Filterbank-based Fast Parallel Algorithms for Real-valued Discrete Gabor Expansion and Transform

Liang Tao School of Computer Science and Technology

Anhui University Hefei, Anhui 230039, China

taoliang@ustc.edu

H. K. Kwan Dept. of Electrical and Computer Engineering

University of Windsor 401 Sunset Avenue, Windsor, Ontario, Canada N9B 3P4

kwan1@uwindsor.ca

Juan-juan Gu Dept. of Electronic Engineering

Hefei University Hefei, Anhui 230022, China

gujj@hfuu.edu.cn

Abstract—Novel and fast parallel algorithms for the DHT-based real-valued discrete Gabor expansion (DGE) and discrete Gabor transform (DGT) are presented based on filterbanks. An analysis filterbank is designed for the DHT-based real-valued DGT and a synthesis filterbank is designed for the DHT-based real-valued DGE. The parallel channel in each of the two filterbanks has a unified structure and can apply the fast DHT (discrete Hartley transform) to reduce its computational load. The computational complexity of each parallel channel is very low and is independent of the oversampling rate. The computational complexity of the proposed parallel algorithms is analyzed and compared with that of the existing major parallel algorithms for the DGT and DGE. The results indicate that the proposed parallel algorithms are attractive for real time signal processing.

I. INTRODUCTION The Gabor transform [1] has been recognized as being

useful in signal processing; however, its real-time applications were limited due to the difficulties associated with computing the Gabor transform coefficients. An analytical solution to this problem in the continuous Gabor transform case, proposed by Bastiaans [2], is to introduce an auxiliary analysis function corresponding to a synthesis function and then compute the Gabor transform coefficients. The transform completeness is equivalent to a continuous biorthogonality relationship between the analysis function and the synthesis function. However, the major problem of Bastiaans’s method is that it is hard to compute the analysis function that satisfies the biorthogonality relationship for a given synthesis function and sampling pattern. Even when the analysis function can be found in a few special cases, as [3] pointed out, it may not localized; thus, in such cases, the transform coefficients can not reflect the local behavior of a signal. To overcome the problem in the continuous Gabor transform case, [4] presented the discrete version of the Gabor expansion (transform) pair for finite and periodic sequence, called the finite discrete Gabor transform (DGT) or the finite discrete Gabor expansion (DGE). For an arbitrary given synthesis function and sampling pattern, the solution of the analysis function is nothing more than solving a linear system. [5] also presented DFT-based fast algorithm for the finite DGT. To compute the finite DGT coefficients even

faster, [6] presented a parallel algorithm for the finite DGT. However, the algorithm was limited to the critically-sampled DGT case. The oversampled DGT and DGE cases are not included in [6]. We showed in [7] that the method used in [7] can be generalized to the oversampled DHT-based real-valued DGT and DGE.

In this paper, filterbank-based fast parallel algorithms for the DHT-based real-valued DGT and DGE are presented. In terms of the similarity between the analysis (and synthesis) filterbanks and the DGT (and DGE), an analysis filterbank will be designed for the finite DGT and a synthesis filterbank will be designed for the finite DGE. Each of the parallel channels in the two filterbanks has a unified structure and can apply the fast DHT to reduce its computational load. The computational complexity of each parallel channel of the filterbanks is independent of the oversampling rate. The computational complexity of the proposed parallel algorithms is analyzed and compared with that of the existing major parallel algorithms for the finite DGT and DGE. The results indicate that the proposed parallel algorithms are attractive for real time signal processing.

II. REVIEW OF DHT-BASED REAL-VALUED DGT AND DGE The DHT-based real-valued DGE [7] for the finite and

periodic discrete input signal x[k] with a period L is defined by

)/π2(cas][],[][1

0

1

0NnkNmkhnmakx

M

m

N

n−= ∑ ∑

−

=

−

=, (1)

where cas(x)=cos(x)+sin(x) is known as Hartley's cas function, and the coefficients a[m,n] can be obtained by

)/π2(cas][][],[1

0NnkNmkkxnma

L

k−= ∑

−

=γ , (2)

where (2) is called the DHT-based real-valued DGT and (1) can also be termed the inverse DGT. In (1) and (2),

MNMNL == , M and N are the Gabor frequency and time sampling intervals, M and N are the numbers of Gabor sampling points in time and frequency domains, respectively. The condition MN ≥ L must be satisfied for a stable reconstruction. The critical sampling occurs when

LMNMN == , and the oversampling occurs when MN>L.

This work was supported in part by the National Natural Science Foundation of China, the Key Project of the Natural Science Research in Anhui Provincial Higher Education Institutions, and the Anhui ProvincialKey Laboratory of Electronic Restriction.

978-1-4244-5309-2/10/$26.00 ©2010 IEEE 2674

The oversampling rate β is defined as LNM /=β . Note that the synthesis window ][kh and the analysis window

][kγ are all real and periodic in L. The Gabor coefficients a[m,n] are also periodic in both m and n with periods M and N, respectively. The completeness condition of the real-valued DGT and DGE is equivalent to the discrete biorthogonality relationship between ][kh and ][kγ , which takes the following form:

nm

L

kMNLkNnkmNkh δδγ )/(][)/π2cas(][

1

0=⋅⋅+∑

−

=, (3)

where 10 −≤≤ Mm , 10 −≤≤ Nn , δk denotes the Kronecker delta.

The DHT-based real-valued DGT coefficients a[m,n] [7] have a simple relationship with the traditional DFT-based complex-valued DGT coefficients C[m,n] [3]-[4] as follows: C[m,n] = (a[m,n]+a[m,N–n])/2 – j· (a[m,n]−a[m,N–n])/2 (4)

where 10 −≤≤ Mm , 10 −≤≤ Nn and 1j −= . Thus, the DHT-based real-valued DGT also offer a faster method to compute the DFT-based complex-valued DGT.

III. DHT-BASED REAL-VALUED DGT IMPLEMENTED BY FILTERBANK

An analysis filterbank can be designed for the finite DGT based on the similarity between the analysis filterbank and the DGT. Let x(t) denote a real finite and periodic continuous-time signal. After sampled uniformly with a sampling interval T1, x(t) becomes a finite and periodic discrete-time signal x(kT1), i.e, x[k] with a period

NMNML == . Rewrite (2) as follows:

)/π2(cas)]([)()( 1

1

012 NnkTNmkkTxmTy

L

kn −= ∑

−

=γ (5)

where ][ Nmk −γ in (2) is rewritten as )]([ 1TNmk −γ , ],[ nma in (2) is rewritten as )( 2mTyn , 12 NTT = . Suppose

that the oversampling rate MMNNLNM /// ===β is taken as an integer, 1−−= ρrNk , Mr , ,2 ,1= ,

1 , ,1 ,0 −= Nρ , and then (5) can be rewritten as

)/)1(π2(cas

)]1([)]1([)( 11

1

012

Nn

TNmrNTrNxmTyM

r

N

n

+⋅

−−−−−= ∑∑=

−

=

ρ

ργρρ (6)

Let βjim += , 1 , ,1 ,0 −= βi , 1 , ,1 ,0 −= Mj , the above equation becomes

)/)1(π2(cas)]1([

)]1([)]([

1

1

1

012

NnTjNNirN

TrNxTjiyM

r

N

n

+−−−−⋅

−−=+ ∑∑=

−

=

ρργ

ρβρ (7)

Again let )]1([)( 12 TrNxrTx −−= ρρ , (8)

)]1([)]([ 12 TjNNirNTrjg i −−−−=− ργρ

)]1)(([ 1TNiNrj −−−−−= ργ (9)

and then (7) becomes

)/)1(π2(cas

)]([)()]([ 21

2

1

02

Nn

TrjgrTxTjiy iM

r

N

n

+⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡−=+ ∑∑

=

−

=

ρ

β ρρρ (10)

Suppose

)]([)()( 21

22 TrjgrTxjTu iM

r

i −= ∑=

ρρρ . (11)

Let 1−−= qNρ , 1 , ,1 ,0 −= Nq , and then (10) becomes

)/π2(cas)()]([1

0212 NnqjTuTjiy

N

q

iqNn ∑

−

=−−=+ β . (12)

We can see from (12) that the DGT coefficients )]([ 2Tjiyn β+ can be obtained by taking an N-point fast

discrete Hartley transform (N-point fast DHT) of )( 2jTuiρ in

the reverse order of ρ . In terms of the above analysis process, an analysis

filterbank can be designed for the DGT as shown in Fig.1, where the outputs of the analysis filterbank are just the DGT coefficients )()]([ 22 mTyTjiy nn =+ β . We see from (9) that the analysis function )( 1kTγ of the DGT has the following

relationship with the unit pulse responses )( 2jTg in of the

analysis filter bank: )]1([)( 12 TNinjNjTg i

n −−−−= γ . (13)

Note that )( 2jTg in is periodic in j with a period M due to

the periodicity of )( 1kTγ . For the fixed i and j, the analysis filterbank in Fig. 1

contains N unified parallel channels. The computational complexity of every single channel corresponding to the output )]([ 2Tjiyn β+ in Fig. 1 is the order of the multiplication number of Equation (11) plus N-point fast DHT, i.e., NNM 2log5.0+ . Therefore, the computational complexity of each channel does not change as the oversampling rate β increases, in fact, it is very low and depends only on the length NML = of the input signal and the number N of the Gabor frequency sampling points. Note that

1 , ,1 ,0 −= βi , 1 , ,1 ,0 −= Mj , so the complete analysis filterbank contains MM =β parallel sub-filterbanks shown in Fig. 1.

IV. DHT-BASED REAL-VALUED DGE IMPLEMENTED BY FILTERBANK

A synthesis filter bank can be designed for the finite DGE based on the similarity between the synthesis filter bank and the DGE. Rewrite (1) as follows:

)/π2(cas)]([)()( 1

1

02

1

01 NnkTNmkhmTykTx

N

nn

M

m−= ∑∑

−

=

−

=, (14)

where ][ Nmkh − in (1) is rewritten as )]([ 1TNmkh − , ],[ nma is rewritten as )( 2mTyn ， 12 NTT = .

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↓ N ∑ −•

r

i Trjg ])[()( 20

N-p

oint

Fas

t DH

T

)( 1kTx )( 20 jTui)( 20 rTx

↓ N ∑ −•r

i Trjg ])[()( 21 )( 21 jTui)( 21 rTx

↓ N ∑ −•r

in Trjg ])[()( 2

)( 2jTuin)( 2rTxn

↓ N ∑ −• −r

iN Trjg ])[()( 21

)( 21 jTuiN −)( 21 rTxN −

1−z

1−z

1−z

1−z

1−z

])[( 20 Tjiy β+

])[( 21 Tjiy β+

])[( 2Tjiyn β+

])[( 21 TjiyN β+−

Rev

erse

Ord

er

)( 21 jTuiN −

)( 22 jTuiN −

)( 21 jTuinN −−

)( 20 jTui

1−z

Fig. 1. An analysis filter bank designed for the finite DGT

N-p

oint

Fas

t DH

T

])[( 20 Tjiy β+

])[( 21 Tjiy β+

])[( 2Tjiyn β+

])[( 21 TjiyN β+−

↑ N ∑ −•j

i Tjrf ])[()( 20

)( 1kTx

)( 20 rTxi)( 20 Tjvi β

∑ −•j

i Tjrf ])[()( 21 )( 21 rTxi)( 21 Tjvi β

∑ −•j

in Tjrf ])[()( 2

)( 2rTxin)( 2Tjvi

n β

∑ −• −j

iN Tjrf ])[()( 21

)( 21 rTxiN −)( 21 Tjvi

N β−

1−z

1−z

1−z

1−z

1−z

∑ −= •1

0 )(βi

)( 20 rTx

↑ N ∑ −= •1

0 )(βi

)( 21 rTx

↑ N ∑ −= •1

0 )(βi

)( 2rTxn

↑ N ∑ −= •1

0 )(βi

)( 21 rTxN −

Rev

erse

Ord

er

)( 21 TjviN β−

)( 22 TjviN β−

)( 21 TjvinN β−−

)( 20 Tjvi β 1−z

Fig. 2. A synthesis filter bank designed for the finite DGE

Suppose that the oversampling rate β is taken as an integer, 1−−= ρrNk , Mr , ,2 ,1= , 1 , ,1 ,0 −= Nρ ,

βjim += , 1 , ,1 ,0 −= βi , 1 , ,1 ,0 −= Mj , and let )()]1([)( 112 kTxTrNxrTx =−−= ρρ , (15)

)]([ )]1([

)]1)([()]([

1

1

12

TNmkhTjNNirNh

TNiNjrhTjrf i

−=−−−−=

−−−−=−

ρ

ρρ

(16)

and then (14) becomes

)/)1(π2(cas

)]([)]([)( 2

1

02

1

0

1

02

Nn

TjrfTjiyrTx iN

nn

M

ji

+⋅

−+= ∑∑∑−

=

−

=

−

=

ρ

β ρ

β

ρ

)]([

)/)1(π2(cas)]([

2

1

02

1

0

1

0

Tjrf

NnTjiy

i

N

nn

M

ji

−⋅

⎥⎦

⎤⎢⎣

⎡++= ∑∑∑

−

=

−

=

−

=

ρ

βρβ

(17)

Assume 1−−= qNρ , 1 , ,1 ,0 −= Nq , and let

)/)1(π2(cas)]([)(1

022 NnTjiyTjv

N

nn

iq ++= ∑

−

=ρββ

)/π2(cas)]([ 1

02 NnqTjiy

N

nn∑

−

=+= β . (18)

Obviously, )( 2Tjviq β can be obtained by taking an N-point

fast DHT of the DGT coefficients )]([ 2Tjiyn β+ . Note that 1−−= qNρ . Substituting (18) into (17) leads to

)()]([)()( 2

1

022

1

01

1

02 rTxTjrfTjvrTx

i

iiM

j

iN

i∑∑∑

−

=

−

=−−

−

==−=

β

ρρρ

β

ρ β

(19) where

)]([)()( 22

1

012 TjrfTjvrTx i

M

j

iN

i −= ∑−

=−− ρρρ β (20)

In terms of the above synthesis process, a synthesis filterbank can be designed for the DGE as shown in Fig. 2, where the output of the synthesis filterbank is the reconstructed signal x(kT1) which is obtained from (15) by upsampling )( 2rTxn by a factor of N and delaying it by time (n+1)T1, where 1 , ,1 ,0 −= Nn . We see from (16) that the synthesis function )( 1kTh of the DGE has the following

relationship with the unit pulse responses )( 2jTf in of the

synthesis filterbank:

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TABLE I. COMPUTATIONAL COMPLEXITY COMPARISON OF SINGLE PARALLEL CHANNEL

References Computational Complexity of Single Parallel Channel Applicability

[6] NLL 2log5.0+ CS, DGT

[7] NNL 2log5.0+ CS, OS, DGT

[7] NLL 2log5.0+ CS, OS, DGE Proposed

algorithms NNM 2log5.0+ CS, OS, DGT, DGE

TABLE II. NUMERICAL COMPARISON ON TOTAL NUMBER OF

MULTIPLICATIONS OF SINGLE PARALLEL CHANNEL

References Total Number of Multiplications of Single Parallel Channel (Applicability)

[6] 6144 (CS, DGT)

[7] 2080 (CS, DGT), 2496 (OS, DGT)

[7] 6144 (CS, DGE), 9216 (OS, DGE) Proposed

algorithms 160 (CS, DGT, DGE), 464 (OS, DGT, DGE)

)]1([)( 12 TNinjNhjTf i

n −−−= . (21)

Note that )( 2jTf in is periodic in j with a period M because

of the periodicity of )( 1kTh . As in the analysis filterbank, the synthesis filterbank is

also designed with large scale unified parallel channels for all the indices i and j ( 1 , ,1 ,0 −= βi , 1 , ,1 ,0 −= Mj ), such that the computational complexity of each parallel channel can be significantly reduced. The computational complexity of every single channel corresponding to the output )( 2rTxn in Fig. 2 is the order of the multiplication number of Equation (20) plus N-point FFT, i.e., NNM 2log5.0+ . Therefore, the computational complexity of each channel also does not change as the oversampling rate β increases.

V. COMPUTATIONAL COMPLEXITY ANALYSIS AND COMPARISON

As shown in Sections III and IV, the computational complexity of every single unified parallel channel corresponding to the output )]([ 2Tjiyn β+ or )( 2rTxn in the analysis or synthesis filterbank is the order of

NNM 2log5.0+ . Therefore, the computational complexity of each channel depends only on the length NML = of the discrete input signal and the number N of the Gabor frequency sampling points, and it does not change as the oversampling rate β increases. The bigger the oversampling rate β , the larger the parallel scale of the filterbanks. Such large scale parallel filterbanks account for the very low computational complexity of each parallel channel.

A computational complexity comparison of single unified parallel channel between the proposed parallel algorithms and the existing major parallel algorithms for the finite DGT and DGE is given in Table I, where the symbols CS, OS, DGT,

and DGE indicate four cases of algorithm applicability which are respectively the critically-sampled case, the oversampled case, the computation of the Gabor transform coefficients, and the reconstruction of the original signal from the coefficients. Since MNML >= and NL > , we can see from the table that the computational complexity related to the total computation time of the proposed parallel algorithms is far less than that of the existing major parallel algorithms for the finite DGT and DGE. Thus, the proposed algorithms are more efficient and faster.

Table II gives a numerical comparison on the total number of multiplications related to the total computational time between the proposed parallel algorithms and the existing major parallel algorithms for the finite DGT and DGE, which further indicates the advantages of the proposed parallel algorithms. In Table II, the parameters L=2048, M=N=128,

16==MN , 8=β for oversampling and L=2048, MM = =128, 16== NN , 1=β for critical sampling are used.

VI. CONCLUSION In terms of the similarity between the analysis (and

synthesis) filterbanks and the DGT (and DGE), we designed an analysis filterbank and a synthesis filterbank for the fast implementation of the finite DGT and the finite DGE, respectively. Each of the parallel channels in the two filterbanks has a unified structure and can apply the fast DHT algorithm to reduce its computational load. The computational complexity of each parallel channel of the filter banks depends only on the length of the discrete input signal and the number of the Gabor frequency sampling points, and is independent of the oversampling rate β . In fact, the bigger the oversampling rate β , the larger the parallel scale of the filterbanks. The computational complexity analysis and comparison showed that the computational complexity related to the total computation time of the proposed parallel algorithms is far less than that of the major parallel algorithms for the finite DGT and DGE. The results indicate that the proposed parallel algorithms are more efficient and faster in computing the DGT coefficients and reconstructing the original signal from the coefficients.

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[7] L. Tao and H. K. Kwan, “Block Time-recursive Real-valued Discrete Gabor Transform Implemented by Unified Parallel Lattice Structures,” IEICE Trans. on Information and Systems, vol. E88-D, no. 7, pp. 1472-1478, 2005.

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