Perfect Reconstruction Filter Banks Using Rational Sampling Rate Changes

Perfect Reconstruction Filter Banks using rational sampling rate changes

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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.


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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).


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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


4

Figure 1. (a) Analysis/Synthesis Filter Banks.

(b) Filter Banks in the polyphase domain.


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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


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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.


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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


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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.


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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.


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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.


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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.


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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).


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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

http://en.wikipedia.org/wiki/Square-integrable

http://en.wikipedia.org/wiki/Real_number

http://en.wikipedia.org/wiki/Complex_number

http://en.wikipedia.org/wiki/Function_%28mathematics%29

http://en.wikipedia.org/wiki/Orthonormal

http://en.wikipedia.org/wiki/Series_%28mathematics%29

http://en.wikipedia.org/wiki/Wavelet

http://en.wikipedia.org/wiki/Integral_transform


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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).


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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.


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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.

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Perfect Reconstruction Filter Banks Using Rational Sampling Rate Changes