On uncertainty principle for signal concentrations with fractional
Fourier transform
Jun Shi
a,b,
n
, Xiaoping Liu
a,
nn
, Naitong Zhang
a,c
a
Communication Research Center, Harbin Institute of Technology, Harbin 150001, China
b
Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA
c
Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China
article info
Article history:
Received 19 May 2011
Received in revised form
10 April 2012
Accepted 12 April 2012
Available online 17 May 2012
Keywords:
Fractional Fourier transform
Fractional domain
Signal concentration
Uncertainty principle
abstract
The fractional Fourier transform (FRFT) – a generalized form of the classical Fourier
transform – has been shown to be a powerful analyzing tool in signal processing. This
paper investigates the uncertainty principle for signal concentrations associated with
the FRFT. It is shown that if the fraction of a nonzer o signal’s energy on a finite interval
in one fractional domain with a certain angle
a
is specified, then the fraction of its
energy on a finite interval in other fractional domain with any angle
b
ð
b
a
a
Þ must
remain below a certain maximum. This is a generalization of the fact that any nonzero
signal cannot have arbitrarily large proportions of energy in both a finite time duration
and a finite frequency bandwidth. The signals which are the best in achievin g
simultaneous concentration in two arbitrary fractional doma ins are derived. Moreover,
some applications of the derived theory are presented.
& 2012 Published by Elsevier B.V.
1. Introduction
The fractional Fourier transform (FRFT) – a generalization
of the classical Fourier transform (FT) – has received much
attention in recent years due to its numerous applications,
including quantum physics, optics, communications, signal
processing [1–8], etc. For more details of the FRFT, see [4,11].
The definition of the FRFT is as follows [2]:
F
a
ðuÞ¼F
a
½f ðtÞð uÞ¼
Z
þ1
1
f ðtÞK
a
ðu, tÞ dt ð1Þ
in which the transform kernel K
a
ðu, tÞ is given by
K
a
ðu, tÞ¼
A
a
e
jððu
2
þt
2
Þ=2Þ cot
a
jut csc
a
,
a
ak
p
d
ðtuÞ,
a
¼2k
p
d
ðt þuÞ,
a
¼ð2k1Þ
p
8
>
<
>
:
ð2Þ
where A
a
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1j cot
a
Þ=2
p
p
and k 2
Z
.Theu axis is
regarded as the fractional domain. The inverse FRFT is the
FRFT at angle
a
,givenby
f ðtÞ¼F
a
½F
a
ðuÞðtÞ¼
Z
þ1
1
F
a
ðuÞK
a
ðu, tÞ du
¼
Z
þ1
1
F
a
ðuÞK
n
a
ðu, tÞ du ð3Þ
where
n
in the superscript denotesthecomplexconjugate.
Whenever
a
¼
p
=2, (1) reduces to the classical FT.
Since the FRFT generalizes the FT, many fundamental
results in Fourier analysis have been extended to signal
analysis associated with the FRFT [2–11], including the
uncertainty principle [12–17].
In [12], it is shown that the uncertainty product of
signal presentations in the time and frequency domains is
not invariant under the FRFT. In [13,14], Ozaktas and
Ayt
¨
ur showed that a lower bound on the uncertainty
product of signal presentations in the
a
th and
b
th frac-
tional domains was given by
1
4
sin
2
ð
a
b
Þ. Shortly after,
Akay and Boudreaux-Bartels [15] also derived the same
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/sigpro
Signal Processing
0165-1684/$ - see front matter & 2012 Published by Elsevier B.V.
http://dx.doi.org/10.1016/j.sigpro.2012.04.008
n
Corresponding author at: Communication Research Center, Harbin
Institute of Technology, Harbin 150001, China.
nn
Corresponding author.
E-mail addresses: j.shi@hit.edu.cn, jshi@udel.edu (J. Shi),
xp.liu@hit.edu.cn (X. Liu), ntzhang@hit.edu.cn (N. Zhang).
Signal Processing 92 (2012) 2830–2836