IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 2, APRIL 2010 375
From Theory to Practice: Sub-Nyquist Sampling of
Sparse Wideband Analog Signals
Moshe Mishali, Student Member, IEEE, and Yonina C. Eldar, Senior Member, IEEE
Abstract—Conventional sub-Nyquist sampling methods for
analog signals exploit prior information about the spectral sup-
port. In this paper, we consider the challenging problem of blind
sub-Nyquist sampling of multiband signals, whose unknown fre-
quency support occupies only a small portion of a wide spectrum.
Our primary design goals are efficient hardware implementation
and low computational load on the supporting digital processing.
We propose a system, named the modulated wideband converter,
which first multiplies the analog signal by a bank of periodic
waveforms. The product is then low-pass filtered and sampled
uniformly at a low rate, which is orders of magnitude smaller than
Nyquist. Perfect recovery from the proposed samples is achieved
under certain necessary and sufficient conditions. We also develop
a digital architecture, which allows either reconstruction of the
analog input, or processing of any band of interest at a low rate,
that is, without interpolating to the high Nyquist rate. Numerical
simulations demonstrate many engineering aspects: robustness
to noise and mismodeling, potential hardware simplifications,
real-time performance for signals with time-varying support and
stability to quantization effects. We compare our system with two
previous approaches: periodic nonuniform sampling, which is
bandwidth limited by existing hardware devices, and the random
demodulator, which is restricted to discrete multitone signals and
has a high computational load. In the broader context of Nyquist
sampling, our scheme has the potential to break through the band-
width barrier of state-of-the-art analog conversion technologies
such as interleaved converters.
Index Terms—Analog-to-digital conversion (ADC), compressive
sampling (CS), infinite measurement vectors (IMV), multiband
sampling, spectrum-blind reconstruction, sub-Nyquist sampling.
I. INTRODUCTION
R
ADIO frequency (RF) technology enables the modulation
of narrowband signals by high carrier frequencies. Con-
sequently, man-made radio signals are often sparse. That is, they
consist of a relatively small number of narrowband transmis-
sions spread across a wide spectrum range. A convenient way to
describe this class of signals is through a multiband model. The
frequency support of a multiband signal resides within several
continuous intervals spread over a wide spectrum. Fig. 1 depicts
a typical communication application, the wideband receiver, in
which the received signal follows the multiband model. The
Manuscript received February 22, 2009; revised October 28, 2009. Current
version published March 17, 2010. Part of this work was presented at the IEEEI,
25th convention of the IEEE, Israel, December 2008. The associate editor co-
ordinating the review of this manuscript and approving it for publication was
Prof. Richard G. Baraniuk.
The authors are with the Technion—Israel Institute of Technology, Haifa
32000, Israel (e-mail: moshiko@tx.technion.ac.il; yonina@ee.technion.ac.il).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JSTSP.2010.2042414
Fig. 1. Three RF transmissions with different carriers
f
. The receiver sees a
multiband signal (bottom drawing).
basic operations in such an application are conversion of the in-
coming signal to digital, and low-rate processing of some or all
of the individual transmissions. Ultimately, the digital product
is transformed back to the analog domain for further transmis-
sion.
Due to the wide spectral range of multiband signals, their
Nyquist rates may exceed the specifications of the best
analog-to-digital converters (ADCs) by orders of magnitude.
Any attempt to acquire a multiband signal must therefore
exploit its structure in an intelligent way. When the carrier fre-
quencies are known, a common practical engineering approach
is to demodulate the signal by its carrier frequency such that the
spectral contents of a band of interest are centered around the
origin. A low-pass filter follows in order to reject frequencies
due to the other bands. Conversion to digital is then performed
at a rate matching the actual information width of the band of
interest. Repeating the process for each band separately results
in a sampling rate which is the sum of the bandwidths. This
method achieves the minimal sampling rate, as derived by
Landau [1], which is equal to the actual frequency occupancy.
An alternative sampling approach that does not require analog
preprocessing was proposed in [2]. In this strategy, periodic
nonuniform sampling is used to directly sample a multiband
signal at an average rate approaching that derived by Landau.
Both conventional demodulation and the method of [2] rely on
knowledge of the carrier frequencies.
In scenarios in which the carrier frequencies are unknown
to the receiver, or vary in time, a challenging task is to de-
sign a
spectrum-blind receiver at a sub-Nyquist rate. In [3] and
[4], a multicoset sampling strategy was developed, indepen-
dent of the signal support, to acquire multiband signals at low
rates. Although the sampling method is blind, in order to re-
cover the original signal from the samples, knowledge of the
frequency support is needed. Recently in [5], we proposed a
fully spectrum-blind system based on multicoset sampling. Our
system does not require knowledge of the frequency support in
either the sampling or the recovery stages. To reconstruct the
signal blindly, we developed digital algorithms that process the
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376 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 2, APRIL 2010
samples and identify the unknown spectral support. Once the
support is found, the continuous signal is reconstructed using
closed-form expressions.
Periodic nonuniform sampling is a popular approach in the
broader context of analog conversion when the spectrum is fully
occupied. Instead of implementing a single ADC at a high-rate
, interleaved ADCs use devices at rate with ap-
propriate time shifts [6]–[8]. However, time interleaving has
two fundamental limitations. First, the
low-rate samplers
have to share an analog front-end which must tolerate the input
bandwidth
. With today’s technology the possible front-ends
are still far below the wideband regime. Second, maintaining
accurate time shifts, on the order of
, is difficult to im-
plement. Multicoset sampling, is a special case of interleaved
ADCs, so that the same limitations apply. In Section II-B we
discuss in more detail the difficulty in implementing interleaved
ADCs and multicoset sampling. In practice, such systems are
limited to intermediate input frequencies and cannot deal with
wideband inputs.
Recently, a new architecture to acquire multitone signals,
called the random demodulator, was studied in the literature of
compressed sensing (CS) [9], [10]. In this approach, the signal
is modulated by a high-rate pseudorandom number generator,
integrated, and sampled at a low rate. This scheme applies
to signals with finite set of harmonics chosen from a fixed
uniform grid. Time-domain analysis shows that CS algorithms
can recover such a multitone signal from the proposed sam-
ples [10]. However, as discussed in Section VI, truly analog
signals require a prohibitively large number of harmonics to
approximate them well within the discrete model, which in
turn renders the reconstruction computationally infeasible and
very sensitive to the grid choice. Furthermore, the time-domain
approach precludes processing at a low rate, even for multitone
inputs, since interpolation to the Nyquist rate is an essential
ingredient in the reconstruction.
In this paper, we aim to combine the advantages of the pre-
vious approaches: The ability to treat analog multiband models,
a sampling stage with a practical implementation, and a spec-
trum-blind recovery stage which involves efficient digital pro-
cessing. In addition, we would like a method that allows low-
rate processing, namely the ability to process any one of the
transmitted bands without first requiring interpolation to the
high Nyquist rate.
Our main contribution is an analog system, referred to as the
modulated wideband converter (MWC), which is comprised of
a bank of modulators and low-pass filters. The signal is mul-
tiplied by a periodic waveform, whose period corresponds to
the multiband model parameters. A square-wave alternating at
the Nyquist rate is one choice; other periodic waveforms are
also possible. The goal of the modulator is to alias the spec-
trum into baseband. The modulated output is then low-pass fil-
tered, and sampled at a low rate. The rate can be as low as
the expected width of an individual transmission. Based on fre-
quency-domain arguments, we prove that an appropriate choice
of the parameters (waveform period, sampling rate) guarantees
that our system uniquely determines a multiband input signal. In
addition, we describe how to trade the number of channels by
a higher rate in each branch, at the expense of additional pro-
cessing. Theoretically, this method allows to collapse the entire
system to a single channel operating at a rate lower than Nyquist.
Our second contribution is a digital architecture which en-
ables processing of the samples for various purposes. Recon-
struction of the original analog input is one possible function.
Perhaps more useful is the capability of the proposed system to
generate low-rate sequences corresponding to each of the bands,
which, in principle, allow subsequent digital processing of each
band at a low rate. This architecture also has the ability to treat
inputs with time-varying support. At the heart of the digital pro-
cessing lies the continuous to finite (CTF) block from our pre-
vious works [5], [11]. The CTF separates the support recovery
from the rest of the operations in the digital domain. In our pre-
vious works, the CTF required costly digital processing at the
Nyquist rate, and therefore provided only analog reconstruction
at the price of high rate computations. In contrast, here, the CTF
computations are carried out directly on the low-rate samples.
The main theme of this paper is going from theory to prac-
tice, namely tying together a theoretical sampling approach with
practical engineering aspects. Besides the uniqueness theorems
and stability conditions, we make use of extensive numerical
simulations, in Section V, to study typical wideband scenarios.
The simulations demonstrate robustness to noise and signal mis-
modeling, potential hardware simplifications in order to reduce
the number of devices, fast adaption to time-varying spectral
support, and the performance with quantized samples. A cir-
cuit-level realization of the MWC is reported in [12].
This paper is organized as follows. Section II describes the
multiband model and points out limitations of multicoset sam-
pling in the wideband regime. In Section III, we describe the
MWC system and provide a frequency-domain analysis of the
resulting samples. This leads to a concrete parameter selection
which guarantees a unique signal matching the digital samples.
We conclude the section with a discussion on the tradeoff be-
tween the number of channels, rate, and complexity. The ar-
chitecture for low-rate processing and recovery, is presented in
Section IV. In Section V, we conduct a detailed numerical eval-
uation of the proposed system. A review of related work con-
cludes the paper in Section VI.
II. F
ORMULATION AND BACKGROUND
A. Problem Formulation
Let
be a real-valued continuous-time signal in .
Throughout the paper, continuous signals are assumed to be
bandlimited to
. Formally, the Fourier
transform of
, which is defined by
(1)
is zero for every
. We denote by the Nyquist
rate of
. For technical reasons, it is also assumed that
is piecewise continuous in . We treat signals from the multi-
band model
defined below.
Definition 1: The set
contains all signals , such that
the support of the Fourier transform
is contained within
a union of
disjoint intervals (bands) in , and each of the
bandwidths does not exceed
.
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MISHALI AND ELDAR: FROM THEORY TO PRACTICE: SUB-NYQUIST SAMPLING OF SPARSE WIDEBAND ANALOG SIGNALS 377
Signals in have an even number of bands due to the
conjugate symmetry of
. The band positions are arbitrary,
and in particular, unknown in advance. A typical spectral sup-
port of a signal from the multiband model
is illustrated in the
example of Fig. 1, in which
and are dictated
by the specifications of the possible transmitters.
We wish to design a sampling system for signals from the
model
that satisfies the following properties:
1) The sampling rate should be as low as possible;
2) the system has no prior knowledge of the band locations;
3) the system can be implemented with existing analog de-
vices and (preferably low-rate) ADCs.
Together with the sampling stage we need to design a recon-
struction scheme, which converts the discrete samples back to
the continuous-time domain. This stage may involve digital pro-
cessing prior to reconstruction. An implicit (but crucial) require-
ment is that recovery involves a reasonable amount of computa-
tions. Real-time applications may also necessitate short latency
from input to output and a constant throughput. Therefore, two
main factors dictate the input spectrum range that the overall
system can handle: analog hardware at the required rate that can
convert the signal to digital, and a digital stage that can accom-
modate the computational load.
In our previous work [5], we proved that the minimal sam-
pling rate for
to allow perfect blind reconstruction is ,
provided that
is lower than the Nyquist rate. The case
represents signals which occupy more than
half of the Nyquist range. No rate improvement is possible in
that case (for arbitrary signals), and thus we assume
in the sequel. Concrete algorithms for blind recovery,
achieving the minimal rate, were developed in [5] based on a
multicoset sampling strategy. The next section briefly describes
this method, which achieves the goals of minimal rate and blind-
ness. However, limitations of practical ADCs, which we detail
in the next section, render multicoset sampling impractical for
wideband signals. As described later in Section III-A, the sam-
pling scheme proposed in this paper circumvents these limita-
tions and has other advantages in terms of practical implemen-
tation.
B. Multicoset Using Practical ADCs
In multicoset sampling, samples of
are obtained on a pe-
riodic and nonuniform grid which is a subset of the Nyquist grid.
Formally, denote by
the sequence of samples taken at the
Nyquist rate. Let
be a positive integer, and be
a set of
distinct integers with . Multicoset
samples consist of
uniform sequences, called cosets, with the
th coset defined by
(2)
Only
cosets are used, so that the average sampling rate
is
, which is lower than the Nyquist rate .
A possible implementation of the sampling sequences (2) is
depicted in Fig. 2(a). The building blocks are
uniform sam-
plers at rate
, where the th sampler is shifted by from
Fig. 2. Schematic implementation of multicoset sampling (a) requires no fil-
tering between the time shifts and the actual sampling. However, the front-end
of a practical ADC has an inherent bandwidth limitation, which is modeled in
(b) as a low-pass filter preceding the uniform sampling.
the origin. Although this scheme seems intuitive and straight-
forward, practical ADCs introduce an inherent bandwidth lim-
itation, which distorts the samples. The distortion mechanism,
which is modeled as a preceding low-pass filter in Fig. 2(b),
becomes crucial for high rate inputs. To understand this phe-
nomenon, we focus on the model of a practical ADC, Fig. 2(b),
ignoring the time shifts for the moment. A uniform ADC at rate
samples/s attempts to output pointwise samples of the input.
The design process and manufacturing technology result in an
additional property, termed analog (full-power) bandwidth [13],
which determines the maximal frequency
that the device can
handle. Any spectral content beyond
Hz is attenuated and dis-
torted. The bandwidth limitation
is inherent and cannot be sep-
arated from the ADC. Therefore, manufacturers usually recom-
mend adding a preceding external anti-aliasing low-pass filter,
with cutoff
, since the internal one has a parasitic response. The
ratio
affects the complexity of the ADC circuit design, and
is typically in the range [14]
(3)
The practical ADC model raises two difficulties in imple-
menting multicoset sampling. First, RF technology allows
transmissions at rates which exceed the analog bandwidth
of
state-of-the-art devices, typically by orders of magnitude. For
example, ADC devices manufactured by Analog Devices Corp.
have front-end bandwidths which reach up to
MHz
[14]. Therefore, any attempt to acquire a wideband signal with
a practical ADC results in a loss of the spectral contents beyond
Hz. The sample sequences (2) are attenuated and distorted
and are no longer pointwise values of
. This limitation is
fundamental and holds in other architectures of multicoset (e.g.,
a single ADC triggered by a nonuniform clock). The second
issue is a waste of resources, which is less severe, but applies
also when the Nyquist rate
for some available
device. For a signal with a sparse spectrum, multicoset reduces
the average sampling rate by using only
out of possible
cosets, where
is commonly used. Each coset in Fig. 2
samples at rate
. Therefore, the ADC samples at rate
, which is far below the standard range (3). This im-
plies sampling at a rate which is much lower than the maximal
capability of the ADC.
As a consequence, implementing multicoset for wideband
signals requires the design of a specialized fine-tuned ADC cir-
cuit, in order to meet the wide analog bandwidth, and still ex-
ploit the nonstandard ratio
that is expected. Though this may
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378 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 2, APRIL 2010
Fig. 3. Modulated wideband converter—a practical sampling stage for multiband signals.
be an interesting task for experts, it contradicts the basic goal
of our design—that is, using standard and available devices. In
[15] a nonconventional ADC is designed by means of high-rate
optical devices. The hybrid optic-electronic system introduces a
front-end whose bandwidth reaches the wideband regime, at the
expense and size of an optical system. Unfortunately, at present,
this performance cannot be achieved with pure electronic tech-
nology.
Another practical issue of multicoset sampling, which also
exists in the optical implementation, arises from the time shift
elements. Maintaining accurate time delays between the ADCs
in the order of the Nyquist interval
is difficult. Any uncer-
tainty in these delays influences the recovery from the sampled
sequences [16]. A variety of different algorithms have been pro-
posed in the literature in order to compensate for timing mis-
matches. However, this adds substantial complexity to the re-
ceiver [17], [18].
III. S
AMPLING
We now present an alternative sampling scheme that uses
available devices, does not suffer from analog bandwidth issues
and does not require nonzero time synchronization. The system,
referred to as the modulated wideband converter (MWC), is
schematically drawn in Fig. 3 with its various parameters. In the
next subsections, the MWC is described and analyzed for arbi-
trary sets of parameters. In Section III-C, we specify a parameter
choice, independent of the band locations, that approaches the
minimal rate. The resulting system, which is comprised of the
MWC of Fig. 3 and the recovery architecture that is presented
in the next section, satisfies all the requirements of our problem
formulation.
A. System Description
Our system exploits spread-spectrum techniques from com-
munication theory [19], [20]. An analog mixing front-end
aliases the spectrum, such that a spectrum portion from each
band appears in baseband. The system consists of several
channels, implementing different mixtures, so that, in principle,
a sufficiently large number of mixtures allows to recover a
relatively sparse multiband signal.
More specifically, the signal
enters channels simul-
taneously. In the
th channel, is multiplied by a mixing
function
, which is -periodic. After mixing, the signal
spectrum is truncated by a low-pass filter with cutoff
and the filtered signal is sampled at rate . The sampling
rate of each channel is sufficiently low, so that existing com-
mercial ADCs can be used for that task. The design parameters
are therefore the number of channels
, the period , the sam-
pling rate
, and the mixing functions for .
For the sake of concreteness, in the sequel,
is chosen as
a piecewise constant function that alternates between the levels
for each of equal time intervals. Formally,
(4)
with
, and for every .
Other choices for
are possible, since in principle we only
require that
is periodic.
The system proposed in Fig. 3 has several advantages for
practical implementation.
A1) Analog mixers are a provable technology in the wide-
band regime [21], [22]. In fact, since transmitters use
mixers to modulate the information by a high-carrier
frequency, the mixer bandwidth defines the input band-
width.
A2) Sign alternating functions can be implemented by a stan-
dard (high rate) shift register. Today’s technology allows
to reach alternation rates of 23 GHz [23] and even 80
GHz [24].
A3) Analog filters are accurate and typically do not require
more than a few passive elements (e.g., capacitors and
coils) [25].
A4) The sampling rate
matches the cutoff of .
Therefore, an ADC with a conversion rate
, and
any bandwidth
can be used to implement this
block, where
serves as a preceding anti-aliasing
filter. In the sequel, we choose
on the order of ,
which is the width of a single band of
.In
practice, this sampling rate allows flexible choice of an
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MISHALI AND ELDAR: FROM THEORY TO PRACTICE: SUB-NYQUIST SAMPLING OF SPARSE WIDEBAND ANALOG SIGNALS 379
ADC from a variety of commercial devices in the low
rate regime.
A5) Sampling is synchronized in all channels, that is there
are no time shifts. This is beneficial since the trigger for
all ADCs can be generated accurately (e.g., with a zero-
delay synchronization device [26]). The same clock can
be used for a subsequent digital processor which re-
ceives the sample sets at rate
.
Note that the front-end preprocessing must be carried out by
analog means, since both the mixer and the analog filter operate
on wideband signals, at rates which are far beyond digital pro-
cessing capabilities. In fact, the mixer output
is not ban-
dlimited, and therefore there is no way to replace the analog
filter by a digital unit even if the converter is used for low-rate
signals. The purely analog front-end is the key to overcome the
bandwidth limitation of ADCs.
B. Frequency Domain Analysis
We now derive the relation between the sample sequences
and the unknown signal . This analysis is used for
several purposes in the following sections. First, for specifying
a choice of parameters ensuring a unique mapping between
and the sequences . Second, we use this analysis to explain
the reconstruction scheme. Finally, stability and implementation
issues will also be based on this development. To this end, we
introduce the definitions
(5a)
(5b)
Consider the
th channel. Since is -periodic, it has a
Fourier expansion
(6)
where
(7)
The Fourier transform of the analog multiplication
is evaluated as
(8)
Therefore, the input to
is a linear combination of
-shifted copies of . Since for , the
sum in (8) contains (at most)
nonzero terms
1
.
1
The ceiling operator
d
a
e
returns the greater (or equal) integer which is
closest to
a
.
The filter has a frequency response which is an ideal
rectangular function, as depicted in Fig. 3. Consequently, only
frequencies in the interval
are contained in the uniform se-
quence
. Thus, the discrete-time Fourier transform (DTFT)
of the
th sequence is expressed as
(9)
where
is defined in (5b), and is chosen as the smallest
integer such that the sum contains all nonzero contributions of
over . The exact value of is calculated by
(10)
Note that the mixer output
is not bandlimited, and, theoret-
ically, depending on the coefficients
, the Fourier transform
(8) may not be well defined. This technicality, however, is re-
solved in (9) since the filter output involves only a finite number
of aliases of
.
Relation (9) ties the known DTFTs of
to the unknown
. This equation is the key to recovery of . For our pur-
poses, it is convenient to write (9) in matrix form as
(11)
where
is a vector of length with th element
. The unknown vector
is of length
(12)
with
(13)
The
matrix contains the coefficients
(14)
where the reverse order is due to the enumeration of
in
(13). Fig. 4 depicts the vector
and the effect of aliasing
in -shifted copies for bands, aliasing rate
and two sampling rates, and . Each
entry of
represents a frequency slice of whose length
is
. Thus, in order to recover , it is sufficient to determine
in the interval .
The analysis so far holds for every choice of
-periodic func-
tions
. Before proceeding, we discuss the role of each pa-
rameter. The period
determines the aliasing of by set-
ting the shift intervals to
. Equivalently, the aliasing
rate
controls the way the bands are arranged in the spectrum
slices
, as depicted in Fig. 4. We choose so that
each band contributes only a single nonzero element to
(referring to a specific ), and consequently has at most
nonzeros. In practice is chosen slightly more than to
avoid edge effects. Thus, the parameter
is used to translate
the multiband prior
to a bound on the sparsity level
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