When a periodogram is used to examine the detailed characteristics of an
FIR filter or
window function, the parameter N is chosen to be several multiples of the non-zero duration of the xn sequence, which is called zero-padding (see
§ Sampling the DTFT).[A] When it is used to implement a
filter bank, N is several sub-multiples of the non-zero duration of the xn sequence (see
§ Sampling the DTFT).
One of the periodogram's deficiencies is that the variance at a given
frequency does not decrease as the number of samples used in the computation increases. It does not provide the averaging needed to analyze noiselike signals or even sinusoids at low signal-to-noise ratios. Window functions and filter impulse responses are noiseless, but many other signals require more sophisticated methods of
spectral estimation. Two of the alternatives use periodograms as part of the process:
The method of averaged periodograms,[8] more commonly known as
Welch's method,[9][10] divides a long x[n] sequence into multiple shorter, and possibly overlapping, subsequences. It computes a windowed periodogram of each one, and computes an array average, i.e. an array where each element is an average of the corresponding elements of all the periodograms. For
stationary processes, this reduces the noise variance of each element by approximately a factor equal to the reciprocal of the number of periodograms.
Smoothing is an averaging technique in frequency, instead of time. The smoothed periodogram is sometimes referred to as a spectral plot.[11][12]
Periodogram-based techniques introduce small biases that are unacceptable in some applications. Other techniques that do not rely on periodograms are presented in the
spectral density estimation article.
^
McSweeney, Laura A. (2004-05-14). "Comparison of periodogram tests". Journal of Statistical Computation and Simulation. 76 (4). online ($50): 357–369.
doi:
10.1080/10629360500107618.
S2CID120439605.
^
Rabiner, Lawrence R.; Gold, Bernard (1975). "6.18". Theory and application of digital signal processing. Englewood Cliffs, N.J.: Prentice-Hall. pp.
415.
ISBN0-13-914101-4.
^Engelberg, S. (2008), Digital Signal Processing: An Experimental Approach, Springer, Chap. 7 p. 56
^Welch, Peter D. (June 1967). "The use of Fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms". IEEE Transactions on Audio and Electroacoustics. AU-15 (2): 70–73.
Bibcode:
1967ITAE...15...70W.
doi:
10.1109/TAU.1967.1161901.
^"DATAPLOT Reference Manual"(PDF). NIST.gov. National Institute of Standards and Technology (NIST). 1997-03-11. Retrieved 2019-06-14. The spectral plot is essentially a "smoothed" periodogram where the smoothing is done in the frequency domain.
Further reading
Box, George E. P.; Jenkins, Gwilym M. (1976). Time series analysis: Forecasting and control. San Francisco: Holden-Day.
Scargle, J.D. (December 15, 1982). "Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data". Astrophysical Journal, Part 1. 263: 835–853.
Bibcode:
1982ApJ...263..835S.
doi:
10.1086/160554.