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* P _{X}*(

where *r _{k}* is the autocorrelation estimate at lag

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* t

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.

Blackman-Tukey correlogram provides a straightforward way to compute the cross-power spectrum *P _{XY}* of the two input signals

** **

where *X*(*f*) and *Y*(*f*)* *are the correlogram estimates of the individual time series

Cross-power spectrum can be used to estimate coherence between the two signals. However it requires averaging of spectral estimates of independent realizations of *x*(*t*) and *y*(*t*). Multi-taper method provides a practical way to compute **Coherence** by averaging the individual spectra given by each
tapered version of the data.

Multi-channel SSA is an advanced, data-adaptive method to analyze oscillatory spatio-temporal modes in multivariate time series. In addition to identifying oscillatory peaks in the cross-spectrum, MSSA allows reconstruction of the multivariate oscillatory modes.

1. Blackman, R.B., and
Tukey, J.,W., 1958: *The measurement of power spectra from the point of
view of communication engineering*. Dover Publications, 190 pp.

2. Kay, S.M., 1988: *Modern spectral estimation--Theory and
application*. Prentice-Hall, 543 pp

3, Percival, D.B., and Walden, A.T.,
1993: *Spectral analysis for physical applications--Multitaper and
conventional univariate techniques*. Cambridge University, 580 pp.

4. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T.,
1989: *Numerical recipes--The art of scientific computing. * Cambridge University Press, 702 pp.