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Here we give a quick description of what is probably the most popular family of filter coefficients h n and g n —those proposed by Daubechies.

Recall the iterated synthesis filterbank. Applying the Noble identities, we can move the up-samplers before the filters, asillustrated in .

The properties of the i -stage cascaded lowpass filter

i 1 H ( i ) z k 0 i 1 H z 2 k
in the limit i give an important characterization of the wavelet system. But how do we know that i H ( i ) ω converges to a response in 2 ? In fact, there are some rather strict conditions on H ω that must be satisfied for this convergence to occur. Without such convergence, we might have a finite-stageperfect reconstruction filterbank, but we will not have a countable wavelet basis for 2 . Below we present some "regularity conditions" on H ω that ensure convergence of the iterated synthesis lowpass filter.
The convergence of the lowpass filter implies convergence of all other filters in thebank.

Let us denote the impulse response of H ( i ) z by h ( i ) n . Writing H ( i ) z H z 2 i 1 H ( i 1 ) z in the time domain, we have h ( i ) n k k h k h ( i 1 ) n 2 i 1 k Now define the function φ ( i ) t 2 i 2 n n h ( i ) n [ n / 2 i , n + 1 / 2 i ) t where [ a , b ) t denotes the indicator function over the interval a b : [ a , b ) t 1 t a b 0 t a b The definition of φ ( i ) t implies

t t n 2 i n 1 2 i h ( i ) n 2 i 2 φ ( i ) t
t t n 2 i n 1 2 i h ( i 1 ) n 2 i 1 k 2 i 1 2 φ ( i 1 ) 2 t k
and plugging the two previous expressions into the equation for h ( i ) n yields
φ ( i ) t 2 k k h k φ ( i 1 ) 2 t k .
Thus, if φ ( i ) t converges pointwise to a continuous function, then it must satisfy the scaling equation, so that i φ ( i ) t φ t . Daubechies showed that, for pointwise convergence of φ ( i ) t to a continuous function in 2 , it is sufficient that H ω can be factored as
P P 1 H ω 2 1 ω 2 P R ω
for R ω such that
sup ω R ω 2 P 1
Here P denotes the number of zeros that H ω has at ω . Such conditions are called regularity conditions because they ensure the regularity, or smoothness of φ t . In fact, if we make the previous condition stronger:
1 sup ω R ω 2 P 1
then i φ ( i ) t φ t for φ t that is -times continuously differentiable.

There is an interesting and important by-product of the preceding analysis. If h n is a causal length- N filter, it can be shown that h ( i ) n is causal with length N i 2 i 1 N 1 1 . By construction, then, φ ( i ) t will be zero outside the interval 0 2 i 1 N 1 1 2 i . Assuming that the regularity conditions are satisfied so that i φ ( i ) t φ t , it follows that φ t must be zero outside the interval 0 N 1 . In this case we say that φ t has compact support . Finally, the wavelet scaling equation implies that, when φ t is compactly supported on 0 N 1 and g n is length N , ψ t will also be compactly supported on the interval 0 N 1 .

Daubechies constructed a family of H z with impulse response lengths N 4 6 8 10 which satisfy the regularity conditions. Moreover, her filters have the maximum possible number of zeros at ω , and thus are maximally regular (i.e., they yield the smoothest possible φ t for a given support interval). It turns out that these filters are the maximally flat filters derived by Herrmann long before filterbanks and wavelets were in vogue. In and we show φ t , Φ Ω , ψ t , and Ψ Ω for various members of the Daubechies' wavelet system.

See Vetterli and Kovacivić for a more complete discussion of these matters.

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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