perfect reconstruction filter banks using rational sampling rate changes
DESCRIPTION
In this paper we are constructing a perfect reconstruction filter bank using rational sampling rate changes.Such filter banks usually will have N branches and each one having a sampling factor of pi/qi and their sum equals one. Here we are trying to extend the concept of uniform filter bank splitting into nonuniform splitting of filter banks. This can be very useful in the analysis of speech and music. The use of unequal subbands was motivated by the experimental evaluation of the instantaneous signal bandwidth of speech frames. A design example showing the advantage of using the direct over the indirect method is given. The theory relies on two transforms, 1 and 2. While Transform 1 when applied, leads to uniform filter banks having polyphase components as individual filters, Transform 2 results in a uniform filter bank containing shifted versions of same filters. This, in turn, results in dependencies in design and is beyond the scope of the paper.TRANSCRIPT
Perfect Reconstruction Filter Banks using rational sampling rate changes
1
1. INTRODUCTION
The frequently studied case of filter banks is the one with irrational or integer sampling
rate changes. Suppose if anyone wants to analyze the signal interms of unequal subbands then
the obvious solution would be to go for rational sampling rate changes. However, the need for a
non-integer sampling rate conversion appears when the two systems operating at different
sampling rates have to be connected, or when there is a need to convert the sampling rate of the
recorded data into another sampling rate for further processing or reproduction. Such
applications are very common in telecommunications, digital audio, multimedia and others.
Usually we use unequal subbands, because during voiced parts of a speech signal, most
of the signal energy is present in the lower frequency region. Therefore it is not necessary to
encode the higher part of the frequency range. Transform coding techniques allocate, in voiced
frames, more bits to code lower frequency components than higher frequency components. So
we give a wide bandwidth range for low frequency component and a comparatively very
narrower bandwidth range for higher frequencies thus resulting in unequal splitting of spectrum.
When we use unequal subbands rational sampling rates has to be allowed. Then each
channel would have a sampling factor pi/qi and their sum equals to one, so as to preserve the
sampling density. We can solve this problem by dividing the spectrum into Q= LCM (qi) parts
and then Resynthesize the appropriate subspectra. Since it is an indirect approach, it is not so
good interms of filter quality and computational complexity. Earlier research in this area was to
aimed at only at alias cancellation. Here we are presenting a direct method to design the perfect
reconstruction filter bank with non integer sampling rates. It depends on two transforms, namely
1 and 2. While Transform 1, when applied, leads to uniform filter banks having polyphase
components as individual filters, Transform 2 results in a uniform filter bank containing shifted
versions of the same filter. This in turn introduces dependencies in design and is beyond the
scope of this paper.
Perfect Reconstruction Filter Banks using rational sampling rate changes
2
1.1 A GLIMPSE AT PERFECT RECONSTRUCTION
FILTER BANKS
Here we are going to revisit some of the fundamental concepts of perfect reconstruction filter
banks.
An Analysis filter bank is a signal processing device that splits the input signal into M
channel signals by means of filtering and downsampling by N ( where N<=M). Here we assume
that the filter bank is critically sampled i.e., N=M. The synthesis filter bank performs the inverse
task (see Fig. 1(a)). Due to the fact that the downsampling is a periodically shift variant operation
with period N (that is, if the input x(n) produces the output y(n), then the input x(n-n0) will
.produce output y(n-n1) only if n0= Nn1), the whole system becomes periodically shift variant. A
way to make the analysis of such a system easier, is to decompose both signals and filters into so
called polyphase components. For a filter, each polyphase component would then represent one
of N impulse responses (at times 0, 1… N-1). Thus a filter can be expressed as
1
0
)(N
i
N
ji
i
j zHzH
Where Hji(z) is the i-th polyphase component of the filter Hj(z), and is given by
n
n
jji zinNhzH ).()(
It turns out that the output of the system can be conveniently expressed in terms of analysis and
synthesis poly-phase matrices (that is, matrices containing polyphase components of analysis
and synthesis filters), as well as forward and inverse polyphase transforms. Forward
polyphase transform inputs the signal and outputs its N polyphase components by means of
shifting and down-sampling by N. The inverse polyphase transform per-forms the inverse
task, that is, the forward and the inverse polyphase transforms are inverses of each other.
Perfect reconstruction is equivalent to forcing the synthesis poly-phase matrix to be the
inverse of the analysis one. A filter bank expressed in the polyphase domain is given in Fig.
1(b). One of the easiest ways to achieve perfect re-construction (i.e., to obtain the output as a
perfect replica of the input), is to construct a paraunitary analysis matrix (or orthogonal,
lossless).
Perfect Reconstruction Filter Banks using rational sampling rate changes
3
In other words, the analysis polyphase matrix has to satisfy the following:
Hp(z-I)
• Hp(z) = I.
Then the synthesis polyphase matrix can be chosen as which in turn yields
filters that are the same as the analysis filters (within shift reversal).
Let us also point out some facts on multirate filtering that are going to be used later.
1) Upsampling by p and downsampling by q can be interchanged if and only if p and q are
relatively prime
2) The output after filtering by H(z) and downsampling by N can be written as
where WN denotes the Nth root of unity, i.e., WN=)/2( Nje .
3) A pair of useful identities known under the name of "noble identities” gives
conditions under which shift-invariant filters can be passed across up- and downsamplers.
They state that any filter in the downsampled domain can be represented in the upsampled
domain by simply upsampling its impulse response. Very similarly, a filter with z-transform
H(z) placed in front of upsampling by N can be moved past the upsampler and represented as
H(zN).
)()(1
)(
11
0
1
Nk
N
N
k
Nk
N zWXzWHN
zY
Perfect Reconstruction Filter Banks using rational sampling rate changes
4
Figure 1. (a) Analysis/Synthesis Filter Banks.
(b) Filter Banks in the polyphase domain.
Perfect Reconstruction Filter Banks using rational sampling rate changes
5
1.2 Filter Banks with Rational Sampling Factors
Here we are investigating the filter banks with rational sampling rates and to realize these
kinds of filters is our paramount objective here. The corresponding depictions are shown in
Fig. 2(a) and 2(b).
Fig. 2 (a) a block diagram
\
Fig. 2 (b) the desired spectrum splitting
Perfect Reconstruction Filter Banks using rational sampling rate changes
6
Here we shall use the notation [p0/q0, p1/q1 … pN-1/qN-1] to denote a filter bank
where the i-th channel (numbered in ascending frequency) has a rate pi/qi and
contains input frequencies ranging over
where the sum is defined to be 0 if the upper bound is negative . We will also assume that the
filter bank is critically sampled, i.e.,
THE INDIRECT METHOD:
One of the methods to achieve the desired above mentioned factors is to identify the least
common multiple of all the downsampling factors and analyze the input into Q=LCM(q0, q1…
qN-1) subbands. To obtain a perfect reconstruction filter bank, one can combine an analysis
filter bank having Q filters with N synthesis filter banks. , if analysis and synthesis banks are
perfect reconstruction, the overall system will be perfect reconstruction as well. This method
however produces frequency shuffling. Even though we try to eliminate shuffling of
frequencies, this method is not so good because of its computational complexity and filter
quality. The equivalent figure, i.e., of the indirect method, is shown in figure 3.
Figure 3.
Filter
designed
indirectly.
Perfect Reconstruction Filter Banks using rational sampling rate changes
7
2. A Direct Design method
Here we are designing a perfect reconstruction filter bank with arbitrary sampling rates
directly. Let us begin our study by considering a critically sampled filter bank as shown in
figure 2. Also we assume that (pi,qi) relatively prime and p > 1.
One can observe that output of the branch is the convolution of the input signal and a
particular polyphase component of the filter, and thus the filter can be expressed as p filters
in parallel, each one of them being
where, , ti=qi mod p ( x denotes the biggest integer not greater than x) and H0,
H1,…Hp-1 are the polyphase components of H with respect to p. It should be noted that for p = 1
there is no transform, i.e. HHzHtzHd
o 0
0
0
' 0 since the only polyphase component with
respect to p = 1 is the filter itself. At this point one could apply Transform 1 to all the branches in
figure 2(a). If q0=…..=qN-1=q, nothing else can be done, since the transform results in
1
0
N
i
i qp
branches followed by downsampling by q. Thus, the problem has been reduced into finding a
perfect reconstruction structure for a q-channel filter bank, with design constraints imposed
on filters H0,…HN-1.
If however, not all qi’s are same, applying the transform for each branch i, produce p i
branches followed by downsampling by qi (note that those with pi = 1 will remain the same).
Thus, what one would like to do is to transform this into a system having Q branches fol-
lowed by downsampling by Q. The relevant figure is shown in figure 4(a).
Now if Q = lcm (q0,q1…qN-1)w e apply transform 2 in each branch with downsampling by q
using p-channel analysis bank with downsampling by Q=PQ and an inverse of polyphase
transform of size p. This method is given in figure 4(b). The filter in the ith branch is just a
shifted version of the original filter )()( zHzzH iq
i .
In Fig. 4(b), move the filter H out of each branch and represent downsampling by Q, as
downsampling by q followed by downsampling by p. Then, using the noble identities, one
)()(' zHzH it
dizi
p
qd
ii
Perfect Reconstruction Filter Banks using rational sampling rate changes
8
can move downsampling by q in front of delays, causing downsampling of the delays by q.
The resulting system is then as follows: filter H followed by downsampling by q, followed
by an identity system, i.e., the starting scheme.
Finally, Fig. 4(c) shows how by using the above transforms one can implement a filter bank
from Fig. 2(a). First Transform 1 is applied in each branch which yields an analysis bank
with
1
0
N
i
ipn
branches and sampling factors q0,q1...qN-1. Now if Q=LCM (q0,q1...qN-1) we apply transform
2 in each branch to obtain analysis bank with
1
0
1
0
.N
i i
N
i i
i Qq
piQ
q
Qpn
branches and downsampling by Q.
It is worth noting here the difference between the indirect and the direct method. In the indirect
one we design the two stages of the analysis bank separately and moreover we have no idea what
kind of characteristics the equivalent filters (H0,H1…HN-1 from figure 2(a) ) are going to have
since we do not know how these filters are related to the filters in the analyzing and
resynthesizing banks. Using a direct method however, allows us to design any filter bank with
rational sampling rates, having at the same time complete control over the desired characteristics
of the filters H0,H1…HN-1.
Perfect Reconstruction Filter Banks using rational sampling rate changes
9
Figure 4(a). Transform 1: : expressing a single branch with upsampling by p and
downsampling by q using a p-channel analysis bank with sampling by q and an inverse
polyphase transform of size p. All the filters involved are just shifted polyphase components of
the original filter. For p 1 there is no transform. Also, p and q are assumed to be coprime.
Perfect Reconstruction Filter Banks using rational sampling rate changes
10
Figure 4(b). Transform 2: expressing a single branch with downsampling by q using a
p-channel analysis bank with sampling by Q = pq and an inverse polyphase transform of size p.
All the filters involved are just shifted versions of the original filter. Note the dependency that
appears in the filter banks after Transform 2.
Perfect Reconstruction Filter Banks using rational sampling rate changes
11
Figure 4(c): To transform any bank we first apply Transform I and then Transform 2 in each
branch. As a result an analysis bank with sampling by Q = lcm (q0,q1...qN-1) and Q branches is
obtained.
Perfect Reconstruction Filter Banks using rational sampling rate changes
12
3. Design Example
As a simple example consider a bank with sampling by 2/3 and 1/3. Let us first construct the
system indirectly, i.e. we design a 3-channel analysis bank and then resynthesize the first two
branches using a 2-channel synthesis bank.
The 3-channel bank contains filters of length 15 and the 2-channel one filters of length 8 with
lattice coefficients a1 = -2.638026, az = 0.7154463, a3 = -0.2598479 and a4 = 0.06388361. As a
result we obtain a lowpass filter of length 14 . 2 + 7 . 3 + 1 = 50 and a highpass filter which is the
third filter from the 3-channel bank. The magnitude response of the lowpass filter is given by the
gray plot in Figure 5. Now instead of this method we first obtain the equivalent directly. In order
to do that we use the 4 lattice parameters from the 2-channel bank as the minimization variables.
We do not touch the 3-channel bank so as not to ruin the highpass filter. The obtained optimized
lattice coefficients are a1 = -0.371151, a2 = 2.732850, a3 = 1.056070 and a4 = 0.664108. The
magnitude response of the resulting filter is given by the black plot in figure 5. As can be seen
from there the improvement is obvious: the passband has been flattened and the stopband has
been greatly reduced.
Figure 5. Magnitudes of the
frequency responses of the lowpass
filters designed using the indirect
(gray plot) and direct method (black plot). Note the improvement obtained
by using the direct design method (the
passband is flattened and the value in
the stopband has been reduced).
Perfect Reconstruction Filter Banks using rational sampling rate changes
13
4. Wavelets with 3/2 dilation factor
A wavelet series is a representation of a square-integrable (real- or complex-valued) function
by a certain orthonormal series generated by a wavelet. Nowadays, wavelet transformation is one
of the most popular candidates of the time-frequency-transformations.
The integral wavelet transform is the integral transform defined as
The wavelet coefficients are then given by
Here, is called the binary dilation or dyadic dilation, and is the binary
or dyadic position.
In this section we address some of the questions that arise when looking at the filter bank
problem from the wavelet theory point of view restricting ourselves at the same time to a
representative case, namely sampling by 2/3 and 1/3. Let us point out that to establish
correspondence between filter banks and wavelets we split the branch with the lowpass filter
using the same filter bank. Repeating this procedure to infinity, the wavelet and the scaling
function can be identified, scaling function as the equivalent filter in the path going through all
lowpass branches, and wavelet in the same path except that in the last stage we go through the
highpass branch.
Let us point out that to establish correspondence between filter banks and wavelets we split the
branch with the lowpass filter using the same filter bank. Repeating this procedure to infinity, the
wavelet and the scaling function can be identified, scaling function as the equivalent filter in the
path going through all lowpass branches, and wavelet in the same path except that in the last
stage we go through
the highpass branch. For sampling by 2, Daubechies gives a sufficient condition for the iterated
filter to converge to a continuous function. It basically states that for a filter to be regular we
have to impose a sufficient number of zeroes at pi (aliasing frequency) and attenuate enough the
remaining factor. Following the same reasoning we conjecture that in this case a filter having
Perfect Reconstruction Filter Banks using rational sampling rate changes
14
sufficient number of zeroes at 3
4,
3
2,
would be regular. To corroborate this statement we
construct a filter having three zeroes at each location, i.e. 32131 )1()1()( zzzzH .
Figure 6 shows graphically how the iterated filter converges to a continuous function. To
complete the perfect reconstruction system we give one of the possible highpass filters as
).2067
4287201()1()( 432121
1
zzzzzzH
Note that the synthesis part of this system would give rise to non-regular filters. Thus we have
constructed a biorthogonal basis with regular analysis.
Figure 6. Fifth iteration of the filter 32131 )1()1()( zzzzH converging to a continuous
function f(z).
Perfect Reconstruction Filter Banks using rational sampling rate changes
15
5. Conclusion
In this paper the solution to the problem of designing perfect
reconstruction filter banks with arbitrary rational sampling rate is given. A
design example showing the advantage of using this method over the
indirect one is presented. And, finally the case with (2/3,1/3) sampling was
examined from the wavelet theory point of view. We conjectured how to
construct regular filters and gave an example.
Perfect Reconstruction Filter Banks using rational sampling rate changes
16
REFERENCES
1. J. Kovacevic and M. Vetterli, "Perfect reconstruction filter banks with rational
sampling rates in one and two dimensions," in Proc. SPIE Corti Visual Comrnun.
Image Processing, Philadelphia, PA, Nov. 1989, pp. 1258-1268.
2. K. Nayehi, T. P. Barnwell, III, and M. J. T. Smith, "The design of perfect reconstruction
nonuniform band filter banks," in Proc. IEEE Int. Conf. Acoust., Speech, Signal
Processing, Toronto, Canada, May, 1991, pp. 1781-1784.
3. P. P. Vaidyanathan, "Multirate digital filters, filter banks, polyphase networks, and
applications: A tutorial," Proc. IEEE, vol. 78, pp. 56-93, Jan. 1990.
4. M. Vetterli, "A theory of multirate filter banks," IEEE Trans. Acoust. , Speech, Signal
Processing, vol. 35, pp. 356-372, Mar. 1987.