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Pulse-amplitude modulation, acronym PAM, is a form of signal modulation where the message information is encoded in the amplitude of a series of signal pulses. | Pulse-amplitude modulation, acronym PAM, is a form of signal modulation where the message information is encoded in the amplitude of a series of signal pulses. | ||
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== Theory== | == Theory== | ||
Revision as of 17:25, 17 November 2008
PAM
Pulse-amplitude modulation, acronym PAM, is a form of signal modulation where the message information is encoded in the amplitude of a series of signal pulses.
Theory
Suppose the baseband data signal (the message) to be transmitted is
$ x_m(t)\, $
and is restricted in amplitude to be
$ \left| x_m(t) \right| \le 1, \, $
and the sinusoidal carrier is
$ x_c(t) = A_c \cos (2 \pi f_c t)\, $
where the parameters fc and Ac describe the carrier sinusoid
The modulator combines the carrier with the baseband data signal to get the transmitted signal,
$ y(t) = A_c \cos \left( 2 \pi \int_{0}^{t} f(\tau) d \tau \right) $ $ = A_{c} \cos \left( 2 \pi \int_{0}^{t} \left[ f_{c} + f_{\Delta} x_{m}(\tau) \right] d \tau \right) $ $ = A_{c} \cos \left( 2 \pi f_{c} t + 2 \pi f_{\Delta} \int_{0}^{t}x_{m}(\tau) d \tau \right). $
In this equation, $ f(\tau)\, $ is the instantaneous phase of the oscillator and $ f_{\Delta}\, $ is the frequency deviation, which represents the maximum shift away from fc in one direction, assuming xm(t) is limited to the range ±1.
Although it may seem that this limits the frequencies in use to fc ± fΔ, this neglects the distinction between instantaneous frequency and spectral frequency. The frequency spectrum of an actual FM signal has components extending out to infinite frequency, although they become negligibly small beyond a point.
The harmonic distribution of a sine wave carrier modulated by a sine wave signal can be represented with Bessel functions - this provides a basis for a mathematical understanding of frequency modulation in the frequency domain.
