Digital PM Spectra by Transform Techniques

The power spectrum of a signal x(t) can be defined in many ways. Every definition, however, has to yield some measure of the expected power at the output of a narrow bandpass filter, as a function of the center frequency of the filter. If x{t) is deterministic then the square of the magnitude of its Fourier transform represents energy density as a function of frequency. If the signal has finite length the energy is finite and we can define the power density to be the energy density divided by the length of the signal. If the signal has infinite length the energy may be infinite, but for realizable signals we can still define the power spectrum by operating on a finite time interval and finding the limit as the interval approaches infinity. This limit may include a set of 8-functions. If x{t) is a random signal the direct way of defining the power spectrum 397 398 T H E BELL SYSTEM TECHNICAL J O U R N A L , FEBRUARY 1969 is to find the Fourier transform of a sample function x0(t) on a finite time interval T0, take the magnitude square, divide by T0, average over all possible x0(t), and finally take the limit as T0 --> °o #l If x(t) is stationary, the power spectrum is proportional to the Fourier transform of the autocorrelation function. This is a very useful property which often simplifies the task of finding the power spectrum. The use of transform techniques can be extended to nonstationary processes by means of a double Fourier transform. 2 There is a very simple relationship between the double Fourier transform of the autocorrelation function R{t1} U) of x{t) and the expected energy (or power) as a function of frequency.