2M-Band Interleaved DFT Modulated Filter Banks With Perfect Reconstruction

8/3/2019 2M-Band Interleaved DFT Modulated Filter Banks With Perfect Reconstruction

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International Journal of Wavelets, Multiresolutionand Information ProcessingVol. 6, No. 4 (2008) 499520c World Scientific Publishing Company

2M-BAND INTERLEAVED DFT MODULATED FILTER

BANKS WITH PERFECT RECONSTRUCTION

PENG-LANG SHUI and XIAO-LONG WANG

National Laboratory of Radar Signal Processing,

Xidian University, Xian, P. R. China

[email protected]@yahoo.cn

Received 24 May 2007

Revised 22 July 2007

In this paper, we propose a new family of perfect reconstruction (PR) complex filter

banks, named interleaved discrete Fourier transform modulated filter banks (InterleavedDFT-FBs). In the filter banks, the analysis filters are generated by interlaced exponentialmodulating two different analysis prototype filters, and the synthesis filters are gener-ated by two different synthesis prototype filters via the same manner. The filter banks

have a simple polyphase structure similar to DFT modulated filter banks (DFT-FBs).More importantly, the proposed Interleaved DFT-FBs can achieve critically sampled PRcomplex filter bank with FIR analysis and synthesis filters, which is impossible for DFT-

FBs. We give and prove the PR condition for 2M-band Interleaved DFT-FBs. Utilizing

the result, the design procedure of the prototype filters is presented. In addition, by thetheoretic analysis and numerical examples, it is shown that the analysis and synthesisfilters cannot simultaneously provide good stopband attenuation for the critically sam-pled PR Interleaved DFT-FBs. Although the limitation always exits, the filter bankscan find applications in some subband coding systems of high bit rate.

Keywords: Interleaved DFT modulated filter bank; prototype filter; biorthogonal.

AMS Subject Classification: 78M50, 94A12

1. Introduction

Modulated filter banks have become an attractive choice in a diverse set of applica-

tions that includes data compression, denoising, multicarrier modulation and adap-

tive filtering.13 Modulated filter banks include two subclasses: discrete Fourier

transform modulated filter banks (DFT-FBs) and cosine modulated filter banks

(CMFBs). Both of them own a simple structure, in which all subband filters are gen-erated by complex exponential or cosine modulating two (or one) prototype filters.

Due to owning many advantages such as perfect reconstruction (PR), finite impulse

response (FIR), and linear phase, M-band CMFBs have been extensively applied

to real-valued signal processing.46 DFT-FBs and some improved versions can split

positive and negative frequency components into different subbands for independent

499


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500 P.-L. Shui & X.-L. Wang

processing. Therefore, they are particularly suitable for processing complex-valued

signals. This property is necessary for some applications. In addition, a poten-

tial application of DFT modulated filter banks is to generate 2D separable DFT

modulated filter banks that can efficiently extract the directional features in 2Dimages. At present, it is an active area to develop 2D filter banks with directional

selectivity.710 However, direction information extracted by 2D wavelet or dual-tree

complex wavelets is limited. The design of non-separable 2D filter banks involved

in large scale optimization problem which may result in longer solution times and

may expose the problem to numerical difficulties. As compared with the dual tree

complex wavelets, it is a more efficient approach to construct 2D filter banks with

directional selevtivity from the tensor products of 1D DFT filter banks. It is noted

that Gabor filter bank (or Gaussian modulated filter bank) belongs to a special sub-

class of exponential modulated filter bank. In Gabor filter bank, prototype filteris a gaussian function, which is modulated by a complex exponential for obtaining

the filter bank. For DFT (or cosine) modulated filter bank, the prototype filters

are unknown, which need to be designed for satisfying the particular requirement

of the filter bank.

In many applications, filter banks are required to satisfying PR, critically sam-

pled, and FIR (all analysis and synthesis filters are FIR). Due to the special

polyphase structure, both analysis and synthesis filter banks cannot be FIR in criti-

cally sampled PR DFT-FBs. The synthesis filters must be infinite impulse response(IIR) when the analysis prototype filter is FIR.2,3 Some improved filter banks based

on DFT modulation have been developed. Typically, oversampled DFT-FBs can

provide most of the desired characteristics except critically sampled.11,12 Based

on stable, causal, IIR analysis and synthesis prototype filters, critically sampled

PR DFT-FBs can be designed.13 Recently, modified DFT modulated filter banks

(MDFT-FBs)14 and modified exponentially modulated filter banks (EMFBs)15 are

proposed, which own many properties such as critically sampled, PR, and FIR. Both

of them are derived from DFT-FBs by introducing some modifications in subbands.

The common breakthrough is owning to their structure-inherent alias cancellation.In this paper, we will design 2M (even)-band PR complex filter banks based on

the interleaved DFT modulation. In the interleaved DFT modulated filter banks

(Interleaved DFT-FBs), the analysis filters are obtained by interlaced exponential

modulating two different analysis prototype filters, and the synthesis filters are

obtained by two different synthesis prototype filters via the same manner. This

new modulation mode offers the potential for canceling alias components from

critically sampling subband signals, and it allows us to design critically sampled

PR complex filter banks with FIR analysis and synthesis filters. However, due to

coupling between both analysis and synthesis prototype filters in the PR conditions,

the analysis prototype filters and synthesis prototype filters cannot simultaneously

achieve high stopband attenuation. In design, we minimize the stopband energy of

two analysis prototype filters such that analysis filter bank is provided with better

frequency selectivity. The Interleaved DFT-FBs with critically sampled and PR can

be applied to high bit rate subband coding of complex-valued signals. This is due to


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2M-Band Interleaved DFT Modulated Filter Banks with Perfect Reconstruction 501

that the performance of a filter bank is often evaluated by the subband coding gain,

which depends on the frequency selectivity of analysis filter bank and the norms of

synthesis filter bank in subband coding of high bit rate.1619 In addition, the filter

banks with causal, stable, IIR synthesis prototype filters are also designed, whichcan further decrease stopband energy of two analysis prototype filters. Moreover,

the orders of recursive parts of synthesis polyphase components are adjustable. This

paper is organized as follows. In Sec. 2, we show the structure of Interleaved DFT-

FBs and derive sufficient and necessary conditions for PR. In Sec. 3, we give design

method of biorthogonal Interleaved DFT-FBs and discuss limitation of Interleaved

DFT-FBs in stopband attenuation. In Sec. 4, the design method of biorthogonal

Interleaved DFT-FBs with stable, causal, IIR synthesis filters is given. In Sec. 5,

some numerical examples are reported.

2. Structure of Interleaved DFT-FBs

Interleaved DFT modulation is a new exponential modulation mode different from

standard DFT modulation and modified DFT modulation. The Interleaved DFT-

FBs use four prototype filters, and must be even band. In the filter banks, the

analysis filters consist of interlaced exponentially modulated versions of two differ-

ent analysis prototype filters and so do the synthesis filters. This structure allows

us to design critically sampled complex filter banks with PR. This property is dif-

ferent from MDFT-FBs and EMFBs. Both MDFT-FBs and EMFBs use a singleprototype filter, which is exactly the same as the case of CMFBs. In addition, 2M-

band Interleaved DFT-FBs decompose a complex-valued signal into 2M complex-

valued subband signals, and 2M-band MDFT-FBs or 2M-band EMFBs decompose

a complex-valued signal into 4M real-valued subband signals with and without a

delay of one sampling period.

2.1. Structure of Interleaved DFT-FBs

Figure 1 shows the structure of 2M-band Interleaved DFT-FB. The analysis filters

are interlaced exponentially modulated versions of two different low-pass analysis

prototype filters H(0)(z) and H(1)(z), and the synthesis filters are interlaced expo-

nentially modulated versions of two different low-pass synthesis prototype filters

G(0)(z) and G(1)(z).

H2k(z) = H(0)(zW2k2M), H2k+1(z) = H

(1)(zW2k+12M ).

G2k(z) = G(0)(zW2k2M), G2k+1(z) = G

(1)(zW2k+12M ).

k = 0, 1, . . . , M 1. (2.1)

where W2M = e(j/M). Apparently, the filters in the even bands come from the

prototype filters H(0)(z) and G(0)(z), while the filters in the odd bands come from

the prototype filters H(1)(z) and G(1)(z).

From the view of the subband segmentation, 2M-band DFT-FBs, 2M-band

MDFT-FBs, 2M-band EMFBs, and 2M-band Interleaved DFT-FBs all divide


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Fig. 1. Structure of 2M-band Interleaved DFT-FBs.

(a)

(b)

(c)

Fig. 2. Subband segmentations of DFT-FBs, MDFT-FBs, EMFBs, and Interleaved DFT-FBs.(a) 2M-band DFT-FBs and 2M-band MDFT-FBs; (b) 2M-band EMFBs; (c) 2M-band InterleavedDFT-FBs.

[, ] into 2M

subbands uniformly. However, their segmentation modes are quitedifferent. Their subband segmentations are illustrated in Fig. 2. In the Interleaved

DFT-FBs, the magnitude responses of the filters in the even bands have the same

shape, and so do the magnitude responses of the filters in the odd bands. We desire

that alias components are canceled by means of alternant change of the filters

shapes.


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2.2. Polyphase structure and implementation

DFT-FBs have a simple polyphase structure, which brings design facility and high

computational efficiency. Similar to DFT-FBs, the Interleaved DFT-FBs can be

designed and realized with the help of its simple polyphase structure and fast DFTtransform.

The type-I polyphase representation of the analysis prototype filters and the

type-III polyphase representation of the synthesis prototype filters are given as

follows, respectively.

H(0)(z) =2M1l=0

zlP(0)l (z

2M), H(1)(z) =2M1l=0

zlP(1)l (z

2M).

G(0)(z) =2M1l=0

zlQ(0)l (z

2M), G(1)(z) =2M1l=0

zlQ(1)l (z

2M).

(2.2)

Let the impulse responses of the four prototype filters be h(0)(n), h(1)(n) and

g(0)(n), g(1)(n), then their polyphase filters are given as follows, respectively.

p(0)l (n) = h

(0)(2M n + l), p(1)l (n) = h

(1)(2M n + l).

q(0)l (n) = g

(0)(2M n l), q(1)l (n) = g

(1)(2Mn l).

l = 0, 1, . . . , 2M 1. (2.3)

Consequently, the type-I polyphase representation of the analysis filter bank and the

type-III polyphase representation of the synthesis filter bank are given as follows,

respectively.

H0(z)

H2(z)

...

H2M2(z)

H1(z)

H3(z)...

H2M1(z)

=

H0(z)

H1(z)

...

HM1(z)

HM(z)

HM+1(z)...

H2M1(z)

=

2M1l=0 zlP

(0)l (z

2M)

2M1l=0 zlWlM P

(0)l (z

2M)

.

..

2M1l=0 zlW

(M1)lM P

(0)l (z

2M)

2M1l=0 zlWl2MP

(1)l (z

2M)

2M1l=0 zlWl2MW

lM P

(1)l (z

2M)

...

2M1l=0 zlWl2MW

(M1)lM P

(1)l (z

2M)

=

WM W

M

P(0)(z2M)

WMD

M W

MD

M

P(1)(z2M)

1

z1

...

z(2M1)

. (2.4)


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G0(z)

G2(z)

...

G2M2(z)

G1(z)

G3(z)...

G2M1(z)

T

=

G0(z)

G1(z)

...

GM1(z)

GM(z)

GM+1(z)...

G2M1(z)

T

T =

2M1l=0 zlQ

(0)l (z

2M)

2M1l=0 zlWlMQ

(0)l (z

2M)

...

2M1l=0 zlW

(M1)lM Q

(0)l (z

2M)

2M1l=0 zlWl2MQ

(1)l (z

2M)

2M1l=0 zlWl2MW

lMQ

(1)l (z

2M)

...

2M1l=0 zlWl2MW

(M1)lM Q

(1)l (z

2M)

T

=

1 z . . . z(2M1)

Q(0)(z2M)

WMWM

Q(1)(z2M)

DMWM

DMWM

. (2.5)

where the superscript denotes the conjugate and transpose, the superscript T

denotes the transpose, and

P(i)(z) = diag[P(i)0 (z), P

(i)1 (z), . . . , P

(i)2M1(z)], i = 0, 1.

Q(i)(z) = diag[Q(i)0 (z), Q(i)1 (z), . . . , Q(i)2M1(z)], i = 0, 1.

DM = diag[1, W12M, W

22M, . . . , W

(M1)2M ].

WM = [WklM]k=0,1,...,M1, l=0,1,...,M1.

2M = []ij

=

1 j = 2i, i = 0, . . . , M 1; j = 2(i M) + 1, i = M , . . . , 2M 1.

0 else.

P(i)(z) and Q(i)(z), i = 0, 1 are two pairs of diagonal matrices consisting of the

polyphase components of analysis and synthesis prototype filters, DM is an M-

by-M diagonal matrix, W is an M-point DFT matrix, and is a 2M-by-2M

permutation matrix. In conclusion, the type-I polyphase matrix of the analysis

filter bank and the type-III polyphase matrix of the synthesis filter bank can be

represented with block matrix form as follows:

H(z) = T W

M W

MP(0)(z)

W

MD

M W

MD

M

P(1)(z) .

G(z) =

Q(0)(z)

WM

WM

Q(1)(z)

DMWM

DMWM

.

(2.6)

The block structure of the polyphase matrices shows that signal decomposition

and reconstruction can be efficiently and fast implemented. The flow-diagram of


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Fig. 3. The flow-diagram of the polyphase implementation for 2M-band Interleaved DFT-FBs.

implementation is illustrated in Fig. 3. It can be seen from Fig. 3 that the analy-sis part requires two M-point Interleaved DFT transforms and the synthesis part

requires two M-point DFT transforms. Consequently, the Interleaved DFT-FBs

have the same computational efficiency as the DFT-FBs, the MDFT-FBs, and the

EMFBs. However, the implementation cost of the MDFT-FBs and EMFBs is twice

that of the Interleaved DFT-FBs respectively, due to the fact that 2M-band MDT-

FBs and 2M-band EMFBs decompose a complex-valued signal into 4M real-valued

subband signals for structure-inherent alias cancellation, respectively.

2.3. PR condition of Interleaved DFT-FBs

Besides high computational efficiency, Interleaved DFT-FBs allow us to design com-

plex filter banks with PR and FIR analysis and synthesis filters. This property can-

not be achieved by critically sampled DFT-FBs with FIR analysis and synthesis

filters except a pure block transform. It is well known that a 2M-band critically

sampled DFT-FB is of perfect reconstruction iff the polyphase components of the

analysis and synthesis prototype filter satisfy:1,13

Pi(z)Qi(z) = czm0, i = 0, 1, . . . , 2M 1 (2.7)

where m0 is an integer, Pi(z) is type-I polyphase components of the analysis pro-

totype filter H(z) and Qi(z) is the type-II polyphase components of the synthesis

prototype filter G(z). Apparently, for FIR analysis and synthesis prototype filters,

it is impossible to achieve PR for DFT-FBs unless all polyphase components are

monomial ofz. Otherwise, either of the analysis and synthesis prototype filters must


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be IIR filter or both of them are IIR filters. PR DFT-FBs with IIR filters have been

investigated thoroughly.13

In what follows, we derive PR conditions of Interleaved DFT-FBs. In terms of

(2.6), the transfer matrix of the multirate system in Fig. 3 isT(z) = G(z)H(z)

= Q(0)(z)

WM

WM

WM W

M

P(0)(z)

+ Q(1)(z)

DMWM

DMWM

WMD

M W

MD

M

P(1)(z).

We use the multiplication of permutation matrix T = I2M in the equa-

tion. Further, using the multiplication of block matrices WMWM = MIM andDMD

M = IM, we can obtain the equation as follows:

T(z) = MQ(0)(z)

IM IM

IM IM

P(0)(z) + MQ(1)(z)

IM IM

IM IM

P(1)(z). (2.8)

The filter bank is PR only when the transformed matrix satisfies: T(z) = zndI2M(nd is an integer ).

1,2 In this way, we obtain the following PR condition.

Theorem 1. Critically sampled interleaved DFT modulated filter bank is PR if and

only if the polyphase components of analysis and synthesis prototype filters satisfy:

P(0)l (z)Q

(0)l (z) + P

(1)l (z)Q

(1)l (z) = z

nd/M, l = 0, 1, . . . , 2M 1,

P(0)l (z)Q

(0)M+l(z) P

(1)l (z)Q

(1)M+l(z) = 0, l = 0, 1, . . . , M 1,

P(0)M+l(z)Q

(0)l (z) P

(1)M+l(z)Q

(1)l (z) = 0, l = 0, 1, . . . , M 1 (2.9)

or

P(0)l (z) P

(1)l (z)

P(0)l+M(z) P(1)l+M(z)Q

(0)l (z) Q

(0)l+M(z)

Q(1)l (z) Q(1)l+M(z) = znd

MI2

,

l = 0, 1, . . . , M 1. (2.10)

Equation (2.10) is the representation of (2.9) in the matrix form. When nd = 0,

the PR conditions degenerate to the biorthogonal conditions. Apparently, an Inter-

leaved DFT-FB is determined by the four prototype filters, and the filter bank is

PR iff their 8M polyphase components satisfy the 4M equations in (2.9) or (2.10).

For convenience, we will focus on design of biorthogonal Interleaved DFT-FBs in

the next section. It is straightforward to transform a biorthogonal filter bank intoa PR filter bank with a system delay.

In the paper, we always assume that the analysis prototype filters are FIR filters,

and the synthesis prototype filters can be FIR or stable IIR filters. In what follows,

we will investigate the design problems of Interleaved DFT-FBs of FIR analysis

and FIR/IIR synthesis filters.


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3. Biorthogonal FIR Interleaved DFT-FBs

When all analysis and synthesis filters of Interleaved DFT-FBs are FIR (that is,

Interleaved DFT-FBs are FIR), all polyphase components of its four prototype

filters are Laurent polynomials about z. Taking the determinants of the matricesin the two sides of Eq. (2.10) (nd = 0 in (2.10)), we obtain

[P(0)l (z)P

(1)l+M(z) + P

(0)l+M(z)P

(1)l (z)][Q

(0)l (z)Q

(1)l+M(z) + Q

(0)l+M(z)Q

(1)l (z)] =

1

M2.

(3.1)

Due to the fact that both factors in the left side of (3 .1) are polynomials of z,

Eq. (3.1) holds iff

P(0)l (z)P(1)l+M(z) + P(0)l+M(z)P(1)l (z) = M

1cz

r,

Q(0)l (z)Q

(1)l+M(z) + Q

(0)l+M(z)Q

(1)l (z) = M

1c1zr,

where c is a nonzero constant and r is an integer. Without loss of generality, we

assume c = 1. In this way, the polyphase components of the two synthesis proto-

type filters are completely determined by that of the two analysis prototype filters,

that is,

Q(0)l

(z) Q(0)

l+M(z)

Q(1)l (z) Q

(1)l+M(z)

= 1M

P(0)l

(z) P(1)

l(z)

P(0)l+M(z) P

(1)l+M(z)

1

= zr

P(1)l+M(z) P

(1)l (z)

P(0)l+M(Z) P

(0)l (z)

. (3.2)

Therefore, an Interleaved DFT-FB with FIR analysis and synthesis filters is

biorthogonal if its polyphase components satisfy:

P

(0)

l (z)P

(1)

l+M(z) + P

(0)

l+M(z)P

(1)

l (z) = z

r

/M.Q(0)l (z) Q

(0)l+M(z)

Q(1)l (z) Q

(1)l+M(z)

= zr

P(1)l+M(z) P

(1)l (z)

P(0)l+M(z) P

(0)l (z)

, l = 0, 1, , M 1. (3.3)

or

G(0)(z) = z2MrM1l=0

[zlP(1)M+l(z

2M) + zM+lP(1)l (z

2M)].

G(1)(z) = z2MrM1l=0

[zlP(0)M+l(z2M) + zM+lP(0)l (z

2M)].

P(0)l (z)P

(1)l+M(z) + P

(0)M+l(z)P

(1)l (z) = z

r/M.

l = 0, 1, . . . , M 1. (3.4)


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An interesting fact is that a 2M-band biorthogonal FIR Interleaved DFT-FB degen-

erates to a two-band biorthogonal FIR filter bank when M = 1.20 When M = 1,

the two-band filter bank derived from (3.4) satisfies:

H0(z) = H(0)(z) = P(0)0 (z2) + z1P(0)1 (z).

H1(z) = H(1)(zW12 ) = H

(1)(z) = P(1)0 (z

2) z1P(1)1 (z

2).

G0(z) = G(0)(z) = z2r+1H1(z).

G1(z) = G(1)(z) = z2r+1H0(z).

(3.5)

3.1. Limitation of FIR analysis and synthesis filters

FIR filter banks are often designed in time domain. Let the analysis prototypefilters h(0)(n), h(1)(n), n = 0, 1, . . . , 2ML 1 be causal and of length 2ML, which

can be written as two column vectors:

h(0) = [h(0)(0), h(0)(1), . . . , h(0)(2ML 2), h(0)(2ML 1)]T.

h(1) = [h(1)(0), h(1)(1), . . . , h(1)(2ML 2), h(1)(2ML 1)]T.

Further, let r = L1 in (3.4), the two synthesis prototype filters can be re-written as

G(0)(z) = z2ML1M1l=0

zlP(1)Ml1(z

2M) + z(M+l)P(1)2Ml1(z

2M).

G(1)(z) = z2ML1M1l=0

zlP(0)Ml1(z

2M) + z(M+l)P(0)2Ml1(z

2M).

(3.6)

Their support in the time domain is the set (2ML + 1, 2ML + 2, . . . , 1, 0) and

the two synthesis prototype filters are re-written as two column vectors:

g(0)

= [g(0)

(2ML + 1), g(0)

(2ML + 2), . . . , g(0)

(1), g(0)

(0)]T

.g(1) = [g(1)(2ML + 1), g(1)(2ML + 2), . . . , g(1)(1), g(1)(0)]T.

It is easily proved that

g(0) = h(1), g(1) = h(0) (3.7)

where is a 2ML-by-2ML permutation matrix and is defined as

JM 0 . . . 0

0 JM . . . 0......

. . ....

0 0 . . . JM

(3.8)

where JM is the M-by-M reversal (or anti-diagonal) matrix. Equation (3.7) indi-

cates that the two synthesis prototype filters are local reversal of the two analysis


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prototype filters, respectively. This special structure in (3.7) results in some limita-

tion in stopband attenuation of filter banks. When M = 1, the matrix degenerates

to the identity matrix and thus G(i)(z) = z(2ML1)H(1i)(z), i = 0, 1. In this case,

the synthesis prototype filters and the corresponding analysis prototype filters areof the identical magnitude responses. However, the situation is quite different when

M 2. The synthesis prototype filters are local reversal of the corresponding anal-

ysis prototype filters. In this case, if the analysis prototype filters are two low-pass

filters with high stopband attenuation, then the corresponding synthesis prototype

filters perforce contain some unwanted jumps from the local reversal introduced by

the matrix . These jumps make the synthesis prototype filters have much stopband

energy or poor stopband attenuation.

In what follows, we will quantitatively analyze the influence of the local reversal

on the stopband energy of the filters. Taking the stopband region of a low-passfilter h(n) of length 2ML as [, (1/2M + ]

[(1/2M + ), ], the stopband

energy of

Es(h(n)) =1

(1/2M+)

|H()|2d = hTT(M,L,)h (3.9)

where T(M,L,) is a symmetric Toeplitz matrix of 2ML-by-2ML whose first row

is the vector [t(0), t(1), . . . , t(2M L 1)] and

t(k) = 1

(1/2M+)

cos(k)d =

1 1/2M k = 0

sin(k(1/2M + ))/(k) k = 0.(3.10)

In this case,

min Es(h(0)(n)) = min(T(M,L,))h

(0)22 (3.11)

where min(T(M,L,)) denotes the minimal eigenvalue of matrix T(M,L,). The

optimal filter is the eigenvector (also called the eigenfilter1) that corresponds to

the minimal eigenvalue ofT(M,L,). Similarly, for the filter pairs (h(0)(n), g(1)(n))

defined by (3.7), the total stopband energy of the two filters is

Es(h(0)(n)) + Es(g

(1)(n)) =1

(1/2M+)

(|H(0)()|2 + |G(1)()|2)d

= (h(0))TF(M,L,)h(0) min(F(M,L,))h(0)22

(3.12)

where F(M,L,) = T(M,L,) + TT(M,L,), and is the permutation matrix

in (3.8). The optimal filter is the eigenvector that corresponds to the minimaleigenvalue of F(M,L,) which minimizes the total stopband energy in (3.12). For

M = 2, 3, 4, 8, L = 2, 3, 4, 5, and = 1/(4M), the minimal eigenvalues of matrix

T(M,L,) and F(M,L,) are listed in Table 1, respectively. From Table 1, it is

shown that, for a single filter, the stopband energy of the optimal filter sharp

decreases by increasing the filters length, whereas for the filter pair, the total


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Table 1. Minimal eigenvalues of the matrices T(M,L,) and

F(M,L,) for different M and L ( = 1/4M).

M L min(T(M,L,)) min(F(M,L,))

2 2 6.9945e04 0.0569092 3 6.0026e06 0.0273592 4 4.7208e08 0.016224

2 5 3.5611e10 0.01083 2 9.1284e04 0.065783 3 9.285e06 0.0319773 4 8.6655e08 0.0192053 5 7.7621e10 0.012684

4 2 9.9544e04 0.0688124 3 1.0702e05 0.0335714 4 1.0561e07 0.019997

4 5 1.0005e9 0.0133388 2 1.079e03 0.0717038 3 1.2214e05 0.0350988 4 1.2696e07 0.020938 5 1.2672e09 0.013967

stopband energy of the optimal filters slowly decreases with increasing of the fil-

ters length. This fact implies that it is difficult to achieve both analysis filters

and synthesis filters with high stopband attenuation in FIR Interleaved DFT-FBs.

Therefore, we only require that analysis prototype filters have high stopband atten-uation in the latter design. This requirement is reasonable in some applications of

signal processing. For example, in subband coders of high bit rate, subband cod-

ing gains are used to evaluate performance of a filter bank. Subband coding gain

depends on the variances of subband signals and the 2-norms of synthesis filters.

In other words, analysis filters are desired to have good frequency selectivity while

the requirements of the frequency selectivity on synthesis filters could be relaxed.

3.2. Design of FIR filter banks with double prototype filters

In design, the stopband energy and passband flatness of the two analysis prototype

filters are used as the objective, and the biorthogonal conditions are used as the

constraints for optimization. Let h(0)(n), h(1)(n), n = 0, 1, . . . , 2ML 1 be the two

causal analysis prototype filters. Due to the fact that two synthesis prototype filters

are determined by the two analysis prototype filters in terms of (3 .7), we call such

a filter bank as one with double prototype filters. The polyphase components of the

analysis prototype filters are written as

P(0)l (z) =

L1n=0

p(0)l (n)z

n =L1n=0

h(0)(2M n + l)zn,

P(1)l (z) =

L1n=0

p(1)l (n)z

n =

L1n=0

h(1)(2M n + l)zn.

(3.13)


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P(0)l (z) and P

(1)l (z), l = 0, 1, . . . , 2M 1 are (L 1)-degree Laurent polynomials,

respectively. Taking r = (L1), the biorthogonal conditions in (3.3) or (3.4) become

P(0)l (z)P(1)M+l(z) + P

(0)M+l(z)P

(1)l (z) = z

(L1)

M, l = 0, 1, . . . , M 1.

It can be transformed into a set of quadratic equations in time domain:

min(k,L1)n=max(0,kL+1)

p(0)l (n)p

(1)M+l(k n) +p

(0)M+l(n)p

(1)l (k n) =

(k L + 1)

M. (3.14)

for k = 0, 1, . . . , 2L 2, l = 0, 1, . . . , M 1 where

(n) =

1 n = 0

0 else.

The constraints are composed of (2L 1)M quadratic equations and thus biorthog-

onality needs to expand (2L 1)M degrees of freedom among 4ML coefficients of

the two analysis prototype filters. The residual (2L + 1)M degrees of freedom are

further used to optimize the passband flatness and stopband attenuation. Let the

region 0 p and s be the passband and the stopband of the filters,respectively, and

p = [1/(2M) p], s = [1/(2M) + s]

are the passband edge and stopband edge respectively, where p, s are two positive

numbers.

The passband flatness can be measured by

Ep(h(0)(n)) =1

p0

[|H(0)()| H(0)(0)]2d (3.15)

and the stopband energy is defined as

Es(h(0)(n)) =

s

|H(0)()|2d = (h(0))TT(M,L,s)h(0). (3.16)

In this way, the design of an Interleaved DFT-FB comes down to the following

optimization:

minh(0)(n),h(1)(n)

[(Es(h(0)(n)) + Es(h

(1)(n))) + (1 )(Ep(h(0)(n)) + Ep(h

(1)(n)))]

s.t. biorthogonal constraints in (3.14) (3.17)

where is a tradeoff parameter between passband and stopband performances.


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3.3. Design of FIR filter banks with single prototype filter

In terms of the biorthogonal conditions in (3.3) or (3.4), a FIR Interleaved DFT-

FB is determined by the two analysis prototype filters. In fact, a FIR Interleaved

DFT-FB can also be determined by a single prototype filter.Let h(0)(n), n = 0, 1, . . . , 2M L1 be a causal filter of length 2ML. Let prototype

filter h(1)(n) be local reversal of the filter h(0)(n), that is

h(1)(n) = h(0)(2M L 1 n), n = 0, 1, . . . , 2ML 1. (3.18)

In this case, the polyphase components of h(1)(n) satisfy:

P(1)l (z) = z

(L1)P(0)2MLl1(z

1), l = 0, 1, . . . , 2M 1. (3.19)

The biorthogonal conditions in (3.3) are simplified to

P(0)l (z)P

(0)Ml1(z

1) + P(0)l+M(z)P

(0)2Ml1(z

1) =1

M,

l = 0, 1, . . . , M/2 (3.20)

or

min(L1,L1+k)n=max(0,k)

p(0)l (n)p(0)Ml1(n k) +p(0)M+l(n)p(0)2Ml1(n k) =(k)

M ,

k = 1 L, 2 L , . . . , L 1, l = 0, 1, . . . , M/2 (3.21)

where x denotes the least integer no less than x. In this case, biorthogonal con-

ditions are composed of (2L 1)M/2 quadratic constraints and available degrees

of freedom for design are 2ML. The single prototype filter h(0)(n) is obtained by

solving the following optimization:

minh(0)(n)

[Es(h(0)(n)) + (1 )Ep(h(0)(n))]

s.t. biorthogonal constraints in (3.21). (3.22)

From the prototype filter h(0)(n), the residual three prototype filters are determined

by the formulae as follows:

h(1) = J2MLh(0), g(1) = h(0), g(0) = J2MLh

(0) (3.23)

where J2ML

is 2ML

-by-2ML

anti-diagonal matrix and is the permutation matrixin (3.8). Therefore, we call such a filter bank as one with a single prototype filter.

It is noted that such a single prototype filter cannot be linear phase. If the single

prototype filter is linear phase, h(0) must be symmetric and then filter h(1) = h(0).

Consequently, the Interleaved DFT-FB degenerates to a DFT-FB. However, it is

well known that there exist no biorthogonal FIR DFT-FBs.


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4. Interleaved DFT-FBs with PR and Causal, Stable IIR

Synthesis Filters

In Sec. 3, we mention that the limitation of Interleaved DFT-FBs with FIR analysis

and synthesis filters. Due to the fact that the synthesis prototype filters are localreversal of the analysis prototype filters, the stopband attenuation of the prototype

filters to be able to achieve is limited. In the section, we substitute the stable IIR

synthesis prototype filters for the FIR synthesis prototype filters. In this way, many

degrees of freedom are released for minimizing the stopband energy of the analysis

prototype filters. Therefore, filter banks with higher stopband attenuation can be

obtained.

Similarly, taking the determinants of the matrices in the two sides of the equa-

tion (2.10) (nd = 0 in (2.10)), we obtain[P

(0)l (z)P

(1)M+l(z) + P

(0)l+M(z)P

(1)l (z)][Q

(0)l (z)Q

(1)l+M(z) + Q

(0)l+M(z)Q

(1)l (z)] =

1

M2.

Let

Rl(z) = P(0)l (z)P

(1)l+M(z) + P

(0)l+M(z)P

(1)l (z).

Since h(0)(n) and h(1)(n) are causal FIR filters of length 2ML, Rl(z) is a Lau-

rent polynomial whose degree is no more than 2L 1. If Rl(z) are stable for

l = 0, 1, . . . , M 1, then the solution of the equations in (3.3) isQ(0)l (z) Q(0)l+M(z)

Q(1)l (z) Q

(1)l+M(z)

= 1

M

P(0)l (z) P(1)l (z)

P(0)l+M(z) P

(1)l+M(z)

1

=1

MRl(z)

P(1)l+M(z) P(1)l (z)

P(0)l+M(z) P

(0)l (z)

. (4.1)

Moreover, all polyphase components of the synthesis prototype filters are also sta-

ble. When Rl(z) are minimum phase for l = 0, 1, . . . , M 1, that is, all zeros strictlylocate inside the unit circle, then all polyphase components of the synthesis proto-

type filters are stable and causal (when an appropriate system delay is allowed).

For the causal FIR filters h(0)(n) and h(1)(n) of length 2ML, we require that

Rl(z) satisfy:

Rl(z) = z(L1)

Nn=0

rl(n)zn,

rl(0)

Nn=1

|rl(n)| > 0, l = 0, 1, . . . , M 1 (4.2)

where N L 1 is a predefined integer and is a positive number less than one.

Theorem 2. If the polyphase components of two analysis prototype filters are given

in (3.13) and Rl(z) satisfy the conditions in (4.2), then Interleaved DFT-FB can


16/22


be perfect reconstruction with a system delay 2ML and causal and stable synthesis

filters.

First, from (4.2), we easily verify that: for arbitrary z = aej , when a 1Nn=0

rl(n)(aej )n

rl(0) Nn=1

an|rl(n)| > rl(0) Nn=1

|rl(n)| > 0.

When the system delay is 2ML, the synthesis prototype filters have the polyphase

components

Q(0)l (z) Q

(0)l+M(z)

Q(1)l (z) Q(1)l+M(z) = zL

M P

(0)l (z) P

(1)l (z)

P(0)l+M(z) P(1)l+M(z)1

=zL

M Rl(z)

P(1)l+M(z) P(1)l (z)

P(0)l+M(z) P

(0)l (z)

=z1

M

Nn=0

rl(n)zn

P(1)l+M(z) P

(1)l (z)

P(0)l+M(z) P

(0)l (z)

. (4.3)

In this way, all polyphase components of the synthesis prototype filters are causal,

stable IIR filters, and thus the two synthesis prototype filters are also causal and

stable IIR filters. Moreover, we can adjust the order of the recursive part in the

synthesis polyphase filters by means of selection of N.

When we allow that synthesis prototype filters are stable IIR filters, the condi-

tions in (4.2) transform into some quadratic equalities and one inequality:

rl(k L + 1) =

min(k,L1)n=max(0,k+1L)

p(0)l (n)p

(1)M+l(k n) +p

(0)M+l(n)p

(1)l (k n).

rl(k) = 0, k = 1 L, 2 L , . . . , 1, N + 1, N + 2, . . . , L 1.

rl(0) Nk=1

|rl(k)| > 0, l = 0, 1, . . . , M 1. (4.4)

Comparing with (3.14), the above constraints on the coefficients of filters are obvi-ously relaxed. Thus, the analysis prototype filters with higher stopband attenuation

can be designed by solving the following optimization problem:

minh(0)(n),h(1)(n)

{(Es(h(0)(n)) + Es(h

(1)(n))) + (1 )(Ep(h(0)(n)) + Ep(h

(1)(n)))}

s.t. constraints in (4.4). (4.5)


17/22


In this case, the two synthesis prototype filters are represented as

G(0)(z) =

M1

l=0

z(l+1)P(1)Ml1(z

2M) + z(M+l+1)P(1)2Ml1(z

2M)

M

Nn=0

rMl1(n)z2Mn

,

G(1)(z) =

M1l=0

z(l+1)P(0)Ml1(z

2M) + z(M+l+1)P(0)2Ml1(z

2M)

MNn=0

rMl1(n)z2Mn

.

(4.6)

It is notable that the Interleaved DFT-FBs with causal and stable IIR synthesis

prototype filters are different in structure from the stable, causal, perfect recon-

struction, IIR uniform DFT-FBs.13 The polyphase components of the synthesisfilter of the latter are the inverses of the corresponding polyphase components of

the analysis filter. When the analysis prototype filter is FIR, synthesis polyphase

components are pure recursive filters and the recursive order equals to the average

order of the FIR analysis polyphase component. In Interleaved DFT filter banks,

the synthesis polyphase components are averaging-recursive filters and the recursive

order is adjustable.

However, for Interleaved DFT-FBs with a single prototype filter, its structure

does not allow us to design filter banks with causal, stable IIR synthesis prototypefilters. Similar to Sec. 3.3, we take the analysis prototype filter h(1)(n) as a reverse

version of the prototype filter h(0)(n), that is, P(1)l (z) = z

(L1)P(0)2Ml1(z

1), l =

0, 1, . . . , 2M 1. In this case, for l = 0, 1, . . . , M/2 if we set

Rl(z) = P(0)l (z)P

(1)l+M(z) + P

(0)l+M(z)P

(1)l (z)

= z(L1)[P(0)l (z)P

(0)Ml1(z

1) + P(0)l+M(z)P

(0)2Ml1(z

1)]

= z(L1)N

n=0

rl(n)zn (4.7)

and rl(0) N

k=1 |rl(k)| > 0, then the polyphase components Q(0)l (z), Q

(0)l+M(z),

Q(1)l (z), Q

(1)l+M(z) for l = 0, 1, . . . , M/2 can be causal, stable IIR filters because

the above conditions ensure that the denominators of the components are of stable

and minimum-phase. However, in terms of the polyphase structure of the filter

bank, we have

RMl1(z) = P(0)Ml1(z)P

(1)2Ml1(z) + P

(0)2Ml1(z)P

(1)Ml1(z)

= z(L1)[P(0)Ml1(z)P

(0)l (z

1) + P(0)2Ml1(z)P

(0)M+l(z

1)]

= z(L1)z(L1)Rl(z1)

= z(L1)Nn=0

rl(n)zn. (4.8)


18/22


Thus, the polyphase components Q(0)l (z), Q

(0)l+M(z), Q

(1)l (z), Q

(1)l+M(z) for l =

M/2 + 1, . . . , M 1 are not causal, stable IIR filters because their denomina-

tors are of maximum-phase rather than the minimum-phase.

5. Numerical Examples

In this section, we give numerical examples of Interleaved DFT-FBs with single

prototype filters in (3.22), Interleaved DFT-FBs with double prototype filters in

(3.17), and Interleaved DFT-FBs with causal, stable IIR synthesis prototype filters

in (4.5). The programs of the examples are designed by using MATLAB6.5. The

design of the prototype filters in example 1, 2, and 3 take about 25 s, 85 s, and 100

s on a 3.0-GHz Pentium IV personal computer, respectively.

Example 1. A FIR Interleaved DFT-FB with a single prototype filter and M = 4,

L = 4. In design, the parameters = 0.5 and s = 1/(4M), p = 1/(8M). The

stopband energy of the analysis prototype filter is 0.031. The stopband energy of

the synthesis prototype filter is 0.242. The coefficients of the analysis prototype filter

h(0)(n) are given in Table 2, and the magnitude responses of h(0)(n) and g(0)(n) are

shown in Fig. 4. It can be seen that, the analysis prototype filters achieve stopband

attenuation about 20 dB, and the synthesis prototype filters have a poor stopband

attenuation. This is owing to the fact that the synthesis prototype filters are local

reversal of the corresponding analysis ones, respectively. Moreover, by increasing

the length of analysis prototype filters, the stopband energy of analysis prototype

filters slowly decreases. However, the sidelobes in the stopband often do not lower.

For example, when M = 4 and L = 10, the stopband energy of analysis prototype

filter is 0.0203 and the magnitude responses of the analysis and synthesis prototype

filters are shown in Fig. 5.

Example 2. A FIR Interleaved DFT-FB with double prototype filters and M = 8,

L = 4. In design, the parameters = 0.5 and s = 1/(4M), p = 1/(8M). Themagnitude responses of the obtained two analysis prototype filters are shown in

Fig. 6. Their stopband energies are 0.0303 and 0.0313, respectively. For the double

Table 2. Coefficients of the prototype filter h(0)(n).

n:07 n:815 n:1623 n:2431

0.01137397755222 0.05289931475008 0.45340779786003 0.08857974978352

0.01487816367138 0.05332450640374 0.42788607640798 0.05209292526068

0.00233925604462 0.07423088239024 0.42242672122903 0.015112789743510.00179393728761 0.10813647488481 0.44019991628710 0.00543140828349

0.00095081056262 0.07372440067066 0.29761867071910 0.00540864800621

0.00498707158950 0.13654915203861 0.23333463834854 0.00646376660236

0.01885258210033 0.23716693555180 0.13585079635128 0.04508669149972

0.02938007919164 0.30470218644620 0.07421699994687 0.06497268832533


19/22


(a) (b)

Fig. 4. Magnitude responses of the analysis and synthesis prototype filters when M= 4, L = 4.The magnitude response of h(0)(n) is shown in Fig. 4(a), the magnitude response of g(0)(n) is

shown in Fig. 4(b).

(a) (b)

Fig. 5. Magnitude responses of the analysis and synthesis prototype filters when M = 4, L = 10.The magnitude response of h(0)(n) is shown in Fig. 5(a), the magnitude response of g(0)(n) isshown in Fig. 5(b).

prototype filters, we can also adjust the stopband attenuation of the two analysis

prototype filters by weighting their stopband energy in the objective function in

(3.17). For example, when the weight of the first filter is two times the weight of

the second filter in (3.17), the magnitude responses of the obtained two analysisprototype filters are shown in Fig. 7. Their stopband energies are 0.0168 and 0.0499,

respectively.

Example 3. An Interleaved DFT-FB with causal, stable IIR synthesis prototype

filters, when M = 8, L = 4, N = 3 (implies that the denominators of all synthesis


20/22


(a) (b)

Fig. 6. Magnitude responses ofh(0)(n) and h(1)(n) when the two prototype filters have the sameweight. The magnitude response ofh(0)(n) is shown in Fig. 6(a), the magnitude response ofh(1)(n)is shown in Fig. 6(b).

(a) (b)

Fig. 7. Magnitude responses ofh(0)(n) and h(1)(n) when the weight ofh(0)(n) is two times thatof h(1)(n). The magnitude response of h(0)(n) is shown in Fig. 7(a), the magnitude response ofh(1)(n) is shown in Fig. 7(b).

polyphase components are minimum-phase Laurent polynomials whose degrees are

no more than 3). In design, = 0.4, = 0.8 and s = 1/(4M), p = 1/(8M).

For the two FIR analysis prototype filters, the stopband energy is 0.013 and 0.01,

respectively. Their stopband attenuations achieve about 23 dB and 26 dB. How-

ever, the synthesis prototype filters suffer from high comb-like stopband sidelobes,

in other words, the stopband attenuations of the analysis prototype filters are

improved at the cost of high stopband sidelobes of the synthesis prototype filters.

The magnitude responses of the obtained two analysis prototype filters are shown

in Fig. 8.


21/22


(a) (b)

Fig. 8. Magnitude responses of h(0)(n) and h(1)(n) when the synthesis prototype filters arecausal, stable, IIR filters. The magnitude response ofh(0)(n) is shown in Fig. 8(a), the magnituderesponse ofh(1)(n) is shown in Fig. 8(b).

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2M-Band Interleaved DFT Modulated Filter Banks With Perfect Reconstruction