(New page: ==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. ==Sinusoidal Freque...) |
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<math> = A_{c} \cos \left( 2 \pi f_{c} t + 2 \pi f_{\Delta} \int_{0}^{t}x_{m}(\tau) d \tau \right). </math> | <math> = A_{c} \cos \left( 2 \pi f_{c} t + 2 \pi f_{\Delta} \int_{0}^{t}x_{m}(\tau) d \tau \right). </math> | ||
| − | In this equation, <math>f(\tau)\,</math> is the | + | In this equation, <math>f(\tau)\,</math> is the instantaneous phase of the oscillator and <math>f_{\Delta}\,</math> is the ''frequency deviation'', which represents the maximum shift away from ''f<sub>c</sub>'' in one direction, assuming ''x''<sub>''m''</sub>(''t'') is limited to the range ±1. |
Although it may seem that this limits the frequencies in use to ''f<sub>c</sub>'' ± ''f''<sub>Δ</sub>, 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. | Although it may seem that this limits the frequencies in use to ''f<sub>c</sub>'' ± ''f''<sub>Δ</sub>, 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. | 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. | ||
Revision as of 16:24, 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.
Sinusoidal Frequency Modulation
Frequency Modulation (FM) is well known as the broadcast signal format for FM radio. It is also the basis of the first commercially successful method for digital sound synthesis. Invented by John Chowning [13], it was the method used in the the highly successful Yamaha DX-7 synthesizer, and later the Yamaha OPL chip series, which was used in all ``SoundBlaster compatible multimedia sound cards for many years. At the time of this writing, descendants of the OPL chips remain the dominant synthesis technology for ``ring tones in cellular telephones.
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.
