a design procedure for oversampled nonuniform filter ... - IEEE Xplore
A DESIGN PROCEDURE FOR OVERSAMPLED NONUNIFORM FILTER BANKS WITH PERFECT-RECONSTRUCTION Mohamed F. Mansour Texas Instruments Inc. Dallas, Texas, USA ABSTRACT We describe a time-domain procedure for designing the synthesis filters of perfect-reconstruction oversampled filter banks. A condition matrix is derived from the basic magnitude distortion and aliasing canceling conditions and we derive the necessary and sufficient conditions for perfect reconstruction. The formulation is general for any decimation scenario, but it is particularly useful with nonuniform decimation. Further, the condition matrix in the proposed design model is in general rectangular and has a nonzero null-space which allows for design optimization. Index Terms— filter banks, oversampling, optimization, nullspace, nonuniform decimation. 1. INTRODUCTION The design of filter banks has been extensively studied in the literature [1], [2] and well-established design procedures have been developed for different categories of filter banks. The main theme in a filter bank design algorithm is to derive conditions on the analysissynthesis pairs of filters so that the aliasing due to decimation is canceled and the overall magnitude and phase transfer function is reduced to a pure delay. In many applications, e.g., subband coding, critically sampled filter banks are used for signal modeling as they usually provide good energy compaction and they are non-expansive transforms where the output size is the same as the input size. The critically sampled filters are usually orthogonal or biorthogonal filters and the decimation factor is the same as the number of analysis filters. For applications that require excessive processing of the subband samples, e.g., subband adaptive filtering [3] and subband dynamic range compression [4], oversampling is necessary to mitigate the aliasing effect of adjacent channels. In this case we have a guardband around the main signal component of each bank after decimation. This relaxes the requirements in the design of the synthesis filter bank for perfect-reconstruction. The oversampling gives extra degrees of freedom that can be exploited in different ways. It results in general in a non-zero null space of the perfect reconstruction condition matrix. This null space can be exploited to optimize objective functions of different criteria without sacrificing the perfect reconstruction [5]. The design of the synthesis filter would be composed in general of the min-norm solution plus some components from the null-space of the condition matrix. The degrees of freedom are proportional to the oversampling factor. The design procedure is in general symmetric, i.e., it could be applied to the analysis filter bank rather than the synthesis filter bank. Nonuniform filter banks refers to filter banks with nonuniform decimation across the channels. It finds many applications in signal processing when unequal processing is needed. For example, in
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some subband adaptive filtering scenarios it may be required to cancel the noise in particular bands. In this case, the frequency bands that are not processed may be maximally decimated for computational saving, whereas the processed bands may not be decimated at all or subsampled to avoid excessive aliasing. In this work, we propose a new approach for the design of oversampled filter banks. The procedure starts from the basic perfect reconstruction conditions of a decimated system. We develop a straightforward condition matrix for the filter bank organization. The condition matrix can accommodate any decimation scenario as long as the resulting number of conditions is less than the number of design variables. We describe necessary and sufficient conditions for both the analysis and synthesis filter banks for perfect reconstruction in the proposed design scenario. The problem is in general solvable if the Z-tranforms of the analysis filters are relatively coprime [1], and the order of the synthesis filters is larger than the number of conditions. The proposed framework gives new insights to this well-established problem. For example, we show a simple procedure for designing a minimal order FIR synthesis filter bank with IIR analysis filter bank. Further, in most cases we have a nonzero null-space of the condition matrix which can be exploited for different optimization criteria. This generalizes the earlier work for optimizing oversampled DFT filter banks in [5]. The basic design procedure does not assume a particular structure of the filter bank, however, it could be customized to reduce the number of unknown parameters if the filter bank has a certain structure, e.g., modulated filter banks. Compared to the earlier approaches for the design of nonuniform filter banks, e.g., [6, 7], the proposed design scheme focuses on the design of the synthesis filter bank for a given design of the analysis filter bank. There are few loose constraints in the design of the analysis filter bank to yield perfect reconstruction that would be discussed in the paper. Moreover, unlike [6], we focus on oversampled filter banks where the condition matrix has a nonzero null-space that could be exploited for optimizing the design of the synthesis filter bank. Further, our design always yields a perfectreconstruction solution. The framework is general to encompass IIR analysis filter banks. 2. SYSTEM CONFIGURATION The basic architecture of the system under study is illustrated in Fig. 1. Note that, the individual banks may have different decimation factor and in all cases we have 1 ≤ Di < M . Note that, each output ˆ i (z); that is the z-transform of x X ˆi (n) in Fig. 1; could be written as [1] Di −1 X ˆ i (z) = 1 Xi (zWik ) (1) X Di k=0
ICASSP 2011
Note that, some of these linear conditions may be dependent according to the choice of the analysis filters. 3. DESIGN PROCEDURE If all the decision variables are arranged in a column vector f of length M (Ns + 1), then perfect reconstruction is attained if f is a solution of the following linear system of equations: Af = b Fig. 1. System configuration of oversampled filter banks with nonuniform decimaion; where 1 ≤ Di < M where Wi = e−j2π/Di . Hence, the input-output relation of the system could be written as Y (z) =
M D i −1 X X 1 X(zWik )Hi (zWik )Fi (z) D i i=1
(2)
k=0
Note that, we do not assume same decimation in all bands and this adds more flexibility in the design of the analysis filter bank. For this general topology we have two sets of conditions for perfect reconstruction. The magnitude constraint is the straightforward relation M X
Hi (z)Fi (z) = cz −Δ
(3)
where A is the condition matrix of size L × M (Ns + 1) (to be described below) and b is a column vector with all zeros except in entry Δ + 1 (where Δ refers to the delay in (3)). If the filter bank is real-valued, then the condition matrix could be reorganized to have all-real entries. In particular, in the aliasing-canceling condition (4), for each distinct decimation factor Dr we could add and subtract the conditions at k and Dr − k to get the equivalent real-valued set of conditions: X (c) Hi,k,r (z)Fi (z) = 0 (8) i
X
i
for all possible distinct value of decimation factors {Dr }. Here the summation is only over the banks with the same decimation. Note that, in the uniform decimation case we have summation over all banks. Assume the orders of the analysis and synthesis filters are Na and Ns respectively; and assume the number of banks is M . Then, the number of decision variables is M (Ns + 1). The number of linear conditions in (3) is Na +Ns +1. For each distinct value Dr of the decimation factors we have (Na + Ns + 1)(Dr − 1) linear conditions. However, if we have more than one even decimation factor, they have the same following condition at k = Dr /2 : X Hi (−z)Fi (z) = 0 (5) i
(s)
Hi,k,r (z)Fi (z)
=
0
(9)
i
for 1 ≤ k ≤ Dr /2, and for all possible values of r. Note that the summation here is over banks that have the decimation factor Dr , and
i=1
with c = 0. The alias cancellation condition is more involved with non-uniform decimation as we would have different modulations of the baseband spectrum. A sufficient condition would be to group bands with similar decimation factor Dr and force the corresponding component to be zero, i.e., we have X Hi (zWrk )Fi (z) = 0 for 1 ≤ k < Dr (4)
(7)
(c)
Hi,k,r (s)
Hi,k,r
Na X
hi (n) cos(2πkn/Dr )z −n
(10)
hi (n) sin(2πkn/Dr )z −n
(11)
n=0 Na X n=0
Note that if Dr is even, then for k = Dr /2, the left hand side of (9) is zero. Denote, (c)
hi,k,r (n)
hi (n) cos(2πkn/Dr )
(12)
(s) hi,k,r (n)
hi (n) sin(2πkn/Dr )
(13)
(s)
(s)
Note that, hi,k,r (0) = 0 and hi,k,r (0) = hi (0). Assume we have R distinct values of the decimation factor, then the condition matrix A in (7) could be expressed as (c)
(c)
(s)
(s)
A = [A0,0 A1,1 . . . AD1 −1,1 A1,1 . . . AD1 −1,1 . . . . . . (c)
(c)
(s)
(s)
A1,R . . . ADR −1,R A1,R . . . ADR −1,R ] (14) The submatrix A0,0 is of size (Na + Ns − 1) × M (N s + 1) and is defined as ` ´ (15) A0,0 A1 A2 . . . . . . AM
More generally, if γ is the great common divisor (gcd) of two decimation factors D1 and D2 at different channels, then the number of where Ai is of size (Na + Ns + 1) × (N s + 1) and is defined as linear equations is reduced by γ − 1. Further, if Δ in (3) is not zero, then for any value of Dr , the first and last terms in (4) are identical 0 1 0 ... 0 0 hi (0) to the corresponding terms in (3). Hence, if we only have nonunity hi (0) ... 0 0 hi (1) B C gcd of two between even decimation factors, and if the number of B C . . B C .. .. distinct odd and even values of the decimation factor is α and β reB C B C C spectively, then the total number of linear conditions is: (N ) h (N − 1) . . . h (0) 0 . . . 0 h Ai B i a i B i a C B B B @
L = (Na + Ns + 1) − (β − 1)(Na + Ns − 1) + α+β
X
(Na + Ns − 1)(Dr − 1)
(6)
r=1
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0
0
hi (Na ) . .. 0
...
...
hi (1) . .. 0
hi (0)
...
0
...
0
C C C A
hi (Na ) (16)
(c)
where we assumed that Ns > Na . The submatrices Ak,r are defined as ” “ (c) (c) (c) Ak,r δ1,r A(c) (17) δ2,r A2,k,r . . . . . . δM,r AM,k,r 1,k,r ” “ (s) (s) (s) Ak,r δ1,r A(s) (18) δ2,r A2,k,r . . . . . . δM,r AM,k,r 1,k,r for 1 ≤ k < Dr , δi,r is one if the decimation in the i-th bank equals Dr , otherwise it is zero; and the submatrices {Aci,k,r , Asi,k,r } are of size (Na + Ns − 1) × (N s + 1) and are defined as Aci,k,r 0 hi (0) B h(c) (1) i,k,r B B B B B (c) B h B i,k,r (Na ) B B 0 B B B B B @ 0 0
0 hi (0) .. . (c) hi,k,r (Na − 1) (c) h1,k,r (Na ) .. . 0 0
... ... ... ... ... ...
0 0 .. . 0 0 .. . (c) h1,k,r (Na ) 0
1
0 0
C C C C C C C 0 C C C 0 C C C C C (c) h1,k,r (Na − 1) A (c) h1,k,r (Na )
to the redundancy of A. If A has a nonzero null space then optimization of the synthesis filters could be obtained by combining the min-norm solution of (7) with components from the null space of A to optimize a certain objective function without sacrificing the perfect reconstruction condition [5]. For example, the null space optimization could be used to minimize the out-of-band power of the synthesis filter bank to yield a large value of c in (3). The previous discussion assumes FIR analysis filter banks. However, the modeling can encompass IIR filter bank or hybrid FIR and IIR filter banks with slight modifications. Assume that the analysis filters are {Pi (z)/Qi (z)}M i=1 and the corresponding synthesis filters are {Fi (z)}. Then in the absence of decimation the perfect reconstruction condition is M X Pi (z) Fi (z) = cz −Δ Q i (z) i=1
Then by multiplying both sides by set of conditions, M X
(19) Asi,k,r 0 (s) hi,k,r (1) B (s) B hi,k,r (2) B B B B B (s) B hi,k,r (Na ) B B 0 B B B @ 0
0
(s)
hi,k,r (1) . .. (s) hi,k,r (Na − 1) (s) h1,k,r (Na ) . .. 0
... ... ... ... ...
0 0 . .. 0 0 . .. (s) h1,k,r (Na )
1
0 0
C C C C C C C C C C C C C A
0 0 (s)
2 4Pi (z)
i=1
The second condition confirms that we have consistent set of equations. In fact the numerical stability of the solution is related to the magnitude of the component of the (Δ + 1)-th row that lies in the row space of the other rows. The first necessary condition can be put in the more restrict form M (N s + 1) > L, where L is as defined in (6). In this case a sufficient conditions on the decimation values would be α+β
2−β+
X
(Di − 1) < M
(21)
i=1
where α and β are as defined in (6). Denote the left hand side in the above inequality by ν. If (21) is satisfied, then a sufficient condition on Ns would be α+β
(M − ν)Ns > β + (2 − β)Na +
X
(Na − 1)(Di − 1)
(22)
i=1
Note that, if (21) is satisfied, then (22) can always be satisfied by increasing Ns . The design degrees of freedom is directly related
3 Qj (z)5 Fi (z) = cz −Δ
M Y
Qj (z)
(24)
j=1
Fi (z) = Fei (z)
M Y
Qj (z)
(25)
j=1
for 1 ≤ i ≤ M . Then the perfect reconstruction conditions become
(20)
2. The (Δ + 1)-th row of A is not in the row-space of the other rows of A.
Qi (z), we get the equivalent
j=i
M X
1. The number of unknowns is bigger than the rank of A.
i
Then we introduce the filters {Fei (z)} as
h1,k,r (Na − 1)
There are in general two necessary conditions for an exact solution of (7):
Y
Q
(23)
i=1
2 4Pi (z)
Y
3 Qi (z)5 Fei (z) = cz −Δ
(26)
j=i
and the problem can be analyzed as the FIR case to find FeQ i (z) (even with nonuniform decimation) with analysis filters {Pi (z) j=i Qi (z)}; then use (25) to compute the synthesis filter bank. 4. DESIGN EXAMPLES In the first example, we evaluate the basic design procedure using 6-channel FIR filter bank. The analysis filter bank is linear phase FIR filters of order 20 and normalized bandwidth of 1/8. The decimation factor is two for odd channels and three for even channels. The synthesis filters have order 60. The degrees of freedom, which equals the dimension of the null-space of the condition matrix A in (7) is 43. The frequency magnitude response of both the analysis and synthesis filter banks is as shown in Fig. 2. The overall input-output delay is 40. Note that, this solution for the synthesis filters is the minimum-norm solution and it is not unique. In the second example, we design an FIR synthesis filters for an IIR analysis filter bank. We use a 4-channel IIR filter banks where each filter is a fourth-order Chebechev filter with a decimation factor of 3 in all channels. The synthesis filter bank is an FIR filter of order 30. In this case there is no degrees of freedom, i.e., the null space of A is zero. The corresponding analysis and synthesis filters are shown in Fig. 3
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(a) analysis filters
(a) analysis filters 20
0 −20
−20
magnitude (dB)
magnitude (dB)
0
−40 −60 −80
−60 −80
−100 −120
−40
0
0.2
0.4 0.6 normalized frequency
0.8
−100
1
0
0.2
50
0
0
−20
magnitude (dB)
magnitude (dB)
0.8
1
0.8
1
(b) synthesis filters
(b) synthesis filters 20
−40 −60 −80
−50 −100 −150 −200
−100 −120
0.4 0.6 normalized frequency
0
0.2
0.4 0.6 normalized frequency
0.8
−250
1
Fig. 2. Example of 6-channel FIR nonuniform filter bank
0
0.2
0.4 0.6 normalized frequency
Fig. 3. Example of 4-channel IIR uniform analysis filter bank with FIR synthesis filters
5. DISCUSSION In this work we develop a new framework for the design of perfectreconstruction nonuniform oversampled filter banks. It is a timedomain framework and it works for both FIR and IIR analysis filters. The framework is based on the modeling in (7) where the condition matrix is directly estimated from the perfect reconstruction conditions. We developed necessary conditions on analysis and synthesis filters for perfect reconstruction. The framework is general to encompass the design of complex-valued filter banks. In general, the condition matrix in the design model is rectangular or can be made rectangular by increasing the order of the synthesis filters. Hence, it has a nonzero null-space which gives degrees of freedom in the design of the synthesis filters. This allows for optimizing the design for a given objective function without affecting the perfect-reconstruction. This result was exploited in [5] for the special case of oversampled DFT filter banks and is generalized by this work.
IEEE Intl. Symp. on Circuits and Systems, ISCAS, vol. 6, pp. 569-572, 1998. [5] M. Mansour, “On the optimization of oversampled DFT filter banks”, IEEE Signal Processing Letters, vol. 14, no. 6, pp. 389392, Jun. 2007. [6] K. Nayebi, T. Barnwell, and M. Smith, “Nonuniform filter banks: a reconstruction and design theory”, IEEE Trans. on Signal Processing, vol. 41, no. 3, pp. 1114-1127,Mar. 1993. [7] S. Akkarakaran, and P.P. Viadyanathan, “New results and open problems in nonuniform filter banks”, Proc. IEEE Intl. conf. on acoust., speech and sign. process., ICASSP, 1999. [8] H. Bolcski, and F. Hlawatsch, “Oversampled Cosine Modulated Filter Banks with Perfect Reconstruction”, IEEE Trans. on Circuits and Systems II, vol. 45, no. 8, pp. 1057-1071, Aug. 1998. [9] J. Kliewer and A. Mertins, “”Oversampled Cosine-Modulated Filter Banks with Arbitrary System Delay”, IEEE Trans. on Signal Processing, vol. 46, no. 4, pp. 941-955, Apr. 1998.
6. REFERENCES [1] P.P. Vaidyantathan, Multirate Systems and Filter Banks, Prentice Hall, 1993. [2] G. Strang, and T. Nguyen, Wavelets and Filter Banks, WellesleCambridge Press, 1997. [3] S. Haykin, Adaptive Filter Theory, fourth edition,Prentice Hall, 2002. [4] R. Brennan, and T. Schneider, “A flexible filterbank structure for extensive signal manipulations in digital hearing aids”,
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some subband adaptive filtering scenarios it may be required to cancel the noise in particular bands. In this case, the frequency bands that are not processed may be maximally decimated for computational saving, whereas the processed bands may not be decimated at all or subsampled to avoid excessive aliasing. In this work, we propose a new approach for the design of oversampled filter banks. The procedure starts from the basic perfect reconstruction conditions of a decimated system. We develop a straightforward condition matrix for the filter bank organization. The condition matrix can accommodate any decimation scenario as long as the resulting number of conditions is less than the number of design variables. We describe necessary and sufficient conditions for both the analysis and synthesis filter banks for perfect reconstruction in the proposed design scenario. The problem is in general solvable if the Z-tranforms of the analysis filters are relatively coprime [1], and the order of the synthesis filters is larger than the number of conditions. The proposed framework gives new insights to this well-established problem. For example, we show a simple procedure for designing a minimal order FIR synthesis filter bank with IIR analysis filter bank. Further, in most cases we have a nonzero null-space of the condition matrix which can be exploited for different optimization criteria. This generalizes the earlier work for optimizing oversampled DFT filter banks in [5]. The basic design procedure does not assume a particular structure of the filter bank, however, it could be customized to reduce the number of unknown parameters if the filter bank has a certain structure, e.g., modulated filter banks. Compared to the earlier approaches for the design of nonuniform filter banks, e.g., [6, 7], the proposed design scheme focuses on the design of the synthesis filter bank for a given design of the analysis filter bank. There are few loose constraints in the design of the analysis filter bank to yield perfect reconstruction that would be discussed in the paper. Moreover, unlike [6], we focus on oversampled filter banks where the condition matrix has a nonzero null-space that could be exploited for optimizing the design of the synthesis filter bank. Further, our design always yields a perfectreconstruction solution. The framework is general to encompass IIR analysis filter banks. 2. SYSTEM CONFIGURATION The basic architecture of the system under study is illustrated in Fig. 1. Note that, the individual banks may have different decimation factor and in all cases we have 1 ≤ Di < M . Note that, each output ˆ i (z); that is the z-transform of x X ˆi (n) in Fig. 1; could be written as [1] Di −1 X ˆ i (z) = 1 Xi (zWik ) (1) X Di k=0
ICASSP 2011
Note that, some of these linear conditions may be dependent according to the choice of the analysis filters. 3. DESIGN PROCEDURE If all the decision variables are arranged in a column vector f of length M (Ns + 1), then perfect reconstruction is attained if f is a solution of the following linear system of equations: Af = b Fig. 1. System configuration of oversampled filter banks with nonuniform decimaion; where 1 ≤ Di < M where Wi = e−j2π/Di . Hence, the input-output relation of the system could be written as Y (z) =
M D i −1 X X 1 X(zWik )Hi (zWik )Fi (z) D i i=1
(2)
k=0
Note that, we do not assume same decimation in all bands and this adds more flexibility in the design of the analysis filter bank. For this general topology we have two sets of conditions for perfect reconstruction. The magnitude constraint is the straightforward relation M X
Hi (z)Fi (z) = cz −Δ
(3)
where A is the condition matrix of size L × M (Ns + 1) (to be described below) and b is a column vector with all zeros except in entry Δ + 1 (where Δ refers to the delay in (3)). If the filter bank is real-valued, then the condition matrix could be reorganized to have all-real entries. In particular, in the aliasing-canceling condition (4), for each distinct decimation factor Dr we could add and subtract the conditions at k and Dr − k to get the equivalent real-valued set of conditions: X (c) Hi,k,r (z)Fi (z) = 0 (8) i
X
i
for all possible distinct value of decimation factors {Dr }. Here the summation is only over the banks with the same decimation. Note that, in the uniform decimation case we have summation over all banks. Assume the orders of the analysis and synthesis filters are Na and Ns respectively; and assume the number of banks is M . Then, the number of decision variables is M (Ns + 1). The number of linear conditions in (3) is Na +Ns +1. For each distinct value Dr of the decimation factors we have (Na + Ns + 1)(Dr − 1) linear conditions. However, if we have more than one even decimation factor, they have the same following condition at k = Dr /2 : X Hi (−z)Fi (z) = 0 (5) i
(s)
Hi,k,r (z)Fi (z)
=
0
(9)
i
for 1 ≤ k ≤ Dr /2, and for all possible values of r. Note that the summation here is over banks that have the decimation factor Dr , and
i=1
with c = 0. The alias cancellation condition is more involved with non-uniform decimation as we would have different modulations of the baseband spectrum. A sufficient condition would be to group bands with similar decimation factor Dr and force the corresponding component to be zero, i.e., we have X Hi (zWrk )Fi (z) = 0 for 1 ≤ k < Dr (4)
(7)
(c)
Hi,k,r (s)
Hi,k,r
Na X
hi (n) cos(2πkn/Dr )z −n
(10)
hi (n) sin(2πkn/Dr )z −n
(11)
n=0 Na X n=0
Note that if Dr is even, then for k = Dr /2, the left hand side of (9) is zero. Denote, (c)
hi,k,r (n)
hi (n) cos(2πkn/Dr )
(12)
(s) hi,k,r (n)
hi (n) sin(2πkn/Dr )
(13)
(s)
(s)
Note that, hi,k,r (0) = 0 and hi,k,r (0) = hi (0). Assume we have R distinct values of the decimation factor, then the condition matrix A in (7) could be expressed as (c)
(c)
(s)
(s)
A = [A0,0 A1,1 . . . AD1 −1,1 A1,1 . . . AD1 −1,1 . . . . . . (c)
(c)
(s)
(s)
A1,R . . . ADR −1,R A1,R . . . ADR −1,R ] (14) The submatrix A0,0 is of size (Na + Ns − 1) × M (N s + 1) and is defined as ` ´ (15) A0,0 A1 A2 . . . . . . AM
More generally, if γ is the great common divisor (gcd) of two decimation factors D1 and D2 at different channels, then the number of where Ai is of size (Na + Ns + 1) × (N s + 1) and is defined as linear equations is reduced by γ − 1. Further, if Δ in (3) is not zero, then for any value of Dr , the first and last terms in (4) are identical 0 1 0 ... 0 0 hi (0) to the corresponding terms in (3). Hence, if we only have nonunity hi (0) ... 0 0 hi (1) B C gcd of two between even decimation factors, and if the number of B C . . B C .. .. distinct odd and even values of the decimation factor is α and β reB C B C C spectively, then the total number of linear conditions is: (N ) h (N − 1) . . . h (0) 0 . . . 0 h Ai B i a i B i a C B B B @
L = (Na + Ns + 1) − (β − 1)(Na + Ns − 1) + α+β
X
(Na + Ns − 1)(Dr − 1)
(6)
r=1
1650
0
0
hi (Na ) . .. 0
...
...
hi (1) . .. 0
hi (0)
...
0
...
0
C C C A
hi (Na ) (16)
(c)
where we assumed that Ns > Na . The submatrices Ak,r are defined as ” “ (c) (c) (c) Ak,r δ1,r A(c) (17) δ2,r A2,k,r . . . . . . δM,r AM,k,r 1,k,r ” “ (s) (s) (s) Ak,r δ1,r A(s) (18) δ2,r A2,k,r . . . . . . δM,r AM,k,r 1,k,r for 1 ≤ k < Dr , δi,r is one if the decimation in the i-th bank equals Dr , otherwise it is zero; and the submatrices {Aci,k,r , Asi,k,r } are of size (Na + Ns − 1) × (N s + 1) and are defined as Aci,k,r 0 hi (0) B h(c) (1) i,k,r B B B B B (c) B h B i,k,r (Na ) B B 0 B B B B B @ 0 0
0 hi (0) .. . (c) hi,k,r (Na − 1) (c) h1,k,r (Na ) .. . 0 0
... ... ... ... ... ...
0 0 .. . 0 0 .. . (c) h1,k,r (Na ) 0
1
0 0
C C C C C C C 0 C C C 0 C C C C C (c) h1,k,r (Na − 1) A (c) h1,k,r (Na )
to the redundancy of A. If A has a nonzero null space then optimization of the synthesis filters could be obtained by combining the min-norm solution of (7) with components from the null space of A to optimize a certain objective function without sacrificing the perfect reconstruction condition [5]. For example, the null space optimization could be used to minimize the out-of-band power of the synthesis filter bank to yield a large value of c in (3). The previous discussion assumes FIR analysis filter banks. However, the modeling can encompass IIR filter bank or hybrid FIR and IIR filter banks with slight modifications. Assume that the analysis filters are {Pi (z)/Qi (z)}M i=1 and the corresponding synthesis filters are {Fi (z)}. Then in the absence of decimation the perfect reconstruction condition is M X Pi (z) Fi (z) = cz −Δ Q i (z) i=1
Then by multiplying both sides by set of conditions, M X
(19) Asi,k,r 0 (s) hi,k,r (1) B (s) B hi,k,r (2) B B B B B (s) B hi,k,r (Na ) B B 0 B B B @ 0
0
(s)
hi,k,r (1) . .. (s) hi,k,r (Na − 1) (s) h1,k,r (Na ) . .. 0
... ... ... ... ...
0 0 . .. 0 0 . .. (s) h1,k,r (Na )
1
0 0
C C C C C C C C C C C C C A
0 0 (s)
2 4Pi (z)
i=1
The second condition confirms that we have consistent set of equations. In fact the numerical stability of the solution is related to the magnitude of the component of the (Δ + 1)-th row that lies in the row space of the other rows. The first necessary condition can be put in the more restrict form M (N s + 1) > L, where L is as defined in (6). In this case a sufficient conditions on the decimation values would be α+β
2−β+
X
(Di − 1) < M
(21)
i=1
where α and β are as defined in (6). Denote the left hand side in the above inequality by ν. If (21) is satisfied, then a sufficient condition on Ns would be α+β
(M − ν)Ns > β + (2 − β)Na +
X
(Na − 1)(Di − 1)
(22)
i=1
Note that, if (21) is satisfied, then (22) can always be satisfied by increasing Ns . The design degrees of freedom is directly related
3 Qj (z)5 Fi (z) = cz −Δ
M Y
Qj (z)
(24)
j=1
Fi (z) = Fei (z)
M Y
Qj (z)
(25)
j=1
for 1 ≤ i ≤ M . Then the perfect reconstruction conditions become
(20)
2. The (Δ + 1)-th row of A is not in the row-space of the other rows of A.
Qi (z), we get the equivalent
j=i
M X
1. The number of unknowns is bigger than the rank of A.
i
Then we introduce the filters {Fei (z)} as
h1,k,r (Na − 1)
There are in general two necessary conditions for an exact solution of (7):
Y
Q
(23)
i=1
2 4Pi (z)
Y
3 Qi (z)5 Fei (z) = cz −Δ
(26)
j=i
and the problem can be analyzed as the FIR case to find FeQ i (z) (even with nonuniform decimation) with analysis filters {Pi (z) j=i Qi (z)}; then use (25) to compute the synthesis filter bank. 4. DESIGN EXAMPLES In the first example, we evaluate the basic design procedure using 6-channel FIR filter bank. The analysis filter bank is linear phase FIR filters of order 20 and normalized bandwidth of 1/8. The decimation factor is two for odd channels and three for even channels. The synthesis filters have order 60. The degrees of freedom, which equals the dimension of the null-space of the condition matrix A in (7) is 43. The frequency magnitude response of both the analysis and synthesis filter banks is as shown in Fig. 2. The overall input-output delay is 40. Note that, this solution for the synthesis filters is the minimum-norm solution and it is not unique. In the second example, we design an FIR synthesis filters for an IIR analysis filter bank. We use a 4-channel IIR filter banks where each filter is a fourth-order Chebechev filter with a decimation factor of 3 in all channels. The synthesis filter bank is an FIR filter of order 30. In this case there is no degrees of freedom, i.e., the null space of A is zero. The corresponding analysis and synthesis filters are shown in Fig. 3
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(a) analysis filters
(a) analysis filters 20
0 −20
−20
magnitude (dB)
magnitude (dB)
0
−40 −60 −80
−60 −80
−100 −120
−40
0
0.2
0.4 0.6 normalized frequency
0.8
−100
1
0
0.2
50
0
0
−20
magnitude (dB)
magnitude (dB)
0.8
1
0.8
1
(b) synthesis filters
(b) synthesis filters 20
−40 −60 −80
−50 −100 −150 −200
−100 −120
0.4 0.6 normalized frequency
0
0.2
0.4 0.6 normalized frequency
0.8
−250
1
Fig. 2. Example of 6-channel FIR nonuniform filter bank
0
0.2
0.4 0.6 normalized frequency
Fig. 3. Example of 4-channel IIR uniform analysis filter bank with FIR synthesis filters
5. DISCUSSION In this work we develop a new framework for the design of perfectreconstruction nonuniform oversampled filter banks. It is a timedomain framework and it works for both FIR and IIR analysis filters. The framework is based on the modeling in (7) where the condition matrix is directly estimated from the perfect reconstruction conditions. We developed necessary conditions on analysis and synthesis filters for perfect reconstruction. The framework is general to encompass the design of complex-valued filter banks. In general, the condition matrix in the design model is rectangular or can be made rectangular by increasing the order of the synthesis filters. Hence, it has a nonzero null-space which gives degrees of freedom in the design of the synthesis filters. This allows for optimizing the design for a given objective function without affecting the perfect-reconstruction. This result was exploited in [5] for the special case of oversampled DFT filter banks and is generalized by this work.
IEEE Intl. Symp. on Circuits and Systems, ISCAS, vol. 6, pp. 569-572, 1998. [5] M. Mansour, “On the optimization of oversampled DFT filter banks”, IEEE Signal Processing Letters, vol. 14, no. 6, pp. 389392, Jun. 2007. [6] K. Nayebi, T. Barnwell, and M. Smith, “Nonuniform filter banks: a reconstruction and design theory”, IEEE Trans. on Signal Processing, vol. 41, no. 3, pp. 1114-1127,Mar. 1993. [7] S. Akkarakaran, and P.P. Viadyanathan, “New results and open problems in nonuniform filter banks”, Proc. IEEE Intl. conf. on acoust., speech and sign. process., ICASSP, 1999. [8] H. Bolcski, and F. Hlawatsch, “Oversampled Cosine Modulated Filter Banks with Perfect Reconstruction”, IEEE Trans. on Circuits and Systems II, vol. 45, no. 8, pp. 1057-1071, Aug. 1998. [9] J. Kliewer and A. Mertins, “”Oversampled Cosine-Modulated Filter Banks with Arbitrary System Delay”, IEEE Trans. on Signal Processing, vol. 46, no. 4, pp. 941-955, Apr. 1998.
6. REFERENCES [1] P.P. Vaidyantathan, Multirate Systems and Filter Banks, Prentice Hall, 1993. [2] G. Strang, and T. Nguyen, Wavelets and Filter Banks, WellesleCambridge Press, 1997. [3] S. Haykin, Adaptive Filter Theory, fourth edition,Prentice Hall, 2002. [4] R. Brennan, and T. Schneider, “A flexible filterbank structure for extensive signal manipulations in digital hearing aids”,
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