Periodogram

 ( Frequency-domain Analysis ) 

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

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

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

$$X\_\{1/T\}\backslash left(\backslash tfrac\{k\}\{NT\}\backslash right)\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; \backslash underbrace\{T\backslash cdot\; x(nT)\}\_\{xn\}\backslash cdot\; e^\{-i\; 2\backslash pi\; \backslash frac\{kn\}\{N\}\},$$

where is an integer. The periodicity of  $e^\{-i\; 2\backslash pi\; \backslash frac\{kn\}\{N\}\}$  allows this to be written very simply in terms of a Discrete Fourier transform:

$$X\_\{1/T\}\backslash left(\backslash tfrac\{k\}\{NT\}\backslash right)\; =\; \backslash underbrace\{\backslash sum\_\{n\}\; x\_\{\_N\}n\backslash cdot\; e^\{-i\; 2\backslash pi\; \backslash frac\{kn\}\{N\}\},\}\_\backslash text\{DFT\}\; \backslash quad\; \backslash scriptstyle\{\backslash text\{(sum\; over\; any\; \}n\backslash text\{-sequence\; of\; length\; \}N)\},$$

where $x\_\{\_N\}$ is a periodic summation:  $x\_\{\_N\}n\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\{\backslash infty\}\; xn.$

When evaluated for all integers, , between 0 and -1, the array: $$S\backslash left(\backslash tfrac\{k\}\{NT\}\backslash right)\; =\; \backslash left|\; \backslash sum\_\{n\}\; x\_\{\_N\}n\backslash cdot\; e^\{-i\; 2\backslash pi\; \backslash frac\{kn\}\{N\}\}\; \backslash right|^2$$ is a periodogram.

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,  more commonly known as Welch's method,  divides a long xn 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.

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.

• Matched filter
• Radon transform (Radon transform)
• Welch's method
• Bartlett's method
• Discrete-time Fourier transform
• Least-squares spectral analysis, for computing periodograms in data that is not equally spaced
• MUltiple SIgnal Classification (MUSIC), a popular parametric superresolution method
• SAMV