Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

8.12 SPECTRUM ESTIMATION

Observations of a random process can be used to estimate its underlying mechanism from which it may be possible to predict future outcomes. Familiar examples for which no satisfactory long-term models have been devised include stock market returns and weather forecasting. In the rest of this chapter, we consider signal models that describe how the power spectrum of a random process is distributed across a range of frequencies. There are basically two approaches to spectrum estimation: (i) parametric and (ii) nonparametric. The parametric approach assumes that a random process has an underlying model represented by filter with a white-noise input. Nonparametric PSD estimators, on the other hand, make no assumption about the underlying structure of the random process. The autocorrelation function is estimated and its Fourier transform is computed to derive the PSD. The basic estimator for this approach is known as the periodogram.

8.12.1 Periodogram

For wide-sense stationary random sequence X[k] with finite variance, the autocorrelation function is

(8.221)

where is the time lag. Consider the following estimate of RXX[m]:

(8.222)

where X[k] is a realization ...