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I have a fairly good understanding of Wavelet Analysis, but what are these bilinear distributions and how do they differ from Wavelet Transform?

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As you mentioned - all these methods share common principle, they allow for representation of our signal in time-frequency domain. First thing to notice is that wavelets are very different from the STFT-like (linear) methods as they provide scale dependent frequency resolution (which in case of STFT is constant and governed by Heisenberg uncertainty principle). Probably you've seen this famous picture:

enter image description here

So called bi-linear transform mostly refers to Wigner Transform which can be considered as the Fourier Transform of signals' instantaneous autocorrelation function. It is governed by following equation:

$$W(t,f)=\int_{-\infty}^{\infty} x \left( t+ \frac{\tau}{2} \right) x^* \left( t-\frac{\tau}{2} \right) e^{-2\pi i f \tau} d\tau$$

where $^*$ denotes complex conjugate and $\tau$ is the time delay.

One might notice that in analogy to STFT, the shifted signal itself is considered as a window. It is also worth of mentioning, that Wigner Transform provides perfect time and frequency resolution at the same time - this is simply great for signals with fast changes of instantaneous frequency. On the other hand WT is the bi-linear transform, which implies major disadvantage: for each linear combination of two signals $x(t)$ and $y(t)$, result consists of two parts: auto-terms and cross-terms. This issue produces interferences in time-frequency representation known as "beats", which can make interpretation of WT difficult. If you are looking for more specific mathematical description, then please refer to article posted below - I see no point in rewriting equations.

Example is shown on a figure below, where two linear chirps were analysed. Cross-term is clearly visible in between, which makes interpretation of results troublesome. Sometimes it happens, that magnitude of cross-term is even higher than auto-terms.

enter image description here

Although some methods were described in literature how to deal with this problem. Please refer to the following article for more details.

R.B. Pachori and P. Sircar - A new technique to reduce cross terms in the Wigner distribution

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