Maximally Concentrated Signals in the Special
Affine Fourier Transformation Domain
Ahmed I. Zayed
Department of Mathematical Sciences
DePaul University
Chicago, IL 60614
Email: azayed@depaul.edu
Abstract—The problem of maximizing the energy of a signal
bandlimited to E1 = [−σ, σ] on an interval T1 = [−τ, τ ] in the
time domain, which is called the energy concentration problem,
was solved by a group of mathematicians, D. Slepian, H. Landau,
and H. Pollak, at Bell Labs in the 1960s.
The goal of this article is to solve the energy concentration
problem for the n-dimensional special affine Fourier transformation which includes the Fourier transform, the fractional Fourier
transform, the Lorentz transform, the Fresnel transform, and the
linear canonical transform (LCT) as special cases. The solution
in dimensions higher than one is more challenging because the
solution depends on the geometry of the two sets E1 and T1 .
We outline the solution in the cases where E1 and T1 are n
dimensional hyper-rectangles and discs.
I. I NTRODUCTION
The problem of maximizing the energy of a signal bandlimited to E1 = [−σ, σ] on an interval T1 = [−τ, τ ] in the time
domain or equivalently, finding a signal that is time limited
to the interval [−τ, τ ] with maximum energy concentration
in [−σ, σ] in the frequency domain, is one of the important
classical problems in signal processing. This problem, which
is called the energy concentration problem, was solved by
a group of mathematicians, D. Slepian, H. Landau, and H.
Pollak, at Bell Labs in the 1960s [10], [11], [22], [23]. The
solution involved the prolate spheroidal wave functions which
are eigenfunctions of a differential and an integral equations.
More recently, S. Pei and J. Ding [19] solved the energy
problem for more general transforms than the Fourier transform, such as the fractional Fourier transform and the linear
canonical transform. Their solution involved what they called
the generalized prolate spheroidal wave functions.
Because bandlimited functions are entire functions, they
cannot vanish outside any interval and as a result the energy
concentration in any interval [−τ, τ ] cannot be 100%. The
percentage of the energy concentration depends on σ and τ and
involves the eigenvalues of a certain integral equation satisfied
by the prolate spheroidal wave functions. The solution of the
problem uses properties of the Fourier transform, among them
is the fact that the Fourier transform of a prolate spheroidal
wave function is a multiple of a scaled version of itself.
We recall that theR energy concentration of a signal f in
τ
2
(−τ, τ ) is given by −τ |f (t)| dt; therefore, the solution of
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
the energy concentration problem can be found by finding the
function f that maximizes the ratio
Rτ
2
|f (t)| dt
2
.
α (τ ) = R −τ
∞
2
|f (t)| dt
−∞
The Fourier transform (FT) has been generalized in a
number of different ways. For example, the fractional Fourier
transform (FrFT) [2], [17], which depends on an angle θ and
reduces to the standard Fourier transform when θ = π/2, has
many applications in signal processing and optics. The FrFT
with an angle θ of a function f is the θ rotation of the Wigner
distribution function Wf (t, ω) of f in phase space [7], [12].
Another integral transform that contains the Fourier transform, the fractional Fourier transform, the Lorentz transform,
and the Fresnel transform as special cases is the linear
canonical transform (LCT) which depends on four parameters
(a, b, c, d), [5], [15]. The LCT has applications in optics, radar
system analysis, filter design, and pattern recognition.
The special affine Fourier transformation (SAFT), which
depends on 6 parameters (a, b, c, d, m, n) is a generalization of
LCT [1]. It offers a unified viewpoint of known optical operations on light waves, such as rotation, lens transformation, and
magnification, and it reduces to the LCT when m = 0 = n.
It was natural to ask whether the energy concentration
problem could be solved for these general transforms. This
question was answered in the affirmative by S.C.Pei and J.J
Ding [19], [20], who introduced a set of functions, called the
generalized prolate spheroidal wave functions (GPSWF) that
played the role of the prolate spheroidal wave functions in the
solution obtained by Landau, Pollak, and Slepian.
They showed that the maximum energy ratios can be
obtained as eigenvalues of an integral equation satisfied by
the GPSWF. A discrete version of these problems were solved
in [28]. The generalized prolate spheroidal wave functions
associated with the fractional Fourier transform and the linear
canonical transform have interesting applications in the analysis of the status of energy preservation of optical systems,
self-imaging phenomenon, and the resonance phenomenon of
finite-sized one-stage and multiple-stage optical systems [21].
The main goal of this article is to solve the energy concentration problem for the special affine Fourier transformation
which includes the Fourier transform, the fractional Fourier
transform, the Lorentz transform, the Fresnel transform, and
the linear canonical transform (LCT) as special cases. Because
of space limitation, we will outline the solution in two special
cases and leave the general case for another publication. It
should be noted that the solution of the energy concentration
problem in dimensions higher than one is more challenging
because the solution depends on the geometry of the two sets
E1 and T1 . We will outline the solution in the cases where
E1 and T1 are n dimensional hyper-rectangles and discs.
A. The Fractional Fourier Transform
The fractional Fourier transform (or FrFT) was first introduced by Namias in 1980 in connection with an application
in quantum mechanics [16]. But since its introduction to the
signal processing community in the early 1990’s, the transform
has become an important tool in signal processing applications
and signal representation in the fractional Fourier transform
domain has been an active area of investigation [2], [6], [9],
[13], [14], [17], [18], [25], [26], [27].
The fractional Fourier Transform or FrFT of a signal or a
function, say f (t) ∈ L2 (IR), is defined by
Z ∞
b
fθ (ω) =
f (t)kθ (t, ω) dt
(1)
−∞
where

2
2
 c(θ) · eja(θ)(t +ω )−jb(θ)ωt , θ 6= pπ
kθ (t, ω) =
δ(t − ω), θ = 2pπ

δ(t + ω), θ = (2p − 1)π
that yields the maximum λ, where
(
ja t2 −ζ 2
Kτ (ω, ζ) =
e
)
sin bτ (t − ζ)
.
π(ω − ζ)
The solutions of the integral equation (3) share similar
properties with the prolate spheroidal wave functions, but
satisfy more general differential and integral equations. For
lack of better terminology, we shall call these new functions
fractional prolate spheroidal wave functions.
B. The Linear Canonical Transform
The linear canonical transform G(a,b,c,d) (u) of a function
f (x), which depends on four parameters a, b, c, d, is defined
as
R∞
K(a,b,c,d) (x, u)f (x)dx, b 6= 0
√−∞(j/2)cdu
G(a,b,c,d) (u) =
2
de
f (ud),
b=0
where
K(a,b,c,d)
1
=√
exp
2πjb
j 2
2
du − 2ux + ax
,
2b
with ad − bc = 1.
For a = cos θ, b = sin θ, c = − sin θ, d = cos θ, the
linear canonical transform reduces to the fractional Fourier
transform.
C. The Special Affine Fourier Transform
In [1] the special affine Fourier transformation (SAFT)
was introduced as a general inhomogeneous lossless linear
mapping in phase space that maps the position x and the wave
is the transformation kernel with
number k into
p
c(θ) = (1 − j cot θ)/2π, a(θ) = cot θ/2, and b(θ) = csc θ.
x
a b
x
m
=
+
,
(4)
k
c d
k
n
The kernel kθ (t, ω) is parameterized by an angle θ ∈ IR and
p is some integer. For simplicity, we may write a, b, c instead with
of a(θ), b(θ), and c(θ). The inverse-FrFT with respect to an
ad − bc = 1.
(5)
angle θ is the FrFT with angle −θ, given by
Z ∞
This transformation, which can model many general optical
f (t) =
fbθ (ω) k−θ (t, ω) dω.
(2) systems [1], [15], maps any convex body into another convex
−∞
body and Eq.(5) guarantees that the area of the body is
When θ = π/2, (1) reduces to the classical Fourier transform, preserved by the transformation. Such transformations form
which will be denoted by fˆπ/2 = fˆ
the inhomogeneous special linear group ISL(2, R). When
Z
m = 0 = n, we obtain the homogeneous special group
1
fˆ(ω) = √
f (t)e−jωt dt.
SL(2, R).
2π IR
The integral representation of the wave-function transforLet kθ (t, ω) be the kernel of FrFT and define the operator Lθ mation associated with the transformation (4) and (5) is given
as
Z ∞
by
j
Lθ [f ] (ω) =
f (t)kθ (t, ω)dt.
exp 2b
dt2 + 2(bn − dm)t
−∞
p
f (t) =
It is easy to see that
2π|b|
Z
j
Lθ (Lφ [f ] (ω)) = Lθ+φ [f ] (ω).
×
exp
ax2 − 2(t − m)x F (x)dx.
2b
R
It can be shown that the solution of the energy concentration
The SAFT offers a unified viewpoint of known optical
problem for the Fractional Fourier transform is the solution of
operations
on light waves. For example, if we denote by g
the integral equation (3)
the matrix
Z σ
a b
F (ω)Kτ (ω, ζ) dω = λF (ζ),
(3)
g=
,
c d
−σ
then we have
g1 (θ) =
g2 (θ) =
g3 (θ) =
g4 (θ) =
g5 (θ) =
It can be shown that the solution may be given by
cos θ
sin θ
(rotation),
− sin θ cos θ
1 0
(lens transformation)
θ 1
1 θ
( free space propagation),
0 1
eθ 0
(magification),
0 e−θ
cosh θ sinh θ
(Hyperbolic transformation).
sinh θ cosh θ
For sampling theorems and other related transforms, see [24]
II. T HE N - DIMENSIONAL S PECIAL A FFINE F OURIER
T RANSFORMATION
In this section we will outline how to solve the energy
concentration problem in two cases: the first case is where
E1 and T1 are n-dimensional hyper-rectangles and the second
where E1 and T1 are discs.
Let T = Rn = E, and T1 , E1 be the hyper cells
T1 = [−τ1 , τ1 ] × · · · × [−τn , τn ]
and
E1 = [−σ1 , σ1 ] × · · · × [−σn , σn ].
Consider the n-dimensional special affine Fourier transformation
j
d|p|2 + 2(bn − dm)p
exp 2b
f (p) =
(2π|b|)n/2
Z
j
2
d|t| − 2p · t − mt F (t)dt,
×
exp
2b
Rn
f (p)
n
X
φnk ,σk /b,τk (pk /b)
k=1
×
exp
−j 2
dpk + 2(bn − dm)pk ,
2b
where φ(n,σ,τ ) is the prolate spheroidal wave function and nk
is a nonnegative integer.
A. Special Affine Fourier Transformation Bandlimited to a
Disc
Now we outline how to solve the energy concentration
problem for a signal bandlimited to a disc in the Special Affine
Fourier Transformation domain.
R2 = E, and T1 be the disc T1 =
Let T =
2
2
2
2
(x, y) ∈ R2 : x2 + y 2 ≤ R2 , and E1 be the disc E1 =
(x, y) ∈ R : x + y ≤ δ . Consider the SAFT in two
dimensions for which
j 2
1
exp
d(p1 + p22 ) + η(p1 , p2 )
h(t, p) =
(2πb)
2b
+ a(t21 + t22 ) − 2(p1 t1 + p2 t2 ) + 2m(t1 , t2 ) ,
where η = 2(bn − dm), p = (p1 , p2 ), t = (t1 , t2 ). After some
calculations, we obtain
Z
K1 (p, q)
=
h(t, q)h̄(t, p)dt
j exp 2b
d |p|2 − |q|2 ) + η(p − q)
=
(2πb)2
Z
−j
×
exp
[(p1 − q1 ) t1 + (p2 − q2 ) t2 ]
b
T1
dt1 dt2 .
T1
where
t = (t1 , · · · , tn ), p = (p1 , · · · , pn ), t · p =
n
Y
=
tk p k ,
k=1
Setting
and
|t|2 =
n
X
t2k ,
dt = dt1 · · · dtn .
(p1 − q1 ) = ρ cos φ, (p2 − q2 ) = ρ sin φ,
k=1
If we denote the kernel of the SAFT by h(t, q), it is easy to
see that
Z
K1 (p, q) =
h(t, q)h̄(t, p)dt = K1 (p1 , q1 ) · · · Kn (pn , qn )
and
t1 = r cos θ, t2 = r sin θ,
and using polar coordinates, we have
j (6)
2
2
+
η(p
−
q)
d
|p|
−
|q|
exp
2b
where
K1 (p, q) =
(2πb)2
j 2
2
Z
Z
R
2π
Kk (pk , qk ) = exp
d(pk − qk ) + 2(bn − dm)(pk − qk )
−j
2b
×
rdr
exp
rρ cos(φ − θ) dθ.
b
0
0
sin [(pk − qk )τ /b]
×
,
π(pk − qk )
j
If we set F (p, q) = exp 2b
d |p|2 − |q|2 ) + η(p − q) ,
k = 1, 2, · · · , n. Solving the energy concentration problem then by using the relations [8, P. 7, Formula (26)]
requires solving the integral equation
∞
Z
X
−jx sin θ
Jk (x)e−jkθ ,
e
=
f (p)K1 (q, p)dp = λf (q).
(7)
T1
E1
k=−∞
R
and xp+1 Jp (x)dx = xp+1 Jp+1 (x), where Jk (x) is the
Bessel function of the first kind, we obtain
Z 2π
Z
−j
F (p, q) R
e b rρ cos(φ−θ) dθ
K1 (p, q) =
rdr
(2πb)2 0
0
Z R
∞
F (p, q) X
k ikφ
=
(i)
Jk (ρr/b)rdr
e
(2πb)2
0
k=−∞
Z 2π
×
e−ikθ dθ
0
Z R
F (p, q)
(2π)
J0 (ρr/b)rdr
=
(2πb)2
0
Z
F (p, q) b2 ρR/b
= (2π)
J0 (x)xdx
(2πb)2 ρ2 0
F (p, q) bR
= (2π)
J1 (ρR/b),
(2πb)2
ρ
where ρ = |p − q| .
or
p
2 + (p − q )2 /b
J
R
(p
−
q
)
1
1
1
2
2
RF (p, q)
p
K1 (p, q) =
.
2πb
(p1 − q1 )2 + (p2 − q2 )2
Setting p = (r, θ) q = (ρ, φ), the integral equation (7) now
takes the form
p
Z δ Z 2π
J1 R r2 + ρ2 − 2rρ cos(θ − φ)/b
p
f (r, θ)
rdrdθ
0
0
r2 + ρ2 − 2rρ cos(θ − φ)
= λf (ρ, φ),
which is a two-dimensional eigenvalue problem that may not
be solved analytically and somewhat challenging to solve
numerically. The solution of this integral equation is beyond
the scope of this article and will be treated in a separate
project.
For simplicity in this article we will solve the problem when
axial symmetry is assumed and relegate the general case to
another publication. It is interesting to note that of the six
parameters of SAFT, only b affects the eigenvalues; the other
parameters affect the phase of the eigenfunctions. For now we
fix b = 1 and take T1 to be the unit disc. If we assume axial
symmetry, the integral equation (7) takes the form
p
Z δ
J1
r 2 + ρ2
f (r) p
rdr = λf (ρ),
r2 + ρ2
0
where
q
ρ = r12 + r22 − 2r1 r2 cos(θ1 − θ2 ),
with (r1 , θ1 ), (r2 , θ2 ) being the polar coordinates of p and q
respectively.
We discretized the above equation and used MATHEMATICA to find the largest eigenvalues and their corresponding
eigenfunctions. The amount of energy concentrated in the disc
E1 of radius δ is proportional to the largest eigenvalue. As
expected from the uncertainty principle, the percentage of the
concentrated energy E increases with δ. For example, when
δ = 0.5 unit, E is only 6%, while for δ = 3, E = 76%.
δ
λ0
0.5
0.0606
1.5
0.4245
2.0
0.6055
3.0
0.7603
TABLE I
L IST OF THE LARGEST FOUR EIGENVALUES FOR N = 10
III. C ONCLUSION
We have outlined how to solve the energy concentration
problem for the SAFT in n-dimensions in two cases: the first
case is where E1 and T1 are n-dimensional hyper-rectangles
and the second case where E1 and T1 are discs. It was also
shown that of the six parameters of SAFT, only b affects
the eigenvalues; the other parameters affect the phase of the
eigenfunctions.
R EFERENCES
[1] S. Abe and J. Sheridan, Optical operations on wave functions as the
Abelian subgroups of the special affine Fourier transformation, Optics
Letters, Vol. 19, No. 22, (1994) pp. 1801-1803.
[2] L. B. Almeida, The fractional Fourier transform and time- frequency
representations, IEEE Trans. on signal processing, vol.42, no.11 (1994)
pp.3084-3091.
[3] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc.,
Vol. 68 (1950), pp. 337-404.
[4] L. Auslander and R. Tolimier, Radar amiguity functions and group theory,
SIAM J. Math. Anal., Vol. 16, pp. 577-601, 1985.
[5] L. M. Bernardo, ABCD matrix formalism of fractional Fourier optics,
Opt. Eng., vol 35 (1996) pp. 732-740.
[6] C. Candan, M. A. Kutay, H. M. Ozakdas, The discrete fractional Fourier
transform, IEEE Trans. on signal processing, vol. 48. no. 5, 2000.
[7] T.A.C.M. Classen and W.F.G. Mecklenbrauker, ”The Wigner distribution”, part 2, Philips research journal, vol. 35, pp. 276-300, 1980.
[8] A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions, Vol. II, Bateman Manuscript, McGraw-Hill, New York
(1955).
[9] M. A. Kutay, H. M. Ozakdas, O. Arikan, and L. Onural , Optimal filtering
in fractional Fourier domains, IEEE Trans. Signal Processing, vol. 45. pp.
1129-1143, 1997.
[10] H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions,
Fourier analysis and uncertainty-II, Bell Syst. Techn. J. ( 1960). pp.6584.
[11] H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions,
Fourier analysis and uncertainty-III; The dimension of the space of
essentially tine and band-limited signals., Bell Syst. Techn. J. ( 1962).
pp.1295-1336.
[12] A. W. Lohmann, Image rotation, Wigner rotation and the fractional
Fourier transform, J. Opt. Soc. Amer. A. , vol. 10, pp. 2181-2186, 1993.
[13] D. Mendlovic, Z. Zalevsky, and H. M. Ozakdas, The applications of
the fractional Fourier transform to optical pattern recognition, in Optical
Pattern Recognition, New York: Academic, 1998, Ch. 3.
[14] D. Mendlovich and H. M. Ozaktas, Fractional Fourier transforms and
their optical implementation 1, J. Opt. Soc. Amer. A. , vol.10. pp. 18751881, 1993.
[15] M. Moshinsky and C. Quesne, Linear canonical transformations and
their unitary representations, J. Math Phys. Vol. 12 pp. 1772-1780, (1971)
[16] V. Namias, The fractional order Fourier transforms and its application
to quantum mechanics, J. Inst. Math. Appl. , vol. 25, pp. 241-265, 1980.
[17] H. M. Ozakdas, M. A. Kutay, and D. Mendlovic, Introduction to the
fractional Fourier transform and its applications, Advances in Imaging
Electronics and Physics. New York: Academic, 1999, ch. 4.
[18] H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, Convolution
filtering, and multiplexing in fractional Fourier domains and their relation
to chirp and wavelet transforms, J. Opt. Soc. Amer. A. , vol. 11, pp. 547559, 1994.
[19] S.c. Pei and J.J. Ding, Generalized prolate spheroidal wave functions for
optical finite fractional Fourier and linear canonical transforms, J. Opt.
Soc. Amer., Vol 22, No. 3, (2005), pp. 460-474.
[20] S.c. Pei and J.J. Ding, Eigenfunctions of the offest Fourier, fractional
Fourier, and linear canonical transforms, J. Opt. Soc. Amer., Vol 20, No.
3, (2003), pp. 522-532.
[21] S-C Pei, M.-H Yeh and T.-L.Luo, Fractional Fourier series expansion
for finite signal and dual extension to discrete-time fractional Fourier
transform, IEEE Trans. on signal processing, vol. 47. no. 10 (1999), pp.
2883-2888.
[22] D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier
analysis and uncertainty-I, Bell System Tech. J., (1961), pp.379-393.
[23] D. Slepian, Prolate spheroidal wave functions, Fourier analysis and
uncertainty–IV: Extensions to many dimensions; generalized prolate
spheroidal functions, Bell System Tech. J. 43 (1964), pp. 3009–3057.
[24] A. Stern, Sampling of linear canonical transformed signals, Signal
Processing, Vol. 86 (2006), pp. 1421-1425.
[25] A. I. Zayed, Hilbert transform associated with the fractional Fourier
transform, IEEE Signal Processing Lett., vol. 5, pp. 206-208, 1998.
[26] A. I. Zayed, Convolution and product theorem for the fractional Fourier
transform, IEEE Signal Processing Lett. , vol. 5, pp. 101-103, 1998.
[27] A. I. Zayed, On the relationship between the Fourier and fractional
Fourier transforms, IEEE Signal Processing Lett. , vol. 3, pp. 310-311,
1996.
[28] H. Zhao, R. Wang, D. Song, and D. Wu, Maximally concentrated
sequences in both time and linear canonical transform domains, Signal
Image Video Process, DOI 10.1007/s11760-012-0309-1.