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# Amplitude Modulation

Module by: Don Johnson. E-mail the author

Summary: Understanding and performing amplitude modulation through Fourier methods.

## Example 1

In communications, a very important operation on a signal st s t is to amplitude modulate it. Using this operation more as an example rather than elaborating the communications aspects here, we want to compute the Fourier transform—the spectrum—of

(1+st)cos2π f c t 1 s t 2 f c t
(1)
Thus,
st=(1+st)cos2π f c t=cos2π f c t+stcos2π f c t s t 1 s t 2 f c t 2 f c t s t 2 f c t
(2)
For the spectrum of cos2π f c t 2 f c t , we use the Fourier series. Its period is 1 f c 1 f c , and its only nonzero Fourier coefficients are c ± 1 =12 c ± 1 1 2 . 1 The second term is not periodic unless st s t has the same period as the sinusoid. Using Euler's relation, the spectrum of the second term can be derived as
stcos2π f c t=Sfei2πftdfcos2π f c t=12Sfei2π(f+ f c )tdf+12Sfei2π(f f c )tdf=12Sf f c ei2πftdf+12Sf+ f c ei2πftdf=Sf f c +Sf+ f c 2ei2πftdf s t 2 f c t f S f 2 f t 2 f c t 1 2 f S f 2 f f c t 1 2 f S f 2 f f c t 1 2 f S f f c 2 f t 1 2 f S f f c 2 f t f S f f c S f f c 2 2 f t
(3)
Exploiting the uniqueness property of the Fourier transform, we have
Fstcos2π f c t=Sf f c +Sf+ f c 2 F s t 2 f c t S f f c S f f c 2
(4)
This component of the spectrum consists of the original signal's spectrum delayed and advanced in frequency. The spectrum of the amplitude modulated signal is shown in Figure 1.

Note how in this figure the signal st s t is defined in the frequency domain. To find its time domain representation, we simply use the inverse Fourier transform.

## Exercise 1

What is the signal st s t that corresponds to the spectrum shown in the upper panel of Figure 1?

### Solution

The signal is the inverse Fourier transform of the triangularly shaped spectrum, and equals st=WsinπWtπWt2 s t W W t W t 2

## Exercise 2

What is the power in xt x t , the amplitude-modulated signal? Try the calculation in both the time and frequency domains.

### Solution

The result is most easily found in the spectrum's formula: the power in the signal-related part of xt x t is half the power of the signal st s t .

In this example, we call the signal st s t a baseband signal because its power is contained at low frequencies. Signals such as speech and the Dow Jones averages are baseband signals. The baseband signal's bandwidth equals W W , the highest frequency at which it has power. Since xt x t 's spectrum is confined to a frequency band not close to the origin (we assume f c W f c W ), we have a bandpass signal. The bandwidth of a bandpass signal is not its highest frequency, but the range of positive frequencies where the signal has power. Thus, in this example, the bandwidth is 2W Hz 2 W Hz. Why a signal's bandwidth should depend on its spectral shape will become clear once we develop communications systems.

## Footnotes

1. From Euler's relation, cos2π f c t=12e(2π f c t)+12e2π f c t 2 f c t 1 2 2 f c t 1 2 2 f c t . This is the Fourier series for cosine!

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