Bandlimited signals have played a fundamental role in the
digital world in the last decades. The Whittaker-Shannon
sampling theorem is the fundamental bridge between analog
and digital signal processings/communications, which has
brought significant interest in both signal processing and
mathematics communities. The sampling theorem is about the
reconstruction of a bandlimited signal from its evenly spaced
samples and an exact reconstruction is possible if the samples
are sampled from the analog signal with a sampling rate not
lower than the Nyquist rate.
Another family of bandlimited signal reconstructions is to
reconstuct a bandlimited signal from its given segment. It is
called bandlimited signal extrapolation and has applications
in, for example, CT imaging, where only limited observation
angles are available. Bandlimited signal extrapolation has also
attracted significant interest in the past, see, for example, [2]-
[9].
In this paper, we introduce a subspace of bandlimited
signals, which is called BT-limited signal space. It consists
of all bandlimited signals such that the non-zero parts of
their Fourier transforms are pieces of bandlimited signals,
which are called BT-limited signals. Note that for a general
bandlimited signal, although its Fourier transform has finite
support, the non-zero spectrum may not be smooth, while the
non-zero spectrum is smooth for a BT-limited signal. It was
found in [9] that BT-limited signals can be characterized by
using prolate spheroidal wavefunctions [1]. In this paper, a
more intuitive and elementary proof for the characterization
is given, which may help to better understand BT-limited
signals. Some new properties about and applying BT-limited
signals are also presented. Interestingly, although there is no
any error estimate existed for a general bandlimited signal
extrapolation from inaccurate data, an analytic error estimate
in the whole time domain was obtained in [9] for a BT-limited
signal extrapolation.
The remainder of this paper is organized as follows. In
Section II, prolate spheroidal wavefunctions are briefly intro-
duced. In Section III, BT-limited signals are introduced and
characterized. In Section IV, a BT-limited signal extrapolation
with analytic error estimate is described. In Section V, some
simulations are presented to verify the theoretical extrapolation
result for BT-limited signals. In Section VI, more properties
on BT-limited signals are presented. In Section VII, this paper
is concluded.
All signals considered in this paper are assumed to have fi-
nite energies. A signal f(t)is called bandlimited of bandwidth
Ω(or Ωbandlimited), if its Fourier transform ˆ
f(ω)vanishes
when |ω|>Ω. Let BLΩdenote the space of all Ωbandlimited
signals. Let T > 0be a constant and Kbe the following
operator defined on L2[−T, T ]:
(Kf)(t) = ZT
−T
sin Ω(t−s)
π(t−s)f(s)ds, for f∈L2[−T, T ].
(1)
Let φkand λk,k= 0,1,2, ..., be the eigenfunctions and the
corresponding eigenvalues of the operator Kwith
ZT
−T
φj(t)φk(t)dt =λkδ(j−k),(2)
where δ(n)is 1when n= 0 and 0otherwise, and 1> λ0>
λ1>···>0with λk→0as k→ ∞.
From (1), for k= 0,1,2, ...,
φk(t) = 1
λkZT
−T
sin Ω(t−τ)
π(t−τ)φk(τ)dτ, for t∈[−T, T ],
(3)
which means that φk(t)can be extended from t∈[−T, T ]to
t∈(−∞,∞). Then,
Z∞
−∞
φj(t)φk(t)dt =δ(j−k)
and {φk(t)}∞
k=0 form an orthonormal basis for space BLΩand
every Ωbandlimited signal fcan be expanded as
f(t) =
∞
X
k=0
akφk(t)(4)
for some constants akwith
∞
X
k=0 |ak|2=kfk2<∞.
The extended eigenfunctions φkare called prolate spheroidal
wavefunctions [1].
On BT-limited Signals
XIANG-GEN XIA
Fellow, IEEE
Department of Electrical and Computer Engineering, University of Delaware, Newark,
DE 19716, USA
Abstract: —In this paper, we introduce and characterize a subspace of bandlimited signals. The subspace consists of all
bandlimited signals such that the non-zero parts of their Fourier transforms are pieces of some T bandlimited signals. The
signals in the subspace are called BT-limited signals and the subspace is named as BT-limited signal space. For BT-limited
signals, a signal extrapolation with an analytic error estimate exists outside the interval [−T, T ] of given signal values with
errors. Some new properties about and applying BT-limited signals are also presented.
Keywords:—Bandlimited signals, BT-limited signals, prolate spheroidal wavefunctions, signal extrapolation
Received: April 28, 2022. Revised: December 22, 2022. Accepted: January 16, 2023. Published: February 24, 2023.
1. Introduction 2. Bandlimited Signal Space
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We next define a subspace of Ωbandlimited signals. An Ω
bandlimited signal is called BT-limited if the non-zero part of
its Fourier transform is a piece of a Tbandlimited signal. In
other words, let f∈ BLΩand its Fourier transform be ˆ
f. If
there exists g∈ BLTand ˆ
f(ω) = g(ω)for ω∈[−Ω,Ω], then
fis called BT-limited. The subspace of all BT-limited signals
in BLΩis denoted as BL0
Ωand called BT-limited signal space.
From the above definition, by taking Fourier transform and
inverse Fourier transfrom, it is not hard to see that f(t)∈ BL0
Ω
if and only if there exists q(t)∈L2[−T, T ]such that
f(t) = ZT
−T
sin Ω(t−s)
π(t−s)q(s)ds, for t∈(−∞,∞).(5)
Thus, from (3), we know that every prolate spheroidal wave-
function φkis BT-limited, φk∈ BL0
Ω, so is any linear
combination of finite many prolate spheroidal wavefunctions.
For a general Ωbandlimited signal, although its Fourier
transform has finite suppport, the non-zero part of the Fourier
transform is only in L2[−Ω,Ω] and may not be smooth. How-
ever, a BT-limited signal is not only smooth (entire function of
exponential type [10]) in time domain but also has the same
smoothness for the non-zero part in frequency domain. To
characterize BL0
Ω, the following result was obtained in [9].
Theorem 1: Let f∈ BLΩwith the expansion (4). Then,
f∈ BL0
Ωif and only if
∞
X
k=0
|ak|2
λk
<∞.(6)
The proof given in [9] is based on a result on operator
theory. Below, we provide an elementary and intuitive proof of
Theorem 1, which may help to understand BT-limited signals
better.
Proof:
We first prove the “if” part. Let f(t)be an Ωbandlimited
signal with the expansion (4) and the property (6) hold. Let
q(t) =
∞
X
k=0
ak
λk
φk(t).(7)
From (2) and (6), we have
ZT
−T|q(t)|2dt =
∞
X
k=0
|a|2
k
λk
<∞.
Thus, q(t)∈L2[−T, T ]. Furthermore, from (1), we have
(Kq)(t) =
∞
X
k=0
akφk(t) = f(t),for t∈(−∞,∞),
which is (5) and therefore, f(t)∈ BL0
Ω. This proves the
sufficiency.
We next prove the “only if” part. If f(t)∈ BL0
Ω, then f(t)
has the form (5) for some q(t)∈L2[−T, T ]. In the meantime,
since f(t)is Ωbandlimited, let f(t)have the expansion (4).
Since {φk(t)}∞
k=0 form an orthogonal basis for L2[−T, T ],
[1], and (2), there exist a sequence of constants {bk}∞
k=0 of
finite energy, i.e., ∞
X
k=0 |bk|2<∞,(8)
such that
q(t) =
∞
X
k=0
bk
φk(t)
√λk
,for t∈[−T, T ].
Thus, from (5) and (1) , we have
f(t) = (Kq)(t) =
∞
X
k=0
bkpλkφk(t),for t∈(−∞,∞).
Therefore, comparing with (4), we obtain ak=bk√λkfor
k= 0,1,2, .... From (8), we then have
∞
X
k=0
|ak|2
λk
=
∞
X
k=0 |bk|2<∞,
which proves (6), i.e., the necessity is proved. q.e.d.
The above result characterizes all BT-limited signals. Since
any linear combinations of finite many prolate spheroidal
wavefunctions are BT-limited, all BT-limited signals are dense
in a bandlimited signal space, i.e., any bandlimited signal can
be approximated by BT-limited signals.
Since {φk}∞
k=0 form an orthonormal basis for space BLΩ,
[1], for any finite energy sequence {ak}∞
k=0, i.e.,
∞
X
k=0 |ak|2<∞,
we know ∞
X
k=0
akφk(t)∈ BLΩ.
On the other hand, from (2),
∞
X
k=0
ak
φk(t)
√λk
=
∞
X
k=0
ak
√λk
φk(t)∈L2[−T, T ].
Since λk→0as k→ ∞, we may have
∞
X
k=0
|ak|2
λk
=∞.
Thus, in general
∞
X
k=0
ak
φk(t)
√λk
/∈L2(−∞,∞),or equivalently, /∈ BLΩ.
However, if
∞
X
k=0
ak
φk(t)
√λk∈L2(−∞,∞),or equivalently, ∈ BLΩ,
then, (6) holds, and from Theorem 1, we obtain
∞
X
k=0
ak
φk(t)
√λk∈ BL0
Ω.
For a general Ωbandlimited signal f∈ BLΩwith expansion
(4) where {ak}∞
k=0 is a general sequence of finite energy, let
fn(t) =
n
X
k=0
akφk(t).(9)
3. BT-limited Signal Space
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From (1) and (3),
fn(t) = K n
X
k=0
ak
λk
φk(t)!.
Let
qn(t) =
n
X
k=0
ak
λk
φk(t).
Clearly, we have fn(t) = (Kqn)(t)∈ BL0
Ω. However, if fis
not BT-limited, i.e., f /∈ BL0
Ω, then, from (2) and Theorem 1,
ZT
−T|qn(t)|2dt =
n
X
k=0
|ak|2
λk→ ∞,as n→ ∞.
This means that qn(t)does not converge in L2[−T, T ], al-
though fn(t)converges to f(t)in L2(−∞,∞), as n→ ∞,
otherwise fwould be BT-limited.
Since Ωbandlimited signal space BLΩand L2[−Ω,Ω] are
isomorphic by using (inverse) Fourier transform, for any signal
g∈L2[−Ω,Ω], let it be the Fourier transform ˆ
fof f∈ BLΩ,
i.e., g=ˆ
fon [−Ω,Ω]. As we can see above, fnapproaches
fin L2(−∞,∞), then ˆ
fnapproaches g=ˆ
fin L2[−Ω,Ω].
Since ˆ
fn=DΩˆqnand ˆqnis Tbandlimited as we can see above
as well, gcan be approximated by a Tbandlimited signal ˆqn
restricted in [−Ω,Ω] in L2[−Ω,Ω], where DΩstands for the
truncation operator from (−∞,∞)to [−Ω,Ω]. Because Tand
Ωare both arbitrary, the above analysis proves the following
corollary.
Corollary 1: Any finite piece signal on [a, b]can be approx-
imated in L2[a, b]by a bandlimited signal restricted in [a, b]
of bandwidth T, where −∞ < a < b < ∞and T > 0are
arbitrary.
Note that the above result does not hold for infinite length
signals. Also, as a comparison, the Weierstrass theorem says
that any finite piece continuous signal can be approximated
by polynomials, which may be thought of as a different
perspective of using smooth/simple signals to approximate
complicated signals.
Bandlimited signal extrapolation had been studied exten-
sively in the 1970s and 1980s, see, for example, [2]- [9]. It
is to extrapolate a bandlimited signal ffrom a given piece
of its values, for example, to extrapolate f(t)for toutside
[−T, T ]when f(t)for t∈[−T, T ]is given. It is possible in
theory since fis bandlimited and thus it is an entire function
[10]. Any entire function is completely determined by its any
segment. However, in practice, a given piece signal f(t)for
t∈[−T, T ]may contain error/noise and in this case, the
extrapolation problem becomes a well-known ill-posed inverse
problem. Any error in a given segment may cause an arbitrary
large error in an extrapolation in general.
However, when fis BT-limited, an extrapolation method
was proposed in [9] and an analytic error estimate for the
extrapolation over the whole time domain was obtained. It
can be described as follows.
Let fǫ(t)be an observation of f(t)for t∈[−T, T ]with
the maximal error magnitude ǫ, i.e., |fǫ(t)−f(t)| ≤ ǫfor
t∈[−T, T ]. Let qǫbe the following minimum norm solution
(MNS) in space L2[−T, T ]:
ZT
−T
|qǫ(t)|2dt = min
q(t)∈L2[−T,T ]ZT
−T
|q(t)|2dt :
ZT
−T
sin Ω(t−s)
π(t−s)q(s)ds −fǫ(t)
≤2ǫ, for t∈[−T, T ].(10)
Let
˜
f(t) = ZT
−T
sin Ω(t−s)
π(t−s)qǫ(s)ds. (11)
One can see that the above ˜
f(t)is obtained from the given
observation segement fǫ(t)of f(t)on [−T, T ]and is called
an extrapolation of f(t). Also, from (5), we have ˜
f(t)∈ BL0
Ω.
For the above extrapolation of f(t), the following result was
obtained in [9].
Theorem 2: If fis BT-limited, i.e., f∈ BL0
Ω, and ˜
fis
defined in (11), then
|˜
f(t)−f(t)| ≤ Cǫ1/3,for all t∈(−∞,∞),(12)
for some constant Cthat is independent of ǫand t.
This result tells that when signal fis BT-limited, i.e.,
not only it is bandlimited but also the non-zero part of its
Fourier transform is a piece of a bandlimited signal, the above
extrapolation (10)-(11) is robust and has an error estimate (12)
for time t. To the author’s best knowledge, no any other error
estimate for a bandlimited signal extrapolation from inaccurate
data on the whole time domain exists in the literature.
In practice, a given observation fǫ(t)for t∈[−T, T ]
is usually discrete in time. A discretization of the above
extrapolation (10)-(11) with a proved convergence was also
given in [9].
More general subspaces BLγ
Ωfor 0≤γ < 1/2in
bandlimited signal space BLΩthan the above BL0
Ωwere
introduced with the corresponding extrapolation, error estimate
and discretization in [9]. It was shown in [9] that, if f(t)is
Ωbandlimited with the expansion (4) and, for 0≤γ < 1/2,
the following inequality holds
∞
X
k=0
|ak|2
λ1−2γ/3
k
<∞,(13)
then, f(t)∈ BLγ
Ω. Clearly, (13) returns to (6) in Theorem 1
when γ= 0, although when γ6= 0, the physical meaning of
signals in subspace BLγ
Ωis not as clear as signals in subspace
BL0
Ωstudied in this paper. For more details, we refer the reader
to [9].
Another comment we want to make here is that, for a
general Ωbandlimited signal f, although it may not be
BT-limited, function fndefined in (9) is BT-limited and
approaches fin L2(−∞,∞)as nbecomes large. Therefore,
for a general Ωbandlimited signal f, from its given segment
˜
fwith errors, we can still apply the MNS extrapolation (10)-
(11). In this case, the analytic error estimate in Theorem 2
may not hold. However, interestingly, it was shown in [9]
that the discretization of the above MNS extrapolation and
its convergence to the analog solution still hold.
4. BT-limited Signal Extrapolation
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We next show some simulation results to verify the above
MNS extrapolation for BT-limited signals. For simplicity, in
this simulation we use Ω = πand T= 1. A BT-limited signal
f(t)is generated by randomly generating q(t)∈L2[−T, T ]in
(5). Its noisy observation fǫ(t)is obtained by adding a random
error with uniform distribution to f(t)so that the maximum
error magnitude not above ǫ.
We sample a noisy analog BT-limited signal fǫ(t)in [−1,1]
with sampling rate 100 Hz, i.e., 201 samples of fǫ(t)in [−1,1]
are used in the MNS in (10)-(11). In Fig. 1, the case of ǫ=
0.0125 is simulated, where Fig. 1(a) shows the true data of a
BT-limited signal f(t)and its noisy data fǫ(t)on [−1,1], and
Fig. 1(b) shows the true signal f(t)and its extrapolation ˜
f(t)
in (11) using the noisy data fǫ(t)shown in Fig. 1(a). Fig. 2
shows the results when ǫ= 0.0031, where one can see that the
error in the extrapolated signal is clearly reduced, comparing
to that in Fig. 1.
Fig. 3 shows the curve (dashed) of the maximum error
magnitude between the true and the extrapolated signals, i.e.,
maxt|f(t)−˜
f(t)|, vs. the maximum error magnitude ǫin the
noisy data over [−1,1], and the curve (dashdot) of the ratio
vs. ǫ:
R(ǫ) = maxt|f(t)−˜
f(t)|
ǫ1/3.
The curves are obtained by using 20 independent trials. From
this figure, one can see that the ratio R(ǫ)is less than a
constant as ǫgets smaller, which verifies the result (12) in
Theorem 2. Note that in Fig. 3, the signal magnitudes are
similar to those in Figs. 1 and 2.
The above definition of a BT-limited signal can be easily
generalized as follows. Let BL[A,B]denote the space of all
finite energy signals f(t)whose Fourier transforms are sup-
ported in the interval [A, B], i.e., ˆ
f(ω) = 0 when ω /∈[A, B].
For real numbers A, B, a, b with −∞ < A < B < ∞
and −∞ < a < b < ∞, if signal f(t)∈ BL[A,B]and
ˆ
f(ω) = g(ω)when ω∈[A, B]for some g(ω)∈ BL[a,b],
then signal f(t)is called BT-limited. The signal space of all
the above BT-limited signals is denoted as BLA,B,a,b. Clearly,
when A=−Ω,B= Ω,a=−T, and b=T, the above
definition for a BT-limited signal returns to that in Section III
and BLA,B,a,b =BL0
Ω.
Let Ω = (B−A)/2and T= (b−a)/2, by some shifts
in frequency and time domains, the representation for a BT-
limited signal in (5) becomes as follows: f∈ BLA,B,a,b if
and only if
f(t) = ej(A+Ω)tZT
−T
sin Ω(t+a+T−s)
π(t+a+T−s)q(s)ds (14)
for any t∈(−∞,∞), for some q(t)∈L2[−T, T ].
Since any bandlimited signal is an entire function when
tis extended to the complex plane [10], it cannot be 0in
any segment of time domain unless it is all 0valued. This
implies that a bandlimited signal fwhose Fourier transform
is supported in two separate bands, for example, ˆ
f(ω)6= 0
for Ai< ω < Bi,i= 1,2, and ˆ
f(ω) = 0 for other ω, where
−∞ < A1< B1< A2< B2<∞, then, signal fis not
BT-limited, i.e., f /∈ BLA1,B2,a,b for any −∞ < a < b <
∞, although in this case, signal fcould be a sum of two
BT-limited signals whose Fourier transforms are supported in
[A1, B1]and [A2, B2], respectively, such that, the non-zero
supports of the two Fourier transforms are the pieces of two
bandlimited signals.
Theorem 3: For two non-zero BT-limited signals fi∈
BLAi,Bi,ai,biwith −∞ < Ai< Bi<∞and −∞ < ai<
bi<∞for i= 1,2, their linear combination f=α1f1+α2f2
with two non-zero complex coefficients α1and α2is BT-
limited if and only if A1=A2and B1=B2.
Proof:
The “if” part is easy to see by setting A=A1=A2,
B=B1=B2,a= min{a1, a2}, and b= max{b1, b2}.
Then, f∈ BLA,B,a,b, i.e., fis BT-limited.
We next prove the “only if” part. Without loss of gen-
erality, we assume A1< A2and fis BT-limited. Then,
f∈ BLA1,max{B1,B2},a,b for some real numbers a, b with
−∞ < a < b < ∞. From the above definition of BT-limited
signals, there exist bandlimited signals gi∈ BLai,bisuch
that ˆ
fi(ω) = gi(ω)for ω∈[Ai, Bi],i= 1,2, and there
exists a bandlimited signal g∈ BLa,b such that ˆ
f(ω) = g(ω)
for ω∈[A1,max{B1, B2}]. On the other hand, we have
ˆ
f=α1ˆ
f1+α2ˆ
f2. This means that ˆ
f(ω) = α1ˆ
f1(ω) = g(ω) =
α1g1(ω)for ω∈[A1,min{A2, B1}]. Since all g, g1, g2are
bandlimited and therefore, entire functions, we must have
ˆ
f(ω) = α1ˆ
f1(ω) = g(ω) = α1g1(ω)for all ω. In other words,
ˆ
f(ω) = α1ˆ
f1(ω)+α2ˆ
f2(ω) = α1ˆ
f1(ω)for all ω. This implies
ˆ
f2(ω) = 0 for all ω, i.e., f2is the 0signal, which contradicts
with the non-zero signal assumption. This proves the necessity.
q.e.d.
In general, for pBT-limited signals fi∈ BLAi,Bi,ai,bi, with
−∞ < Ai< Bi<∞and −∞ < ai< bi<∞,i=
1,2, ..., p, their non-zero linear combination f=Pp
i=1 αifi
for non-zero complex coefficients αiis not BT-limited, unless
Ai=Bifor all i= 1,2, ..., p. It is easy to see that if Ai=Bi
for all i= 1,2, ..., p, then the above linear combination f
is indeed BT-limited and f∈ BLA,B,a,b, where A=Ai,
B=Bi,a= min{a1, a2, ..., ap}and b= max{b1, b2, ..., bp}.
Although the above linear combination fof pBT-limited
signals is generally not BT-limited, from (14), we have
f(t) =
p
X
i=1
αiej(Ai+Ωi)tZTi
−Ti
sin Ωi(t+ai+Ti−s)
π(t+ai+Ti−s)qi(s)ds
(15)
for any t∈(−∞,∞), where Ωi= (Bi−Ai)/2,Ti= (bi−
ai)/2, and qi∈L2[−Ti, Ti]for i= 1,2, ..., p.
In this paper, BT-limited signal space was introduced and
characterized. It is a subspace of bandlimited signals where the
non-zero parts of their Fourier transforms are also pieces of
bandlimited signals. Some new properties about and applying
BT-limited signals were also presented. For BT-limited signals,
an extrapolation from inaccurate data with an analytic error
estimate in the whole time domain exists. Some simulations
5. Simulations
6. More Properties of B7-limited Signals
7. Conclusion
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were presented to verify the theoretical extrapolation results
for BT-limited signals.
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[9] X.-G. Xia and M. Z. Nashed, “A method with error estimates for band-
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[10] R. P. Boas, Entire Functions, Academic, New York, 1954.
-1 -0.5 0 0.5 1
-0.1
-0.05
0
0.05
0.1
0.15
=0.0125
true signal
given noisy signal
(a)
-6 -4 -2 0 2 4 6
-0.1
-0.05
0
0.05
0.1
0.15
=0.0125
true signal
extrapolated signal
(b)
Fig. 1. BT-limited signal extrapolation from noisy data with the maximum
error magnitude ǫ= 0.0125: (a) given noisy data on [−1,1] and (b)
extrapolated signal using the MNS method.
References
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.2
Xiang-Gen Xia
E-ISSN: 2224-3488
17
Volume 19, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
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https://creativecommons.org/licenses/by/4.0/deed.en
_US
-1 -0.5 0 0.5 1
-0.1
-0.05
0
0.05
0.1
0.15
=0.0031
true signal
given noisy signal
(a)
-6 -4 -2 0 2 4 6
-0.1
-0.05
0
0.05
0.1
0.15
=0.0031
true signal
extrapolated signal
(b)
Fig. 2. BT-limited signal extrapolation from noisy data with the maximum
error magnitude ǫ= 0.0031: (a) given noisy data on [−1,1] and (b)
extrapolated signal using the MNS method.
0 20 40 60 80 100
10-6
10-5
10-4
10-3
10-2
10-1
100
max extrapolation error
ratio of max extrapolation error over 1/3
Fig. 3. The maximum error between extrapolated and true signals vs. the
maximum magnitude ǫof the errors in the given data over the time interval
[−1,1], and its ratio, R(ǫ), over ǫ1/3.
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.2
Xiang-Gen Xia
E-ISSN: 2224-3488
18
Volume 19, 2023
