FAST RECOVERY ALGORITHS FOR TIE ENCODED BANDLIITED SIGNALS Aurel A Lazar Dept of Electrical Engineering Columbia University, New York, NY 27, USA e-mail: aurel@eecolumbiaedu Ernő K Simonyi National Council of Hungary for Information and Communication Technology, H-73 Budapest, Hungary e-mail: simonyi@nhithu László T Tóth Dept of Telecom and edia Informatics, Budapest Univ of Technology and Economics, H-7 Budapest, Hungary e-mail: tothl@tmitbmehu ABSTRACT Time encoding is a real-time asynchronous mechanism of mapping amplitude information into a time sequence We investigate fast algorithms for the recovery of time encoded bandlimited signals and construct an algorithm that has provably low computational complexity We also devise a fast algorithm that is parameter-insensitive INTRODUCTION Time encoding is a real-time asynchronous mechanism of mapping amplitude information into a time sequence In [4] time encoding and irregular sampling were shown to be largely equivalent modalities of information representation and recovery For communications applications, however, irregular sampling requires the transmission of both amplitude and sampling time information Time encoding requires only the transmission of the time sequence Thus, capacity requirements for time encoded versus irregular sampled bandlimited signals are lower by a factor of two [5] For the case of irregular sampling, fast algorithms for signal recovery have been extensively studied in the literature (eg, [2], [3], [6]) While these algorithms are an excellent starting point to search for fast algorithms for recovery of time encoded bandlimited signals, they can not be directly applied to the time encoding case The reasons are purely technical: the method developed for the irregular sampling case calls for reducing the solution of a linear systems of equations to a Toeplitz matrix inversion Our method for obtaining a fast algorithm for the time encoding case is based on the observation that the indefinite integral of the same signal can be directly recovered from its amplitude values sampled at instances provided by the time sequence In the process we find that the complexity of recovering the indefinite integral of an arbitrary bandlimited signal from irregular samples is essentially the same as the complexity of recovering the same signal from its time encoded sequence In addition, we find that the condition numbers of the pseudo-inverse matrices that arise in both formulations can be chosen to fall in the same range This paper is organized as follows A brief overview of the classical recovery algorithm for irregular sampling followed by a review of an efficient recovery for irregular sampling is presented in section 2 In section 3 we derive a fast recovery algorithm for time encoded bandlimited signals In section 4, a parameter-insensitive reconstruction algorithm is devised Simulation results are presented in section 5 2 FAST RECOVERY ALGORITH FOR IRREGULAR SAPLING 2 The Classical Recovery Algorithm for Irregular Sampling Given a set of irregular sampling times (t k ), k Z, we shall represent the Ω-bandlimited signal x = x(t), t R, as x(t) = g T c = g(t t k ), () where c = [ ] is a vector of weights, T denotes the transpose and g = [g(t t k )] with g(t) = sinωt πt is the vector of t k -shifted impulse response of an ideal lowpass filter with bandwidth Ω Also, R and Z above denote the set of real numbers and integers, respectively Denoting by: [q] l = x( ) and [G] l,k = g( t k ), (3) for all k, k Z, and l, l Z, it is easy to see that (2) q = Gc (4) Proceedings of ICASSP 25, arch 8-23, 24, Philadelphia c IEEE 25
If the average density of the s k s is at or above the Nyquist rate then, c can be evaluated as c = G + q, (5) where G + is the pseudo-inverse (oore-penrose) of G The practical solution, however, is challenging because G is typically ill-conditioned, and, q, c and G are infinite dimensional In summary, the bandlimited signal x can be approximately represented by equation (3) with the weighting coefficients evaluated following () The low complexity of the algorithm () is due to the fact that T is a Hermitian Toeplitz matrix 3 FAST RECOVERY ALGORITH FOR TIE ENCODING 22 A Fast Recovery Algorithm for Irregular Sampling For irregular sampling, let us consider with T = π/ω and α = (2 + )T g(t) = α e jn Ω t = α sin ( 2+ 2 Ωt) sin ( ) (6) Ωt 2 instead of the impulse response defined in (2) When tends to infinity this function converges to the original impulse response At the same time, the latter choice of g(t) is both Ω-bandlimited and periodic with a period 2 T In analogy to (), we define the function f = f(t), t R, by f(t) = α e jn Ω (t t k) (7) Assuming an appropriate norm, f is an approximation of x as provided that f( ) = x( ) for all l, l Z [3] Evaluating the above equality at t = we obtain f( ) = α e jn Ω ( t k ), (8) for all l, l Z By denoting [q] l = x( ) and [S] m,l = e jm Ω, equation (8) above after multiplication with S becomes Sq = αss H Sc, (9) or d = αt + Sq, () 3 Classical Recovery Algorithm for Time Encoding In the time encoding case an Ω-bandlimited signal x = x(t), t R, is represented as a discrete strictly increasing time sequence (t k ), k Z The time sequence (t k ), k Z, is generated using a Time Encoding achine (TE) [4] An example of a TE is depicted in Figure [4] It consists of an ideal integrator and a noninverting Schmitt trigger in a feedback arrangement The output z(t) takes the values b or b at transition times denoted by t k It can be shown [4] that this circuit is described by for all l, l Z, by the recursive equation x(u)du = ( ) k [2κδ b(t k+ t k ] (4) Integrator Z dt κ where d = αsc and T = αss H () [q] l = x(u)du and [G] l,k = g(u s k )du, For the record (6) [d] n = α e jn Ω t k and [T] m,n = α where s k = (t k+ + t k )/2, it is easy to see that e j(n m) Ω q = Gc and c = G + q (7) l Z (2) Note that T is both Toeplitz and Hermitian since [T] m,n = 32 Reformulation of the Classical Recovery Algorithm [T] m n and T H = T (the superscript H indicates conjugate transposition) Finally with [d] n = d n, x(u)du, t If x is a bandlimited function, then so is t R, and therefore: t f(t) = d n e jn Ω t, (3) x(u)du = g T c = g(t t k ), (8) 2 x(t) + y(t) Noninverting Schmitt trigger δ b z b Fig An Example of a Time Encoding achine δ y z(t) The recovery of the signal x is achieved via the representation x(t) = g T c = g(t s k ) (5) with an appropriate set of weights, k Z Denoting by:
where g = [g(t t k )], g(t) is given by (2) and c = [ ] is an appropriate set of coefficients Since we have x(u)du = [g(+ t k ) g( t k )], (9) q = PGc, (2) where [P] l,k = δ l+,k δ l,k (using Kronecker s notation) and [G] l,k = [g( t k )] (same as in (3)) and thus with [P ] i,k = c = G + P q (2) { if i k if i > k (22) Remark Note that by multiplying both sides of equation (2) with +, we obtain: (+ )[P q] l = (+ ) sin Ω( t k ) π( t k ) }{{} [ ] l,k (23) In the time encoding case, [q] l = g(u s k )du = (+ ) sin Ω(ξ (24) l s k ) π(ξ l s k ) for some ξ l [t tl, + ] Therefore, the elements of the matrix identified by the lower brace in equation (23) are approximately equal to the elements of the G matrix for time encoding This points to the close relationship between the recovery algorithm of a time encoded bandlimited signal x and the recovery of an irregularly sampled integrated signal t x(u)du For both recovery methods the same time sequence is used 33 Fast Recovery Algorithm Now, (23) can be transformed into a Toeplitz system in the same way as carried out in Sec 22 Replacing the sinc term by the approximation introduced in (6) gives: (+ )[P q] l = α(+ ) e jn Ω ( t k ) Denoting by D = diag(+ ), l Z, we have in matrix form DP q = αds H Sc ultiplying both sides by S from the left gives: Equivalently with we have SDP q = αsds H Sc T = αsds H, d = αsc, (25) d = αt + SDP q (26) The matrix T is both Toeplitz and Hermitian since [T] n,m = α (t k+ t k )e j(m n) Ω t k Finally, the original signal x is approximated the function f given by f(t) = jω nd n e jn Ω t, (27) with the vector d given by (26) 4 PARAETER-INSENSITIVE RECONSTRUCTION The original reconstruction of a time encoded bandlimited function depends on the TE parameters κ and δ In [4] the Compensation Principle was used to address the insensitivity of the recovery algorithm with respect to the parameters of the TE A simple argument shows, however, that the method employed in [4] does not directly apply here Therefore, a different technique is needed Parameter insensitivity can be achieved if the termp q in equation (25) does not depend on κ and δ Carrying out P q by using (22) gives: [P q] l = κδ [ + ( ) i] + b L ( ) k (t k+ t k ) k=i where L denotes the size of G (ideally L ) Therefore, as seen, P q will be independent of κδ for odd values of i If - in the last column of every second line of P is changed to zero then each integral is carried out over even number of subintervals This can be achieved by adding the vector q = }{{} a [ ] q }{{} b T 3
to P q and defining p = P q + ab T q, (28) where for illustration purposes and in finite dimensions q q 2 p = q 3 q 4 q 5 q 6 Note that p can be calculated without using parameters δ and κ Since p = P q + ab T PP q and P q = αs H Sc, we have: p = αs H Sc + αab T PS H Sc ultiplying both sides above by SD gives: SDp = αsds H Sc + αsdab T PS H Sc Using the matrix T and vector d defined in (25) and (26), respectively, and denoting by we have: and therefore u = αsda, v T = b T PS H, (29) (T + uv T )d = SDp, d = (T + uv T ) + SDp (3) Finally, employing the result of [] (page 5, Corollary 33), the pseudo-inversion in (3) can be determined based on the pseudo-inverse of T as (T + uv T ) + = T + T+ uv T T + + v T T + u provided that +v T T + u Thus, to solve (3) we again have to calculate the pseudo- inverse of a Hermitian Toeplitz matrix and the parameter-insensitivity is also guaranteed The reconstructed signal is again given by (27) 5 SIULATION RESULTS With Ω = π 8 khz and the corresponding Nyquist period T = π/ω = 25 µs the bandlimited signal x(t) was generated by its Shannon-representation x(t) = 65 n= 35 x(nt)sinc(ω(t nt)), 4 3 2 - -2 x(t) -5-25 25 5 75 t (ms) Fig 2 Overall input signal x(t) and the simulation range (dashed box) for time encoding where the samples x( 35T),,x(65T) were randomly selected in the range ( 3, 3) and sinc(t) = sin t/t if t and sinc() = This is shown in Fig 2 in the range t [ 5T, 8T] The dashed box in the figure shows the simulation range t [ µs, 2538 µs] for the TE With parameters δ = 7 6, κ = /2, and c = 3 the TE simulation produced 35 trigger times t =, t,, t 35 Fig 3 shows the error signal defined as the difference between the right-hand-side and the left-hand-side of (27) in an even further reduced range t [2525 µs, 22538 µs] to decrease the boundary effects [4] 75 5 25-25 -5-75 e(t) 9 e RS = 543 db 5 5 2 t (µs) Fig 3 Error signal using the original formulation in the reduced range to reduce the boundary effects Finally, Fig 4 shows the results for the RS error and the condition number of T (based on using 2 ) in terms of different values of by using the new reconstruction technique It can be seen that as the condition number increases the accuracy of the reconstruction improves Fig 5 shows the simulation results with the original (squares) and the parameter-insensitive reconstruction technique (stars) The error with the original method is the same as that shown in Fig 4 4
db 4 3 2 - cond(g) e RS original cond(t) e RS () 2 4 6 8 2 Fig 4 Simulation results for the condition numbers and RS error e RS (db) -2-4 -6-8 - -2-4 2 4 6 8 2 Fig 5 Simulation results for the RS error 7 REFERENCES [] Campbell, SL and eyer, CD Jr, Generalized Inverses of Linear Transformations, Dover Publications, 979 [2] Feichtinger, HG, Gröchenig, K and Strohmer, T, Efficient Numerical ethods in Non-uniform Sampling Theory, Numerische athematik, Vol 69, pp 423-44, 995 [3] Gröchenig, K, Irregular Sampling, Toeplitz atrices, and the Approximation of Entire Functions of Exponential Type, athematics of Computation, Vol 68, #226, pp 749-765, April 999 [4] Lazar, AA and Tóth, LT, Perfect Recovery and Sensitivity Analysis of Time Encoded Bandlimited Signals, IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol 5, No, pp 26-273, October 24, 26-273 [5] Lazar, AA, Roska, T, Simonyi, EK and Toth, LT, A Time Decoding Realization with a CNN, Proceedings of Neurel 24, pp 97-2, September 23-25, 24, Belgrade [6] Strohmer, T, Numerical Analysis of the Non-uniform Sampling Problem, J Comp Appl ath, Vol 2, #-2, pp 297-36, 2 Using the same parameters as before we found that increasing the accuracy improves The nonzero eigenvalues of T are approaching to those of G in (23) and practically zero eigenvalues are introduced if is increased In particular, having 27 trigger times the error for = (2 + = 23), = 3 (2 + = 27), = 5 (2 + = 3), the corresponding error turned out to be -86 db, -3996 db, -4628 The accuracy cannot be improved below -543 db corresponding to the case when the original g(t) is used 6 CONCLUSIONS We have shown that the indefinite integral of an arbitrary bandlimited signal can be directly recovered from the time encoded sequence associated with the same bandlimited signal This simple observation enabled us to devise a fast algorithm for signal recovery We have also demonstrated that our approach can be extended to parameter insensitive signal recovery Taken together, these results shed further light on the close relationship between irregular sampling and time encoding 5