Amplitude Adaptive ASDM without Envelope Encodin Kaspars Ozols and Rolands Shavelis Institute of Electronics and Computer Science 14 Dzerbenes Str., LV-16, Ria, Latvia Email: kaspars.ozols@edi.lv, shavelis@edi.lv Abstract In this paper a method of encodin the sinals by usin Amplitude Adaptive Asynchronous Sima-Delta modulator (AA-ASDM) scheme without an additional envelope encodin of the sinal is proposed. Accordin to AA-ASDM, the timevaryin envelope function of the input sinal is used in the feedback loop to reduce the switchin rate of the output trier and thus the power consumption of the circuit. In previous work, the sinal and its envelope function were encoded and transmitted separately, thus resultin in inefficiency, since two sinals instead of one were required to be transmitted. In order to solve this inefficiency, in this paper it is proposed to select such a time-varyin envelope function which does not require additional encodin and transmission, and still be able to recover the oriinal sinal from the obtained time sequence. The proposed method is particularly advantaeous for sinals with wide dynamic rane. I. INTRODUCTION As wireless technoloies evolve, more and more new enery constrained sensin applications appear, such as wearables, portable medical devices, wireless sensor networks and others. One of the most important issue in such applications is enery consumption and manaement of the battery life. In order to deal with this issue, researchers are tryin to find new and effective solutions to reduce the power consumption of wireless devices and thus prolon the life of the battery. One essential part of all wireless sensin devices is an analo to diital converter (ADC), the enery consumption of which can be sinificantly reduced. As shown in [1], [2], [3], asynchronous desins, instead of synchronous, in ADCs exhibit better properties such as exclusion of electromanetic interference, immunity to metastable behaviour, absence of clock jitter and most importantly lower enery consumption. One such asynchronous ADC is Asynchronous sima-delta modulator (ASDM), which converts amplitude information into time sequence in a very enery efficient way. The latest implementations show that it is possible to create ASDM with power consumption less than 28nW [4]. The fundamentals of ASDM are iven in [5]. In previous paper [6], an improved version of ASDM, called Amplitude Adaptive Asynchronous Sima-Delta modulator (AA-ASDM) was presented, where by usin time-varyin envelope of the input sinal in the feedback loop of ASDM, it was possible to reduce the switchin activity of ASDM and thus the power consumption of the circuit by up to 59.86%. Reardless of this reduction the perfect recovery of the oriinal sinal from the obtained time sequence was still possible. [6] However, as concluded in [6], the efficiency of AA-ASDM could still be improved if there was no necessity to additionally encode and transmit the time-varyin envelope function of the sinal in order to recover it at a receiver. Further studies have led us to a solution which not only solves the above-mentioned problem, but also further reduces the power consumption of the circuit. Therefore, in this paper an improved version of AA-ASDM is presented in Section III, which requires the time-varyin envelope function of the input sinal as proposed in Section IV. The method is particularly advantaeous for sinals with wide dynamic rane, as shown by numerical simulations usin electroencephaloram (EEG) sinals in Section V. II. ASYNCHRONOUS SIGMA-DELTA MODULATOR The block diaram of classical Asynchronous Sima-Delta modulator (ASDM), which consists of interator and Schmitt trier is shown in Fi.1. ASDM circuit is defined by κ, δ and b parameters, which determine the averae switchin rate of the Schmitt trier and thus the enery consumption of the circuit. [7] Fi. 1. ASDM block diaram. The input sinal x(t) is bounded in amplitude as x(t) c < b (1) Since output of the Schmitt trier has either z(t) = b or z(t) = b value, the interator input is either x(t) b or x(t)+b. By considerin (1), it follows that the interator output y(t) is increasin or decreasin function for t [, +1 ], thus y( ) = ( 1) k δ and x(t)dt = ( 1) k [2κδ b(+1 )] (2) 978--9928-6265-7/16/$31. 216 IEEE 165
for all k, where k Z. [7] The sinal from the obtained time sequence can be perfectly reconstructed as described in [7] if sup k Z (+1 ) π Ω, (3) where Ω is the bandwidth of the sinal. Due to (1) the distances between consecutive trier switchin points and +1 are bounded as τ min = 2κδ b + c +1 2κδ b c = τ max (4) Assumin that C is a maximum value of x(t) and c(t) x(t) is a time-varyin envelope of the sinal, the maximum and minimum distances τ max and τ min are time-varyin as well: τ max1 (t) = 2κδ b c(t) = T (5) 1 + 1/α c(t)/(αc) τ min1 (t) = 2κδ b + c(t) = T 1 + 1/α + c(t)/(αc), (6) where coefficient α > and parameters b and δ are chosen as b = (1 + α)c and δ = αct/(2κ). [6] As shown in [6], if sinals with wide dynamic rane are encoded, both distances τ min1 and τ max1 for small c(t) values are considerably less than T and thus over-trierin occurs, which is not necessary for reconstruction of the oriinal sinal. This is due to ASDM circuit parameters are chosen considerin only the maximum value which is never exceeded by the sinal. [6] Since dynamic power consumption of the trier is directly proportional to its switchin activity, the enery consumption of the whole ASDM circuit reduces, if the averae switchin rate of the Schmitt trier decreases. This also leads to a decreased number of time codes needed to be transmitted. The number of trierins can be reduced, if the coefficient α is increased, however, too hih α values lead to small differences τ max1 τ min1 and thus more precision is needed to encode the distances between consecutive trier times. [6] III. AMPLITUDE ADAPTIVE ASYNCHRONOUS SIGMA-DELTA MODULATOR Since the envelope of the sinal chanes over time, in [6] it was proposed to chane the constant parameter b accordin to the time-varyin maximum value c(t) x(t) to ensure τ max2 (t) = 2κδ 2 = const. = T. (7) b 2 (t) c(t) In this case the difference b 2 (t) c(t) must be constant and thus b 2 (t) can be written as b 2 (t) = c(t) + βc, (8) where β >. The second parameter δ 2 = βct/(2κ) follows from (7) and the minimum distance becomes [6] τ min2 (t) = 2κδ 2 b 2 (t) + c(t) = T 1 + 2c(t)/(βC). (9) As shown in [6], in this case τ min2 > τ min1 and τ max2 > τ max1 for all c(t) and β = α values, which means the over-trierin reduces and the power consumption as well in comparison to the classical b = const. case. Also the differences τ max2 τ min2 are larer for all c(t) < C values. The block diaram of the Amplitude Adaptive Asynchronous Sima-Delta Modulator (AA-ASDM) is shown in Fi.2. In addition to ASDM circuit (Fi. 1) there is an envelope Fi. 2. AA-ASDM block diaram. detector with output c(t) connected in the feedback loop, therefore now instead of (2) the followin equation holds [6]: t k+1 x(t)dt = ( 1) k [2κδ βc(+1 ) t k+1 c(t)dt], (1) where κ, δ, β, C are iven coefficients. A method of recovery of the oriinal sinal is described in [6], where the envelope function of the sinal is also needed for reconstruction of x(t), therefore c(t) is encoded by another ASDM. This is the drawback of AA-ASDM in [6]. In order to increase the efficiency, it is necessary to find such a timevaryin envelope function which does not require additional encodin and transmission in order to recover the oriinal sinal. IV. PROPOSED METHOD In order to solve the weak point of AA-ASDM, in this paper it is proposed to choose the time-varyin envelope function as follows. A. Envelope Function The time-varyin envelope can be chosen as: c(t) = const. + x 2 (t), (11) where the input sinal x(t) is bounded in amplitude: x(t) [ 1, 1]. By choosin const. = 5, the inequality c(t) x(t) holds for all x(t) [ 1, 1], therefore the time-varyin envelope c(t) of the input sinal x(t) is selected as: c(t) = 5 + x 2 (t) x(t), (12) which does not require any additional encodin scheme (see Section IV-B). 166
B. Sinal Recovery By insertin (12) into (1), a relationship between the output sequence { } k=1,2,...,k and the input sinal x(t) of AA- ASDM can be obtained: By denotin x(t) = ( 1) k [2κδ βc(+1 ) (5 + x 2 (t))dt]. (13) q k = ( 1) k [2κδ (βc + 5)(+1 )], (14) the equation (13) becomes: x(t) + ( 1) k x 2 (t)dt = q k. (15) By assumin that the input sinal x(t) can be represented as x(t) = N 1 d n n (t), (16) where d n are unknown coefficients and n (t) are chosen base functions, the first term of (15) can be written as x(t)dt = N 1 d n n (t)dt = d T k, (17) where d = [d, d 1,, d N 1 ] T and (t)dt k = 1 (t)dt.. (18) N 1 (t)dt The second term of (15) can be written as x 2 (t)dt = ( N 1 d n n (t)) 2dt = = d T Ĝk d, (19) where Ĝk mn = +1 m (t) n (t)dt. From (15), (17) and (19) the final equation correspondin to the time interval t [, +1 ] follows: d T k + ( 1) k d T Ĝk d = q k. (2) As there are total K 1 time intervals, then K 1 equations are obtained, and the unknown coefficients d are found by minimizin the error value: K 1 k=1 ( d T k + ( 1) k d T Ĝk d q k ) 2. (21) The question is what base functions n (t) to choose for representin the input sinal x(t), the bandwidth of which is limited to ω [ Ω, Ω]. The classical answer would be to select n (t) = sinc(ω(t t n )), however, two problems exist: 1) these functions are well suited for representin only time-unlimited sinals; 2) calculation of k and Ĝk is both time consumin and not perfectly precise since no analytical solutions of the interals exist. Therefere, iven the output sequence { } k=1,2,...,k with the correspondin time period Θ = t K t 1, the input sinal for t [t 1, t K ] is expressed in Fourier series as M x(t) = d + (d m cos(m 2π Θ t) + d m+m sin(m 2π ) Θ t), m=1 (22) where the upper limit M follows from the bandwidth Ω of the sinal: ΩΘ M =. (23) 2π Such a representation of x(t) is both well suited for expressin time-limited sinals of lenth Θ and allows fast and precise calculation of k and Ĝk. C. Real-Time Sinal Recovery In order to reconstruct the sinal in real-time, the reconstruction must be carried out in short time intervals t [t γj, t γj+l ], where γ =, 1, 2,... desinates the order number of the interval, but J determines the number of switchins atfer which the reconstruction of the next interval can start [8]. Since the precision of the reconstructed sinal frament at the beinnin and at the end of the frament is low, the reconstructed sinal ˆx γ (t) is multiplied by a correspondin window function [8] :, if t / (τ γ, σ γ+1 ], θ γ (t), if t (τ γ, σ γ ], w γ (t) = (24) 1, if t / (σ γ, τ γ+1 ], 1 θ γ+1 (t), if t (τ γ+1, σ γ+1 ], where τ γ = t γj+i, σ γ = t γj+i+λ and θ γ (t) = sin 2 π(t τ γ ) (25) 2(σ γ τ γ ) After the multiplication, the sinal frament is different from zero only in the middle part of the interval, therefore the next interval is chosen after receivin J = L 2I Λ switchin time instants, thus ensurin overlappin of the intervals. By combinin all intervals, it is possible to obtain the whole reconstructed sinal ˆx(t) = γ Z ˆx γ (t)w γ (t), (26) which can futher be low pass filtered in order to reconstruct the initial sinal frequency bandwidth before it was multiplied by the window function. V. SIMULATION RESULTS The dynamic power consumption of the trier is directly proportional to the switchin activity [9]. It means that the enery consumption of the whole ASDM circuit reduces, if the averae switchin rate of the Schmitt trier decreases. 167
This leads also to a decreased number of time codes needed to be transmitted. 1.8.5.7.8.9 1 1.1 1.2 1.3 1.4.8 1.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (s) Fi. 3. Oriinal sinal x(t) (ray solid line); error sinal x(t) ˆx(t) (black solid line) after reconstruction; positive and neative envelopes c(t) and c(t) (dotted lines) of x(t). The non-adaptive ASDM and adaptive AA-ASDM were tested on up to 49 Hz bandlimited EEG sinals. One example of such a test sinal x(t) and its envelope function c(t) = 5 + x 2 (t) is shown in Fi.3. The averae amounts of trierin events per second for different α = β values are shown in Table 1. In both cases the obtained maximum distance between consecutive trier times was close and did not exceed the Nyquist step π/ω. The enery savin in case of AA-ASDM in comparison to ASDM was calculated as E = (1 N AA-ASDM N ASDM ) 1%. (27) Table 1 : Comparison of ASDM and AA-ASDM α = β N ASDM N AA-ASDM Enery savin, %.1 99 283 71.4.3 44 171 57.7.7 231 134 42. 1 192 124 35.4 1.3 171 119 3 1.9 148 113 23.7 2.5 136 19 19.9 As it follows from the table, AA-ASDM is more enery efficient than ASDM at low α = β values, however, too low α and β are not recommended since hih switchin rates are obtained. The optimal choice could be α = β = 1, when the minimum distance between consecutive trier times in both cases is T/3. After obtainin the output sequence { } k=1,2,...,k of AA- ASDM, the oriinal sinal x(t) is reconstructed from the iven.5.7.8.9 1 1.1 1.2 1.3 1.4.5.7.8.9 1 1.1 1.2 1.3 1.4.5.7.8.9 1 1.1 1.2 1.3 1.4 Time (s) Fi. 4. Consecutive sinal reconstruction: the upper three fiures correspond to three consecutive sinal framents (dashed line oriinal sinal, solid line reconstructed sinal), the lower fiure corresponds to the reconstructed sinal by combinin all three upper framents. time codes as descibed in Section IV. For real-time recovery, the number L of consecutive trierins per sinal frament was chosen to be L = 5, while the parameters I and Λ of the window function were chosen to be I = 6 and Λ = 1. In this case, after receivin every new J = L 2I Λ = 28 switchin time instants in at least JT/3 =.95 seconds, every new sinal frament was reconstructed. It was accomplished in less than.9 seconds confirmin that real-time sinal recovery is possible. The reconstructed three framents and their combined sinal are shown in Fi.4, while the error sinal x(t) ˆx(t), which is the difference between the oriinal sinal x(t) and the recontructed combined sinal ˆx(t), of more loner duration is shown in Fi.3 by the black solid line. VI. CONCLUSIONS In this paper it is shown that it is possible to considerably reduce the switchin activity of ASDM, if the classical circuit of ASDM is supplemented with the time-varyin envelope function of the input sinal in the feedback loop of the circuit. Since the maximum distance between consecutive 168
trier times does not exceed the Nyquist step, it is still possible to reconstruct the sinal from the obtained reduced time sequence. For different α = β values, it is possible to achieve up to 71% enery savin compared to non-adaptive ASDM. As the α = β values row, the advantae (in switchin activity) of amplitude adaptive ASDM over non-adaptive ASDM becomes less, however, it is not recommended to choose too hih these values since the variance of distances between consecutive trier times reduces and more precision (more bits) is needed to measure the distances. Too low α = β values are not recommended as well since hih switchin activity appears in both cases. ACKNOWLEDGMENT This work is supported by the Latvian National research proram SOPHIS under rant areement Nr.1-4/VPP-4/11. REFERENCES [1] Marc Renaudin, Asynchronous circuits and systems: a promisin desin alternative, Microelectronic enineerin, vol. 54, no. 1, pp. 133 149, 2. [2] Tomislav Matic, Tomislav Svedek, and Marijan Herce, Limit cycle frequency jitterin of an asynchronous sima-delta modulator, in Telecommunication in Modern Satellite, Cable, and Broadcastin Services, 29. TELSIKS 9. 9th International Conference on. IEEE, 29, pp. 198 21. [3] David Kinniment, Alex Yakovlev, and Bo Gao, Synchronous and asynchronous ad conversion, Very Lare Scale Interation (VLSI) Systems, IEEE Transactions on, vol. 8, no. 2, pp. 217 22, 2. [4] Luís HC Ferreira and Sameer R Sonkusale, A 5-v 28-nw 58-db dynamic rane asynchronous delta sima modulator in 13-nm diital cmos process,. [5] CJ Kikkert and DJ Miller, Asynchronous delta sima modulation, in Proc. IREE, 1975, vol. 36, pp. 83 88. [6] K Ozols, Rolands Shavelis, and Modris Greitans, Amplitude adaptive asynchronous sima-delta modulator, in Imae and Sinal Processin and Analysis (ISPA), 213 8th International Symposium on. IEEE, 213, pp. 467 47. [7] Aurel A Lazar, László T Tóth, et al., Time encodin and perfect recovery of bandlimited sinals., in ICASSP (6), 23, pp. 79 712. [8] Aurel A Lazar, Erno K Simonyi, and László T Tóth, A real-time alorithm for time decodin machines, in Sinal Processin Conference, 26 14th European. IEEE, 26, pp. 1 5. [9] Binu K Mathew, The perception processor, Ph.D. thesis. The University of Utah, 24. 169