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Research Article
Poisson Summation Formulae Associated with the Special Affine
Fourier Transform and Offset Hilbert Transform
Zhi-Hai Zhuo
Beijing Information Science & Technology University, Beijing, China
Correspondence should be addressed to Zhi-Hai Zhuo; zhuozhihai@bistu.edu.cn
Received 7 May 2017; Accepted 16 July 2017; Published 15 August 2017
A
cademicEditor:AlessandroLoSchiavo
Copyright © Zhi-Hai Zhuo. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
is paper investigates the generalized pattern of Poisson summation formulae from the special ane Fourier transform (SAFT)
and oset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the
relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. en, the generalized Poisson sum formulae are
obtained based on above relationships. e novel results can be regarded as the generalizations of the classical results in several
transform domains such as FT, fractional Fourier transform, and the linear canonical transform.
1. Introduction
e classical Poisson summation formula, which demon-
strates that the sum of innite samples in the time domain
of a signal ()is equivalent to the sum of innite samples of
(⋅)in the Fourier domain, is of importance in theories and
applications of signal processing []. e traditional Poisson
sum formula can be represented as follows []:
+∞
𝑘=−∞(+)=1
+∞
𝑛=−∞
𝑗(𝑛𝑡/𝜏) ()
or
+∞
𝑘=−∞()=1
+∞
𝑛=−∞
, =0, ()
where (⋅)denotes the Fourier transform (FT) of a signal ()
and stands for the sampling interval. Not only does Poisson
summation formula play a key role in various branches of the
mathematics,butalsoitndsnumerousapplicationsinlotsof
elds, for example, mechanics, signal processing community,
and many other scientic elds. e Poisson summation
formula is related to the Fourier transform, and, with the
development of modern signal processing technologies, there
are many other kinds of transforms that have been proposed,
it is therefore worthwhile and interesting to investigate the
Poissonsumformulaindeepassociatedwiththesekindsof
new integral transforms.
e special ane Fourier transform (SAFT) [, ], also
known as the oset linear canonical transform [, ] or the
inhomogeneous canonical transform [], is a six-parameter
(,,,,0,0)class of linear integral transform. Many
well-known transforms in signal processing and optic sys-
tems are its special cases such as Fourier transform (FT),
fractional Fourier transform (FRFT), the linear canonical
transform (LCT), time shiing and scaling, frequency mod-
ulation, pulse chirping, and others [, ]. SAFT can be inter-
preted as a time shiing and frequency modulated version of
LCT [–], that is much more exible because of its extra
parameters (0,0). Recently, it has been widely noticed in
many practical applications along with the rapid development
of LCT [–]. us, developing relevant theorems for SAFT
is of importance and necessary in optical systems and many
signal processing applications as well.
In addition, the generalized Hilbert transform closely
relatedtoSAFT,calledosetHilberttransform(OHT),is
anotherpowerfultoolintheeldsofopticsandsignalpro-
cessing community []. It has been presented recently and
widely used for image processing, especially for edge detec-
tion and enhancement, because it can emphasize the deriva-
tives of the image [, ]. In recent decades, many essential
theories and useful applications of SAFT and OHT have
been derived from in-depth researching on it [, , , ].
Hindawi
Mathematical Problems in Engineering
Volume 2017, Article ID 1354129, 5 pages
https://doi.org/10.1155/2017/1354129
Mathematical Problems in Engineering
To the best of our knowledge, Poisson sum formula has
been generalized with many transforms such as FRFT, LCT,
fractional Laplace transform and fractional Hilbert transform
[,,,].However,noneoftheresearchpapersthrow
light on the study of the traditional Poisson sum formula
associated with the SAFT and OHT yet. Based on the existing
results, the motivation of this paper is to generalize the
above-mentioned Poisson sum formula into SAFT and OHT
domains.
e rest of this paper is organized as follows. Section
gives some fundamental knowledge of SAFT and OHT. In
Section , we give the relationships between SAFT/OHT
and FT in detail. Some novel Poisson summation formulae
associated with SAFT are presented in Section . Section
concludes the paper.
2. Preliminaries
2.1. e Special Ane Fourier Transform. e special ane
Fourier transform (SAFT) with real parameters A= (,,,
,0,0)of a signal ()is dened by the following [, ]:
A()=A
𝐿()()
=
+∞
−∞ ()A(,) =0
𝑗(𝑐𝑑/2)(𝑢−𝑢0)2+𝑗𝑤0𝑢−0 =0,
()
where
A(,)=A(𝑗/2𝑏)[𝑎𝑡2+2𝑡(𝑢0−𝑢)−2𝑢(𝑑𝑢0−𝑏𝑤0)+𝑑𝑢2],
A=1
2𝑗(𝑑/2𝑏)𝑢2
0
()
and − = 1.Notethat,for=0, the SAFT of a signal
is essentially a chirp multiplication and it is of no particular
interest for our objective in this work. Hence, without loss
of generality, we set =0in the following section unless
stated otherwise. e inverse of an SAFT with parameter
A= (,,,,0,0)isgivenbyanSAFTwithparameter
A−1 =(,−,−,,0−0,0−0),whichis
()=A−1
𝐿A()()=+∞
−∞ A()A−1 (,), ()
where =
𝑗(1/2)(𝑐𝑑𝑢2
0−2𝑎𝑑𝑢0𝑤0+𝑎𝑏𝑤2
0).iscanbeveriedby
the denition of SAFT. Most of important transforms can be
its special cases when parameter Ais replaced with specic
parameters. For example, when A= (0,1,−1,0,0,0),SAFT
coincides with FT; when A=(cos ,sin ,−cos ,sin ,0,0),
SAFT is FRFT; when A= (,,,,0,0),SAFTequalsLCT.
Furthermore, many important theories on SAFT have been
investigated [, , , ].
f(t)
Hilbert lter
fA
O(t)
ej(1/2b)(at2+2u0t) e−j(1/ 2b)(at2+2u0t)
F : Oset Hilbert lter.
2.2. Oset Hilbert Transform. e oset Hilbert transform
(OHT) of a signal ()is dened as follows []:
A
𝑂()=A
𝑂()
=p.v.−𝑗((𝑎𝑡2+2𝑢0𝑡)/2𝑏)
+∞
−∞ ()
−𝑗((𝑎𝑥2+2𝑢0𝑥)/2𝑏). ()
It should be noted that the above denition uses the Cauchy
principal value of the integral (denoted here by p.v.). To
obtain the relationship between the stand and HT and OHT,
we can rewrite () as
A
𝑂()=A
𝑂()
=−𝑗((𝑎𝑡2+2𝑢0𝑡)/2𝑏) ()𝑗((𝑎𝑡2+2𝑢0𝑡)/2𝑏)∗().()
Notice that computing the OHT of a signal is equivalent
to multiplying it by a chirp, 𝑗((𝑎𝑡2+2𝑢0𝑡)/2𝑏),thenpassingthe
product through a standard Hilbert lter and nally multiply-
ing the output by the chirp, −𝑗((𝑎𝑡2+2𝑢0𝑡)/2𝑏).isrelationship
between OHT and the classical HT can be shown in Figure .
3. The Relationships between SAFT/OHT and
Fourier Transform
In order to derive novel Poisson summation formulae based
on SAFT and OHT, some relationships between SAFT/OHT
and FT are obtained in this section rstly.
Lemma 1. Suppose the SAFT of a signal ()with parameters
A= (,,,,0,0)is A(⋅),andset() = ()𝑗(𝑎/2𝑏)𝑡2,
and then the following relations hold:
A()
=1
2𝑗(1/2𝑏)[𝑑(𝑢2
0+𝑢2)−2𝑢(𝑑𝑢0−𝑏𝑤0)]−0
,
(V)=2−𝑗(1/2)[𝑑𝑏V2+2𝑏V𝑤0+2𝑢0𝑤0]AV+0,
()
where (⋅)is the FT of signal ().
Proof. It is easy to verify Lemma by the denitions of SAFT
and FT.
Mathematical Problems in Engineering
Lemma 2. Suppose the SAFT of a signal ()with parameters
A= (,,,,0,0)is A(⋅),andset() = (( −
)/)𝑗(𝑎/2𝑏)𝑥2, and then the following relations hold:
A
𝑂()=1
+0
,
(V)=A
𝑂V−0
, ()
where (⋅)is the FT of signal ().
Proof. According to the denition of OHT,
A
𝑂()=−𝑗((𝑎𝑡2+2𝑢0𝑡)/2𝑏)
+∞
−∞ ()𝑗((𝑎𝑥2+2𝑢0𝑥)/2𝑏)
−
=−𝑗((𝑎𝑡2+2𝑢0𝑡)/2𝑏)
⋅+∞
−∞ (−)𝑗((𝑎(𝑡−𝑥)2+2𝑢0(𝑡−𝑥))/2𝑏)
= 1
⋅+∞
−∞ (−)
𝑗(𝑎/2𝑏)𝑥2−𝑗((𝑎𝑡+𝑢0)/𝑏)𝑥= 1
⋅+∞
−∞ ()−𝑗((𝑎𝑡+𝑢0)/𝑏)𝑥= 1
+0
.
()
By replacing V=(+0)/,()canberewrittenas
A
𝑂V−0
=1
(V).()
is completes the proof of Lemma .
4. Main Results
BasedontherelationshipsinLemmasand,thegeneralized
Poisson summation formulae associated with SAFT and
OHT are derived in following subsections, respectively.
4.1. e Poisson Sum Formula Based on SAFT
eorem 3. e Poisson summation formulae of a signal ()
in the SAFT domain with parameter Aare
+∞
𝑘=−∞(+)𝑗(𝑎/2𝑏)(2𝑘𝑡𝜏+𝑘2𝜏2)=2
⋅−𝑗(𝑢0𝑤0+(𝑎/2𝑏)𝑡2)+∞
𝑛=−∞−𝑗(𝑑𝑏𝑛2/2𝜏2+𝑏𝑛𝑤0/𝜏)A
+0𝑗(𝑛𝑡/𝜏),
()
+∞
𝑘=−∞()𝑗(𝑎/2𝑏)𝑘2𝜏2=2
⋅−𝑗𝑢0𝑤0+∞
𝑛=−∞−𝑗(𝑑𝑏𝑛2/2𝜏2+𝑏𝑛𝑤0/𝜏)A
+0,
=0.
()
Proof. If we set () = ()𝑗(𝑎/2𝑏)𝑡2, from the traditional
Poisson sum formula for ()in the Fourier domain, that is,
(), we obtain
+∞
𝑘=−∞(+)=1
+∞
𝑛=−∞
𝑗(𝑛𝑡/𝜏).()
By directly using Lemma , we derive that
+∞
𝑘=−∞(+)𝑗(𝑎/2𝑏)(𝑡+𝑘𝜏)2=1
⋅+∞
𝑛=−∞2−𝑗(1/2𝑏)[𝑑𝑏2(𝑛/𝜏)2+2𝑏2(𝑛/𝜏)𝑤0+2𝑏𝑢0𝑤0]A
⋅
+0𝑗(𝑛𝑡/𝜏).
()
eorem is proved by simple calculation on ().
Equations () and () can be regarded as the general-
ization of classical Poisson sum formula based on SAFT. It
should be noticed that when the parameters of the SAFT are
chosen to be the special cases of the SAFT, the derived results
reduce to the classical results of Fourier transform domain,
fractional Fourier transform domain, and linear canonical
transform domains. It clearly demonstrates that the innite
sum of periodic phase-shied replica of a signal ()in the
time domain is equivalent to the innite sum of periodic
phase-shied replica A(⋅)in the SAFT domain.
In addition, it is of importance to investigate the Poisson
sum formula of signals with compact support in SAFT
domain. A signal () is said to have compact support in
SAFT domain if its SAFT A()>A,whereA>0is some
real number. Without loss of generality, let >0,>0,0>
0in the following analysis.
Corollary 4. Suppose a signal () is band-limited in SAFT
domain with a compact support A; then the Poisson sum
formula derived in eorem 3 can be rewritten as the following
forms according to the replica period :
(a) When />A+0and A>0,
+∞
𝑘=−∞(+)𝑗(𝑎/2𝑏)(2𝑘𝑡𝜏+𝑘2𝜏2)
=2
−𝑗(𝑎/2𝑏)𝑡2−𝑗𝑢0𝑤0A0.
()
Mathematical Problems in Engineering
(b) When (A+0)/2</<A−0and A>30,
+∞
𝑘=−∞(+)𝑗(𝑎/2𝑏)(2𝑘𝑡𝜏+𝑘2𝜏2)=2
⋅−𝑗(𝑎/2𝑏)𝑡2−𝑗𝑢0𝑤0×A0
+−𝑗(1/2)[𝑑𝑏/𝜏2+2𝑏𝑤0/𝜏]A
+0𝑗(𝑡/𝜏)
+−𝑗(1/2)[𝑑𝑏/𝜏2−2𝑏𝑤0/𝜏]A−
+0−𝑗(𝑡/𝜏).
()
(c) When (A+0)/( +1) < / < (A−0)/and
A>(2+1)0,
+∞
𝑘=−∞(+)𝑗(𝑎/2𝑏)(2𝑘𝑡𝜏+𝑘2𝜏2)=2
⋅−𝑗(𝑎/2𝑏)𝑡2−𝑗𝑢0𝑤0𝑚
𝑛=−𝑚−𝑗(𝑑𝑏𝑛2/2𝜏2+𝑏𝑛𝑤0/𝜏)A
+0𝑗(𝑛𝑡/𝜏).
()
Proof. (a) Since ()isaband-limitedsignalinSAFTdomain
with a compact support A, it is easy to derive the right hand
A(/+0)of () that is equal to zeros when =0.atis,
from />A+0,wederivethat/+0>/−0>A
and −2/+0<−/+0<−A. us, it is easy to derive
that
2
−𝑗(𝑢0𝑤0+(𝑎/2𝑏)𝑡2)+∞
𝑛=−∞−𝑗(𝑑𝑏𝑛2/2𝜏2+𝑏𝑛𝑤0/𝜏)A
+0𝑗(𝑛𝑡/𝜏) =2
−𝑗(𝑎/2𝑏)𝑡2−𝑗𝑢0𝑤0A0. ()
Substituting () into () yields the nal results.
(b)Itiseasytoprovethatonlywhen=−1,0,+1,
A(/+0)is nonzero. e right hand of () is
2
⋅−𝑗(𝑢0𝑤0+(𝑎/2𝑏)𝑡2)+∞
𝑛=−∞−𝑗(𝑑𝑏𝑛2/2𝜏2+𝑏𝑛𝑤0/𝜏)A
+0𝑗(𝑛𝑡/𝜏) =2
−𝑗(𝑎/2𝑏)𝑡2−𝑗𝑢0𝑤0A0
+−𝑗(1/2)[𝑑𝑏/𝜏2+2𝑏𝑤0/𝜏]A
+0𝑗(𝑡/𝜏)
+−𝑗(1/2)[𝑑𝑏/𝜏2−2𝑏𝑤0/𝜏]A−
+0−𝑗(𝑡/𝜏).
()
Substituting () into () yields the nal results.
(c) e proof of this situation is similar to the proof of (a)
and (b), and we omit it here.
4.2. e Poisson Sum Formula Based on OHT
eorem 5. e Poisson sum formula of a signal ()in the
OHT domain with parameter Ais as follows:
+∞
𝑘=−∞−−
+ 𝑗(𝑎/2𝑏)(2𝑘𝑦𝜏+𝑘2𝜏2)
=
−𝑗(𝑎/2𝑏)𝑦2+∞
𝑛=−∞A
𝑂−0
𝑗(𝑛𝑦/𝜏).()
Proof. If we set ()=((−)/)𝑗(𝑎/2𝑏)𝑥2,itiseasytoverify
eorem via () and Lemma :
+∞
𝑘=−∞+=1
+∞
𝑛=−∞
𝑗(𝑛𝑦/𝜏).()
at is
+∞
𝑘=−∞−−
+ 𝑗(𝑎/2𝑏)(𝑦+𝑘𝜏)2
=
+∞
𝑛=−∞A
𝑂−0
𝑗(𝑛𝑦/𝜏)
()
By simple calculation, eorem is completed.
Equation()canbeseenasthePoissonsumformula
associated with oset Hilbert transform. Furthermore, it is
worthwhile and interesting to study the signals with compact
support in oset Hilbert transform domain. Let A
𝑂() be
the OHT of a signal ().en()is said to have compact
support in OHT domain, if A
𝑂() = 0for || > A,where
A>0is some real number.
Corollary 6. Suppose signal () is band-limited in OHT
domain with a compact support A; then the Poisson sum
formula derived in eorem 5 can be rewritten as the following
forms according to the replica period :
(a) When />A+0/and A>0/,
+∞
𝑘=−∞−−
+ 𝑗(𝑎/2𝑏)(𝑦+𝑘𝜏)2=
A
𝑂−0
. ()
(b) When (A+0/)/2 < / < A−0/and A>
3(0/),
+∞
𝑘=−∞−−
+ 𝑗(𝑎/2𝑏)(𝑦+𝑘𝜏)2
=
A
𝑂−−0
−𝑗(𝑦/𝜏) +A
𝑂−0
+A
𝑂−0
𝑗(𝑦/𝜏).
()
Mathematical Problems in Engineering
(c) When (A+0/)/(+1) < / < (A−0/)/
and A>(2+1)(0/),
+∞
𝑘=−∞−−
+ 𝑗(𝑎/2𝑏)(𝑦+𝑘𝜏)2
=
𝑚
𝑛=−𝑚A
𝑂−0
𝑗(𝑛𝑦/𝜏).()
Proof. It is easy to verify this corollary using eorem and
the similar method in Corollary .
5. Conclusion
In this paper, the traditional Poisson summation formula has
been generalized into SAFT and OHT domain. eorems
and are the generalizations of Poisson summation formulae
based on SAFT and OHT, respectively. In addition, signals
withcompactsupportaremostlyusedinsignalprocessing
and considered in this paper as well. Some novel results asso-
ciated with Poisson summation formula have been derived in
the form of Corollaries and .
Conflicts of Interest
e author declares that there are no conicts of interest
regarding the publication of this paper.
Acknowledgments
is work was supported in part by the National Natural
Science Foundation of China (no. ) and Beijing
City Board of Education Science and Technology Plan (no.
KM).
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