Let be a stationary Gaussian time series with (one-sided) power spectral density , where is even and samples are taken at constant sampling intervals .
Let be the (complex-valued) discrete Fourier transform (DFT) of the time series. Then for the Whittle likelihood one effectively assumes independent zero-mean Gaussian distributions for all with variances for the real and imaginary parts given by
where is the th Fourier frequency. This approximate model immediately leads to the (logarithmic) likelihood function
In case the noise spectrum is assumed a-priori known, and noise properties are not to be inferred from the data, the likelihood function may be simplified further by ignoring constant terms, leading to the sum-of-squares expression
The Whittle likelihood in general is only an approximation, it is only exact if the spectrum is constant, i.e., in the trivial case of white noise.
The efficiency of the Whittle approximation always depends on the particular circumstances.
Note that due to linearity of the Fourier transform, Gaussianity in Fourier domain implies Gaussianity in time domain and vice versa. What makes the Whittle likelihood only approximately accurate is related to the sampling theorem—the effect of Fourier-transforming only a finite number of data points, which also manifests itself as spectral leakage in related problems (and which may be ameliorated using the same methods, namely, windowing). In the present case, the implicit periodicity assumption implies correlation between the first and last samples ( and ), which are effectively treated as "neighbouring" samples (like and ).
Whittle's likelihood is commonly used to estimate signal parameters for signals that are buried in non-white noise. The noise spectrum then may be assumed known,
or it may be inferred along with the signal parameters.
Signal detection is commonly performed utilizing the matched filter, which is based on the Whittle likelihood for the case of a known noise power spectral density.
The matched filter effectively does a maximum-likelihood fit of the signal to the noisy data and uses the resulting likelihood ratio as the detection statistic.
The matched filter may be generalized to an analogous procedure based on a Student-t distribution by also considering uncertainty (e.g. estimation uncertainty) in the noise spectrum. On the technical side, this entails repeated or iterative matched-filtering.
The Whittle likelihood is also applicable for estimation of the noise spectrum, either alone or in conjunction with signal parameters.
^Choudhuri, N.; Ghosal, S.; Roy, A. (2004). "Contiguity of the Whittle measure for a Gaussian time series". Biometrika. 91 (4): 211–218. doi:10.1093/biomet/91.1.211.
^Countreras-Cristán, A.; Gutiérrez-Peña, E.; Walker, S. G. (2006). "A Note on Whittle's Likelihood". Communications in Statistics – Simulation and Computation. 35 (4): 857–875. doi:10.1080/03610910600880203.