Discrete wavelet transform relation to sampling frequency of the signal

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.

Answer 1

[Wayne King]

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');

Answer 2

[Wayne King]

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)

[chirag]

Thankx Wayne for the reply,

Actually i took a Pure 50 Hz sine function with high frequency about 5000 Hz sampled with 1Mhz and decomposed it using Wavedec up to 13 th level. i am getting correct energy at D7 as it contains 5000 Hz... but at the same time D13 is giving high energy value. D13 should contain 60 to 122 Hz components which are not present in my signal.

is that because of aliazing..???

i thought only Apporimation shd contain 0- 60 Hz and hence high energy value. There is no 60-120 Hz component in my signal. Please suggest.

[Wayne King]

which scaling/wavelet filter are you using?

[chirag]

i am using 'db6' wavelet.

[chirag]

Thankx wayne for the answer.

I will try and use higher order wavelet

:) Thankx again./..