1D Discrete wavelet transform(DWT)

H0(z), analysis scaling filter (ie. S)
H1(z) = (-z)^-N*H0(-z^-1), analysis wavelet filter (ie. W)
F0(z) = H1(-z), synthesis scaling filter
F1(z) = -H0(-z), synthesis wavelet filter

Now lets say that the scaling coefficients are h0,h1,h2, and h3. 
Then the z-transform of the filter is

H0(z) = h0 + h1*z^-1 + h2*z^-2 + h3*z^-3.

Yes. Oh, and if you're confused why db4 has more than 4 coefficients (as
Daubechies 4 should have) it is because in matlab the number after db
refers to the number of vanishing moments of the wavelet.
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As in matlab we have more than four coefficients , how then do we extend
this rule to more than four coefficients.
S = (h0,h1,h2,h3)
W = (h3,-h2,h1,-h0)

I have tried to find out the relationship between the coefficients of the
synthesis filters, I got the following results(?):
synthesis scaling filter ( h3,h2,h1,h0)
synthesis wavelet filter (-h0,h1,-h2,h3)

By the way, if I recall correctly, from the z transform we get the
frequency response by just replacing z = e^jw where w is the angular
frequency. Do we have to worry about the cutoff frequency of these
filters? 

Accocrding the the decomposition rule these filters work as halfband
filter which means that after the first level I get a lowfrequency part (0
to1khz) and highfrequency part(1kHz to 2kHz) for a 2kHz signal for
instance. It could be quite diff�cult to desing own quadrature mirror
filters, so i hope if i use these matlab coefficients  it could do the
same job!
Thanks!

Umutesi Faith wrote:

> As in matlab we have more than four coefficients , how then do we extend
> this rule to more than four coefficients.

By using the four equations I gave you. Generally H0(z) is

H0(z) = h0 + h1*z^-1 + h2*z^-2 + ... + h{N-1}*z^-(N-1)

> 
> I have tried to find out the relationship between the coefficients of the
> synthesis filters, I got the following results(?):
> synthesis scaling filter ( h3,h2,h1,h0)
> synthesis wavelet filter (-h0,h1,-h2,h3)

Looks ok to me.

> 
> By the way, if I recall correctly, from the z transform we get the
> frequency response by just replacing z = e^jw where w is the angular
> frequency. Do we have to worry about the cutoff frequency of these
> filters?

No. 

Only thing one has to be concerned of is perfect reconstruction and this is
preseverved if the scaling filter (lowpass filter) has certain properties
and the other three filters are calculated from the scaling filters trought
the simple equations I gave you. When matlab or some other source gives you
the coefficients for the scaling function you don't have to worry a thing.

> 
> Accocrding the the decomposition rule these filters work as halfband
> filter which means that after the first level I get a lowfrequency part (0
> to1khz) and highfrequency part(1kHz to 2kHz) for a 2kHz signal for
> instance. It could be quite diff�cult to desing own quadrature mirror
> filters, so i hope if i use these matlab coefficients  it could do the
> same job!

Actually designing new wavelets is quite easy, but finding wavelets that
perform well for a given task is hard. If you are really intrested in
wavelets I propose you read papers by Wim Sweldens
(http://cm.bell-labs.com/who/wim/papers/papers.html) and especially this: 
http://cm.bell-labs.com/who/wim/papers/athome.pdf. Lifting is a way to
implement dwt efficiently and losslessly (even in the precense of
quantization errors) and Sweldens papers gets you started. 


-- 
Jani Huhtanen
Tampere University of Technology, Pori

>Umutesi Faith wrote:
>
>> As in matlab we have more than four coefficients , how then do we
extend
>> this rule to more than four coefficients.
>
>By using the four equations I gave you. Generally H0(z) is
>
>H0(z) = h0 + h1*z^-1 + h2*z^-2 + ... + h{N-1}*z^-(N-1)
>
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Ok ,i'll use this formula and see what results i get for more than 4
coefficients, and apply it to the matlab (db4) coefficients!
Thanks a lot for your time!!