Amplitude Modulation 2.5 2000/08/14 2004/08/04 15:20:22.538 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Joe Montgomery montgom@rice.edu Understanding and performing amplitude modulation through Fourier methods. In communications, a very important operation on a signal 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 s t 2 f c t Thus, s t 1 s t 2 f c t 2 f c t s t 2 f c t For the spectrum of 2 f c t , we use the Fourier series. Its period is 1 f c , and its only nonzero Fourier coefficients are c ± 1 1 2 . From Euler's relation, 2 f c t 1 2 2 f c t 1 2 2 f c t . This is the Fourier series for cosine! The second term is not periodic unless s t has the same period as the sinusoid. Using Euler's relation, the spectrum of the second term can be derived as 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 Exploiting the uniqueness property of the Fourier transform, we have F s t 2 f c t S f f c S f f c 2 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 .

modulated A signal which has a triangular shaped spectrum is shown in the top plot. Its highest frequency—the largest frequency containing power—is W  Hz. Once amplitude modulated, the resulting spectrum has "lines" corresponding to the Fourier series components at ± f c and the original triangular spectrum shifted to components at ± f c and scaled by 1 2 .

Note how in this figure the signal s t is defined in the frequency domain. To find its time domain representation, we simply use the inverse Fourier transform. What is the signal s t that corresponds to the spectrum shown in the upper panel of ? The signal is the inverse Fourier transform of the triangularly shaped spectrum, and equals s t W W t W t 2 What is the power in x t , the amplitude-modulated signal? Try the calculation in both the time and frequency domains. The result is most easily found in the spectrum's formula: the power in the signal-related part of x t is half the power of the signal s t . In this example, we call the signal 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 , the highest frequency at which it has power. Since x t 's spectrum is confined to a frequency band not close to the origin (we assume ≫ 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 2 W Hz . Why a signal's bandwidth should depend on its spectral shape will become clear once we develop communications systems.