Bilinear time–frequency distribution

Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub-field of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data. Such methods are used where one needs to deal with a situation where the frequency composition of a signal may be changing over time; this sub-field used to be called time–frequency signal analysis, and is now more often called time–frequency signal processing due to the progress in using these methods to a wide range of signal-processing problems. Methods for analysing time series, in both signal analysis and time series analysis, have been developed as essentially separate methodologies applicable to, and based in, either the time or the frequency domain. A mixed approach is required in time–frequency analysis techniques which are especially effective in analyzing non-stationary signals, whose frequency distribution and magnitude vary with time. Examples of these are acoustic signals. Classes of "quadratic time-frequency distributions" (or bilinear time–frequency distributions") are used for time–frequency signal analysis. This class is similar in formulation to Cohen's class distribution function that was used in 1966 in the context of quantum mechanics. This distribution function is mathematically similar to a generalized time–frequency representation which utilizes bilinear transformations. Compared with other time–frequency analysis techniques, such as short-time Fourier transform (STFT), the bilinear-transformation (or quadratic time–frequency distributions) may not have higher clarity for most practical signals, but it provides an alternative framework to investigate new definitions and new methods. While it does suffer from an inherent cross-term contamination when analyzing multi-component signals, by using a carefully chosen window function(s), the interference can be significantly mitigated, at the expense of resolution. All these bilinear distributions are inter-convertible to each other, cf.