Re: First post for me
Sorry if my reply was confusing. All of these methods of analysis are related at a fundamental level. Their algorithmic application to actual signals just differs significantly. Hilbert Decomposition and TDS don't determine the response of the system from H(z) = Y(z)/X(z) the way FFT/Convolution based methods do.
Time Frequency Representations are not really measuring the response of a system in their typical application. Though there is a paper on AES showing the operations using the Wigner-Ville distribution which replicate TDS and many more on IEEE Signal Processing proving the equivalence.
More commonly in the audio industry the TFR is applied to the impulse response measured by another method, or directly to the recorded time waveform. A real time Spectrogram is commonly applied to the recorded time waveform. The Wavelet Distribution TFR images Bennett posted were from measured impulse responses.
Just to add to what you were saying;
The uncertainty principle for sinusoidal signal processing is very simply stated:
T = std dev of duration of the signal
B = std dev of bandwidth of the signal
TB >= 0.5
Note there is nothing in this statement regarding the resolution of an FFT, this is a common misconception. It just says you can't have a signal which is arbitrarily small in either time or frequency. Or more exactly, you can't have a Fourier pair x(t) and X(w) that have support which is arbitrarily small. The Gaussian pulse has optimal support in both time and frequency.
Since our whole idea of time and frequency is based on Fourier mathematics, all TFRs try to achieve this same level of concentration in the t-f plane at the expense of other properties. The other properties might be really important, like being positive, not being complex valued, not working on multi-component signals (ie. music), or like the WVD producing interference from nonlinear instantaneous frequency.
The reason why the STFT has poor T-F performance is not because of the uncertainty statement above. It is because when you make a very small time window of a signal you have a whole new uncertainty principle based on the size of that window. The STFT works by slicing up the signal into small time segments and taking the FFT; ie apply the above uncertainty statement to the new windowed signal.
The wavelet TFR works in a similar way except that the window is also a function of frequency. So a Wavelet TFR is like a sliding STFT where the frequency concentration changes with the frequency.
I put the source code for Octave or MATLAB for WVD in the DIY audio section.
http://www.soundforums.net/live/threads/3087-A-couple-of-Time-Frequency-Analysis-MATLAB-programs
