Blackman-Tukey Correlogram Analysis

The correlogram constructs an estimate of the power spectrum using a windowed fast Fourier transforms (FFT) of the autocorrelation function of the time series. It was developed by Blackman and Tukey (1958) and is based on the Wiener-Khinchin theorem, which states that if the Fourier transform of a series g(t) is G(w), and if the autocorrelation function of the series is R, then the Fourier transform of R is |G(w)|2 or the power spectrum of g (e.g., Press et al., 1989). The resulting power-spectrum estimate is called a correlogram. An alternative that is not included in the Toolkit is direct or windowed FFT of the time series itself, called a periodogram.

Both periodograms and correlograms are usually performed on weighted versions of the time series or autocorrelation functions in order to reduce power leakage (artificially high power estimates at frequencies away from the true peak frequencies). Press et al. (1989, pp. 423-424) note that "when we select a run of N sampled points for periodogram spectral estimation, we are in effect multiplying an infinite run of ... data ... by a window function in time, one which is zero except during the total sampling time [NDt], and is unity during that time." The sharp edges of this window function contain much power at highest frequencies, which is imparted to the windowed signal and leads to power leakage. A similar argument can be made for correlograms. Weighting the data or correlation function by various tapered shapes (high in center and falling off to sides) is an accepted traditional approach to reducing power leakage. In the Blackman-Tukey approach, the power spectrum P(w) is estimated by

(1)

where rj is the autocorrelation function, M is the maximum lag considered and window length, and wj is the windowing function. Several window shapes are available in the Toolkit: Bartlett (triangular), Hamming (cosinusoidal), Hanning (slightly different cosinusoidal), and none.

You may find that the various windows of the same widths give similar results. The more important choice is how wide the windows should be. The averaging associated with windowing a series reduces the resolution of the methods, from the frequency intervals of 1/N, to a windowed frequency intervals of about 1/M (e.g., Kay 1988, p. 81). Thus, wider windows yield higher spectral resolution, and vice versa.

However, there is a trade-off between higher resolution and increasing variance of the spectral estimate. At the extreme, a single (M=N) direct application of FFT to an unwindowed time series results in a periodogram with a theoretical standard deviation of the estimates equal to the estimates at each frequency, regardless of the number of observations in the time series (Press et al. 1989, p. 423). Averaging the results from many short data windows throughout the series (or autocorrelation) effectively increases the number of independent samples used in estimation and thereby reduces the estimation variance. Kay (1988, section 4.5) shows that the variance of a power spectrum obtained by a windowed correlogram is 2M/3N of the estimated power at each frequency. Thus a narrower window should be used to smooth the spectrum and reduce the sampling errors on the estimate. In practice, Kay (1988) recommends that windows should be no more than one-fifth to one-tenth the total number of data points (to obtain desired estimate-variance reductions) and not too much smaller (in order to retain the ability to distinguish between powers at neighboring frequencies and to obtain the desired leakage reductions).

Theoretical estimates of variance for Blackman-Tukey power spectra are available (e.g., Kay, 1988) and the Toolkit provides error bars constructed from them. These can either be plotted about the estimates themselves, or as a red-noise uncertainty interval. In the latter case, an AR(1) process is fitted to the data, and the the error bars are centered on the theoretical AR(1) spectrum.

As a "traditional" method, the correlogram is intended to provide a familiar benchmark against which the other more modern methods provided in the Toolkit can be judged.