18th European Signal Processing Conference (EUSIPCO-2010) Aalborg, Denmark, August 23-27, University of Pittsburgh Pittsburgh, PA, USA

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1 8th European Signal Processing Conference (EUSIPCO-) Aalborg, Denmar, August 3-7, ADAPTIVE LEVEL-CROSSING SAMPLING AND RECONSTRUCTION Seda Senay, Luis F. Chaparro, Mingui Sun and Robert J. Sclabassi, Department of Electrical and Computer Engineering, Laboratory for Computational Neuroscience University of Pittsburgh Pittsburgh, PA, 56 USA ABSTRACT We propose an adaptive level crossing approach for the sampling and reconstruction of s for applications where cloc-free and low-power data gathering are required. Due to the lac of a priori information on the statistics, uniform levels are typically chosen in level-crossing (LC) sampling. Our approach uses as reference levels the local means obtained from an asynchronous sigma delta modulator (ASDM). This -dependent sampling constitutes non-uniform sampling with local means and their times of occurrence. The local means can be seen as optimal estimators in a mean-square sense. The reconstruction of the is approached using Prolate Spheroidal Wave functions which performs well when the is timelimited and essentially band-limited. The optimality of the levels can be illustrated when comparing our procedure with uniform LC sampling and reconstruction. The proposed procedure is especially suited for applications where the occurs in bursts, and data gathering is done under cloc free, low power and low transmission conditions.. INTRODUCTION Although non-uniform sampling is not a preferred method for data acquisition, since both samples and their corresponding sample times are needed for reconstruction, in many applications it is an alternative to conventional timedriven sampling. That is the case whenever clocs are not desirable, and high energy efficiency and low data transmission are required. In biomedical monitoring [9], for brain or heart computer interfaces, speech processing and networed control [], level crossing (LC) or Lebesgue sampling have been considered as an alternative to conventional time-driven sampling. In LC sampling the is not sampled but directly quantized whenever the crosses a certain level. As indicated in [] this approach avoids frequency aliasing and could be applied to s that are not band-limited. LC is particularly useful in dealing with s that deliver the information in bursts rather than in a constant stream. Such s appear in biomedical applications [9] and in sensor networ transmissions [3]. Because of their short duration and bursty nature, these s are not necessarily band-limited. Level crossing (LC) [4] is a form of nonuniform sampling that can be used for bursty or sparse s. Given the economical sampling, LC sampling may be used to obtain a discrete representation for sparse s in the compressive sensing [5]. The advantages of LC sampling in both data transmission and reconstruction depend on the proper placement of reference levels within the dynamic range of the input. Typically, the levels have been treated as uniform quantization levels [4, 6, 7] instead of optimal level allocation [3], where the dictates the rate of data collection and quantization. In this paper, we propose a novel approach to determine dependent reference levels for the LC sampling. These adaptive levels are obtained by using an asynchronous sigma delta modulator (ASDM) which is a nonlinear feedbac system that performs time encoding of the [9]. The output of an ASDM is a binary that is continuous in time. The ASDM provides a cloc-less data acquisition and reconstruction from the sample times [9,, ]. Moreover, it enables the computation of the local means of the. It is these local means that we will use as the levels for the LC sampling. The local means require that more samples be taen when the is bursty and fewer otherwise. Considering s that are time limited and essentially band-limited (a high percentage of the energy is concentrated in a certain bandwidth) the reconstruction is approached using prolate spheroidal wave functions (PSWFs). In biomedical data monitoring, ASDM-based adaptive LC sampling can be used efficiently and implemented as in []. As illustrated by the simulations, the ASDM-based adaptive LC sampling and reconstruction using PSWFs show a lot of promise in the processing of time-limited and essentially band-limited s, including bursty s [3, 4, 5].. ASYNCHRONOUS SIGMA DELTA MODULATORS An Asynchronous Sigma Delta Modulator (ASDM), Fig. (), is a nonlinear feedbac system consisting of an integrator and a non-inverting Schmitt trigger [8]. In the ASDM, amplitude information of a x(t) is transformed into time information. x(t) y(t) δ y(t) dt + κ Integrator b b δ Schmitt Trigger Figure : Asynchronous sigma delta modulator. Duty-cycle Modulation and Time-encoding Time-encoding can be seen as duty-cycle modulation, where a sequence of binary rectangular pulses (Fig.) is characterized by the duty-cycle defined for two consecutive pulses of duration T = α + β, α being the duration of the pulse of amplitude and β the duration of the other pulse of amplitude. For x(t), t +, the duty-cycle is defined as < α T = + x(t) < t + () Thus, if the is zero for all times, x(t) =, < t <, it is modulated into a train of square pulses with t t t3 t EURASIP, ISSN

2 α = β or with duty cycle of α /T =.5. If x(t) = A, A <, t +, then the two pulses for that time are rectangular where α = ( + A)T and β = T α. If the is not constant in a time segment, the duty-cycle is not clearly defined with respect to the amplitude. + + t α β Figure : Duty-cycle modulation. Suppose x is the local average of the in t +, from the above we can see that it can be obtained from the α and β in the duty cycle modulation, i.e., x = α β α + β α = +, β = + () Time encoding has been proposed in [9] for representation of a bandlimited using ASDMs. The relationship between the binary output and the input x(t) of the ASDM for + >, and integers, is given by the integral equation Z t+ x(u)du = ( ) [ b(+ ) + κδ] (3) Thus from the duty-cycle modulation, or the output of the ASDM, giving the time-sequence { } we are only able to recover local averages in each segment. If the x(t) is continuous in t +, the local average coincides with one of the values x(ζ) for ζ + and one could thin then of a non-uniformly sampled for which we would lie to interpolate the rest of the values in that segment. Thus, equation (3) provides a way to obtain that interpolation as we will see in the next section. The train of rectangular pulses displays non-uniform zero-crossing times that depend on the input amplitude. The reconstruction of the x(t) can be done by approximating the integral in (3), by the trapezoidal rule. Using an increment = (+ )/D for an integer D > (the larger this value the better the approximation), we have that " D x( ) X x(τ)dτ + x( + l ) l= + x(t +) Z t+ We then obtain the following reconstruction algorithm: (i) (ii) (iii) v = Qx = QPγ γ = [QP] v x = Pγ where v is the right term in (3), Q is the matrix for the trapezoidal approximation, P is either a matrix with sinc functions or PSW functions [9, 5]. The symbol represents pseudo-inverse. Thus the x(t) can be reconstructed from the zero crossings { } of the output of the ASDM. The accuracy of the reconstruction however depends on the approximation of the integral, and on the being bandlimited. Our approach strives to avoid these two constrains.. Calculation of Local Averages Assuming the output x(t) of the ASDM is bounded as x(t) c < b, the output of the integrator at time + > is y(+ ) = y( ) + κ Z t+ [x(u) z(u)]du. If the Schmitt trigger is in the state ( b, δ) at t =, (y( ) = δ and z( ) = b), at some time + > we have that δ = δ + κ = δ + κ Z t+ Z t+ [x(u) + b]du x(u)du + b(+ ) Right after +, the trigger switches to a (b, δ) state so that for some time + > + δ = δ + κ = δ + κ which when added gives Z t+ Z t+ + Z t+ [x(u) + b]du + x(u)du b(+ + ) x(τ)dτ = = α β (4) where α and β are defined as above. To obtain the local average we need T = α + β, which are obtained from the derivative of the binary, d dt = X ( ) δ(t ) (5) which in practice can be obtained using a time-to-digital converter []. The value chosen for κ is of great significance in the computation of the local averages. If the value of κ is chosen appropriately, the difference of the output of the integrator at times and + is y(+ ) y( ) = δ = κ If we let δ =.5, κ = Z t+ Z t+ x(u)du + α [x(u) z(u)]du 97

3 and since x(t) < c we obtain the following upper bound for κ > Z t+ κ x(u)du + α α (c + ) indicating that it depends on the local variation of the amplitude or local frequency. If x(t) is bandlimited, its reconstruction from the time sequence {, =,, } is possible if [9]: max (α ) = max (+ ) T N (6) where T N = π/ω max is the Nyquist sampling period. Thus for bandlimited s, we have κ ( + c)t N (7) Figure 3 shows the operation of ASDM on an arbitrary. x(t) y(t) To minimize the mean-square error ε with respect to the levels {q }, with no additional information that the local averages provided by the ASDM, the local average x in [, + ] constitutes an optimal estimate of the in the interval. This is so since no second-order statistics is available. These local averages are completely determined by the time-codes {, =,, }. To insure that the reconstruction is possible, we assume the s are not only time-limited, but also essentially band-limited or that most of its energy is within a certain frequency band [5]. The essential bandwidth can be used to determine the value of κ for the ASDM (c) time (sec) Figure 4: Adaptive LC sampling ( and ) and reconstruction error (c) of a smooth time ( sec) Figure 3: Operation of ASDM 3. ADAPTIVE LEVEL-CROSSING SAMPLING Level-crossing (LC) sampling is threshold based: the x(t) is compared with a set of reference levels and only when the exceeds one of the reference levels the sample is taen. Thus, the determines when samples are taen and what the quantization level is, i.e., it is non-uniform sampling. For a bursty more samples are taen during the burst and fewer otherwise. For a smooth the samples are randomly but uniformly distributed. The reconstruction of the depends on this distribution even in the case of bandlimited s [7]. A piecewise constant reconstruction of x(t), is obtained for a set of reference levels {q } and non-uniform sample times {ζ } as [3]: ˆx(t) = X where u(t) is the unit-step function. minimize a mean-square error q [u(t ζ ) u(t ζ + )] (8) Assume we wish to ε = E[x(t) ˆx(t)] (9) by choosing the levels {q }. It is not possible to obtain optimal levels without statistical nowledge of the and choosing uniform levels does not provide the optimal solution, either. In [3], the authors propose a sequential algorithm to obtain optimal levels. 4. SIGNAL RECONSTRUCTION USING PSWF The PSWFs, {ϕ n(t)}, are real-valued functions with finite time support that maximize their energy in a given bandwidth [3]. These functions provide an interpolation of a continuous similar to the sinc interpolation in the sampling theory [4, 5]. Indeed, the sinc function S(t) can be expanded in terms of the basis {ϕ n(t)}, with T s as the Nyquist period, as S(t T s) = X ϕ m(t s)ϕ m(t) () m= allowing us to write the sinc interpolation of a band-limited x(t) as " # X X x(t) = x(t s)ϕ m(t s) ϕ m(t) = m= X γ mϕ m(t) () m= which is an infinite dimensional interpolation of the continuous in terms of PSWFs [4]. The finite reconstruction for the above at uniform sampling times is: 98 ˆx( ) = where the coefficients are γ M,m = N X = M X m= γ M,mϕ m( ) () x(t s)ϕ m(t s)

4 and the value of M is obtained when eigenvalues {λ n} associated with the length N PSWFs are approximately zero for n M (c) time(sec) Figure 5: Uniform LC sampling ( and ) and reconstruction error (c) of a smooth. The matrix form of the projected at the uniform times { = T s} is ˆx( ) = Φ( )γ M (3) where ˆx( ), N n is a length N n vector containing samples {x(t s)}, γ M is the vector formed by the coefficients of the projection and the matrix Φ( ) is composed of PSWFs. The LC-sampling using the local means { x } gives a set of non-uniform times ζ + (which we assume is a subset of a uniform sample values) where the continuous equals the local mean, i.e., x(ζ ) = x Thus, the couples {x(ζ ), ζ } =,,..., N l (4) provide the necessary data for the PSWF interpolation: x(ζ ) = Φ(ζ )γ M (5) Since the values {ζ } occur within [ + ] in a nondeterministic way, the matrix Φ(ζ ), of dimension M N l, can be considered non deterministic. Using its pseudoinverse we obtain the coefficients γ M = [Φ(ζ )] x(ζ ), which can be used to obtain the reconstructed x r(t) = Φ(t) [Φ(ζ )] x(ζ ) = Θ x(ζ ) (6) Considering the {x(ζ )} the measurements, the reconstructed resembles the results obtained in compressive sensing. Figure 4 shows adaptive LC sampling together with reconstruction error for a smooth, while Fig. 5 shows the results using uniform LC sampling with 8 levels and the accompanying normalized. We chose 8 uniform levels, because we thought it would be a fair comparison in terms of the step size of the levels. Although the uniform LC has 4 samples compared to samples of the adaptive LC, the normalized for the uniform LC is 4 4 compared to 7 4 of the adaptive LC sampling. It is possible to obtain a lower reconstruction error for the uniform LC sampling but that would require many more samples. Notice the different distribution of the crossing times, in the adaptive case they are more evenly distributed. 5. SIMULATIONS The significance of our procedure is highlighted when sampling and reconstructing bursty s. In the simulations we used a bursty which is from the wavelet decomposition of an EEG with a sampling period of 5 3 provided by [6]. We used.64 seconds of that which would require 3 samples in uniform sampling case. We converted the into continuous form by using sinc interpolation. An ASDM is then used to find the values of the local means. For the uniform sampling the dynamic range is divided into equal levels. As shown in Figs. 6 and 8, the quantization is finer with the uniform levels compared to the adaptive one, but the distribution of the samples is more uniform for the adaptive case. There are certain intervals where the uniform sampling does not have any samples and as indicated in Fig. 9, it is in those segments where the reconstruction is the worst. Despite the fewer samples in the adaptive sampling, the reconstruction is almost perfect as in Fig.7. In fact, we found it better than using the trapezoidal approximation of the integral resulting from the ASDM processing, shown in Fig adaptive levels samples (n) samples Figure 6: Adaptive LC of a bursty : quantization with local means, times of local averages reconstructed time (second) Figure 7: PSWF reconstruction from local levels and their times,. 6. CONCLUSIONS We propose a novel approach for the sampling and reconstruction of s using level crossing. In particular, our method is shown to perform well for bursty s, commonly found in biomedical applications and sensor networ transmission. Using local mean values as the levels for LC sampling, the dictates the rate of data collection. The 99

5 .. uniform levels...4 recosntructed samples samples(n) Figure 8: Uniform LC of a bursty : quantization with uniform levels, times of uniform levels x 7 reconstructed time (seconds) Figure 9: PSWF reconstruction from uniform levels and their times,. reconstruction is done using PSW functions under assumptions of finite time support and essentially band-limited for the s. The simulations indicate it is the distribution of the missing samples that determines the performance of the proposed method. The advantage of our method over conventional uniform sampling is in demonstrated in applications where s do not satisfy the band limited conditions, or where high cloc frequencies could cause physical complications. Using ASDM and level-crossing sampling devices provide a lowpower performance and do not cause frequency aliasing. Further study is needed to establish the frequency performance of the proposed method. REFERENCES [] E. Kofman and J. H. Braslavsy, Level crossing sampling in feedbac stabilization under data rate constraints, Proc. IEEE Conf. Decision and Control, pp , Dec. 6. [] Y. Tsividis, Mixed-domain and systems processing based on input decomposition, IEEE Trans. on Circ. and Syst., pp , Oct. 6. [3] K. M. Guan, S. Kozat and A. C. Singer, Adaptive reference levels in a Level-Crossing Analog-to-Digital converter, EURASIP J. on Advances in Sig. Proc., 8. [4] N. Sayiner, A level crossing sampling scheme for A/D conversion, IEEE Trans. on Circ. and Syst., pp , time(second) Figure : Time-decoding using trapezoidal approximation of the integral. [5] R. G. Baraniu, Compressive sensing, in IEEE Signal Processing Magazine, pp. 8, Jul. 7. [6] J. Mar and T. Todd, A nonuniform sampling approach to data compression, IEEE Trans. Comm., pp. 4-3, Jan. 98. [7] F. Aopyan, R. Manohar, and A. B. Apsel, A level-crossing flash asynchronous analog-to-digital converter, Proc. IEEE Intl. Symp. on Asynchronous Circ. and Syst., Mar. 6. [8] E. Roza, Analog-to-digital conversion via duty-cycle modulation, IEEE Trans. Circ. and Syst., pp , Nov [9] A. A. Lazar and L.T. Toth, Perfect recovery and sensitivity analysis of time encoded bandlimited s, IEEE Trans. on Circ. and Syst., pp. 6-73, 4. [] S. Senay, L. F. Chaparro, R. Sclabassi, and M. Sun, Time-encoding and reconstruction of multichannel data by brain implants using asynchronous sigma delta modulators, IEEE Intl. Conf. Engineering in Medicine and Biology (EMBC 9), Sep. 9. [] S. Senay, L. F. Chaparro, R. Sclabassi, and M. Sun, Asynchronous Sigma Delta Modulator Based Subdural Neural Implants for Epilepsy Patients, IEEE North East Bioengineering Conf. (NEBEC), Apr. 9. [] J. Daniels, W. Dehaene, M. Steyart, and A. Weisbauer, A/D conversion using an Asynchronous Delta-Sigma Modulator and a time-to-digital converter, IEEE Int. Symp. on Circ. and Syst., ISCAS 8, pp , May 8. [3] D. Slepian, H. O. Polla, Prolate spheroidal wave functions, Fourier analysis and uncertainty, Bell Syst. Tech. J, 96. [4] G. Walter and X. Shen, Sampling with PSWFs, Sampling Theory in Signal and Image Processing, pp. 5-5, 3. [5] S. Senay, L.F. Chaparro, and L. Dura, Reconstruction of nonuniformly sampled time-limited s using prolate spheroidal wave functions, Signal Processing, vol. 89, no., pp , 9. [6] M. Sun and R. J. Sclabassi, Precise determination of starting time of epileptic seizures using subdural EEG and wavelet transforms, Proc. of IEEE-SP Intl. Symp. on Time-Freq. and Time-Scale Analysis, pp. 57-6, Oct [7] P. Ferreira, The stability of a procedure for the recovery of lost samples in band-limited s, IEEE Signal Processing, pp. 95-5, Dec