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

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.

855

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

856

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).

857

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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