2864 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 10, OCTOBER 2000
Filterbank Reconstruction of Bandlimited Signals
from Nonuniform and Generalized Samples
Yonina C. Eldar, Student Member, IEEE, and Alan V. Oppenheim, Fellow, IEEE
Abstract—This paper introduces a filterbank interpretation of
various sampling strategies, which leads to efficient interpolation
and reconstruction methods. An identity, which is referred to as
the Interpolation Identity, is developed and is used to obtain par-
ticularly efficient discrete-time systems for interpolation of gener-
alized samples as well as a class of nonuniform samples, to uni-
form Nyquist samples, either for further processing in that form
or for conversion to continuous time. The Interpolation Identity
also leads to new sampling strategies including an extension of Pa-
poulis’ generalized sampling expansion.
Index Terms—Filterbanks, generalized sampling, interpolation,
nonuniform sampling, sampling.
I. INTRODUCTION
D
ISCRETE-TIME signal processing (DSP) inherently re-
lies on sampling a continuous time signal to obtain a dis-
crete-time representation of the signal. The most common form
of sampling used in the context of DSP is uniform (periodic)
sampling. However, there are a variety of applications in which
data is sampled in other ways, such as nonuniformly in time
or through multichannel data acquisition. Examples in which
nonuniform sampling may arise include data loss due to channel
erasures and additive noise. Multichannel data can arise in dig-
ital flight control, where the velocity as well as the position are
recorded. There are also applications where we can benefit from
deliberately introducing more elaborate sampling schemes. Po-
tential applications include data compression, efficient quanti-
zation methods [10], and flexible A/D converters.
Several extensions of the uniform sampling theorem are well
known [5]. Specifically, it is well established that a bandlim-
ited signal is uniquely determined from its nonuniform samples,
provided that the averagesampling rate exceeds the Nyquist rate
[1]. However, in contrast to uniform sampling, reconstruction of
a continuous-time signal from its nonuniform samples using the
direct interpolation procedure is computationally difficult. Sev-
eral alternative reconstruction methods from nonuniform sam-
ples have been previously suggested. These methods involve it-
erative algorithms (e.g., [2], [6], [14]), which are computation-
ally demanding and have potential issues of convergence.
Manuscript received July 12, 1999; revised June 1, 2000. This work was sup-
ported in part through collaborative participation in the Advanced Sensors Con-
sortium sponsored by the U.S. Army Research Laboratory under Cooperative
Agreement DAAL01-96-2-0001 and supported in part by the Texas Instruments
Leadership University Program. Y. Eldar is supported by an IBM Research
Fellowship. The associate editor coordinating the review of this paper and ap-
proving it for publication was Dr. Xiang-Gen Xia.
The authors are with the Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, MA 02139 USA (e-mail: yonina@mit.edu;
avo@mit.edu).
Publisher Item Identifier S 1053-587X(00)07683-2.
Another well-known sampling theorem by Papoulis [8],
which generalizes uniform sampling of a signal, states that
a bandlimited signal can be reconstructed from uniformly
spaced samples of the outputs of
linear time-invariant (LTI)
systems with the signal as their input sampled at one-
th
of the Nyquist rate. However, the reconstruction from these
generalized samples is again computationally complex. In order
to exploit alternative sampling methods in various applications,
practical, efficient reconstruction algorithms are required.
Recently, there has been some work on sampling theorems for
nonbandlimited signals [3], [13] and on nonuniform and gen-
eralized sampling theorems for discrete-time signals [12, Sec.
10.2].
Many of the algorithms for processing and analyzing a
discrete-time signal assume that the signal corresponds to uni-
formly spaced samples of a continuous-time signal. When other
sampling procedures are employed, a common approach is to
interpolate to uniform Nyquist samples of the continuous-time
signal prior to processing. Existing interpolation methods
include approximate polynomial interpolation and iterative
procedures [9]. Here again, practical, efficient interpolation
algorithms are desirable.
Inthis paper, wederiveanidentity thatleads toefficientrecon-
struction methods from generalized samples, as well as efficient
interpolation to uniformly spaced samples. We then develop a
new noniterative approach to reconstruction from recurrent and
th-order nonuniform samples. The resulting procedure con-
sists of processing the samples with a bank of LTI filters, either
toreconstruct the original bandlimitedcontinuous-time signal or
to interpolate the nonuniform samples to uniformly spaced sam-
ples. In addition to offering efficient implementations, the filter-
bankframeworkleadstoanewclassofsamplingstrategies.Asan
example,weshowthat applyingthe identity derivedinthis paper
to perfectreconstruction filterbanks resultsin a generalization of
Papoulis’ sampling theorem [8].
The organization of this paper is as follows. In Section II, we
formulate the Interpolation Identity. Section III illustrates an ap-
plication of the identity to samplingof asignal andits derivative.
In Section IV, we describe recurrent nonuniform sampling and
arrive at a continuous-time filterbank implementation of the re-
construction. We then apply the Interpolation Identity to convert
the continuous-time filterbank to an equivalent discrete-time
filterbank. The resulting discrete-time filterbank inherently in-
terpolates the uniform samples of the continuous-time signal. In
SectionV,weintroduceatypeofnonuniformsamplingreferredto
as
th-ordernonuniformsampling.Wethendevelopanefficient
interpolation and reconstruction method from these samples
usingarationaldiscrete-timefilterbank.SectionVIdemonstrates
1053–587X/00$10.00 © 2000 IEEE
ELDAR AND OPPENHEIM: FILTERBANK RECONSTRUCTION OF BANDLIMITED SIGNALS 2865
Fig. 1. Converting the sequence of samples
x
(
nT
)
to a continuous-time
impulse train
y
(
t
)
.
how the filterbank framework leads to new sampling strategies.
In particular, we present a generalization to Papoulis’ sampling
theorem. In the various sections, key results are stated and their
detailedderivationisincludedintheappropriateappendix.
II. I
NTERPOLATION IDENTITY
Throughout this paper, we use the variables and to de-
note frequency variables for continuous-time and discrete-time,
respectively. Capital letters are used to denote the Fourier trans-
form, e.g.,
and denote the continuous-time and
discrete-time Fourier transforms of
and , respectively.
Parentheses are used for continuous-time signals and brackets
for discrete-time signals. To further distinguish between contin-
uous-time and discrete-time signals, we will usually denote the
former with a subscript, e.g.,
. We assume that all signals
have finite energy and are bandlimited to
, i.e., their Fourier
transform is zero for
. denotes the Nyquist period
given by
. We use the notation in the block dia-
gram of Fig. 1 to denote conversion of the sequence of sam-
ples
to a continuous-time impulse train , where
. We refer to this operation
as impulse modulation.
The following equivalence, which we refer to as the Interpo-
lation Identity, will be used in subsequent sections to arriveat ef-
ficient implementations of the reconstruction from generalized
and nonuniformly spaced samples. The proof of this identity is
given in Appendix A.
Interpolation Identity: Let
be a finite energy con-
tinuous-time signal bandlimited to
, and let
denotethe impulseresponses ofthe
continuous-time filters with corresponding frequency responses
bandlimitedto .Forany and
such that for some integer , the block
diagrams depicted in Fig. 2(a)and (b) areequivalent for
(1)
The block diagram of Fig. 2(b) consists of expanding a se-
quence of samples by a factor of
and then filtering by
a discrete-time filter with frequency response given by (1). The
filtered output is then decimated by a factor of
followed by
impulse modulation and lowpass filtering. The input–output re-
lation for the expander is given by
otherwise.
(2)
The input–output relation for the decimator is given by
(3)
For the case
and , the Interpolation Identity
reduces to the equivalence depicted in Fig. 3(a) and (b), where
(4)
Note that (4) implies that
, where is
the discrete-time impulse response with frequency response
, and is the continuous-time impulse response with
frequency response
. Since , the sequence
is, in general, an undersampled representation of ,
and consequently,
is, in general, a filtered and aliased
version of
.
As an illustration of the use of the identity, we apply it in
the next section to a well-known sampling theorem associated
with the reconstruction from uniform samples of a signal and its
derivative. Through the use of the identity, we obtain a particu-
larly efficient system for interpolation of the samples to uniform
Nyquist samples, either for further processing in that form or for
conversion to continuous time. In Sections IV and V, we apply
the identity to two classes of nonuniform sampling strategies
referred to as recurrent and
th-order nonuniform sampling. In
Section VI, the Interpolation Identity is used to generate new
classes of sampling theorems.
III. I
NTERPOLATION AND RECONSTRUCTION FROM SAMPLES
OF A
SIGNAL AND ITS DERIVATIVE
As an example of the application of the Interpolation Identity,
consider sampling a signal and its derivative. It is well known
that a bandlimited signal can be reconstructed from uniform
samples of the signal and its derivative at half the Nyquist rate
[4] using the reconstruction formula
sinc
(5)
where
, and . Note
that the sequences
and are undersampled representa-
tions of
and , respectively.
Equation (5) can be implemented using the continuous-time
filterbank depicted in Fig. 4, with
sinc and
sinc . Note that both filters in Fig. 4 are
bandlimited to
. If, instead of reconstructing ,
we are interested in interpolating the uniform Nyquist samples
of
from and , the interpolation formula obtained
from substituting
in (5) is
sinc
(6)
2866 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 10, OCTOBER 2000
Fig. 2. Interpolation Identity.
Fig. 3. Interpolation Identity for the case
M
=
1
and
T
=
NT
.
Reconstruction of using (5) or interpolation using (6)
are both difficult to implement directly. However, both inter-
polation and reconstruction can be implemented in a simpler
form by applying the Interpolation Identity to the system in
Fig. 4. Specifically, the continuous-time filterbank of Fig. 4
can be converted to a discrete-time filterbank followed by a
continuous-time lowpass filter (LPF). Applying the equivalence
of Fig. 3 to each branch in Fig. 4 and moving the identical
impulse train modulation and LPF in each branch outside the
summer, we obtain the equivalent implementation in Fig. 5,
where
for . As with the contin-
uous-time filterbank, the overall output of Fig. 5 is the orig-
inal continuous-time signal
. Since is reconstructed
through lowpass filtering of a uniformly spaced impulse train
with period
, the impulse train values must correspond
to uniformly spaced samples of
at the Nyquist rate. Thus,
we conclude that the discrete-time filterbank provides a dis-
Fig. 4. Reconstruction from samples of a signal and its derivative at half the
Nyquist rate.
crete-time mechanism for converting the uniform generalized
samples of the signal and its derivative to uniform Nyquist sam-
ples. The filterbank can be implemented very efficiently, ex-
ploiting the many known results regarding efficient implemen-
tation of the filters comprising a discrete-time filterbank (see,
e.g., [12]).
By following an analogous procedure, we can arrive at effi-
cient interpolation and reconstruction methods for other forms
of generalized samples. In the next section, we focus on efficient
implementation of the reconstruction from recurrent nonuni-
form samples using a bank of continuous-time and discrete-time
filters.
IV. R
ECURRENT NONUNIFORM SAMPLING
It is well established that a continuous-time signal
can be reconstructed from its samples at a set of sampling
times
if the average sampling period is smaller than the
ELDAR AND OPPENHEIM: FILTERBANK RECONSTRUCTION OF BANDLIMITED SIGNALS 2867
Fig. 5. Interpolation and reconstruction using a discrete-time filterbank.
Nyquist period, where the average sampling period is defined
as
. The essential result is incorporated in the
following theorem by Yao and Thomas [15].
Theorem 1: Let
be a finite energy bandlimited signal
such that
for for some .
is uniquely determined by its samples if
(7)
The reconstruction is given by
(8)
where
(9)
is the derivative of evaluated at , and if
for some , then .
Reconstruction from nonuniform samples using (8) directly
is considerably more complex than reconstruction from uniform
samples. In this section, we focus on an efficient implementation
of (8) for the case of recurrent nonuniform sampling. In this
form of sampling, the sampling points are divided into groups
of
points each. The groups have a recurrent period, which
is denoted by
, that is equal to times the Nyquist period
. Each period consists of nonuniform sampling points.
Denoting the points in one period by
,
the complete set of sampling points are
(10)
where
. Without loss of generality, we will assume
throughout that
.
Recurrent nonuniform samples can be regarded as a combi-
nation of
sequences of uniform samples taken at one th of
the Nyquist rate. An example of a sampling distribution for the
case
is depicted in Fig. 6.
Recurrent nonuniform sampling arises in a broad range of ap-
plications. For example, we might consider converting a con-
tinuous-time signal to a discrete-time signal using a series of
A/D converters, each operating at a rate lower than the Nyquist
rate, such that the average sampling rate is equal to the Nyquist
Fig. 6. Sampling distribution for
N
=
3
.
rate. This may be beneficial in applications where high-rate A/D
converters are required. Typically, the cost and complexity of a
converter will increase (more than linearly) with therate. In such
cases, we can benefit from converting a continuous-time signal
to a discrete-time signal using
A/D converters, each operating
at one
th of the Nyquist rate. Since the converters are typically
not synchronized, the resulting discrete-time signal is a com-
bination of
sequences of uniform samples, where each se-
quence corresponds to samples at one
th of the Nyquist rate of
a time delayed version of the continuous-time signal. Thus, the
resulting discrete-time signal corresponds to recurrent nonuni-
form samples of the continuous-time signal.
Dividingthe time axis into nonoverlapping intervalsof length
, every interval contains sampling points, which implies
that the average sampling rate is the Nyquist rate. Based on The-
orem 1, we can therefore reconstruct a continuous-time signal
from its recurrent nonuniform samples , where the
sampling times
are givenby (10). In particular, substituting
(10) in (8) and (9), we obtain the following reconstruction for-
mula (see Appendix B):
(11)
where
(12)
As with (8), direct implementation of (11) is computationally
difficult. We will nowdevelopnew, efficient,noniterativeimple-
mentations of (11). In Section IV-A,we develop an implementa-
tionthatconsistsofprocessingthesampleswithabankofcontin-
uous-time LTI filters. In Section IV-B, we develop an alternative
implementationusing a bank of discrete-time LTI filters.
2868 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 10, OCTOBER 2000
Fig. 7. Reconstruction from recurrent nonuniform samples using a con-
tinuous-time filterbank.
A. Reconstruction from Recurrent Nonuniform Samples Using
a Continuous-Time Filterbank
In this subsection, we develop a continuous-time filterbank
representation of (11). To this end, we interchange the order of
summations in (11) and denote the inner sum by
, i.e.,
(13)
Using the relation
, we can express
as a convolution. Specifically
(14)
where
(15)
and
is an impulse train of samples, i.e.,
(16)
given by (11) can now be expressed as a sum of con-
volutions:
(17)
Equation (17) can be interpreted as a continuous-time fil-
terbank as depicted in Fig. 7. The signals
are formed
according to (16), i.e., the samples are divided into
subse-
quences, where each subsequence corresponds to samples at
one-
th of the Nyquist rate of a time-shifted version of .
Each subsequence is converted toa continuous-time signal
using a shifted impulse train. The signal is then filtered
Fig. 8. Alternative form of Fig. 7.
by a continuous-time filter with impulse response given
by (15). Summing the outputs of the
branches results in the
reconstructed signal
.
Note that each one of the subsequences corresponds to uni-
form samples at one-
th of the Nyquist rate. Therefore, the
output of each branch of the filter bank is an aliased and fil-
tered version of
. The filters, as specified by (15), have the
inherent property that the aliasing components of the filter out-
puts cancel in forming the summed output
.
An alternative form of Fig. 7, which we will find useful in
Section IV-B, is shown in Fig. 8. This form follows in a straight-
forward way by simply noting that the delay of
in the impulse
train of the
th branch can be incorporated into the filter .
To determine the frequency responses of the filters in Figs. 7
and 8, we note that the impulse response given by (15) can be
expressed as
(18)
(19)
where the complex coefficients
are the result of expanding
the product of sines in (18) into complex exponentials.
The first term in (19)
corresponds to an
ideal LPF with cut-off frequency
, which we denote as
. The effect of the summation is to create
shifted and scaled versions of the LPF, i.e.,
(20)
Hence, we conclude that the filters
in Figs. 7 and 8 have
the properties that
for , i.e., the filters
are bandlimited to the same bandwidth as the continuous-time
signal, and each filter
is piecewise constant over fre-
quency intervals of length
.
In the next subsection, we will derive a discrete-time filter
bank implementation of the reconstruction, which also provides
efficient interpolation to uniform samples.
