Periodogram
Encyclopedia
The periodogram is an estimate of the spectral density
of a signal. The term was coined by Arthur Schuster
in 1898 as in the following quote:
Note that the term periodogram may also be used to describe the quantity
, which is its common meaning in astronomy (as in "the modulus-squared of the discrete Fourier transform
of the time series (with the appropriate normalisation)"). See Scargle (1982) for a detailed discussion in this context.
A spectral plot refers to a smoothed version of the periodogram. Smoothing
is performed to reduce the effect of measurement noise.
In practice, the periodogram is often computed from a finite-length digital sequence using the fast Fourier transform
(FFT). The raw periodogram is not a good spectral estimate because of spectral bias and the fact that the variance at a given frequency
does not decrease as the number of samples used in the computation increases.
The spectral bias problem arises from a sharp truncation of the sequence, and can be reduced by first multiplying the finite sequence by a window function
which truncates the sequence gradually rather than abruptly.
The variance problem can be reduced by smoothing the periodogram. Various techniques to reduce spectral bias and variance are the subject of spectral estimation.
One such technique to solve the variance problems is also known as the method of averaged periodograms or as Bartlett's method. The idea behind it is, to divide the set of N samples into L sets of M samples, compute the DFT of each set, square it to get the power spectral density and compute the average of all of them. This leads to a decrease in the standard deviation as
Spectral density
In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...
of a signal. The term was coined by Arthur Schuster
Arthur Schuster
Sir Franz Arthur Friedrich Schuster FRS was a German-born British physicist known for his work in spectroscopy, electrochemistry, optics, X-radiography and the application of harmonic analysis to physics...
in 1898 as in the following quote:
Note that the term periodogram may also be used to describe the quantity
, which is its common meaning in astronomy (as in "the modulus-squared of the discrete Fourier transformDiscrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
of the time series (with the appropriate normalisation)"). See Scargle (1982) for a detailed discussion in this context.
A spectral plot refers to a smoothed version of the periodogram. Smoothing
Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. Many different algorithms are used in smoothing...
is performed to reduce the effect of measurement noise.
In practice, the periodogram is often computed from a finite-length digital sequence using the fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
(FFT). The raw periodogram is not a good spectral estimate because of spectral bias and the fact that the variance at a given frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
does not decrease as the number of samples used in the computation increases.
The spectral bias problem arises from a sharp truncation of the sequence, and can be reduced by first multiplying the finite sequence by a window function
Window function
In signal processing, a window function is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation...
which truncates the sequence gradually rather than abruptly.
The variance problem can be reduced by smoothing the periodogram. Various techniques to reduce spectral bias and variance are the subject of spectral estimation.
One such technique to solve the variance problems is also known as the method of averaged periodograms or as Bartlett's method. The idea behind it is, to divide the set of N samples into L sets of M samples, compute the DFT of each set, square it to get the power spectral density and compute the average of all of them. This leads to a decrease in the standard deviation as
See also
- Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
- Least-squares spectral analysisLeast-squares spectral analysisLeast-squares spectral analysis is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis...
- SeasonalitySeasonalityIn statistics, many time series exhibit cyclic variation known as seasonality, periodic variation, or periodic fluctuations. This variation can be either regular or semi regular....
- Spherical harmonicsSpherical harmonicsIn mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
- Welch methodWelch methodIn physics, engineering, and applied mathematics, Welch's method, named after P.D. Welch, is used for estimating the power of a signal at different frequencies: that is, is is an approach to spectral density estimation. The method is based on the concept of using periodogram spectrum estimates,...