Approximate reconstruction of bandlimited functions for the integrate and fire sampler
Hans G. Feichtinger, José C. Príncipe, José Luis Romero, Alexander Singh Alvarado, Gino Angelo Velasco
aa r X i v : . [ phy s i c s . d a t a - a n ] D ec APPROXIMATE RECONSTRUCTION OF BANDLIMITEDFUNCTIONS FOR THE INTEGRATE AND FIRE SAMPLER
HANS G. FEICHTINGER, JOS´E C. PR´INCIPE, JOS´E LUIS ROMERO,ALEXANDER SINGH ALVARADO, AND GINO ANGELO VELASCO
Abstract.
In this paper we study the reconstruction of a bandlimited signalfrom samples generated by the integrate and fire model. This sampler allowsus to trade complexity in the reconstruction algorithms for simple hardwareimplementations, and is specially convenient in situations where the samplingdevice is limited in terms of power, area and bandwidth.Although perfect reconstruction for this sampler is impossible, we give ageneral approximate reconstruction procedure and bound the correspondingerror. We also show the performance of the proposed algorithm through nu-merical simulations.
Keywords: integrate and fire, non-uniform sampling, bandlimited func-tion. Introduction
The integrate and fire (IF) model is well known in computational neuroscienceas a simplified model of a neuron [8, 12] and is typically used to study the dynam-ics of large populations. The model consists of a leaky integrator followed by acomparator. The leak corresponds to a gradual loss of the value of the integral.More recently, the IF model has also been considered as a sampler [4, 16, 10, 11],where the sampler output is tuned to the variation of the integral of the signal.This feature can be exploited when sampling neural recordings, for which relevantinformation is localized in small intervals where the signal has a high amplitude [3].The block diagram of the sampler is presented in Figure 1. At every instant s ,the continuous input x ( t ) is integrated against an averaging function u k,s ( t ) andthe result is compared to a positive and negative threshold. When either of these isreached, a pulse is created at time t k = s representing the threshold value (positiveor negative), the value of the integrator is then reset and the process repeats. The x ( α ) u ( α ) d α ∫ > < θ p θ n !" x ( t ) Figure 1.
Block diagram for the BIF model.output is a nonuniformly spaced pulse train, where each of the pulses is either1 or -1. The averaging function u k,s ( t ) is defined by e α ( t − s ) X [ t k ,s ] , where X I isthe characteristic function of I and α > of the integrator due to practical implementations. The precise firing conditiondetermining the pulses is:(1) ± θ = Z t k +1 t k f ( t ) e − ( tk +1 − t ) α dt =: h f, u k i . The simplicity of the sampler translates into an efficient hardware implementationwhich saves both power and area when compared to conventional analog-to-digitalconverters (ADC) [4]. These constraints are severe, in the case of wireless brainmachine interfaces [13], for which the entire system has to be embedded inside thesubject. Hence, the IF sampler allows us to move the complexity of the design intothe reconstruction algorithm while providing a simple front end at the samplingstage.The problem of reconstructing a signal from the IF output should be distin-guished from the study of the dynamics of a population of neurons, when somestochastic assumption is made on the firing parameters [9]. In this article we studythe deterministic reconstruction of a bandlimited signal from the integrate and fireoutput. Part of the challenge of this stems from the fact that the sampling map thatassociates a function to its samples is non-linear. Indeed, we see from Equation (1)that the magnitude of the samples is always θ . Moreover, exact reconstruction forthe IF sampler is impossible since the output of the sampler does not completelydetermine the signal (see Example 1 below.)In this article, we will show that it is however possible to approximately recon-struct a bandlimited signal in L ∞ norm with an error comparable to the threshold θ .Moreover, we give a concrete reconstruction procedure which is of course non-linearbut, nevertheless, easy to implement. Since in many situations the IF sampler is somuch more convenient to implement than conventional analog-to-digital converters,the loss of accuracy in the reconstruction is a very reasonable trade-off [4], speciallyif the final analysis of the reconstructed data tolerates some small error [3].The methods considered so far [11] reconstruct the signal f from the systemof equations h f, u k i = ± θ (cf. Equation (1)), thus treating the reconstruction asa (linear) average sampling problem (see [7, 1, 15, 14]). These approaches imposedensity restrictions on the set of sampling functions { u k } k (cf. Equation (1).) Sincethese sampling functions depend on the signal, the density constraints on them aresomehow unnatural.The key for the reconstruction method that we develop lies in the observationthat the information derived from the IF output is much richer than the meresystem of equations h f, u k i = ± θ . It also contains the information that no propersubinterval [ t k , t ′ ] of [ t k , t k +1 ] satisfies Equation (1). We will exploit this extra infor-mation to give an approximate reconstruction procedure for a general bandlimitedfunction. Since the sampling process starts at a certain instant t , an additionalassumption on the size of f before t is required in order to fully reconstruct f .Roughly speaking, the assumption means that the sampling scheme would not haveproduced any pulse before t .In Section 2 we formally describe the output of the IF sampling scheme. Thisoutput depends on an initial time t when the process is started and two parameters:the threshold θ and the constant α > HE INTEGRATE AND FIRE SAMPLER 3 The integrate and fire sampling problem
We now define precisely the integrate and fire sampling scheme. Throughout thearticle we will assume the following.
Assumption 1.
A bandlimited function f ∈ P W Ω and numbers t ∈ R , α, θ > are given. Here,
P W Ω is the Paley-Wiener space P W Ω := (cid:8) f ∈ L ( R ) (cid:12)(cid:12) supp( ˆ f ) ⊆ [ − Ω , Ω] (cid:9) , of (complex-valued) bandlimited functions and ˆ f ( w ) := R R f ( x ) e − πiwx dx is theFourier transform of f . We call t the initial time , α the firing parameter and θ the threshold . Using these parameters we formally define the output of the sampler.We first define recursively a finite or countable sequence t < . . . < t j . . . called thetime instants . Suppose that the instants t < . . . < t j have already been definedand consider the function F j : [ t j , + ∞ ) → C given by F j ( t ) := Z tt j f ( x ) e α ( x − t ) dx. Observe that F j is continuous and F j ( t j ) = 0. If | F j ( t ) | < θ , for all t ≥ t j , then theprocess stops. If | F j ( t ) | ≥ θ , for some t ≥ t j , by the continuity of F j , we can define t j +1 as the minimum number satisfying the equation(2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t j +1 t j f ( x ) e α ( x − t j +1 ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = θ. Clearly, in this case t j +1 > t j .We have defined a finite or countable sequence of points t < . . . < t j . . . . Wewill prove in Proposition 1 that this sequence is in fact finite. Let us assume thetime instants { t , . . . , t n } and define the samples { q , . . . , q n } by,(3) q j := Z t j t j − f ( x ) e α ( x − t j ) dx, (1 ≤ j ≤ n ) . Observe that, by the definition of the time intervals, | q j | = θ .The output of the sampler is formally given by the time instants { t , . . . , t n } and the numbers { q , . . . , q n } . We say that this output has been produced bythe integrate and fire . The succeeding results apply generally to complex-valuedfunctions, but in the case of the application that motivated this sampling scheme,the signal is taken to be real-valued, and the output of the sampler is encoded as atrain of impulses, where only the sign of the samples q j is stored.3. Some remarks on the IF output
First we note that bandlimited functions are not completely determined by theoutput of the IF sampler.
Example 1.
There are non-zero bandlimited signals that will never produce anoutput from the sampler. Take for instance f θ ( x ) = θ sin ( π x )2 π x . Since ˆ f θ ( ω ) = θ max { −| ω | , } , f θ is bandlimited. We have for any t ∈ R , (cid:12)(cid:12)(cid:12)R tt f θ ( x ) e α ( x − t ) dx (cid:12)(cid:12)(cid:12) ≤ R tt (cid:12)(cid:12)(cid:12) θ sin ( π x )2 π x (cid:12)(cid:12)(cid:12) dx ≤ θ R R sin ( π x ) π x dx = θ < θ, for all t ≥ t . H.G. FEICHTINGER, J. C. PRINCIPE, J.L. ROMERO, A. S. ALVARADO, AND G. VELASCO
We now prove that the set of time instants produced by the IF sampler is indeedfinite and give some bounds on its distribution. To this end we introduce someauxiliary functions that will be used throughout the remainder of the article.Consider the function g : R → R given by g ( x ) = e − αx χ [0 , ∞ ] and define, v ( t ) := ( f ∗ g )( t ) := Z t −∞ f ( x ) e α ( x − t ) dx. (4)Since g ∈ L ( R ), v ∈ P W Ω . In the Fourier domain, v and f are related byˆ f ( w ) = (2 πiw + α ) ˆ v ( w ) . (5)In the time domain, this can be expressed as f ( t ) = ∂v ( t ) ∂t + αv ( t ) . (6) Observation 1.
The function v is continuous and v ( t ) −→ , when t −→ ±∞ .Proof. We have already observed that v ∈ P W Ω . Since ˆ v ∈ L and supp(ˆ v ) ⊆ [ − Ω , Ω], we have that ˆ v ∈ L and the conclusion follows from the Riemann-LebesgueLemma. (cid:3) The following straightforward equation relates v to the integrate and fire process.(7) Z ts f ( x ) e α ( x − t ) dx = v ( t ) − e α ( s − t ) v ( s ) , s ≤ t. We can now prove that the output of the IF process is finite.
Proposition 1.
Under Assumption 1, the following holds. (a)
The set of time instants produced by the integrate and fire scheme is a finiteset { t , . . . , t n } . (b) The numbers of time instants t j in a given finite interval [ a, b ] is boundedby k f k θ ( b − a ) / + 1 . (c) If f is integrable, the total number of time instants is bounded by k f k θ + 1 . Proof.
We first prove (b) and (c). Let [ a, b ] be an interval and let { t j , . . . , t j + m − } be m consecutive time instants contained in [ a, b ]. If m ≤ m ≥
2. For each 0 ≤ k ≤ m −
2, using Equation (2) we have, θ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t j + k +1 t j + k f ( x ) e α ( x − t j + k +1 ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t j + k +1 t j + k | f ( x ) | dx. Summing over the m − { t j , . . . , t j + m − } wehave, ( m − θ ≤ Z ba | f ( x ) | dx. (8) HE INTEGRATE AND FIRE SAMPLER 5
Letting a = −∞ and b = + ∞ yields (b). For (a), H¨older’s inequality gives,( m − θ ≤ k f k ( b − a ) / , and the conclusion follows.Now we prove (a). Assume on the contrary that the IF process goes on foreverproducing an infinite set of instants { t j : j ≥ } . Given s > t , by part (b), onlya finite number of instants t j belong to [ t , s ]. Therefore t n → + ∞ , as n → + ∞ .Using Equations (7) and (2) it follows that, θ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t j +1 t j f ( x ) e α ( x − t j +1 ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) v ( t j +1 ) − e α ( t j − t j +1 ) v ( t j ) (cid:12)(cid:12)(cid:12) ≤ | v ( t j +1 ) | + | v ( t j ) | . This contradicts Observation 1. (cid:3) The reconstruction
We now address the problem of approximately reconstructing a bandlimitedfunction from the integrate and fire output. Since the samples are taken in thehalf-line [ t , + ∞ ) we will make some assumption about the size of f before theinitial instant. Roughly speaking, the integrate and fire process would not haveproduced any sample in the interval ( −∞ , t ]. Assumption 2.
The function defined in Equation (4) satisfies, | v ( t ) | ≤ θ, for all t ≤ t . Note that by Observation 1, any t ≪ f we will first approximately reconstruct v from the integrateand fire output and then derive information about f by means of Equation (5).We will use the structure of the IF process to produce a number of approximatesamples for v .First we argue that, from the output of the IF process, we have enough infor-mation to approximate v on the time instants { t , . . . , t n } . Rewriting Equation (3)in terms of v (cf. Equation (7)) we have,(9) v ( t j +1 ) = e α ( t j − t j +1 ) v ( t j ) + q j +1 , (0 ≤ j ≤ n − . Since the value v ( t ) may not be exactly known we cannot determine from this re-currence relation all the values v ( t j ). However, we can construct an approximationto these values. Let w := 0 and define recursively,(10) w j +1 = e α ( t j − t j +1 ) w j + q j +1 , (0 ≤ j ≤ n − . Observe that Assumption 2 implies that | w − v ( t ) | ≤ θ . Using this estimate as astarting point we can iterate on Equation (9) and (10) to get,(11) | w j − v ( t j ) | ≤ θ, (0 ≤ j ≤ n ) . Consequently, using only the output of the IF sampling scheme, we have constructeda set of values { w , . . . , w n } that approximates v on the instants { t , . . . , t n } . Thesecond step is to approximate v on an arbitrary point of R . H.G. FEICHTINGER, J. C. PRINCIPE, J.L. ROMERO, A. S. ALVARADO, AND G. VELASCO
To this end observe that, according to the definition of t j as the minimum numbersatisfying Equation (2), we have that, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z tt j f ( x ) e α ( x − t ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ θ, for all t ∈ [ t j , t j +1 ].Rewriting this inequality in terms of v (cf. Equation (7)) gives,(12) (cid:12)(cid:12)(cid:12)(cid:12) v ( t ) − e α ( t j − t ) v ( t j ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ, for all t ∈ [ t j , t j +1 ].Combining this last inequality with (11) yields,(13) (cid:12)(cid:12)(cid:12)(cid:12) v ( t ) − e α ( t j − t ) w j (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ, for all t ∈ [ t j , t j +1 ].We now show that this inequality allows us to approximate v anywhere on the line. Claim 1.
Given an arbitrary time instant t ∈ R , choose x ∈ R d in the followingway: (a) if t < t , let x := 0 , (b) if t belongs to some (unique) interval [ t j , t j +1 ) , let x := e α ( t j − t ) w j , (c) if t ≥ t n , let x := e α ( t n − t ) w n .Then, | v ( t ) − x | ≤ θ . Remark 1.
Observe that the procedure to obtain x from t depends only on theoutput of the IF process.Proof. For case (a), the conclusion follows from Assumption 2. For case (b), theconclusion follows from Inequality (13). For case (c), the fact that the fire conditionis never satisfied after t n gives,(14) (cid:12)(cid:12)(cid:12)(cid:12) v ( t ) − e α ( t n − t ) v ( t n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ θ. Combining this estimate with Inequality (11), the conclusion follows. (cid:3)
We will now choose a window function.
Assumption 3.
A Schwartz class function ψ such that • ˆ ψ ≡ on [ − Ω , Ω] , and, • ˆ ψ is compactly supported,has been chosen. Since v ∈ P W Ω , the classic oversampling trick for bandlimited functions (see forexample [6] or [7]) implies that there exists a number 0 < β < (2Ω) − such that(15) v = X k ∈ Z v ( βk ) ψ ( · − βk ) . Using the procedure described in Claim 1, we produce a set { s k } k ∈ Z such that(16) | v ( βk ) − s k | ≤ θ, for all k ∈ Z . Let ϕ be the function defined by,(17) ˆ ϕ ( w ) = (2 πiw + α ) ˆ ψ ( w ) . HE INTEGRATE AND FIRE SAMPLER 7
It follows that ϕ is also a Schwartz function. Moreover, using Equation (5) we havethat,(18) f = X k ∈ Z v ( βk ) ϕ ( · − βk ) . Observe that, since v ∈ P W Ω , the sequence { v ( βk ) } k ∈ ℓ and the series in Equa-tion (18) converges in L and uniformly - in fact, it converges in the Wiener amal-gam norm W ( C , L ), see for example [6], [7] and [2].)Now we can define the approximation of f constructed from the IF samples. Let,(19) ˜ f := X k ∈ Z s k ϕ ( · − βk ) . Since, by Inequality (16), the sequence { s k } k is bounded and ϕ is a Schwartzfunction, it follows that Equation (19) defines a bounded function and that theconvergence is uniform (see [6] or [2].)The reconstruction algorithm consists then of calculating the approximated sam-ples { s k } k following Claim 1 and then convolving them with the kernel ϕ , that canbe pre-calculated.We now give a precise error bound for the reconstruction. Theorem 1.
Under Assumptions 1, 2 and 3, the function defined by Equation (19) satisfies, k f − ˜ f k ∞ ≤ Cθ, for some constant C that only depends on Ω and the window function chosen inAssumption 3.Proof. According to Equations (18), (19) and Inequality (16), k f − ˜ f k ∞ ≤ supess X k ∈ Z | v ( βk ) − s k | | ϕ ( · − βk ) |≤ θ supess X k ∈ Z | ϕ ( · − βk ) | . It suffices to define C := 2 sup P k ∈ Z | ϕ ( · − βk ) | . Since ϕ is a Schwartz function, C < + ∞ (see for example [5]). (cid:3) Remark 2.
We currently do not know what choice of the window function ψ min-imizes the constant in the theorem. A more detailed study of the choice of thewindow function should not only consider the size of that constant but also the rateof convergence of the series in Equation (19) . Numerical experiments
We study the behavior of the reconstruction algorithm under variations in thethreshold and the oversampling period for a specific choice of reconstruction ker-nel ψ . The test signal f is of finite length and real valued, produced as a linearcombination of five ‘sinc’ kernels (sin( πx ) / ( πx )) at a 1Hz frequency, with randomlocations and weights. The amplitude of the input has been normalized to 1. Al-though the theory covers infinite dimensional spaces, our simulations are limitedby the practical implementations of the sampler and algorithms. The effects oftruncation and quantization are not considered here. H.G. FEICHTINGER, J. C. PRINCIPE, J.L. ROMERO, A. S. ALVARADO, AND G. VELASCO
This signal is encoded by the IF sampler with α = 1 and recovered using theprocedure described in Section 4. The reconstruction kernel ψ is a raised cosine,defined by,(20) ψ ( t ) = sinc( t/T s ) cos( πγt/T s ) (cid:18) − γ t T s (cid:19) − , where γ = 0 . T s = 0 .
25 are determined by the maximum input frequencyΩ and the desired oversampling period β (cf. Equation (15).) Figure 2(a) showsthe raised cosine ψ in the time and frequency domain. Observe that the spectrumof ψ is constant for frequencies less than the input bandwidth and then decayssmoothly towards zero. The corresponding kernel ϕ (cf. Equation (17)) is shownin Figure 2(b). Using ϕ we recover ˜ f (cf. Equation (19).), an approximation of f M agn i t ude (a) Kernel ψ . M agn i t ude (b) Kernel ϕ . Figure 2.
Reconstruction kernels.as shown in Figure 3. As expected the error decreases in regions with high densityof samples. This behavior is evident from Figure 4, the dense regions imply thatthe uniform samples will most likely coincide with the estimated values of v ( t )at the sample locations. On the other hand, for samples that are far apart theapproximation follows a exponential decay from its original value which is not thenatural trend in the signal. Figure 4 shows v ( t ) (solid line), and the approximatedsamples of v on the lattice β Z , constructed using the procedure described in Claim1 called { s k } k and the envelope v ( t ) ± θ where these samples are known to lie(dashed line.)Currently the reconstruction algorithm uses the approximated samples of v ( t )at the pulse locations to define the piecewise exponential bound and estimate thereconstruction coefficients on the uniform lattice. Based on the numerical experi-ments the algorithm can be improved by including the estimated value of v ( t ) atthe pulse locations although it implies reconstruction on a nonuniform grid. Forboth cases similar error bounds can be defined as in Theorem 1. The variation ofthe error in relation to the threshold (pulse rate) is shown in Figure 5. The errordepends on the choice of generator and the oversampling period β , as seen in Fig-ure 6. The relationship between the kernels and the optimal oversampling periodis still not evident. HE INTEGRATE AND FIRE SAMPLER 9 Acknowledgements
The second and fourth authors were supported by NINDS (Grant Number:NS053561). The third author was partially supported by the following grants:PICT06-00177, CONICET PIP 112-200801-00398 and UBACyT X149. The thirdand fourth authors’ visit to the Numerical Harmonic Analysis Group (NuHAG)of the University of Vienna was funded by the European Marie Curie ExcellenceGrant EUCETIFA FP6-517154.
OriginalRecovered0 2 4 6 8 10−0.0500.05 Time [s]
Figure 3.
Reconstruction of f ( t ) from the impulse train. v(t)approx Samplessamplesv(t) bounds Figure 4.
Reconstruction of v ( t ) with β = 1 / θ = 0 . E rr o r Figure 5.
Variation of the error || f − b f || ∞ in relation to thethreshold and pulse rate (dotted line) with β = 0 . β ] E rr o r θ = 1e−3 θ = 1e−2 θ = 5e−2 Figure 6.
Variation of the error in relation to the oversamplingperiod for different thresholds. The error is defined as || f − b f || ∞ . HE INTEGRATE AND FIRE SAMPLER 11
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Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, Vienna, Austria
E-mail address , Hans G. Feichtinger: [email protected]
Department of Electrical and Computer Engineering, University of Florida, Gainesville,FL 32611, USA
E-mail address , J. C. Pr´ıncipe: [email protected]
Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Univer-sidad de Buenos Aires, Ciudad Universitaria, Pabell´on I, 1428 Capital Federal, Ar-gentina, and CONICET, Argentina
E-mail address , Jos´e Luis Romero: [email protected]
Department of Electrical and Computer Engineering, University of Florida, Gainesville,FL 32611, USA
E-mail address , Alexander Singh Alvarado: [email protected]
Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, Vienna, Aus-tria, and Institute of Mathematics, University of the Philippines Diliman, Quezon City,Philippines
E-mail address , Gino Angelo Velasco:, Gino Angelo Velasco: