Periodogram

Last updated November 15, 2024

In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898.^[1] Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most common tool for examining the amplitude vs frequency characteristics of FIR filters and window functions. FFT spectrum analyzers are also implemented as a time-sequence of periodograms.

Definition

There are at least two different definitions in use today.^[2] One of them involves time-averaging,^[3] and one does not.^[4] Time-averaging is also the purview of other articles (Bartlett's method and Welch's method). This article is not about time-averaging. The definition of interest here is that the power spectral density of a continuous function, $x(t),$ is the Fourier transform of its auto-correlation function (see Cross-correlation theorem, Spectral density, and Wiener–Khinchin theorem): ${\mathcal {F}}\{x(t)\circledast x^{*}(-t)\}=X(f)\cdot X^{*}(f)=\left|X(f)\right|^{2}.$

Computation

A power spectrum (magnitude-squared) of two sinusoidal basis functions, calculated by the periodogram method. Periodogram.svg — A power spectrum (magnitude-squared) of two sinusoidal basis functions, calculated by the periodogram method.

For sufficiently small values of parameter $T,$ an arbitrarily-accurate approximation for $X (f)$ can be observed in the region $-{\tfrac {1}{2T}}<f<{\tfrac {1}{2T}}$ of the function:

$X_{1/T}(f)\ \triangleq \sum _{k=-\infty }^{\infty }X\left(f-k/T\right),$

which is precisely determined by the samples $x (nT)$ that span the non-zero duration of $x (t)$ (see Discrete-time Fourier transform).

And for sufficiently large values of parameter $N$ , $X_{1/T}(f)$ can be evaluated at an arbitrarily close frequency by a summation of the form:

$X_{1/T}\left({\tfrac {k}{NT}}\right)=\sum _{n=-\infty }^{\infty }\underbrace {T\cdot x(nT)} _{x[n]}\cdot e^{-i2\pi {\frac {kn}{N}}},$

where $k$ is an integer. The periodicity of $e^{-i2\pi {\frac {kn}{N}}}$ allows this to be written very simply in terms of a Discrete Fourier transform:

$X_{1/T}\left({\tfrac {k}{NT}}\right)=\underbrace {\sum _{n}x_{_{N}}[n]\cdot e^{-i2\pi {\frac {kn}{N}}},} _{\text{DFT}}\quad \scriptstyle {{\text{(sum over any }}n{\text{-sequence of length }}N)},$

where $x_{_{N}}$ is a periodic summation: $x_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }x[n-mN].$

When evaluated for all integers, $k$ , between 0 and $N$ -1, the array: $S\left({\tfrac {k}{NT}}\right)=\left|\sum _{n}x_{_{N}}[n]\cdot e^{-i2\pi {\frac {kn}{N}}}\right|^{2}$ is a periodogram.^[4]^[5]^[6]

Applications

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 $x [n]$ 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 $x [n]$ 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.

Notes

↑ $N$ is designated $NFFT$ in the Matlab and Octave applications.

Related Research Articles

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous, and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

<span class="mw-page-title-main">Nyquist–Shannon sampling theorem</span> Sufficiency theorem for reconstructing signals from samples

The Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample rate must be at least twice the bandwidth of the signal to avoid aliasing. In practice, it is used to select band-limiting filters to keep aliasing below an acceptable amount when an analog signal is sampled or when sample rates are changed within a digital signal processing function.

<span class="mw-page-title-main">Fourier transform</span> Mathematical transform that expresses a function of time as a function of frequency

In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions is the product of their Fourier transforms. More generally, convolution in one domain equals point-wise multiplication in the other domain. Other versions of the convolution theorem are applicable to various Fourier-related transforms.

<span class="mw-page-title-main">Fourier series</span> Decomposition of periodic functions into sums of simpler sinusoidal forms

A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.

In signal processing, the power spectrum $of a continuous time signal describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of any sort of signal as analyzed in terms of its frequency content, is called its spectrum.$

<span class="mw-page-title-main">Window function</span> Function used in signal processing

In signal processing and statistics, a window function is a mathematical function that is zero-valued outside of some chosen interval. Typically, window functions are symmetric around the middle of the interval, approach a maximum in the middle, and taper away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions.

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.

In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.

In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values.

In digital signal processing, downsampling, compression, and decimation are terms associated with the process of resampling in a multi-rate digital signal processing system. Both downsampling and decimation can be synonymous with compression, or they can describe an entire process of bandwidth reduction (filtering) and sample-rate reduction. When the process is performed on a sequence of samples of a signal or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate.

In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. Upsampling can be synonymous with expansion, or it can describe an entire process of expansion and filtering (interpolation). When upsampling is performed on a sequence of samples of a signal or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate. For example, if compact disc audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125.

In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectral density of that process.

The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by this formula:

The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval. It has several applications in electrical communication.

In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density of a signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

In digital signal processing, a discrete Fourier series (DFS) is a Fourier series whose sinusoidal components are functions of discrete time instead of continuous time. A specific example is the inverse discrete Fourier transform.

In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.

Multidimension spectral estimation is a generalization of spectral estimation, normally formulated for one-dimensional signals, to multidimensional signals or multivariate data, such as wave vectors.

References

↑ Schuster, Arthur (January 1898). "On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena" (PDF). Terrestrial Magnetism. 3 (1): 13–41. Bibcode:1898TeMag...3...13S. doi:10.1029/TM003i001p00013. It is convenient to have a word for some representation of a variable quantity which shall correspond to the 'spectrum' of a luminous radiation. I propose the word periodogram, and define it more particularly in the following way.
↑ 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. S2CID 120439605.
↑ "Periodogram—Wolfram Language Documentation".
1 2 "Periodogram power spectral density estimate - MATLAB periodogram".
↑ Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). Discrete-time signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. p. 732 (10.55). ISBN 0-13-754920-2.
↑ Rabiner, Lawrence R.; Gold, Bernard (1975). "6.18" . Theory and application of digital signal processing. Englewood Cliffs, N.J.: Prentice-Hall. pp. 415. ISBN 0-13-914101-4.
↑ "Do-it-yourself Science — is Proxima c hiding in this graph?". www.eso.org. Retrieved 11 September 2017.
↑ 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.
↑ "Welch's power spectral density estimate - MATLAB pwelch".
↑ Spectral Plot, from the NIST Engineering Statistics Handbook.
↑ "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.