# Intuitively explain Bi-linear time frequency distributions, someone please?

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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## 1 Answer

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:

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.

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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