IEICE TRANS. INF. & SYST., VOL.E88–D, NO.7 JULY 2005
1511
PAPER
Special Section on Recent Advances in Circuits and Systems
Noise Reduction for NMR FID Signals via Oversampled
Real-Valued Discrete Gabor Transform
Liang TAO†a) and Hon Keung KWAN††b) , Nonmembers
SUMMARY An efficient algorithm to reduce the noise from the Nuclear Magnetic Resonance Free Induction Decay (NMR FID) signals is
presented, in this paper, via the oversampled real-valued discrete Gabor
transform using the Gaussian synthesis window. An NMR FID signal in
the Gabor transform domain (i.e., a joint time-frequency domain) is concentrated in a few number of Gabor transform coefficients while the noise
is fairly distributed among all the coefficients. Therefore, the NMR FID
signal can be significantly enhanced by performing a thresholding technique on the coefficients in the transform domain. Theoretical and simulation experimental analyses in this paper show that the oversampled Gabor transform using the Gaussian synthesis window is more suitable for the
NMR FID signal enhancement than the critically-sampled one using the exponential synthesis window, because both the Gaussian synthesis window
and its corresponding analysis window in the oversampling case can have
better localization in the frequency domain than the exponential synthesis
window and its corresponding analysis window in the critically-sampling
case. Moreover, to speed up the transform, instead of the commonly-used
complex-valued discrete Gabor transform, the real-valued discrete Gabor
transform presented in our previous work is adopted in the proposed algorithm.
key words: oversampled discrete Gabor transform, nuclear magnetic resonance free induction decay signals, signal enhancement, Gaussian synthesis window
1. Introduction
Nuclear Magnetic Resonance Free Induction Decay (NMR
FID) signals are very useful in chemical and biomedical applications and research [1]. They are usually modeled as
a finite mixture of modulated exponential functions or sequences. The estimation of parameters in the model, including amplitudes, phases, frequencies and exponent constants
becomes an important issue in model-fitting. The main obstacle of the parameter estimation in the model is their very
low signal to noise ratio and their overlapped resonances
with different exponent constants. Enhancing the NMR FID
signals was the topic of many papers [2]–[5]. One of the
typical methods for reducing the noise in the NMR FID
signals is the algorithm presented by [2] via the criticallysampled discrete Gabor expansion (or transform), which
demonstrated that using the expansion and exponential prototype sequences for FID model-fitting, an NMR FID signal
Manuscript received September 27, 2004.
Manuscript revised February 5, 2005.
†
The author is with the Dept. of Computer Science & Technology, Anhui University, Hefei, Anhui 230039, P.R. China.
††
The author is with the Dept. of Electrical & Computer Engineering, University of Windsor, 401 Sunset Avenue, Windsor,
Ontario, N9B 3P4, Canada.
a) E-mail: [email protected]
b) E-mail: [email protected]
DOI: 10.1093/ietisy/e88–d.7.1511
can be well represented by the Gabor transform coefficients
distributed in the Gabor transform domain (i.e., a joint timefrequency domain). The NMR FID signal in the transform
domain is concentrated in a few number of Gabor transform
coefficients while the noise is fairly distributed among all
the coefficients. Therefore, performing a thresholding technique on the coefficients in the transform domain, one can
significantly enhance the NMR FID signal. However, our research in this paper will show that in the algorithm presented
by [2], the Gabor transform coefficients obtained by the
critically-sampled discrete Gabor transform using the exponential synthesis window can not best represent the NMR
FID signals. Also results of theoretical analyses and simulations obtained in this paper will show that the oversampled
discrete Gabor transform using the Gaussian synthesis window is more suitable for the NMR FID signal enhancement
than the critically-sampled one using exponential synthesis
window. It is because the Gaussian synthesis window and
its corresponding analysis window in the oversampling case
can have better localization in the frequency domain than the
exponential synthesis window and its corresponding analysis window in the critically-sampling case. Besides, to
speed up the transform, instead of the complex-valued discrete Gabor transform (CDGT) used in [2], the real-valued
discrete Gabor transform (RDGT) presented in our previous
papers [6]–[8] will be adopted in this paper.
The paper is organized as follows. Section 2 reviews
the RDGT. It will be shown that, due to the real operations,
the RDGT can save significant computation as compared
with the CDGT. The similarity between the RDGT and the
discrete Hartley transform (DHT) allows the RDGT to utilize the fast DHT algorithms for fast computation. In addition, the RDGT has a simple relationship with the CDGT
such that the CDGT coefficients can be directly computed
from the RDGT coefficients. Therefore, the RDGT offers
a faster and more efficient method to compute the CDGT.
Section 2 also presents a discussion of why the oversampled discrete Gabor transform with the Gaussian synthesis
window is supposed to give, theoretically, better results than
critically-sampled one with the exponential synthesis window used in [2] for the NMR FID signal enhancement. In
Sect. 3, the proposed approach used to reduce the noise is
discussed with some comparison results on a typical NMR
FID signal.
c 2005 The Institute of Electronics, Information and Communication Engineers
Copyright IEICE TRANS. INF. & SYST., VOL.E88–D, NO.7 JULY 2005
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where νT = [L/(MN) 0 0 . . . 0] is a vector of length M N,
γT = [γ̃(0) γ̃(1) · · · γ̃(L − 1)] = [γ(0) γ(1) · · · γ(L − 1)],
and H is an (M N) × L real matrix constructed by
2. Real-Valued Discrete Gabor Transform (RDGT)
2.1 Definition of the RDGT
H(mN + n, k) = h̃(k + mN) · cas(2πnk/N)
Let x(k) denote a real finite and periodic sequence with a period L, the real-valued discrete Gabor expansion is defined
by
x(k) =
M−1
N−1
amn h̃mn (k)
(1)
m=0 n=0
amn =
L−1
x(k)γ̃mn (k)
(2)
k=0
where
h̃mn (k) = h̃(k − mN) · cas (2πnk/N)
γ̃mn (k) = γ̃(k − mN) · cas (2πnk/N)
(3)
(4)
where cas(x) = cos(x) + sin(x) is known as Hartley’s cas
function [9]. (2) defines the RDGT and (1) defines the inverse RDGT. In the transform, L = N M = N M, M and N
are the numbers of sampling points in time and frequency
domains. M and N are the frequency and time sampling intervals, respectively. The condition N M ≤ L (or MN ≥ L)
must be satisfied for a stable reconstruction. The critical
sampling occurs when N M = N M = L. There may be a loss
of information in an undersampling condition (MN < L).
Note that h̃(k) and γ̃(k) are all real and periodic extensions
of the synthesis window h(k) and the analysis window γ(k),
respectively, i.e.,
h(k + iL) = h̃(k + L)
(5)
h̃(k) =
i
γ̃(k) =
γ(k + iL) = γ̃(k + L)
(6)
i
The Gabor coefficients amn ’s in this case are also real and
periodic in both m and n,
a(m+iM)(n+ jN) = amn
for i, j = 0, ±1, ±2, ±3, . . .
(7)
2.2 Biorthogonality of the RDGT
It is proved in Appendix A that the biorthogonality between
h̃(k) and γ̃(k) in finite discrete case is equivalent to
L−1
2πnk
L
δm δn
· γ̃(k) =
h̃(k + mN) · cas
(8)
N
M
N
k=0
where 0 ≤ m ≤ M − 1, 0 ≤ n ≤ N − 1, and δk denotes
the Kronecker delta. (8) can also be written in the following
matrix form,
H·γ=ν
Now γ̃(k) becomes the solution of a linear system given by
(9). For critical sampling, i.e., M = M and N = N (or
L = M N), γ̃(k) is unique if H is nonsingular. In the oversampling case, i.e., M N < L, γ in (9) is not unique. It is
proved in Appendix C that the minimum norm solution of
(9) is as follows:
γ0 = HT (HHT )−1 ν
and the coefficients amn, s can be obtained by
(9)
(10)
(11)
However, we should note that the computation of γ0 in the
CDGT case is much more complicated because H in the
CDGT case is a complex matrix [11].
For example, given an exponential synthesis window
(Fig. 1 (a)),
h(k) = 0.5 × exp{−π(k − 511.5)/200}u(k − 511.5),
where u(k) is a unit step function. Let L = 1024, M =
16, N = 64. This corresponds to the critical-sampling case
in the sense that MN = L. Using (11), the corresponding
analysis window can be computed as shown in Fig. 1 (b).
The Fourier spectrums of h(k) and γ(k) are also calculated
and shown in Fig. 1 (c) and Fig. 1 (d), respectively.
Here is another example: given a Gaussian synthesis
window (Fig. 2 (a)),
h(k) = 0.01 × exp {−0.5π[(k − 511.5) × 0.03]2 },
let L = 1024, M = 256, N = 512. This corresponds to the
oversampling case due to MN > L. Using (11), the corresponding analysis window γ(k) can be computed as shown
in Fig. 2 (b). The Fourier spectrums of h(k) and γ(k) are
plotted in Fig. 2 (c) and Fig. 2 (d), respectively.
Comparing Fig. 1 with Fig. 2, one can conclude that
the Gaussian synthesis window h(k) and its corresponding
analysis window γ(k) in the oversampling case have better frequency concentration than the exponential synthesis
window h(k) and its corresponding analysis window γ(k)
in the critical sampling case. For NMR FID signals which
have few frequency components, localized γ(k) has implications that the NMR FID signals will span a fewer number of
Gabor transform coefficients and hence the energy in these
coefficients will be high. Thus, the difference between the
energy of coefficients containing signal plus noise and the
energy of coefficients containing noise only will increase.
2.3 Fast RDGT Algorithms
Once γ(k) is determined, it is rather trivial to compute amn
in (2) by a fast discrete Hartley transform (DHT) which is
faster and simpler than the FFT [10]:
amn =
L−1
k=0
x(k) · γ̃(k − mN) · cas(2πnk/N)
TAO and KWAN: NOISE REDUCTION FOR NMR FID SIGNALS VIA OVERSAMPLED RDGT
1513
(a)
(b)
(c)
(d)
Fig. 1 (a) Exponential synthesis window h(k), L = 1024, (b) the corresponding analysis window γ(k),
L = 1024, M = 16, N = 64, critical sampling, (c) Fourier spectrum of h(k), (d) Fourier spectrum of
γ(k).
(a)
(b)
(c)
(d)
Fig. 2 (a) Gaussian synthesis window h(k), L = 1024, (b) the corresponding analysis window γ(k),
L = 1024, M = 256, N = 512, oversampling, (c) Fourier spectrum of h(k), (d) Fourier spectrum of γ(k).
IEICE TRANS. INF. & SYST., VOL.E88–D, NO.7 JULY 2005
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
N−1 
 M−1

=
Rm (iN +

j=0
i=0


j) · cas(2πn j/N)
(12)
where Rm (k) = x(k) · γ̃(k − mN), k = iN + j. Obviously the
first summation is an N-point DHT. The total computation
time required to compute amn ’s through the above equation
is M × (N point 1-D fast DHT) computation time + M × L
real multiplication time + (M − 1) × M × N real addition
time, while the typical CDGT algorithm proposed in [11]
for computing CDGT coefficients needs M × M × (N point
1-D FFT) computation time + M ×L real multiplication time
+ (M − 1) × M × N complex addition time.
The method can also be used in the reconstruction of
the original signal x(k) (i.e., the inverse RDGT). (1) can be
rewritten as
M−1
N−1
2πnk
amn cas
xo (k) =
h̃(k − mN)
N
m=0
n=0
Let k = qN + k0 , q = 0, 1, . . ., M − 1, k0 = 0, 1, . . . , N − 1,
the above equation becomes
xo (qN + k0 ) =
M−1
h̃(qN + k0 − mN)
m=0
N−1
amn cas
n=0
2πnk0
N
(13)
where the second summation is an N-point DHT. The total
computation time of the reconstruction is M × (N point 1D fast DHT) computation time + M × L real multiplication
time + (M − 1) × L real addition time, while the typical
CDGT algorithm proposed in [11] for the reconstruction of
the original signal needs total computation time is 2M×N×L
complex multiplication time + (M × N − 1) × L complex
addition time.
From the above comparison of the computation time
between the RDGT and the CDGT, it can be seen that the
RDGT is obviously simpler and faster than the CDGT.
2.4 Relationship between RDGT and CDGT
The traditional complex-valued discrete Gabor transform [11] for a finite sequence x(k) can be written as follows:
x(k) =
M−1
N−1
m
bmn =
L−1
bmn · h mn (k)
(14)
n=0
∗
x(k) · γ mn (k)
(15)
k=0
h mn (k) = h(k − mN) · WNnk
∗
γ mn (k) = γ(k − mN) · WNnk
WNnk = exp( j · 2πnk/N)
(16)
(17)
(18)
√
where j = −1, the asterisk denotes complex conjugation. Comparing with the proposed RDGT, one can easily
prove the relationship between amn and the CDGT coefficients bmn = Re(bmn ) + j · Im(bmn ) as:
Re(bmn ) = (amn + am(N−n) )/2
Im(bmn ) = −(amn − am(N−n) )/2
(19)
(20)
In addition, it can easily be found that both the RDGT and
the CDGT have the same analysis window γ(k) if these two
transforms use the same synthesis window h(k). Therefore,
the RDGT also offer a faster and more efficient method to
compute the CDGT.
By amn or bmn , one can compute the Gaborgram [12]
which is defined as:
Cm,n = |bmn |2
= [(amn + am(N−n) )/2]2 + [(amn − am(N−n) )/2]2
(21)
where m = 0, 1, 2, · · · , M − 1; n = 0, 1, 2, · · · , N/2 − 1.
3. NMR FID Signal Enhancement via Oversampled
RDGT
The Ref. [3] indicated that any NMR FID signal can be well
modeled as a finite mixture of modulated exponential functions plus noise, i.e.,
x(k) = s(k) + nw (k)
Q
∆t
bq exp −
k + j(2π fq ∆t k + ϕq + nw (k))
=
T 2q
q=1
(22)
√
where j = −1, bq , fq , ϕq , and T 2q are the amplitude, frequency, phase and exponent constant of the q-th component,
respectively, ∆t is the sampling period and 0 ≤ k ≤ L − 1.
nw (k) is mainly the thermal noise in the receiver coil and
it can be well approximated by an additive white Gaussian
noise with zero mean and standard deviation σ.
An example, which has been widely applied as a test
data, is the simulated phosphorus FID sequence [4], [5] with
length L = 1024 sampling points and sampling period ∆t =
1/12000 s. This sequence is composed of the reference
signal and the required FID signal which consists of five
peaks. The frequencies, exponent constants, amplitudes,
and phases are given in Table 1 and the Fourier spectrum
without noise is shown in Fig. 3. Suppose that an additive white Gaussian noise with zero mean and σ = 4000
Table 1
Parameters of the phosphorus FID sequence.
TAO and KWAN: NOISE REDUCTION FOR NMR FID SIGNALS VIA OVERSAMPLED RDGT
1515
Fig. 3 Fourier spectrum of the simulated phosphorus FID sequence without noise.
Fig. 4
4000).
Fig. 5 Gaborgram of the phosphorus FID sequence in the oversampling
case using the Gaussian synthesis window.
Fourier spectrum of the noisy phosphorus FID sequence (σ =
is added to the sequence. Since the NMR FID sequence
is time varying, the SNR is defined as the signal energy
(the reference signal is not included) over the energy of the
noise in the observation period (0 ≤ k ≤ L − 1), i.e., SNR
= 20 log(||s||/||nw ||) = −3.33 dB. The corresponding Fourier
spectrum in Fig. 4 shows that the Pl , γ, α, and β components
are totally distorted by the noise.
Using the Gaussian synthesis window, the corresponding analysis window in Fig. 2 (the oversampling rate
MN/L = 128), and the proposed RDGT, the resulting
Gaborgram of the phosphorus FID sequence is shown in
Fig. 5, where it is obvious that the noise is fairly distributed
among all the Gabor transform coefficients in contrast to the
sequence components which are concentrated in a few number of the coefficients. Thus, it suggests that to reduce the
noise, one can retain only the coefficients which their magnitudes are above a certain threshold. The threshold value can
be optimized according to the noise level. Then, performing
the inverse Gabor transform to the retained coefficients to
obtain the noise-reduced sequence x0 . The resulting Fourier
spectrum of the noise reduced sequence is shown in Fig. 6.
It is now easy to recognize the six peaks. The SNR after process is 20 log(||s||/||x0 − s||) = 4.43 dB, i.e., a gain of 7.76 dB
in the SNR. By changing the oversampling rate MN/L from
128 to 32 (i.e., M = 64, N = 512), the SNR after processing
Fig. 6 Fourier spectrum of the enhanced phosphorus FID sequence using
the oversampled Gabor transform and the Gaussian synthesis window (σ =
4000).
increases to 3.79, i.e., a gain of 7.12 dB in the SNR. However, if using the Gaussian synthesis window in the criticalsampling case (the oversampling rate MN/L = 1), we can
find that the SNR after processing can not be improved at
all.
By altering σ from the original 4000 to 4500 and still
using the windows in Fig. 2 (L = 1024, M = 256, N =
512) and the proposed RDGT, the SNR before processing
is −4.23 dB and the SNR after processing is 3.12 dB, i.e., a
gain of 7.35 dB in the SNR. The resulting Fourier spectrum
of the noise reduced sequence is shown in Fig. 7.
For comparison, the same procedure is applied using the algorithm presented in [2] based on the criticallysampled CDGT, where the exponential synthesis window
and its corresponding analysis window in Fig. 1 are used.
The resulting Gaborgram of the phosphorus FID sequence
is shown in Fig. 8, and the resulting Fourier spectrum of the
noise reduced sequence is shown in Fig. 9. The SNR increases from −3.33 dB (σ = 4000) to −2.48 dB, i.e., a gain
of only 0.85 dB in the SNR. Obviously, one can not recognize the six peaks in Fig. 9 as easily as in Fig. 6. This
is because both the Gaussian synthesis window and its corresponding analysis window in the oversampling case can
IEICE TRANS. INF. & SYST., VOL.E88–D, NO.7 JULY 2005
1516
Fig. 7 Fourier spectrum of the enhanced phosphorus FID sequence using
the oversampled Gabor transform and the Gaussian synthesis window (σ =
4500).
Fig. 8 Gaborgram of the phosphorus FID sequence in the criticalsampling case using the exponential synthesis window.
Fig. 10 Fourier spectrum of the enhanced phosphorus FID sequence using the oversampled Gabor transform and the exponential synthesis window (σ = 4000).
By using the exponential synthesis window in the oversampling case (L = 1024, M = 256, N = 512, the oversampling rate MN/L = 128), and letting σ = 4000, the resulting
Fourier spectrum of the noise reduced sequence is shown in
Fig. 10. The SNR after processing is 3.65 dB, i.e., a gain
of 6.98 dB in the SNR. Obviously, the result obtained by
using the Gaussian synthesis window in the oversampling
case is still better than that by using the exponential synthesis window in the same oversampling case. In spite of the
fact that the localization property of the exponential analysis window in the oversampling case with the high oversampling rate obtains much larger improvement than that in
the critical-sampling case, the Gaussian synthesis window
and its corresponding analysis window in the oversampling
case still have better localization in the frequency domain
than the exponential synthesis window and its corresponding analysis window in the same oversampling case.
4. Conclusions
Fig. 9 Fourier spectrum of the enhanced phosphorus FID sequence obtained by the algorithm in [2] using the critically-sampled Gabor transform
and the exponential synthesis window (σ = 4000).
have better localization in the frequency domain than the exponential synthesis window and its corresponding analysis
window in the critical-sampling case. This affects the frequency resolution and hence the signal energy spans among
more coefficients. As a result, this reduces the effect of
thresholding on the NMR FID sequence.
This paper has presented an efficient algorithm to reduce the
noise from the NMR FID signals via the oversampled realvalued discrete Gabor transform using the Gaussian synthesis window. Theoretical and simulation experimental analyses show that the oversampled Gabor transform using the
Gaussian synthesis window is more suitable for the NMR
FID signal enhancement than the critically-sampled one using the exponential synthesis window in the algorithm presented by [2]. This is because both the Gaussian synthesis
window and its corresponding analysis window in the oversampling case can have better localization in the frequency
domain than the exponential synthesis window and its corresponding analysis window in the critical-sampling case.
The RDGT proposed in our previous work, which is simpler and faster than the commonly-used CDGT, is adopted
in this paper to speed up the Gabor transform.
Acknowledgments
The authors would like to thank the anonymous reviewers
TAO and KWAN: NOISE REDUCTION FOR NMR FID SIGNALS VIA OVERSAMPLED RDGT
1517
for their constructive comments. The first author would
like to acknowledge the supports of the Excellent Young
Teachers Program of the Ministry of Education, and the
Anhui Provincial Natural Science Foundation under Grant
No. 01042210. The first author would also like to acknowledge J.J. Gu for technical assistance.
2πnk
2πnk
· cas
=N·
cas
δk−k −iN
N
N
i
n=0
N−1
(A· 3)
and substituting (A· 3) into (A· 2) leads to
f (k, k ) = N ·
δk−k −iN
i
References
M−1
h̃(k + iN − mN) · γ̃(k − mN)
m=0
(A· 4)
[1] J.D.D. Certaines, W.M.M.J. Bovee, and F. Podo, Magnetic Resonance Spectroscopy in Biology and Medicine, Pergamon Press, Oxford, 1993.
[2] Y. Lu, S. Joshi, and J.M. Morris, “Noise reduction for NMR FID
signal via Gabor transform,” IEEE Trans. Biomed. Eng., vol.44,
no.6, pp.512–528, 1997.
[3] M. Joliot, B.M. Mazoyer, and R.H. Heusman, “In vivo NMR spectral parameter estimation: A comparison between time and frequency domain methods,” Magnetic Resonance in Medicine, vol.18,
pp.358–370, 1991.
[4] J.H.J. Leclerc, “Time-frequency representation of damped sinusoids,” Journal of Magnetic Resonance, vol.95, pp.10–31, 1991.
[5] P.A. Angelidis and G.D. Sergiadis, “Time-frequency representation
of damped sinusoids using the Zak transform,” Journal of Magnetic
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[6] L. Tao and G.J. Chen, “Real-valued discrete Gabor transforms for
discrete signal and image representation,” Chinese Journal of Electronics, vol.10, no.4, pp.444–449, 2001.
[7] L. Tao, J.J. Gu, J.A. Yang, and Z.Q. Zhuang, “Fast algorithms for
1-D and 2-D real-valued discrete Gabor transforms,” Proc. SPIE,
Second International Conference on Image and Graphics, vol.4875,
pp.227–242, 2002.
[8] J.J. Gu, L. Tao, and H.K. Kwan, “NMR FID signal enhancement via
the oversampled Gabor transform using the Gaussian synthesis window,” Proc. 47th IEEE Midwest Symposium on Circuits and Systems, vol.3, pp.183–186, 2004.
[9] R.M. Bracewell, “The fast Hartley transform,” Proc. IEEE, vol.72,
no.8, pp.1010–1018, 1984.
[10] R.M. Bracewell, The Fourier Transform and Its Applications, 2nd
ed., McGraw-Hill, New York, 1986.
[11] S. Qian and D. Chen, “Discrete Gabor transforms,” IEEE Trans. Signal Process., vol.21, no.7, pp.2429–2438, 1993.
[12] B. Friedlander and A. Zeira, “Oversampled Gabor representation
for transient signals,” IEEE Trans. Signal Process., vol.43, no.9,
pp.2088–2094, 1995.
Applying the (A· 11) to the m-summation in (A· 4) yields
f (k, k ) = N ·
δk−k −iN
i





N−1 L−1







2πk
j
j
2πk

 −1  
 cas
h̃(k + iN) · γ̃(k ) · cas
·
N









N
N 
j=0 k =0
(A· 5)
and substituting the biorthogonality condition (8) for the
rectangular bracketed term in (A· 5) leads to



N−1





2πk
L
j
−1


δ
δk−k −iN 
N
·
δ
·
cas
f (k, k ) = N ·

i
j




MN


N
i
j=0
= δk−k
(A· 6)
We conclude that indeed the biorthogonality condition (8)
implies (A· 1) if γ(k) exists. Obviously, this conclusion remains valid under the condition of critical sampling.
Appendix B
Suppose {a(n)} is a periodic sequence with period L = MN.
Define its discrete Hartley transform (DHT) by
L−1
2πnk
A(k) = DHT[a(n)] =
a(n)cas
(A· 7)
L
n=0
and its periodic extension {â(n)} by
â(n) =
M−1
a(n − mN),
â(n) = â(n + N)
(A· 8)
m=0
Appendix A
Also, let
Substituting (2) into (1) yields
M−1
N−1
Â(k)
h̃mn (k) · γ̃mn (k ) = δk−k
(A· 1)
m=0 n=0
Using f (k, k ) to denote the left-hand side of (A· 1) we may
recast it into the form
f (k, k ) =
M−1

 N−1  M−1

 2πn k

= DHT [â(n)] =
a(n − mN)cas
N
=



N−1  M−1
n =0 m=0
2πnk
2πnk
cas
·
· cas
N
N
n=0
N−1
n =0 m =0
(A· 2)
=
L−1
n=0
recalling
 
  2π [n +(M − m)N]k 

a[n +(M − m)N]cas 
N




N−1  M−1

  2π[n + m N]k 


a[n + m N] cas 
=
N
h̃(k − mN) · γ̃(k − mN)
m=0
n =0 m=0
a(n) · cas
2πnk
N
(A· 9)
IEICE TRANS. INF. & SYST., VOL.E88–D, NO.7 JULY 2005
1518
Taking the inverse DHT (IDHT) of (A· 9) yields
â(n) = IDHT Â(k) = N
−1
N−1
Â(k) · cas
k=0
2πnk
N
(A· 10)
Substituting (A· 9) into (A· 10), we obtain
â(n)=
M−1
a(n − mN)
m=0
=N
−1
 L−1
 N−1  


2πn
k
2πnk


a(n ) · cas
cas
N 
N
k=0 n =0
(A· 11)
Appendix C
In the oversampling case, i.e., M N < L, γ in (9) is not
unique. Suppose that γ in is arbitrary one of the solutions of
(9) and note that
||γ||22 = ||γ0 + γ − γ0 ||22
= ||γ0 ||22 + ||γ − γ0 ||22 + 2(γ0 )T (γ − γ0 )
(A· 12)
using (9) and (11), we obtain
2(γ0 )T (γ − γ0 ) = 0
(A· 13)
Therefore, (A· 12) can be written as
||γ||22 = ||γ0 ||22 + ||γ − γ0 ||22
(A· 14)
Because ||γ − γ0 ||22 ≥ 0, we obtain
||γ||22 ≥ ||γ0 ||22
(A· 15)
We conclude that indeed γ0 given by (11) is the minimum
norm solution of (9).
Liang Tao
received his Ph.D. degree in
Information and Communication Engineering
from the University of Science and Technology
of China, in June, 2003. From Aug. 1998 to
Aug. 1999, supported by the China Scholarship
Council, he was a visiting scholar at the University of Windsor, Windsor, Ontario, Canada.
Currently he is a professor with the School of
Computer Science and Technology, Anhui University. Dr. Tao has published over 40 papers.
His main research interests include digital signal/image processing, and pattern recognition.
Hon Keung Kwan
received his D.I.C. and
Ph.D. in Electrical Engineering in 1981 from
Imperial College, University of London, England, UK. He joined the University of Windsor, Canada in 1988 and holds the rank of Professor in Electrical and Computer Engineering
since 1989. His previous experience includes
a design engineer in electronics and computer
memory industry for one year; and serving as a
faculty member in the Hong Kong Polytechnic
University for one year, and in the University
of Hong Kong for seven years. He has been active in the Digital Signal
Processing Technical Committee and the Neural Systems and Applications
Technical Committee of the IEEE Circuits and Systems Society. His research interests include architecture and design of digital filters and neural
networks. Dr. Kwan is a Professional Engineer (Ontario, Canada), a Chartered Electrical Engineer (UK), and a Fellow of the Institution of Electrical
Engineers (UK).