Discrete wavelet transform relation to sampling frequency of the signal

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chirag
chirag 2012-5-17
Dear all, I am trying to decompose a signal which is sampled at (1e-6) second sampling rate. that comes out to be 1Mhz Frequency. According to Nequist sampling theorem the highest freq in the signal shd be , 1Mhz / 2 = 500000 Hz = 500 KHz Please suggest me what is the relation between the sampling frequency and DWT, (i am usind WAVEDEC function of MATLAB) Does it mean that D1 should contain 250Khz to 500 Khz frequency. or it does not follow the nequist theorem as it is.

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Wayne King
Wayne King 2012-5-17
Yes, what you are seeing is that the filters are not perfect bandpass filters, but they are better if they are longer. For example, compare the 'db20'
Fs = 1e6;
t = 0:1/Fs:1-1/Fs;
x = cos(2*pi*50*t);
[C,L] = wavedec(x,15,'db20');
details = detcoef(C,L,'cells');
d14recon = wrcoef('d',C,L,'db20',14);
plot(d14recon,'k'); % approx 30-60 Hz amplitude nearly 1
d13recon = wrcoef('d',C,L,'db20',13);
hold on;
plot(d13recon,'r'); %look how small the amplitude is
Now compare against the approximation at level 13
a13recon = wrcoef('a',C,L,'db20',13);
plot(a13recon,'b');

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Wayne King
Wayne King 2012-5-17
Hi Chirag, Yes, in general you can say that D1 contains information in the signal from 250 kHz to 500 kHz. However, two things to keep in mind
1.) The approximation of D1 to a bandpass filtering of the signal depends on the wavelet filter you use. In general the longer the wavelet filter, the better the approximation.
2.) Keep in mind that there is downsampling by a factor of two involved here. Downsampling can introduce aliasing. There is often aliasing in the wavelet details (see point 1 above). That aliasing is canceled on reconstruction (the DWT is a perfect reconstruction filter bank)
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chirag
chirag 2012-5-17
Thankx wayne for the answer.
I will try and use higher order wavelet
:) Thankx again./..

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