Periodogram

In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most common tool for examining the amplitude vs frequency characteristics of FIR filters and window functions. FFT spectrum analyzers are also implemented as a time-sequence of periodograms. In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most common tool for examining the amplitude vs frequency characteristics of FIR filters and window functions. FFT spectrum analyzers are also implemented as a time-sequence of periodograms. There are at least two different definitions in use today. One of them involves time-averaging, and one does not. Time-averaging is also the purview of other articles (Bartlett's method and Welch's method). This article is not about time-averaging. The definition of interest here is that the power spectral density of a continuous function, x ( t ) , {displaystyle x(t),}   is the Fourier transform of its auto-correlation function (see Cross-correlation theorem): For sufficiently small values of parameter T, an arbitrarily-accurate approximation for X(f) can be observed in the region   − 1 2 T < f < 1 2 T {displaystyle -{ frac {1}{2T}}<f<{ frac {1}{2T}}}   of the function: which is precisely determined by the samples x(nT) that span the non-zero duration of x(t)  (see Discrete-time Fourier transform). And for sufficiently large values of parameter N,   X 1 / T ( f ) {displaystyle X_{1/T}(f)} can be evaluated at an arbitrarily close frequency by a summation of the form: where k is an integer. The periodicity of   e − i 2 π k n N {displaystyle e^{-i2pi {frac {kn}{N}}}}   allows this to be written very simply in terms of a Discrete Fourier transform: where x N {displaystyle x_{_{N}}} is a periodic summation:   x N [ n ]   ≜ ∑ m = − ∞ ∞ x [ n − m N ] . {displaystyle x_{_{N}} riangleq sum _{m=-infty }^{infty }x.} When evaluated for all integers, k, between 0 and N-1, the array: is a periodogram.