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ASYNCHRONOUS SAMPLING AND RECONSTRUCTION OF SPARSE SIGNALS
Azime Can, Ervin Sejdi´c and Luis Chaparro
Department of Electrical and Computer Engineering1140 Benedum
Hall, University of Pittsburgh,
Pittsburgh, PA, 15261, USA
ABSTRACT
Asynchronous signal processing is an appropriate
low-powerapproach for the processing of bursty signals typical
inbiomedical applications and sensing networks. Differentfrom the
synchronous processing, based on the Shannon-Nyquist sampling
theory, asynchronous processing is freeof aliasing constrains and
quantization error, while allowingcontinuous-time processing. In
this paper we connect level-crossing sampling with time-encoding
using asynchronoussigma delta modulators, to develop an
asynchronous decom-position procedure similar to the Haar transform
waveletdecomposition. Our procedure provides a way to
reconstructbounded signals, not necessarily band-limited, from
relatedzero-crossings, and it is especially applicable to
decomposesparse signals in time and to denoise them. Actual and
syn-thetic signals are used to illustrate the advantages of
thedecomposer.
Index Terms— Continuous-time digital signal process-ing,
time-encoding of signals, level-crossing sampling, asyn-chronous
sigma delta modulators, asynchronous signal pro-cessing.
1. INTRODUCTION
The recent interest in asynchronous processing of signalsis due
to applications where low power consumption andcontinuous-time
processing are essential. The range of ap-plications of
asynchronous processing goes from biomedicalimplants [1, 2, 3] to
sensor networks in health, military andhome [4], for which
processing and communication is limitedby power consumption.
Asynchronous processing outweighsthe traditional Shannon-Nyquist
synchronous processing notonly in power consumption but in the
possible continuous-time signal processing [5, 6, 7]. Given the
bursty nature ofmany biomedical signals, signal-dependent sampling
proce-dures are more appropriate than uniform-sampling.
Uniform sampling resulting from the Shannon-Nyquistsampling
theory cannot be implemented in many situations,for instance when
the nodes of a sensor network have limitedsensing and processing
power, or due to sensor problems. Inother situations uniform
sampling is not desirable due to the
required high sampling rates and complex processing. Sig-nals
collected from sensor nodes or health monitoring devicesexhibit
sparse nature in time in many applications. They arealmost zero
most of the time and changes occur on brief inter-vals, which
challenge the analog to digital conversion giventhe high sampling
rates required. The asynchronous samplingmethods are an efficient
alternative.
Uniform sampling approximates a signal by a Riemannsum, while
level-crossing (LC) — a non-uniform method thatreverses the roles
of amplitude and time in the sampling —does the approximation by a
Lebesgue sum. The significanceof LC is that it follows the signal
by sampling more oftenwhenever the signal varies rapidly and less
otherwise. The“opportunistic nature” of LC [8, 9, 13] is similar to
the waycompressive sensing deals with sparse signals [10]. Both
lookfor compression or sparseness in the representation. AlthoughLC
sampling requires a-priori a set of quantization levels,typically
uniform, and the samples need to be coupled withthe times at which
they occur, it provides a representation freeof aliasing and
quantization error.
A different approach, based on time-encoding, is providedby an
asynchronous sigma delta modulator (ASDM), a non-linear feedback
system, that represents the signal amplitudeby a binary signal with
zero-crossing times at different scaleparameters. When comparing
the LC and the ASDM sam-pling schemes, it can be shown that the
ASDM is a LC sam-pler with quantization levels given by local
estimates of thesignal average for a certain scale. The information
availablein the binary signal can only provide a multi-level
approxima-tion to the signal for any particular scale setting in
the ASDM.As we will show, using different scales it is possible to
getrepresentations that closely approximate the signal. Thus
theidea of a decomposition procedure using ASDMs — eachwith
different scales — is similar to the wavelet decompo-sition. The
multilevel signals at each scale are represented bysequences of
local averages and their location times provid-ing a compressed
representation of the signal. The proposeddecomposition can be
related to the Haar transform wavelet,which is very appropriate for
multi-level signals.
Advantages of the asynchronous decomposition are: ana-log in
nature, uses scale instead of frequency for the decom-position, and
it does not suffer from aliasing — it thus applies
20th European Signal Processing Conference (EUSIPCO 2012)
Bucharest, Romania, August 27 - 31, 2012
© EURASIP, 2012 - ISSN 2076-1465 854
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to non band-limited signals. Moreover, it provides a
recursivesignal reconstruction of the signal from zero-crossings
[11].In this paper we illustrate the sampling, and reconstruction
ofsparse continuous-time signals using the proposed procedure.
2. ASYNCHRONOUS SAMPLING ANDRECONSTRUCTION
The complexity of reconstructing a signal in an interval[ta, tb]
can be measured using the number of degrees offreedom in the
sampled signal xs(t) in the interval [12]. Innon-uniform sampling,
it is not only necessary to have theamplitude of the samples but
also their occurrence times, andas such compared with uniform
sampling the reconstructionof the original signal is more complex.
Level crossing (LC)sampling is a non-uniform procedure that for a
given set ofquantization levels it generates a non-uniform sampled
signalwhere each sample is taken whenever the signal attains oneof
these quantization levels (See Fig. 1). Although recon-struction of
the original signal from such sampled signal ismore complex than
that of a uniform sampled signal, LC sam-pling is less restrictive
in other ways. It does not require theband-limited condition of
uniform sampling and is also freeof quantization. More importantly
LC is signal dependent —samples are only taken when the signal is
significant.
t0 t1 t2 t3
x(t)
t
q0
q1
q2
q3
q4
q�1
Fig. 1. Level crossing for fixed quantization levels.
A related sampling system is obtained with the asyn-chronous
sigma delta modulator (ASDM) shown in Fig. 2,which is a nonlinear
feedback system consisting of an in-tegrator and a Schmitt trigger
[2]. The ASDM maps theamplitude information of a bounded input
signal x(t) into atime sequence tk, or the zero crossings of the
binary outputz(t) of the ASDM. The bounded signal x(t) and the
zero-crossing times {tk} of the ASDM output z(t) are related bythe
integral equation [2]
Z tk+1
tk
x(⌧)d⌧ = (�1)k[�b(tk+1 � tk) + 2�] (1)
where b, � and are parameters of the ASDM. A
multi-levelapproximation for x(t), that depends on the scale
parameter, is obtained by connecting the width of the pulses in
z(t)
+ �
1
Zdt
x(t) y(t) z(t)b
�b
��
�
�1
1
t1
t2
t3
tSchmitt TriggerIntegrator
z(t)
y(t)
t
Fig. 2. Asynchronous sigma delta modulator
with local averages of the signal. This multi-level
approxi-mation can be seen as the output of a zero-order hold
non-uniform sampler.
Letting � = 0.5, b = 1 and some , adding two consecu-tive
integral equations as in (1), we haveZ t,k+2
t,k
x(⌧)d⌧ = [(t,k+2 � t,k+1)| {z }�,k
� (t,k+1 � t,k)| {z }↵,k
]
where ↵,k and �,k are defined as in Fig. 3. If we then letT,k =
�,k + ↵,k, then the local average
x̄,k =1
T,k
Z t,k+1
t,k
x(⌧)d⌧ +1
T,k
Z t,k+2
t,k+1
x(⌧)d⌧
=↵,k � �,k↵,k+�,k
(2)
Thus, x̄,k or the local average in [t,k, t,k+2] correspondsto
the difference of the areas under two consecutive pulsesin z(t)
divided by the length of the two pulses. Using theseconnection
between z(t) and the local averages, we can obtaina multi-level
approximation of x(t) that would be equivalentto one using a
level-crossing sampler with quantization levels{x̄k}. If we
consider the x̄k the best linear estimator of thesignal in [t,k,
t,k+2] when no data is provided, the time-encoder can be thought of
an optimal LC sampler. This wouldrequire to process the signal
first with an ASDM and then touse the obtained local averages as
the quantization levels forthe LC.
The scale parameter relates to the maximum frequencyof the
signal. Indeed, using that |x(t)| c and that b > c, we
z(t)
t
1
�1
· · · · · ·
↵,k
�,k
t,k
t,k+1 t,k+2
Fig. 3. The parameters ↵,k and �,k in z(t) for some scale.
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obtain
�c(tk+1 � tk) Z tk+1
tk
x(⌧)d⌧ c(tk+1 � tk)
Replacing (1), and solving for we have that
(b � c)(tk+1 � tk)2�
< <
(b + c)(tk+1 � tk)2�
(3)
In the case of non-uniform sampling, a sufficient condition
forreconstruction of band-limited signals is that the maximum
of{tk+1 � tk} should be less the sampling period Ts. In such acase
letting � = 0.5, b = 1 and b = c+�, positive � ! 0, therelationship
with the maximum frequency fmax of the signalis
(2c + �)Ts 1 � 0.5�
fmax⇡ 1
fmax. (4)
To obtain an expansion of the signal for different scales,just
as in wavelet analysis we generate a basis for z(t). Themost
suitable is the Haar basis which is generated from thenormalized
mother function
(t) =
8<
:
1 0 t < 12�1 12 t < 1
0 otherwise(5)
that generates the contracted and shifted wavelet family
forintegers m � 0 and k � 0,
m,k(t) = 2(m/2)
(2mt � k)
forming an orthonormal basis in the square–integrable space.The
indice m is scaling index, k time translation and the term2m/2
maintains a constant norm independent of scale m. Thispermit us to
expand any signal ⇣(t) in that space as partialsum that converges
in the mean square metric,
⇣̂(t) =X
m,k
�m,k m,k(t)
where the expansion coefficients are averages of the signal
fordifference scales {1/2m}, m = 0, 1, · · · :
�m,k =1
2m
"Z tm,k+1
tm,k
⇣(t)dt �Z tm,k+2
tm,k+1
⇣(t)dt
#(6)
where tm,k+2 � tm,k = 1/2m and tm,k+1 � tm,k = 1/2m+1.There is
clearly similarity between the local averaging forsome in equation
(2) and these equations. The local av-erages obtained from the
output of the ASDM for differentscales provide an approximation to
the signal. In the follow-ing section we propose a decomposition
procedure similar tothe Haar transform wavelet that uses the
ASDM.
3. ASYNCHRONOUS DECOMPOSITION
Figure 4 displays the decomposer for three levels. At an
ini-tial scale 0 the output of the ASDM is used to find the
corre-sponding local averages from which we obtain a smooth
outmultilevel signal using an averager and a low-pass filter. ForL
decomposition levels the detail signals are
f0,1(t) = x(t) � d0,1(t)f1,2(t) = d0,1(t) � d1,2(t)
...fL�1,L(t) = dL�2,L�1(t) � dL�1,L(t) (7)
with scale factors
` = 0/2`
` = 1, · · · , L
From (7) we obtain the following expansion for the signal
x(t) =LX
`=1
f`�1,`(t) + dL�1,L(t) (8)
corresponding to the different levels with different scales.
3.1. Representation of sparse signals in time
The above decomposition is especially appropriate for
sparsesignals in time. Modeling a sparse signal as
s(t) =X
k
↵kp�(t � tk) + ⌘(t)
where p�(t) = u(t) � u(t � �), � ! 0
that is, s(t) is a sequence of very narrow pulses located
atarbitrary times tk and embedded in noise ⌘(t) with a variancemuch
smaller than that of the signal.
Assuming the noise ⌘(t) is zero mean, the above de-composition
for an appropriate scale 0 will pick the narrowpulses as
d1(t) =X
k
↵kp�(t � tk)
and the detail signal f1(t) = ⌘(t) would be the noise. Fora real
sparse signal, it would be necessary to consider morethan one
decomposition level with different scales.
4. SIMULATIONS
To illustrate the performance of the decomposer we apply it
toheart sound records [14, 15] which are inherently sparse.
Theanalyzed signal and the decomposed parts for the first
threelevels are shown in the left plot of Fig. 5. As we can seefrom
the resulting spectra in the right plot of the same figure,d(t) and
f(t) waveforms correspond to slowly and rapidly
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ASDM Averager LPFx(t)
+
�
d1(t)
f1(t)
ASDM Averager LPF
f2(t)
d2(t)
�
+
· · ·
m1(t) m2(t)
z2(t)z1(t)0 1
Fig. 4. Decomposer
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
d 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2
0
2
f 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
d 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
f 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
d 3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
time [sec]
f 3
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
50
|X(Ω
)|
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.502040
|D1(Ω
)|
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
50
|F1(Ω
)|
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5050100
|D2(Ω
)|
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
50|F
2(Ω
)|
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5024
|D3(Ω
)|
−fs/2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 fs/2050100
Normalized frequency [Hz]
|F3(Ω
)|
Fig. 5. Left plot: outputs of the first three levels of the
decomposer to a heart sound signal (top). Right plot:
Correspondingspectra.
changing parts of the signal. The component d1(t) is the lo-cal
average approximation of the input signal while f1(t) isthe error
of this approximation. The initial value of the scaleparameter, 0,
used in the first level of the decomposer leadsto averaging over a
narrow window of the signal, while thesmaller {`} in the subsequent
levels provide averaging overa wider window for the {d`}. The
component d3(t) indicatesthe point at which the decomposition is
terminated, as feedingd3(t) into another level reveals no further
information. Thereconstruction error, using only these three
levels, is smallerthan 10�15.
In a second simulation, meant to stress the denoising be-havior
of the proposed scheme, we consider a sparse signalembedded in
noise. The reconstruction performance under30dB SNR is shown in
Fig. 6. It takes only one level ofdecomposition with a proper value
to accurately recoverthe sparse signal. This also means that the
signal can be re-constructed from the resulting local averages with
their widthlengths. For this highly sparse signal 6.6% compression
is ob-tained. Also approximation by using Haar wavelet resulted
in21% compression. We anticipate that this compression andsignal
dependent noise canceling feature is highly promis-
ing for continuous processing of bursty signals embedded
innoise.
To simulate the denoising behavior for different levelsof noise,
300-trial Monte Carlo simulation was performed.Noise with SNR
values between �10 to 15 dB was addedto the original signal and
reconstructed using our algorithm.Computing the average mean-square
error for each SNR we
0 0.5 1 1.5−1
0
1
x 0(t)
0 0.5 1 1.5−1
0
1
x n(t)
0 0.5 1 1.5−1
0
1
t [sec]
x r(t)
Fig. 6. Reconstruction under noise: original noise-free
signal(top), noisy signal with SNR 30dB (middle), and
denoisedsignal (bottom).
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obtain the results shown in Fig. 7. The performance fromSNR �10
dB and zero is significant, and for additive noisewith SNR higher
than 10 dB the performance of the algorithmlevels off.
−10 −5 0 5 10 150
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
SNR in dB
Mea
n Sq
uare
Err
or
Fig. 7. Mean Square Error of the reconstructed signal
fordifferent SNR values
5. CONCLUSION
In this paper we consider asynchronous processing of
sparsesignals appearing in biomedical and wireless sensor
applica-tions. A scale–based representation is suggested for
enablingefficient transmission of spiky or bursty data in
biomedicalimplants or sensor networks that run on batteries. The
alias-free continuous scheme exploiting the sparse behavior
resultsin remarkable compression while asynchronous design
signi-fies low-power dissipation. The proposed algorthm is robustto
noise while having low computational complexity. The re-covery
results are promising given the obtained high degree ofcompression
with adequate accuracy. Integrating our proce-dure with wavelets
allows us to investigate the realizability ofa simple yet efficient
transmission and retrieval of such preva-lent signals.
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