2008 ISECS International Colloquium on Computing, Communication, Control, and Management
DCT -based Real-Valued Discrete Gabor Transform and Its Fast Algorithms
Juan-juan Gu 1, Liang Tao 2
1 Dept. of Electronic Information and Electrical Engineering,
Hefei University, Hefei, Anhui, 230022, China
2 School of Computer Science and Technology,
Anhui University, Hefei, Anhui, 230039, China
[email protected],
[email protected]
Abstract
Also, complex-valued Gabor transforms are complex to
be implemented in software or hardware. For real-valued
signals, such as sampled speech signals, real-valued
Gabor transform permits a computationally faster
implementation. In 1946, Gabor [lJ had a clear insight to
see the advantage of the real transform and introduced a
continuous-time real-valued Gabor transform together
with the traditional continuous-time complex-valued
Gabor transform. Based on it, in 1995, Stewart, et al. [8J
proposed a DCT-based real-valued discrete Gabor
transform (RDGT). In 1997, Lu, et al. [9] presented a
parallel algorithm for the DCT-based RDGT. However,
The DCT-based RDGT was limited to the critical
sampling case. The biorthogonality relationship (similar to
that in [3]) between the analysis window and the synthesis
window for the transform was not unveiled. To overcome
those drawbacks, this paper proposes a novel DCT-based
RDGT, which can be applied under both the critical
sampling case and the over-sampling case. And the
biorthogonality relationship between the analysis window
and the synthesis window for the transform is also proved
by applying the biorthogonal analysis method used in [3].
Because it only involves real operations and can utilize
fast DCT algorithms for fast computation, it makes
computation and implementation easier by hardware
and/or software compared to the traditional complexvalued discrete Gabor transform.
The traditional DCT-based real-valued discrete Gabor
transform (RGDT) was limited to the critical sampling
case. The biorthogonality relationship between the
analysis window and synthesis window for the transform
was not unveiled. To overcome those drawbacks, this
paper proposes a novel DCT-based real-valued discrete
Gabor transform, which can be applied under both the
critical sampling condition and the over-sampling
condition, and the biorthogonality relationship between
the analysis window and synthesis window for the novel
transform can be derived. Because the proposed DCTbased RDGT only involves real operations and can utilize
fast DCT algorithms for fast computation, it is easier in
computation and implementation by hardware or software
compared to the traditional complex-valued discrete
Gabor transform.
1. Introduction
The Gabor transform was first proposed by Gabor [1] to
represent a signal in both time and frequency domains.
Although the Gabor transform has 'been recognized as
being useful in diverse areas such as communications;
speech and image processing; radar, sonar and seismic
data processing and interpretation, its real-time
applications are limited 'due to difficulties associated with
the high complexity of the computation of the discrete
Gabor transform coefficients and the reconstruction of the
original signal from the transform coefficients. Therefore,
a number of approaches have been proposed to solve the
problem [2.7], such as the analytical solution method by
Bastiaans [2J, Wexler and Raz [3], Qian and Chen [4J, [5J, the
DFT-based method by Qiu, et al. [6], the frame theory by
Morris and Uu [7] and so on. Generally these methods for
computing the Gabor transforms all involve complex
operations. The Gabor basis functions, the auxiliary
biorthogonal functions and the Gabor transform
coefficients are all complex. Complex operations require
significant computation compared with real operations.
978-0-7695-3290-5/08 $25.00 © 2008 IEEE
DOIIO.II09/CCCM,2008.14
2. Novel neT-based
RDGT
Let x(k) denote a real finite and periodic discrete-time
signal with a period L, the novel DCT-based real-valued
discrete Gabor expansion is defined by
M-1N-I
x(k) =
2:
2:a(m,n)lIm.n(k)
(I)
m=O n=O
and the coefficients a(m,n) can be obtained by
L-I
a(m,n) = 2:x(k)Ym,n(k)
(2)
k=O
where
99
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'.'
society
,S;=
0.5
=
+ J'C
I
£... £...
~]~IR
j=O i=O
m ('N
')
n
(15)
fE-cos---[j + 1I2]nn
N
N
where Rm(k)=x(k)·Y(k-mN),
k=iN+j. Obviously
the
first summation is an N-point DCT (Type II [101).
The method can also be used in the reconstruction
of
0.4
~ 0.3
'cil
0.2
0.1
the original signal x(k). Equation (1) can be rewritten as
M-I
N-]
x(k) = L h(k-mN)
00
La(m,n)
m=O
n=O
16
32
48
64
k
(16)
Figure 1 Two-side exponential synthesis window g(k), L=64.
nfN
("2 cos-[m_o_d_(
k_,_N_)_+_1
N/_2_]_m_o_d(_n_,
N_)_n
Let k=qN+ko, q=O, I, ...,M -I, ko=O, I, ..., N-l,
•C
the
20
above equation becomes
M-I
N-]
10
x(qN + ko) = L h(qN + ko - mN) La(m,n)
m=O
g
n=O
a
'"
.cnfN
("2 cos-----N----[mod(ko,N) + 1/2]mod(n,N)n
= L g(qN + ko - mN) La(m,n).
M-I
- N-I
~
~
Cn
-10
fE-N cos_o--[k + N
1/2]nn
-200
16
32
48
64
k
(17)
where the second summation is an N-point IDCT (Type II
Figure 2. Analysis window y(k) , L=64, N=16,
critical sampling.
N =16, /3=1,
(101).
3
4. Numerical experiments
2
In the first experiment, we use (14) to compute five
biorthogonal functions or analysis windows (Fig. 2 - Fig.
6) corresponding
to a two-side exponential synthesis
window (Fig. I),
g(k) = exp( -D.2
I
k-31.5
I
g1
,..,
o
-1
)/2.2286
a
16
critical
sampling
case,
L=64,
N=16,
N
=16,
where
the
oversampling
rates
fl=N/
N
48
Figure 3 Analysis window y(k) , L=64, N=16,
oversampling.
the
oversampling rate fl=N/N =1. The analysis windows ](k)
in Fig. 3 - Fig. 6 are computed in the four oversampling
cases,
32
64
k
The analysis window ](k) in Fig. 2 is computed in the
N =8, /3=2,
3
=2
2.5
(N=16, N =8), 4 (N=16, N =4), 8 (N=32, N =4), 16
2
(N=32, N =2), respectively. The similarity between the
synthesis window g(k) and its corresponding
analysis
window ](k) is proportional to the oversampling rate fl.
The reason is similar to that given in [4].
The five analysis windows corresponding
to the
synthesis window in Fig. I were used in the RDGT of
several signals such as rectangular pulses and chirp
signals. Reconstruction errors (MSE) were all around 10]5, which were virtually error-free reconstruction.
g 1.5
'"
0.5
a
-0.50
16
32
48
64
k
Figure 4 Analysis window y(k) , L~64,
oversampling.
101
N=16,
N =4, /3=4,
Appendix B
References
Suppose {a(n)} is a periodic sequence of period
L = MN . Define its discrete cosine transform (DCT) by
L-l
A(k) = DCT[a(n)] = Ia(n)
n=O
(B.l)
[1] D. Gabor, "Theory of communication," J. Inst. Electr. Eng.,
vol. 93, no. 3, pp. 429-457, ]946.
[2] M. Bastiaans, "Gabor's expansion of a signal into Gaussian
elementary signals," Opt. Eng., vol. 20, no. 4, pp. 594-598,
1981.
[3] J. Wexler and S. Raz, "Discrete Gabor Expansions," Signal
Processing, vol. 21, no. 3, pp. 207-220, ]990.
[4] S. Qian and D. Chen. "Discrete Gabor transforms," IEEE
Transactions on Signal Processing, vol. 41, no. 7, pp. 24292438,1993.
[5] S. Qian and D. Chen, "Joint time-frequency analysis,"
IEEE Signal Processing Magazine, vol. 6, no. 2, pp. 52-67,
1999.
[6] Sigang Qiu, Feng Zhou, and Phyllis E. Crandall, "Discrete
Gabor transforms with complexity O(N1ogN)," Signal
Processing, vol. 77, no. 2, pp. ]59-]70, ]999.
[7] J. M. Morris and Y. Liu, "Discrete Gabor expansion of
discrete-time signals in 12(Z) via frame theory," IEEE
Signal Processing Magazine, vol. 40, no. 2, pp. 151-181,
1994.
[8] D. F. Stewart, 1. C. Potter and S. C. Ahalt,
"Computationally Attractive Rea] Gabor Transforms,"
IEEE Trans. on Signal Processing, vo1.43, no.l, pp. 77-84,
]995.
[9] C. Lu, S. Joshi and Joel M. Morris, "Parallel Lattice
Structure of Block Time-recursive Generalized Gabor
Transforms," Signal Processing, vol. 57, no. 2, pp. 195203, 1997.
'Ck
L
If-cos--------[mod(n,L)+
L
]/2]mod(k,L)n
and its periodic extension {a(n) } by
M-l
a(n)=
Ia(n-mii),
m=O
(B.2)
a(n)=a(n+N)
Also, let
A(j)=DCT[a(n)]=
~I~:a(n'-mN)J
~ cos [mod(n',N)+
r{N
.c.
=
I
Ia(n'
n'=o
m=O
N-l[M-l
1/2]mod(J,N)n
N
+(M -m)N)
_}
j
[moden' + (M - m)N,N)
·cos------------N
m'=O
= n'=o
~1[1:1a(n'
+ m'N)}j
-=
~N
+ 1/2] mod(J,
fN
~
+ m'N,N) + 1/2] mod(j,
·cos----------N
[moden'
~.cos[mod(n,N)+
I:
N)n
= n=O a(n)cj V"N
JiT)n
1/2]mod(j,N)n
N
(B.3)
Taking the inverse DCT (IDCT) of (B.3) yields
~I.1A(j)
Ii(n) = IDCT[A(J)] = V"N
j=O
·c·
J
(BA)
[mod(n,N) + ] /2]mod(J,N)n
cos--------N
Substituting (B.3) into (BA), we obtain
M-l
a(n) = Ia(n
m'=O
- m' N)
_~~l[~l(') ~-=
- -= ~
~a n . C j
N j=O n'=O
N
cos------
[mod(n',N)N + 112]Jn
.cj cos
Jl
+ 1I2]Jn
[mod(n,N)
N
(B.5)
103