Easy Start on Wavelets

In this example we will generate a wavelet CSD plot

Also note that only real part of the wavelet is shown. Imaginary part can be included if and when phase information is required.

A wavelet in the time domain looks for example like this:

What we are interested in wavelets is the time-frequency information so we want to see the amplitude versus time. Amplitude of the wavelet can be seen in the envelope like this:

To analyse a whole frequency band we generate a family of wavelets. Here is seen from 100Hz to 10kHz in 5 steps:

And for the ampitude information we calculate the envelope for each like this:

From now on in this case we are only interested of the envelope so we select only that and we have:

Because a human is a logarithmic device, we take LOG of the above and see the envelope in 20dB scale here:

Now we have some fun and visit the Indian bazaar and get one of those colorful blankets and throw it on the top:

And now we are almost done! For better visibility we choose the view angle at the zenith and finally we have this:

And thus we have generated a family of wavelets that can be used to analyse systems impulse response amplitude and phase bahaviour in time-frequency domain. The color bar indicates the amplitude scale in dB.

Next thing to do is to combine the generated wavelets with the system inpulse response.

In the beginning there was an impulse response:

And there was a wavelet:

We combine them through the transform:

And have this:

where
blue = wavelet
green = transform of the wavelet and the impulse response
red = envelope of the wavelet
magenta = envelope of the transform

Now the envelope is the interesting one which is better to see on the dB scale:

Next thing to do is to repeat the above at every frequency of interest. And after making another visit to the Indian bazaar one has for example this plot: