Compressive Sampling of Stochastic Multiband
Signal Using Time Encoding Machine
Dominik Rzepka, Dariusz Kościelnik, Marek Miśkowicz
Department of Electronics
AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
Email: drzepka, koscieln, miskow @agh.edu.pl
Abstract—In this paper we present an improvement of architecture of the Asynchronous Sigma-Delta time encoding machine,
which makes it suitable for sampling the wideband sparse
analog signals. Since the values of the samples are encoded into
sampling instants, this is a signal-dependent sampling scheme.
By superseding the commonly used multitone model of the input
signal with multiband stochastic model, the randomness of the
sampling instants is obtained. This allows for the reconstruction
of signal using compressive sampling methods without additional
randomization hardware. The simulation results validate the
presented approach.
I. I NTRODUCTION
The introduction of Shannon-Nyquist theorem in the middle
of 20 century paved a way for the digital processing of analog signals and inspired a research on the effective hardware
architectures for analog-to-digital conversion. The beginning
of the 21 century came with the introduction of Compressive
Sampling (CS) paradigm, which opened possibility for the
reduction of sampling rate for the redundant signals, and posed
a challenge of the analog-to-information converter design.
The mandatory requirement for the compressive sampling
sampling scheme is known as a Restricted Isometry Principle (RIP) [1]. For the signals expressed using orthogonal
expansions, RIP is equivalent to maintaining the energy of
the samples taken at sub-Nyquist rate, close to the energy of
the sampled signal. This can be realized using the generalized
random measurements, as in the multiband wideband converter
and random demodulator, where the signal is convolved with
a pseudorandom sequence or sequences [2], [3]. Another
option is to sample the signal nonuniformly, according to the
deterministic periodic sequence (multicoset sampling [4]) or
with the random locations of samples [5], [6]. An interesting approach was presented in [7], where the time encoding paradigm was used, resulting with the signal-dependent
nonuniform sampling. In this setup, the input signal values are
encoded using Asynchronous Sigma Delta Modulator (ASDM)
time encoding machine into the train of pulses that changes
sign at the times
(quasi-digital signal). The advantage of
the ASDM is its simplicity and ultralow power consumption
[8]. The disadvantage, which is particularly visible when the
This work was supported from Statutory Activity & Dean’s Grant (AGH
University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunication). 978-1-4673-7353-1/15/$31.00 c 2015 IEEE
signal is sampled under the Nyquist rate, is that the dynamics
of the output pulse lengths depends not only on the input signal
dynamics, but also on its spectrum. This results with increased
quantization noise. The second disadvantageous effect was
observed in the ASDM architecture for the multitone input
signals. Since such signals are periodic, also the periodicity
(cycles) of the sampling pattern appeared, which had a negative influence on the fulfilling the RIP property. This problem
was solved in [7] by introducing the random disturbances of
the input signal, which reduced the effect of cycles.
In this paper the improved architecture of ASDM is presented to solve the problem of dependence between signal
spectrum and the dynamics of the output pulses. This was
achieved by adding the simple sample-and-hold circuit before
the input of the ASDM and triggering it at the edges of output
quasi digital signal. Furthermore, instead of the multitone
input, the multiband stochastic signal is used as the model of
input signal. This class of signals is well suited for the radio
signals, where the spectrum divided into channels is often
sparse, because only some part of them is used at once. The
randomness of the input signal results with the randomness of
the sampling instants. We show, that such sampling scheme
becomes equivalent to additive random sampling, which was
successfully used as the compressive sampling scheme [5], [6].
Therefore for this class of signals the additional randomization
hardware is not necessary. The satisfaction of RIP is provided
by theory of structured random matrices [1]. The simulation
results validate the presented approach.
II. C OMPRESSIVE SAMPLING
Let us consider the multiband signal
, whose energy is
present only in the
of the
of total subbands
,
each of width Hz. The maximum frequency component can
be at most
Hz. Do we need to sample the signal at the
rate
S/s to be able for its reconstruction? Even though
it is required by a Shannon-Nyquist theorem, such standard
approach does not take into account the sparsity of the signal
. Landau proved [9], that it is sufficient to sample such
signal at the
, that is twice the summary width of nonempty subbands. The reconstruction of signal is possible using
the variant of bandlimited interpolation [10] providing that
the location non-empty subbands is known. Unfortunately,
this knowledge is not always available. This is where the
compressed sensing comes with help.
In the original finite vector space formulation, compressed
sensing guarantees with high probability that -dimensional
vector
containing only
non-zero components can be
recovered from the vector composed of
measurements,
where
. Relation between signal and
measurements is given by
consumption and simple architecture [8]. The architecture of
the ASDM time encoding machine consists of an integrator
and the Schmitt trigger operating in a negative feedback loop
(Fig. 1). The integrator is fed by a difference of the input
and the output of the Schmitt trigger
, where
signal
. The ASDM operation is described by the
equation
(1)
(5)
is
matrix. Equation (1) constitutes underwhere
determined system of linear equations with infinite number of
solutions. However, if the matrix is designed properly, there
exists an unique, the sparsest solution (where the sparsity
denotes the number of non-zero components in ). This
solution can be obtained by solving the optimization problem
( -minimization)
subject to
where is an initial observation time. As long as the ASDM
output is in the high state (
), the voltage of the
integrator output
grows. When
reaches a lower
), the modulator
threshold of the Schmitt trigger (
output is switched to
, and
starts to increase.
The waveforms on the integrator and ASDM outputs,
and
, for a given signal
are presented in the Fig. 2 During
(2)
x(t)
Unfortunately (2) is computationally unfeasible, since direct
minimization of
is NP-hard problem. But the same solution (with high probability) can be obtained by minimization
of the -norm
, known as a Basis Pursuit
algorithm
subject to
y(t)
z(t)
Fig. 1. Architecture of the Asynchronous Sigma-Delta Modulator
(3)
which is soluble using linear programming methods in the
polynomial time. Another effective way to find minimum norm solution of (1) are greedy algorithms [1]. The stability of
the recovery depends on the structure of the matrix , which
should satisfy the Restricted Isometry Property (RIP)
(4)
for all -sparse vectors , with small value of
for reasonably large
[1]. In general, proving the RIP for
the deterministic construction of
matrix is a challenging
problem. The most common strategy to circumvent it is to
consider the ensemble of random matrices and prove that given
its probabilistic properties, the RIP is satisfied with the high
probability.
CS framework can be used also to recover perfectly the
continuous-time signals, if they can be expressed using finite
expansion (multitone signals, finite rate of innovation signals
[3]). In the case of the signals represented by an infinite
expansion, such as multiband signals, the truncated models
are used in practice [4]. Despite of such approximation it
is still possible to obtain the high quality reconstruction of
continuous-time signal with acceptable computational requirements [3], [5], [6], [11].
III. T IME E NCODING M ACHINE
A. Asynchronous Sigma-Delta Modulator
Conception of time encoding is an interesting alternative for
classical analog to digital converters, providing the low power
Fig. 2. Example of the signals in the ASDM: input
and the Schmitt trigger output
, integrator output
the time interval
,
is changed by
, so according to (5)
while
(6)
Finally, the width of -th pulse
of
is
(7)
where
is the mean value of
in the interval
.
The pulse widths
on the ASDM output can be considered
as samples of the mean value
of the input signal
,
as shown explicitly by (7). For zero input signal, the ASDM
produces the waveform of
duty cycle with pulse width
.
is defined as the self-oscillation period of the
ASDM. To assure the correct operation of the ASDM, the
input signal must be bounded by
, which limits
the pulse widths to
. In order to guarantee
finite pulse widths, the input signal must obey even stronger
limit
pulse widths
, where
. This bounds the
by
(8)
where
is the modulation depth.
B. Digitalization of the pulse length
The information about the source signal
is encoded into
lengths
of the pulses in quasi-digital signal
. To process
this data digitally, the values
must be quantized. The
dynamic range of of
should correspond to the range of
used pulse lengths, and signal
reaching the values
should result with the values
reaching the
and
.
However in the ASDM, according to the (7) the pulse length
depends not on the maximal or minimal
value in
, but rather on the average of
in this
interval
interval. Since
, then even
full scale signal will be encoded into the narrowed interval
where
.
Averaging
in the interval
can be represented in
a frequency domain as a filtering with transfer function
. Then for the input
the signal average is bounded by
(9)
C. Sample-and-Hold Asynchronous Sigma-Delta Modulator
The problem introduced in the previous subsection can be
solved by adding a sample and hold circuit coupled with
the ASDM as shown in Fig. 4. This relevant modulation
and corresponding architecture will be now called SH-ASDM
(Sample-and-Hold ASDM). At the time instant , the sample
of the input signal is latched in the sample-and-hold
circuit. The value
provided to the integrator input causes
the generation the pulse of length , which is a function of the
value of
instead of average , like it was in the case of
ASDM. After the time , the next sample
is taken,
and the procedure of time-encoding of an instantaneous signal
value repeats. The relation between the sample value and the
pulse length is determined by equation
(10)
which gives
(11)
The relationship (11) clearly indicates, that length of intervals
does not depend on the signal average, but on the signal
value. The output
is also a square wave, like in classic
ASDM, which provides a backward-compatibility.
x(t)
xq(t)
y(t)
z(t)
yielding
Fig. 4. ASDM extended with sample-and-hold circuit
The result of this effect is shown in the Fig. 3 For the increasing frequency of signal the width of the interval
decreases. As the digitization of the signal requires quantization with the fixed dynamics and quantization step, the higher
frequencies are encoded into the significantly smaller number
of bits than the full-scale. Furthermore, for some frequencies
we have
, which makes such timeencoding ambiguous. As a result, time encoding with ASDM
is inapplicable to the sampling the signal with frequency
considerably larger than
, which is a common situation
in the compressed sensing. The modification of the ASDM
structure is therefore indispensable.
IV. C OMPRESSIVE SAMPLING WITH SH-ASDM
A. Model of the input signal
Let us define the wide-sense stationary stochastic Gaussian
process
with the autocorrelation
and
corresponding power spectral density
with frequencies
limited to the -band
of the width
Hz. Such process can be represented using series
(12)
where function
(13)
1.6
1.4
1.2
1
0.8
0
5
10
15
20
25
30
Fig. 3. Shrinking of the output pulse range
35
40
forms the orthogonal basis
for the space
of bandpass signals contained in -band. Discrete stochastic
process
comprises of the normally
distributed random variables with autocorrelation function
. The desired multiband stochastic process
is
obtained by summing the processes
according to
(14)
where the set
with cardinality
determines,
which subbands are occupied by the signal. The autocorrelation function of
is
. In the
compressed sensing scenario, the
and
are given, but
the is unknown and has to be determined. Such model is a
stochastic counterpart of the union of subspaces [3].
The signal
is to be sampled by SH-ASDM. However,
due to the conditions described in Section III, the signal should
be bounded by
. This is not fulfilled by the
marginal distribution of
, which is Gaussian and have
infinite tails. Note that even if the distribution of
was compactly supported, the (12) does not have to be
bounded, because of the infinite sum (15). The boundedness of
the stochastic process can be assured by transforming Gaussian
process using nonlinear function
(15)
By setting
we get the
stationary stochastic process with marginal distribution
[13]. Unfortunately, the nonlinear transformation does not preserve autocorrelation
of the process
. The resulting autocorrelation
of
does not
have exactly multiband power spectral density. Despite of it,
the multiband model can be still used for the reconstruction,
as the samples of
acquired using SH-ASDM can be
transformed back using the function
, and
processed as the samples of the original multiband signal
.
B. Structured random sensing matrix
As mentioned in Section II, the solution of the underdetermined linear system (1) is possible for the matrix fulfilling
the RIP condition (4), at least with the high probability.
This requires the proper choice of basis for reconstruction
and the sampling scheme (usually randomized) allowing for
the operation with finite number of samples. One of the
random matrices class satisfying RIP are the structured random
matrices [1]. Assume, that
is a set of
functions orthonormal in the interval
, that is for every
we have
(16)
and there exists a constant , such that every function
follows inequality
. This conditions
define the bounded orthonormal system. Now suppose, that
random sampling points
are chosen independently
and distributed uniformly in the interval
. Then
(17)
has the unique solution minimizing
with probability
at least
[1] if the number of measurements follows
, for some universal constant
. In the case of the reconstruction of multiband signal, the
set of functions
defined as (13) is
in the interval
also bounded orthonormal system for
, with
.
C. Distribution of sampling points
The distance
between consecutive sampling instants is
given by (11). Let be the random variable, describing this
distance for the full scale input stochastic signal with uniform
distribution
. Then the probability density
function
can be calculated as a function of (11) as
(18)
with expected value
. Location of -th
sampling instant is a sum of
random variables ,
, falling into the class of additive random sampling
[12]. Sampling point density follows
(19)
In the limit we have
which is also
mean sampling rate [12]. Fig. 5 shows the convergence of
to the limit for the different values of . Parameters
and were set to maintain
.
3
3
2
2
1
1
0
0
0
5
10
15
20
3
3
2
2
1
1
0
0
5
10
15
20
0
5
10
15
20
0
0
5
10
15
20
Fig. 5. Sampling point density function for different modulation depths .
Mean sampling interval is equal to 1.
The rate of convergence is affected by the
and cor. For the normally
relation between consecutive intervals
distributed intervals it can be shown that positive correlation
increases the rate of convergence, while negative correlation
slows it down [12]. In the case of sampling with SH-ASDM,
the consecutive sampling intervals
depend on the
which are correlated
consecutive samples
according to autocorrelation function
of process
.
For coarse analysis it can be assumed, that
,
which is justified by the small influence of transformation
on the autocorrelation function [13]. Therefore for -th sample
we have autocorrelation
(20)
The minus in (20) is a result of sign inversion in every second
sample, stemming from the principle of ASDM operation.
Since the difference
in (20) is itself a random
variable, then correlation
is also random. The probability
density function of correlation
the can be derived for
the given
and
. To demonstrate the effect of the
correlation, the example of the sampling points distribution for
the multiband signal is given in the Fig. 6. Despite of the fact
K
3
2
K
1
0
0
5
10
15
20
25
30
Fig. 6. Sampling point density function for the
, and normalized
histogram for SH-ASDM samples location, for multiband input signal with
two active subbands,
Hz.
V. N UMERICAL EXPERIMENTS
The theoretical framework from Section IV.B provides the
method for recovery of sparse signal for the dimensionality of
sample vector approaching infinity and uniformly distributed,
independent sampling instants. When SH-ASDM is used, the
sampling instants are random but not independent and not
exactly uniformly distributed. Furthermore, in practice the
number of samples is finite, which implies also finite so that
orthogonality condition (16) is fulfilled only approximately. To
validate the compressed sampling algorithm in realistic setup,
a few experiments were conducted.
The input signal was constructed using to the truncated
equation (15) with
,
using random,
uncorrelated Gaussian vectors
. The
support
was chosen at random, with the sparsity in the
range
and fixed number of bands
. This
resulted with the signal energy concentrated in the interval
. Samples were acquired according to the (11)
using SH-ASDM sampling scheme. To reduce the effects of
the truncation of infinite series (15), only the middle part of
signal
was used for the recovery, and only the
samples from this interval were processed. Consequently, the
subject of recovery were the vectors
. The
number of measurements
was chosen in the range from
to
by adjusting the parameters of the SH-ASDM, so that
samples would fall in the desired interval
. For
the recovery the Block Orthogonal Matching Pursuit was used
[11]. The complete signal processing chain is shown in the Fig.
7.
Fig. 7. Signal processing chain for the compressive sampling using SHASDM
For the each settings of
and , the 10 random signals
were generated and the recovery of the frequency support was
attempted. The statistics of the valid recoveries is shown in
the Fig. 8 . It is visible, that success of the recovery depends
on the modulation depth . This remains in agreement with
9
8
7
6
5
4
3
2
1
40 80 120 160 200 240 280 320 360 400
M
9
8
7
6
5
4
3
2
1
40 80 120 160 200 240 280 320 360 400
M
K
the conclusions of the Section IV.C - the deeper modulation,
the faster the convergence to the uniform distribution of the
sampling instants.
K
of the slower convergence than in the uncorrelated case, the
distribution of sampling points approaches uniform for
,
which is desired to fulfill the RIP for the structured random
sensing matrix.
9
8
7
6
5
4
3
2
1
40 80 120 160 200 240 280 320 360 400
M
9
8
7
6
5
4
3
2
1
40 80 120 160 200 240 280 320 360 400
M
Fig. 8. Empirical success of the support recovery. Black corresponds to 100%
empirical success probability, white - 0% empirical success probability
VI. C ONCLUSION
Compressive sampling using the Asynchronous SigmaDelta Modulator architecture is possible at the small cost
of additional sample-and-hold device. Furthermore, for the
input signal modeled as the stochastic process, the SH-ASDM
device alone is sufficient to provide compressed samples of
multiband signal, offering a practical, simple and low power
architecture of the analog-to-information converter.
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