Spectrum Analysis Basic Notation and Principles - Data Windows and Spectral Density Estimates

In practice, when analyzing actual data, it is not usually of crucial importance to identify exactly the frequencies for particular underlying sine or cosine functions. Rather, because the periodogram values are subject to substantial random fluctuation, one is faced with the problem of very many "chaotic" periodogram spikes. In that case, one would like to find the frequencies with the greatest spectral densities, that is, the frequency regions, consisting of many adjacent frequencies, that contribute most to the overall periodic behavior of the series. This can be accomplished by smoothing the periodogram values via a weighted moving average transformation. Suppose the moving average window is of width m (which must be an odd number); the following are the most commonly used smoothers (note: p = (m-1)/2).

Daniell (or equal weight) window. The Daniell window (Daniell 1946) amounts to a simple (equal weight) moving average transformation of the periodogram values, that is, each spectral density estimate is computed as the mean of the m/2 preceding and subsequent periodogram values.

Tukey window. In the Tukey (Blackman and Tukey, 1958) or Tukey-Hanning window (named after Julius Von Hann), for each frequency, the weights for the weighted moving average of the periodogram values are computed as:

wj = 0.5 + 0.5*cos(p*j/p)    (for j=0 to p)
w
-j = wj    (for j 0)

Hamming window. In the Hamming (named after R. W. Hamming) window or Tukey-Hamming window (Blackman and Tukey, 1958), for each frequency, the weights for the weighted moving average of the periodogram values are computed as:

wj = 0.54 + 0.46*cos(p*j/p)    (for j=0 to p)
w
-j = wj    (for j 0)

Parzen window. In the Parzen window (Parzen, 1961), for each frequency, the weights for the weighted moving average of the periodogram values are computed as:

wj = 1-6*(j/p)2 + 6*(j/p)3    (for j = 0 to p/2)
w
j = 2*(1-j/p)3    (for j 0)

Bartlett window. In the Bartlett window (Bartlett, 1950) the weights are computed as:

1-(j/p)    (for j = 0 to p)
w
-j = wj    (for j 0)

With the exception of the Daniell window, all weight functions will assign the greatest weight to the observation being smoothed in the center of the window, and increasingly smaller weights to values that are further away from the center. (Note also that the Time Series module standardizes the weights so that they sum to 1.) In many cases, all of these data windows will produce very similar results; in the Time Series module it is very easy to experiment with different windows and different widths, and to see the effect on the interpretability of results.