Sampling Random Signals in a Fractional Fourier tao Zhang, Wang, 2010)

8/6/2019 Sampling Random Signals in a Fractional Fourier tao Zhang, Wang, 2010)

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Sampling random signals in a fractional Fourier domain

Ran Tao n, Feng Zhang, Yue Wang

Department of Electronic Engineering, Beijing Institute of Technology, Beijing 100081, China

a r t i c l e i n f o

Article history:

Received 21 February 2010Received in revised form

12 November 2010

Accepted 20 November 2010

Keywords:

Random signals

Fractional Fourier transform

Fractional power spectrum

Fractional correlation

Multi-channel sampling

Periodic nonuniform sampling

a b s t r a c t

In this paper, we consider thesamplingand reconstruction schemesfor randomsignals in

the fractional Fourier domain. We define the bandlimited random signal in the fractionalFourier domain, and then propose the uniform sampling and multi-channel sampling

theorems forthe bandlimited randomsignal in thefractional Fourierdomain by analyzing

statistical properties of the input and the output signals for the fractional Fourier filters.

Our formulation and results are general and include derivative sampling and periodic

nonuniform samplingin thefractional Fourier domainfor randomsignalsas special cases.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Digital signal processing relies on sampling a signal and

reconstructing it from its samples. Consequently, sampling

theory lies at the heart of signal processing as the digital

applications have developed rapidly over the last few

decades. Shannon sampling theory (also attributed to

Nyquist, Whittaker and Kotelnikov) is the milestone both

in terms of achievement and conciseness, which states that

for a complete reconstruction of an original bandlimited

signal the sampling rate must be at least twice the

maximum frequency present in the signal. (This is the

so-called Nyquist rate.) [1,2] The reconstruction formula

that complements the sampling theorem is

xðt Þ ¼X1

n ¼ À1 xðnT Þ sinpðt =T ÀnÞ

pðt =T ÀnÞ

where T is the sampling interval. In this widely used theorem,

the signal is assumed to be bandlimited or compact in the

Fourier domain. Thus, the sampling theorem associated with

the Fourier transform (FT) represents a signal in terms of

sinusoidals.

In fact, many natural signals are better represented in

alternative bases other than the Fourier basis. As the

fractional Fourier transform (FRFT) is a generalization of

the conventional Fourier transform and has found many

applications in optics and signal processing [3–11], the

study of the sampling theorems associated with the FRFT

has blossomed in recent years [12–18]. In [12], Xia firstly

shows that if a nonzero signal x(t ) is bandlimited in the

fractional Fourier domain with angle a, then it cannot be

bandlimited in the fractional Fourier domain with respect to

another angle b where ba7a+np for any integer n. Then,

the sampling expansion and spectral properties for a uni-

formly sampled signal bandlimited in the fractional Fourier

domain have been derived from different ways [12–15]. In

[16], the spectral analysis and reconstruction of a periodic

nonuniformly sampled signal bandlimited in the fractional

Fourier domain are presented. A more general sampling

theorem is considered in [17], where the multi-channel

sampling theorem in the fractional Fourier domain is also

studied. Recently, sampling and reconstruction of sparse

signals in thefractional Fourier domainhave been presented

[18]. The above sampling theorems assert that if a signal has

a narrower bandwidth or compact support in a fractional

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/sigpro

Signal Processing

0165-1684/$- see front matter & 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.sigpro.2010.11.006

n Corresponding author.

E-mail addresses: [email protected] (R. Tao),

[email protected] (F. Zhang).

Signal Processing ] (]]]]) ]]]–]]]

Please cite this article as: R. Tao, et al., Sampling random signals in a fractional Fourier domain, Signal Process. (2010),doi:10.1016/j.sigpro.2010.11.006

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Fourierdomain,thenwe can usetheFRFTinstead of the FT to

sample a signal with a larger sampling interval. Therefore,

the classical sampling methods are not always efficient with

the possibility of unnecessary computational cost.

In practice, signals are often of random character. While

all previous generalized sampling approaches have typi-

cally dealt with the class of deterministic signals of known

finite spectral support in the fractional Fourier domain, wewill consider sampling methods for random signals with

known spectral densities. Firstly, we define a bandlimited

random signal in the fractional Fourier domain using the

fractional power spectrum. Then, we address the problem

of reconstructing the random signal with finite power

spectral support in the fractional Fourier domain from

uniform samples and multi-channel samples. Instead of

perfect reconstruction of the original signal in terms of the

samples, we determine the reconstruction through mini-

mization of the mean squared error (MSE) between the

signal and its reconstructed version. In particular, we

construct two kinds of multi-channel sampling structures

for fractional correlation functions. Our formulation andproof are general and include derivative sampling and

periodic nonuniform sampling in the fractional Fourier

domain for random signals as special cases.

2. Preliminaries

2.1. The fractional Fourier transform

The FRFT with angle a of a signal x(t ) is defined as [3,4]

X aðuÞ ¼ F a xðt Þ½ �ðuÞ ¼ exp À jpsgnðsinaÞ=4Â Ã

exp ja=2Â Ã

ffiffiffiffiffiffi2pp

ffiffiffiffiffiffiffiffiffiffiffiffiffisina q ÂZ 1

À1e jðu2 þ t 2=2ÞcotaÀ jut csca xðt Þdt ð1Þ

where sgnðUÞ is the sign function. For a=0 and a=p/2, the

FRFT reduces to the identity transform and the conven-

tional FT, respectively. Note that the factor exp

À jpsgnðsinaÞ=4Â Ã

exp ja=2Â Ã

= ffiffiffiffiffiffi

2pp ffiffiffiffiffiffiffiffiffiffiffiffiffi

sina q

is sometimes

simplified as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1À j cotaÞ=2p

p .

The fundamental property of the FRFT is the angle

additivity property, which can be given by

F aþb xðt Þ½ �ðuÞ ¼ F a F b xðt Þ½ �Â ÃðuÞ ð2ÞBesides, the FRFT has the following important space

shift and phase shift properties [3]:

F a xðt ÀtÞ½ � ¼ X aðuÀtcosaÞe jðt2=2ÞsinacosaÀ jut sina ð3Þ

F a xðt Þe jvt h i

¼ X aðuÀv sinaÞeÀ jðv2=2Þsinacosaþ juv sina ð4Þ

where t and v represent the space and phase shift para-

meters, respectively. More details on the FRFT can be found

in [3,4].

2.2. The fractional power spectral density

For a random signal { x(t ),ÀNot oN}, its auto-correla-

tion function is defined by R xx(t 1,t 2)= R xx(t 2+t,t 2)= E [ x(t 1) xn

(t 2)]where E d½ � indicates the statistical expectation, t=t 1Àt 2, and

superscriptn is thecomplex conjugation.Motivated bythefact

that theFRFT generalizes theFT in a rotational manner, theath

fractional auto-correlation function of x(t ) is defined as [19]

Ra xxðtÞ ¼ limT -1

1

2T

Z T

ÀT

R xxðt 2 þt,t 2Þe jt 2tcota dt 2 ð5Þ

Then, the ath fractional power spectral density can be

given by [19]

P a xxðuÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ j cota

2p

r F a Ra

xxðtÞÂ ÃðuÞeÀ jðu2=2Þcota ð6Þ

From(5) and (6) wecan see that botha timeaverage and

an ensemble average are in the fractional correlation func-

tion, and the fractional power spectrum is expressed in

terms of thefractional correlation function. Whena=p/2,(6)

becomes the Wiener–Khinchine theorem.

Filteringin a fractional Fourier domaincan be expressed

as [7,9]

s1ðt Þ ¼ F Àa F a s0ðt Þ½ � H aðuÞ

È Éwhere H a(u) is the transfer function, s0(t ) is the determi-nistic input signal and s1(t ) is the corresponding output

signal.

Let x(t ) and y(t ) be the random input and output of a

fractional Fourier filterH a(u), respectively; P a xxðuÞ and P a yyðuÞbe the fractional auto-power spectra of x(t ) and y(t ),

respectively; and P a yxðuÞ be the fractional cross-power

spectrum. Then the input–output relationships of the

fractional power spectral density for H a(u) are expressed

as [19]

P a yxðuÞ ¼ H aðuÞP a xxðuÞ ð7Þ

and

P a yyðuÞ ¼ H aðuÞ 2P a xxðuÞ ð8Þ

3. Sampling random signals in the fractional Fourier

domain

In this section, we treat the problem of reconstructing a

random signal that has compact support in the fractional

Fourier domainfrom a sequence of its uniform samples and

multi-channel samples. To analyze the sampling of random

signals in the fractional Fourier domain, we firstly give the

definition of a bandlimited random signal in the fractional

Fourier domain.

Definition 1. A randomsignal x(t ) is said tobe bandlimited

in the ath fractional Fourier domain if its fractional power

spectral density satisfies

P a xxðuÞ ¼ 0, 9u94ur ð9Þwhere ur is the smallest number such that (9) holds true

and is called the bandwidth of the random signal x(t )inthe

fractional Fourier domain.

3.1. Uniform sampling theorem for bandlimited random

signals in the fractional Fourier domain

Let a deterministic signal x(t ) be bandlimited in theath fractional Fourier domain with the bandwidth ur ,


R. Tao et al. / Signal Processing ] (]]]]) ]]]–]]]2





i.e., F a[ x(t )]=0 when 9u94ur . Then, the signal x(t ) can be

reconstructed from its uniform samples [12]:

xðt Þ ¼ eÀ jðt 2=2ÞcotaX1

n ¼ À1 xðnT Þe jðn2T 2=2Þcota sin ur cscaðt ÀnT Þ½ �

ur cscaðt ÀnT Þð10Þ

where the sampling interval T satisfies psina=T ¼ ur .It should be pointed that ur cscaðt ÀnT Þ ¼ pðt =T ÀnÞ.

Now, let us consider the case of uniform sampling for a

bandlimited random signal in the fractional Fourier

domain. We have the following sampling theorem:

Theorem 1. Let a random signal x(t) be bandlimited in the

ath fractional Fourier domain with the bandwidth ur . If its

chirped form, i.e., xðt Þe jðt 2=2Þcota, is stationary in the wide sense,

then x(t) can be reconstructed as

xðt Þ ¼ l:i:m: eÀ jðt 2=2ÞcotaX1

n ¼ À1 xðnT Þe jðn2 T 2=2Þcota

Âsin ur

ðt À

nT Þcsca

½ �ur ðt ÀnT Þcsca ð11Þwhere l:i:m: stands for limit in the mean square sense or

convergence in probability as well, i.e.,

limN -1

E xðt ÞÀeÀ jðt 2=2ÞcotaXN

n ¼ ÀN

xðnT Þe jðn2T 2=2Þcota

"(

Â sin ur ðt ÀnT Þcsca½ �ur ðt ÀnT Þcsca

#2)

¼ 0

and sampling interval T ¼ ðpsina=ur Þ .

Proof. Let the estimate ^ xðt Þ be

^ xðt Þ ¼ eÀ jðt 2=2ÞcotaX1

n ¼ À1 xðnT Þe jðn2T 2=2Þcota sin ur ðt ÀnT Þcsca½ �

ur ðt ÀnT Þcsca

ð12Þ

Then, we have

E xðt ÞÀ^ xðt ÞÂ Ã xÃðmT ÞÈ É¼ R xxðt ,mT ÞÀeÀ jðt 2=2Þcota

ÂX1

n ¼ À1R xxðnT ,mT Þe jðn2T 2=2Þcota sin ur ðt ÀnT Þcsca½ �

ur ðt ÀnT Þcscað13Þ

Since xðt Þe jðt 2=2Þcota is wide-sense stationary (WSS), we

have that

E xðt 1Þe jðt 21=2Þcota xÃðt 2ÞeÀ jðt 2

2=2Þcota

h i¼ e jððt 2 þtÞ2Àt 2

2=2ÞcotaE xðt 2 þtÞ xÃðt 2Þ½ �

¼ e jðt2=2Þcotaþ jt 2tcotaR xxðt 2 þt,t 2Þ ð14Þis only a function of the variable t where t=t 1Àt 2.

That is to say e jt 2tcotaR xxðt 2 þt,t 2Þ is only a function of the

variable t, then the fractional correlation function of x(t )

can be expressed as

Ra xxðtÞ ¼ limT -1

1

2T

Z T

ÀT

R xxðt 2 þt,t 2Þe jt 2tcota dt 2

¼R xx

ðr

þt,r

Þe jrtcota

ð15

Þwhere the equation is valid for all r.

Substituting (15) into (13), we can deduce that

E xðt ÞÀ^ xðt ÞÂ Ã xÃðmT ÞÈ É¼ Ra xxðt ÀmT ÞeÀ jmT ðt ÀmT ÞcotaÀeÀ jðt2=2Þcota

ÂX1

n ¼ À1Ra xxðnT ÀmT Þe jðn2T 2=2ÞcotaÀ jmT ðnT ÀmT Þcota

(

Â

sin ur ðt ÀnT Þcsca½ �

ur ðt ÀnT Þcsca):

ð16

ÞAccording to (3), (4) and (6), the FRFT of the fractional

correlation function with space shift and phase shift can be

written as

F a Ra xxðtÀt0ÞeÀ jt0tcotah i

¼ F a Ra xxðtÞÂ Ã

eÀ jðt20=2ÞcotaÀ jut0 csca

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2p

1þ j cota

s P a xxðuÞe jðu2=2ÞcotaÀ jðt2

0=2ÞcotaÀ jut0 csca

Note that the random signal x(t ) is bandlimited in the ath

fractional Fourier domain, i.e., the ath fractional power

spectral density of the signal x(t ) obeys (9). Thus theshifted

fractional correlation function Ra xxðtÀt0ÞeÀ jt0tcota is ban-dlimited in the ath fractional Fourier domain with the

bandwidth ur . Applyingthe uniform sampling expansionto

the deterministic function Ra xxðtÀt0ÞeÀ jt0tcota yields

Ra xxðtÀt0ÞeÀ jt0tcota ¼ eÀ jðt2=2ÞcotaX1

n ¼ À1e jðn2T 2=2ÞcotaRa xxðnT Àt0Þ

ÂeÀ jt0 nT cota sin ur ðtÀnT Þcsca½ �ur ðtÀnT Þcsca

ð17Þ

Interchanging the variables t=t and t0=mT in (17), we

obtain

Ra xx

ðt À

mT ÞeÀ jmTt cota

¼eÀ jðt 2=2Þcota X

1

n ¼ À1e jðn2 T 2=2ÞcotaRa

xx

ðnT

ÀmT

ÞÂeÀ jmnT 2 cota sin ur ðt ÀnT Þcsca½ �

ur ðt ÀnT Þcscað18Þ

Substituting (18) into (16), we deduce

E xðt ÞÀ^ xðt ÞÂ Ã xÃðmT ÞÈ É¼ 0 ð19Þ

This means that, for every m, xðt ÞÀ^ xðt ÞÂ Ãis orthogonal to

x(mT ). Since ^ xðt Þ is a linear summation of x(mT ), xðt ÞÀ^ xðt ÞÂ Ãis also orthogonal to ^ xðt Þ, i.e.,

E xðt ÞÀ^ xðt ÞÂ Ã^ xÃðt ÞÈ É¼ 0 ð20Þ

On the other hand,

E xðt ÞÀ^ xðt ÞÂ Ã xÃðt ÞÈ É¼ R xxðt ,t ÞÀeÀ jðt2=2Þcota

X1n ¼ À1

R xxðnT ,t Þ

Âe jðnT 2=2Þcota sin ur ðt ÀnT Þcsca½ �ur ðt ÀnT Þcsca

¼ Ra xxð0ÞÀX1

n ¼ À1Ra xxðnT Àt Þe jððnT Àt Þ2=2Þcota

Â sin ur ðt ÀnT Þcsca½ �ur ðt ÀnT Þcsca

ð21Þ

Similarly, choosing the variables t=t0=t in (17), we

obtain

Ra xxð0ÞeÀ jt

2cota ¼ eÀ jðt

2

=2Þcot

aX1

n ¼ À1 e jðn

2

T

2

=2Þcot

aRa xxðnT Àt Þ


R. Tao et al. / Signal Processing ] (]]]]) ]]]–]]] 3





ÂeÀ jtnT cota sin ur ðt ÀnT Þcsca½ �ur ðt ÀnT Þcsca

ð22Þ

Substituting (22) into (21) yields

E xðt ÞÀ^ xðt ÞÂ Ã xÃðt ÞÈ É¼ 0 ð23Þ

Therefore, combining (20) and (23), we can obtain

E xðt ÞÀ^ xðt Þ 2h i¼ E xðt ÞÀ^ xðt ÞÂ ÃxÃðt ÞÀ^ xÃðt ÞÂ ÃÈ É

¼ E xðt ÞÀ^ xðt ÞÂ Ã xÃðt ÞÈ ÉÀE xðt ÞÀ^ xðt ÞÂ Ã

^ xÃðt ÞÈ É¼ 0:

This concludes the proof of the theorem. ’

Remarks. Theorem 1 can be developed and understood

from another aspect.

Let xc (t ) denote the chirped form of the signal x(t ), i.e.,

xc ðt Þ ¼ xðt Þe jðt 2=2Þcota ð24ÞSince the random signal x(t ) is bandlimited in the ath

fractional Fourier domain with the bandwidth ur ,by(6)and

(9) we can derive

Ra xxðtÞ ¼Z ur

Àur

P a xxðuÞe jðu2=2Þcotah i

eÀ jðu2 þt2Þ=2 cotaþ jutcscadu

¼Z ur

Àur

P a xxðuÞeÀ jðt2=2Þcotaþ jutcscadu ð25Þ

Since xc (t ) is WSS, we can obtain its auto-correlation

function from (14) and (15):

R xc xc ðtÞ ¼ e jt2

2cotaþ jt 2tcotaR xxðt 2 þt,t 2Þ ¼ e jt

2

2cotaRa xxðtÞ ð26Þ

Combining (25) and (26), we have

R xc xc ðtÞ ¼ Ra xxðtÞe jðt2=2Þcota ¼Z

ur

Àur

P a xxðuÞe jutcscadu ð27Þ

Wecan see that xc (t ) is conventionally bandlimited with the

bandwidth ur csca. Therefore, applying the classical recon-

structionto the bandlimited random signal xc (t ), wecan obtain

(11). In fact, we can utilize the chirp multiplication (Q) and

FRFT (F a) operators for the development [4].

Eq. (10)establishes the relationship between an original

random signal and its uniform samples. From Theorem 1,

for a bandlimited random signal in the fractional Fourier

domain, we can reconstruct the original signal in terms of

its uniform samples x(nT ) in mean square sense, provided

the sampling interval satisfies T rðpsina=ur Þ.

3.2. Multi-channel sampling theorem for bandlimited

random signals in the fractional Fourier domain

There is a variety of applications in which a signal is

sampled in other ways, such as derivative sampling or

periodic nonuniform sampling[20,21]. In [22], Papoulis has

introduced the multi-channel sampling (generalized sam-

pling) theorem, and the derivative sampling and the

periodic nonuniform sampling are typical instances of it.

In [17], the generalized sampling theorem for deterministic

signals bandlimited in the fractional Fourier domain is

obtained. Multi-channel sampling theorem in the frac-

tional Fourier domain is a generalized sampling theorem.Many sampling strategies in the fractional Fourier domain

can be regarded as special cases of this generalized

sampling method such as derivative sampling and periodic

nonuniform sampling in the fractional Fourier domain. In

this part, we consider this generalized sampling for the

bandlimited random signal in the fractional Fourier

domain. We have the following theorem:

Theorem 2. Let a random signal x(t) be bandlimited in theath fractional Fourier domain with the bandwidth ur , and its

chirped form, i.e., xðt Þe jðt 2=2Þcota, be stationary in the wide

sense.If therandom signalx(t) is processedby M ath fractional

Fourier filters H a,k(u) resulting M outputs g k(t), k ¼ 1,. . .,M ,

then x(t) canbe reconstructed in terms of the samples g k(nT) in

the mean square sense:

xðt Þ ¼ l:i:m:eÀ jðt 2=2ÞcotaX1

n ¼ À1

XM

k ¼ 1

g kðnT Þe jðn2T 2=2Þcota ykðt ÀnT Þ

ð28Þwhere l:i:m: stands for limit in the mean square sense or

convergence in probability as well, i.e.,

limN -1

E xðt ÞÀeÀ jðt 2=2ÞcotaXN

n ¼ ÀN

XM

k ¼ 1


" #2

8<:

9=;¼ 0

and sampling interval is T ¼ ðM psina=ur Þ .

Here, reconstruction kernel functions ykðt Þ ¼ ð1=c Þ R Àur þ c Àur

Y kðu,t Þe jut cscadu, k ¼ 1,. . .,M , and M functions Y k(u,t) are

obtained by solving the following M linear equations:

H 1ðuÞY 1ðu,t Þ þ Á Á Á þH M ðuÞY M ðu,t Þ ¼ 1

H 1ðuþc ÞY 1ðu,t Þ þ Á Á Á þ H M ðuþc ÞY M ðu,t Þ ¼ e jct csca

Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁH 1

½u

þðM

À1Þc �Y 1

ðu,t

Þ þ Á Á Á þH M

½u

þðM

À1Þc �Y M

ðu,t

Þ ¼e jðM À1Þct csca

8>>>><

>>>>:ð29Þ

where Àur rurÀur +c and c ¼ ð2ur =M Þ.

Proof. Rewriting the fractional Fourier filter in the form

H a,kðuÞ ¼ H !

a,kðuÞeÀ jðu2=2Þcota, we have

Ga,kðuÞ ¼ X aðuÞH a,kðuÞ ¼ X aðuÞ H !

a,kðuÞeÀ jðu2=2Þcota ð30Þ

where Ga,k(u) is the ath FRFT of g k(t ).

From the definition of the FRFT, we have

Ga,kðuÞ ¼ eÀ jðu2=2Þcota

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1À jcota

2pr Z 1

À1

e jðu2 þ t 2=2ÞcotaÀ jut csca xðt Þdt

Â ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1À jcota

2p

r Z 1

À1e jðu2 þv2=2Þ cotaÀ juvcsca h

!kðvÞdv

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1À jcota

2p

r 2 Z 1

À1

Z 1

À1e jðu2 þ t 2 þv2=2ÞcotaÀ juðt þ vÞcsca

Â xðt Þ h!

kðvÞdtdv

By making the change of variable v = z Àt , wecanobtain a

convolution form of (30):

g kðt Þe jðt 2=2Þcota ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1À j cota

2p

r xðt Þe jðt 2=2ÞcotaÃ h

!kðt Þe jðt 2=2Þcota

ð31

Þwhere n denotes the conventional convolution operator.







Note that xðt Þe jðt 2=2Þcota is a wide-sense stationary random

signal. Then, from (31) we canobtain that g kðt Þe jðt 2=2Þcota is also

wide-sense stationary, and random signals xðt Þe jðt 2=2Þcota and

g kðt Þe jðt 2=2Þcota are jointly wide-sense stationary.

From (6) and (7), we have

P a g k , xðuÞ ¼ H a,kðuÞP a xxðuÞ ð32Þ

and

F a Ra g k , xðtÞh i

¼ H a,kðuÞF a Ra x, xðtÞÂ Ã

: ð33Þ

From (33) we can see that, the deterministic fractional

correlation function Ra x, xðtÞ can beseen astheinputof the M

fractional Fourier filters H a,k(u), k ¼ 1,. . .,M , and conse-

quently results in M deterministic outputs Ra g k , xðtÞ. We

show this important relation in Fig. 1, where T a,kðuÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1À j cota=2pÞ

p R 1À1 ykðt ÞeÀ jut cscadt .

Therefore, applying the generalized sampling theorem in

the fractional Fourier domain to the fractional correlationfunction Ra x, xðtÞ, we can obtain

Ra x, xðtÞ ¼ eÀ jðt2=2ÞcotaX1

n ¼ À1

XM

k ¼ 1

Ra g k , xðnT Þe jðn2T 2=2Þcota ykðtÀnT Þ

ð34Þ

From (31), we can obtain its time-delay version:

g kðt Àt0Þe jððt Àt0Þ2=2Þcota


1À j cota

2p

r xðt Àt0Þe jððt Àt0Þ2=2ÞcotaÃ h

!kðt Þe jðt 2=2Þcota ð35Þ

which can be further written as

g kðt Àt0Þe jððt20À2t t0Þ=2Þcota

h ie jðt 2=2Þcota


1À jcota

2p

r xðt Àt0Þe jðt2

0À2t t0Þ=2Þcota

h iÂe jðt 2=2ÞcotaÃ h

!kðt Þe jðt 2=2Þcota ð36Þ

According to (36), we can rewrite (34) as

Ra x, xðtÀt0Þe jððt20À2tt0Þ=2Þcota ¼ eÀ jðt2=2Þcota

X1n ¼ À1

XM

k ¼ 1

Ra g k , xðnT Àt0Þ

Âe jððt20À2nT t0Þ=2Þcotae jðn2T 2=2Þcota ykðtÀnT Þ ð37Þ

Let the estimate

^ xðt Þ ¼ eÀ jðt 2=2ÞcotaX1

n ¼ À1

XM

k ¼ 1


ð38ÞAccording to the joint stationarity of the random signals

xðt Þe j

ðt 2=2

Þcota and g

kðt Þe j

ðt 2=2

Þcota, we can derive

E xðt ÞÀ^ xðt ÞÂ Ã xÃðt ÞÈ É¼ R x, xðt ,t ÞÀeÀ jðt 2=2Þcota

ÂX1

n ¼ À1

XM

k ¼ 1

R g k , xðnT ,t Þe jððnT Þ2=2Þcota ykðt ÀnT Þ

¼ Ra x, xð0ÞÀeÀ jðt 2=2Þcota

ÂX1

n ¼ À1

XM

k ¼ 1

R g k , xðnT ,t Þe jððnT Þ2=2Þcota ykðt ÀnT Þ

¼ Ra x, xð0ÞÀeÀ jðt 2=2Þcota

ÂX1

n ¼ À1

XM

k ¼ 1

Ra g k , xðnT Àt ÞeÀ jt ðnT Àt Þe jððnT Þ2=2Þcota ykðt ÀnT Þ:

ð39

ÞLet t=t0=t in (37) and then substitute the result into (39):

E xðt ÞÀ^ xðt ÞÂ Ã xÃðt ÞÈ É¼ 0 ð40Þ

On the other hand, according to (38), for the samples

g l(mT ) of the lth output we have

E xðt ÞÀ^ xðt ÞÂ Ã g l

ÃðmT ÞÈ É¼ R x, g l ðt ,mT ÞÀeÀ jðt 2=2Þcota

X1n ¼ À1

XM

k ¼ 1

R g k , g l ðnT ,mT Þ

Âe jððnT Þ2=2Þcota ykðt ÀnT Þ ð41ÞAccording to the properties of the fractional correlation

function Ra x, g lðtÞ, we have

F a Ra g k , g lðtÞ

h i¼ H a,kðuÞF a Ra x, g l

ðtÞh i

, k ¼ 1,. . .,M ð42Þ

and

P a g k , g lðuÞ ¼ H a,kðuÞP a x, g l

ðuÞ, k ¼ 1,. . .,M ð43Þ

From (43) we can see that the deterministic fractional

correlation function Ra x, g lðtÞ can beseen asthe input ofthe M

fractional Fourier filters H a,k(u), k ¼ 1,. . .,M , and results in

M deterministic outputs Ra g k , g l

ðtÞ. We show this important

relation in Fig. 2.

Fig. 1. The generalized sampling configuration where the input is Ra x, xðtÞ. Fig. 2. The generalized sampling configuration where the input is Ra x, g lðtÞ.







Therefore, applying the generalized sampling theorem in

the fractional Fourier domain to the fractional correlation

function Ra x, g lðtÞ, we can obtain

Ra x, g lðtÞ ¼ eÀ jðt2=2Þcota

X1n ¼ À1

XM

k ¼ 1

Ra g k , g lðnT Þe jðn2 T 2=2Þcota ykðtÀnT Þ

ð44

ÞSimilar to (37), we can further obtain

Ra x, g lðtÀt0Þe jððt2

0À2tt0Þ=2Þcota ¼ eÀ jðt2=2Þcota

ÂX1

n ¼ À1

XM

k ¼ 1

Ra g k , g lðnT Àt0Þe jððt2

0À2nT t0Þ=2Þcotae jðn2T 2=2Þcota ykðtÀnT Þ

ð45ÞReplacing t=t and t0=mT in (45), and then substituting

the result into (41) yields

E xðt ÞÀ^ xðt ÞÂ Ã

g lÃðmT Þ

È É¼ R x, g l ðt ,mT ÞÀeÀ jðt 2=2Þcota

ÂX1

n ¼ À1

XM

k ¼ 1

R g k , g l ðnT ,mT Þe jððnT Þ2=2Þcota ykðt ÀnT Þ

¼ Ra x, g lðt ÀmT ÞeÀ jmT ðt ÀmT ÞcotaÀeÀ jðt 2=2Þcot a

ÂX1

n ¼ À1

XM

k ¼ 1

Ra g k , g lðnT ÀmT ÞeÀ jmT ðnT ÀmT Þcota

Âe jððnT Þ2=2Þcota ykðt ÀnT Þ ¼ 0 ð46ÞFrom (46) we can obtain that for every m and l, 1r lrM ,

xðt ÞÀ^ xðt ÞÂ Ãand g l(mT ) are orthogonal. Since ^ xðt Þ is a linear

summation of the signals g l(mT ), we have

E xðt ÞÀ^ xðt ÞÂ Ã^ xÃðt ÞÈ É¼ 0 ð47Þ

Combining (40) and (47), we can obtain

E xðt ÞÀ^ xðt Þ 2h i

¼ E xðt ÞÀ^ xðt ÞÂ ÃxÃðt ÞÀ^ x

Ãðt ÞÂ ÃÈ É¼ E xðt ÞÀ^ xðt ÞÂ Ã

xÃðt ÞÈ ÉÀE xðt ÞÀ^ xðt ÞÂ Ã^ xÃðt ÞÈ É¼ 0

This completes the proof. ’

From Theorem 2, for a bandlimited randomsignal in the

fractional Fourier domain, we can reconstruct the original

signal in terms of generalized samples g l(nT ), 1r lrM in

the sense of mean square.

Note that Theorem 2 has a similar form with the

classical result in Papoulis’ work [Eq. (39), 22].Since derivative sampling and periodic nonuniform

sampling strategies in the fractional Fourier domain can

be seen as the special cases of the generalized sampling

method in the fractional Fourier domain by choosing

special fractional Fourier filters H a,k(u), we can obtain

the reconstruction methods for bandlimited random sig-

nals in the fractional Fourier domain from derivative

samples and periodic nonuniform samples that are listed

as follows.

Corollary 1. Let a random signal x(t) be bandlimited in the

ath fractional Fourier domain with the bandwidth u r . If its

chirped form, i.e., xðt Þe jðt 2=2

Þcota

, is stationary in the wide sense,then x(t) can be reconstructed from the samples of the signal

and its derivatives:

xðt Þ ¼ l:i:m: eÀ jðt 2=2ÞcotaX1

n ¼ À14sin

2 saðt ÀnT Þcsca

2

!

Âe jððnT Þ2=2Þcota xðnT Þðsaðt ÀnT ÞcscaÞ2

þ xuðnT Þsinaþ j nTxðnT Þcosa

s2aðt ÀnT Þcsca

" #

ð48Þ

where the sampling interval is T ¼ ð2psina=ur Þ.

Proof. Since the FRFT has the following properties [3]:

F a xuðt Þ½ � ¼ X uaðuÞcosaþ juX aðuÞsina

F a txðt Þ½ � ¼ uX aðuÞcosaþ jX uaðuÞsina

then

F a xuðt Þsinaþ j txðt Þcosa½ � ¼ juX aðuÞHence, by considering the case of Theorem 2 for M =2

with H a,1(u)= 1 and H a,2(u)= ju, and after some derivations,

we can obtain the result. ’

Corollary 2. Consider a periodic nonuniform sampling scheme: the sampling points are divided into groups of M

points with each group having a period of T. Denoting the

points in one period by t k, k ¼ 1,. . .,M , the complete set of

sample points can be written as nT+t k, n ¼ . . .,À1,0,1,. . .,

k ¼ 1,. . .,M , where T ¼ M psina=ur . Let a random signal x(t)

be bandlimited in the ath fractional Fourier domain with the

bandwidth ur . If its chirped form, i.e., xðt Þe jðt 2=2Þcota, is

stationary in the wide sense, then x(t) can be reconstructed

from its periodic nonuniform samples:

xðt Þ ¼ l:i:m: eÀ jðt 2=2ÞcotaXM

k ¼ 1

X1n ¼ À1

xðnT þt kÞe jððnT þ t kÞ2=2Þcota

Â ðÀ1ÞnM QM

q ¼ 1 sin½pðt Àt qÞ=T �ðpðt ÀnT Àt kÞ=T ÞQM

q ¼ 1,qak sin½pðt kÀt qÞ=T �ð49Þ

where nT+t k are sampling points and T ¼ ðM psina=ur Þ.Proof. According to (3) and (4), we have

F a xðt þt kÞe jt k cota t h i

¼ X aðuÞeÀ jðt 2k=2Þcotaþ jut k csca

Then, by choosing H a,kðuÞ ¼ eÀ jðt 2k=2Þcotaþ jut k csca in the case

of Theorem 2 we can obtain the result. ’

4. Conclusion

In this paper, we have treated the problem of samplingand reconstruction of random signals in the fractional

Fourier domain. We have shown that for bandlimited

random signals in the fractional Fourier domain, the

original signal can be reconstructed from its uniform

samples and multi-channel samples in MSE sense. Our

formulation and proof are general, and include derivative

sampling and periodic nonuniform sampling in the frac-

tional Fourier domain for random signals as special cases.

Acknowledgements

This work was supported in part by the National ScienceFoundation of China for Distinguished Young Scholars under







Grant 60625104, the National Natural Science Foundation of

China under Grants 60890072 and60572094 andtheNational

Key Basic Research Program Founded by MOST under Grant

2009CB724003.

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