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Thread Subject:
signal reconstruction using wavelets

Subject: signal reconstruction using wavelets

From: FAISAL PEER MOAHMED

Date: 9 May, 2010 19:46:03

Message: 1 of 4

Hi

I am using wavelets to denoise a noisy signal. Perfect reconstruction depends upon the
wavelet basis functions used. cross correlation of the signal with wavelts basis can help us to select the optimum wavelet. Comeback to wavelet decomposition , signal is convoluted with filter coefficients.

Can any one please help me the relation between cross corelation and convolution in wavelet decomposition ?

Regards

Faisal

Subject: signal reconstruction using wavelets

From: Wayne King

Date: 9 May, 2010 21:27:03

Message: 2 of 4

"FAISAL PEER MOAHMED" <pfaisalbe@gmail.com> wrote in message <hs73dr$6ku$1@fred.mathworks.com>...
> Hi
>
> I am using wavelets to denoise a noisy signal. Perfect reconstruction depends upon the
> wavelet basis functions used. cross correlation of the signal with wavelts basis can help us to select the optimum wavelet. Comeback to wavelet decomposition , signal is convoluted with filter coefficients.
>
> Can any one please help me the relation between cross corelation and convolution in wavelet decomposition ?
>
> Regards
>
> Faisal

Hi Faisal, the cross correlation of a signal with an FIR filter is the same as convolving the signal with the complex-conjugated and time-reversed filter. At the decimation, the wavelet coefficients are obtaining by cross-correlating the signal with the wavelet filter at even lags, but that is equivalent to downsampling the convolution of the signal with the time-reverse and complex-conjugate of the wavelet filter.

Note for

[LoD,HiD,LoR,HiR] = wfilters('db1');

that the difference between HiD and HiR is just that they are time-reversed versions of each other. Since the filters are real-valued, you obviously don't see the effect of taking the conjugate. Next note that:

dwtmode('per','nodisp');
reset(RandStream.getDefaultStream);
[ca,cd] = dwt(x,'db1');

produces wavelet coefficients, cd, that are equivalent to:

downsample(conv(x,HiD,'valid'),2)

Hope that helps,
Wayne

Subject: signal reconstruction using wavelets

From: FAISAL PEER MOAHMED

Date: 10 May, 2010 12:52:23

Message: 3 of 4

Many Thanks.

As you mentioned "At the decimation, the wavelet coefficients are obtaining by cross-correlating the signal with the wavelet filter at even lags".

So in which domain this cross correlation is taking place ? i mean in time domain or frequency domain

Regards

Faisal





"Wayne King" <wmkingty@gmail.com> wrote in message <hs79b7$k5i$1@fred.mathworks.com>...
> "FAISAL PEER MOAHMED" <pfaisalbe@gmail.com> wrote in message <hs73dr$6ku$1@fred.mathworks.com>...
> > Hi
> >
> > I am using wavelets to denoise a noisy signal. Perfect reconstruction depends upon the
> > wavelet basis functions used. cross correlation of the signal with wavelts basis can help us to select the optimum wavelet. Comeback to wavelet decomposition , signal is convoluted with filter coefficients.
> >
> > Can any one please help me the relation between cross corelation and convolution in wavelet decomposition ?
> >
> > Regards
> >
> > Faisal
>
> Hi Faisal, the cross correlation of a signal with an FIR filter is the same as convolving the signal with the complex-conjugated and time-reversed filter. At the decimation, the wavelet coefficients are obtaining by cross-correlating the signal with the wavelet filter at even lags, but that is equivalent to downsampling the convolution of the signal with the time-reverse and complex-conjugate of the wavelet filter.
>
> Note for
>
> [LoD,HiD,LoR,HiR] = wfilters('db1');
>
> that the difference between HiD and HiR is just that they are time-reversed versions of each other. Since the filters are real-valued, you obviously don't see the effect of taking the conjugate. Next note that:
>
> dwtmode('per','nodisp');
> reset(RandStream.getDefaultStream);
> [ca,cd] = dwt(x,'db1');
>
> produces wavelet coefficients, cd, that are equivalent to:
>
> downsample(conv(x,HiD,'valid'),2)
>
> Hope that helps,
> Wayne

Subject: signal reconstruction using wavelets

From: Wayne King

Date: 10 May, 2010 13:00:25

Message: 4 of 4

"FAISAL PEER MOAHMED" <pfaisalbe@gmail.com> wrote in message <hs8vi7$6jl$1@fred.mathworks.com>...
> Many Thanks.
>
> As you mentioned "At the decimation, the wavelet coefficients are obtaining by cross-correlating the signal with the wavelet filter at even lags".
>
> So in which domain this cross correlation is taking place ? i mean in time domain or frequency domain
>
> Regards
>
> Faisal
>
>
>
>
>
> "Wayne King" <wmkingty@gmail.com> wrote in message <hs79b7$k5i$1@fred.mathworks.com>...
> > "FAISAL PEER MOAHMED" <pfaisalbe@gmail.com> wrote in message <hs73dr$6ku$1@fred.mathworks.com>...
> > > Hi
> > >
> > > I am using wavelets to denoise a noisy signal. Perfect reconstruction depends upon the
> > > wavelet basis functions used. cross correlation of the signal with wavelts basis can help us to select the optimum wavelet. Comeback to wavelet decomposition , signal is convoluted with filter coefficients.
> > >
> > > Can any one please help me the relation between cross corelation and convolution in wavelet decomposition ?
> > >
> > > Regards
> > >
> > > Faisal
> >
> > Hi Faisal, the cross correlation of a signal with an FIR filter is the same as convolving the signal with the complex-conjugated and time-reversed filter. At the decimation, the wavelet coefficients are obtaining by cross-correlating the signal with the wavelet filter at even lags, but that is equivalent to downsampling the convolution of the signal with the time-reverse and complex-conjugate of the wavelet filter.
> >
> > Note for
> >
> > [LoD,HiD,LoR,HiR] = wfilters('db1');
> >
> > that the difference between HiD and HiR is just that they are time-reversed versions of each other. Since the filters are real-valued, you obviously don't see the effect of taking the conjugate. Next note that:
> >
> > dwtmode('per','nodisp');
> > reset(RandStream.getDefaultStream);
> > [ca,cd] = dwt(x,'db1');
> >
> > produces wavelet coefficients, cd, that are equivalent to:
> >
> > downsample(conv(x,HiD,'valid'),2)
> >
> > Hope that helps,
> > Wayne

Time domain, but it is equivalent a sum of multiplications in the frequency domain ( a sum of multiplications and not just one because of the decimation).

Wayne

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