈ 200 π

That is, it remains almost constant for all times. For the narrowband FM, the spectrum of the

modulated signal remains the same for all times. To illustrate this we computed the spectrogram

ofx(t). Simply, the spectrogram can be thought of as the computation of the Fourier transform as

the signal evolves with time (see Figure 6.19(c)).

To illustrate the wideband FM, we consider two messages,

m 1 (t)=80 sin( 20 πt)u(t)

m 2 (t)= 2000 tu(t)

giving FM signals,

xi(t)=cos( 2 πfcit+ 50 π

∫t

−∞

mi(τ)dτ) i=1, 2

wherefc 1 =2500 Hz andfc 2 =25 Hz. In this case, the instantaneous frequency is

IFi(t)= 2 πfci+ 50 πmi(t) i=1, 2