WO2006102178A2

WO2006102178A2 - System and method for time encoding analog signals

Info

Publication number: WO2006102178A2
Application number: PCT/US2006/009916
Authority: WO
Inventors: Aurel A. Lazar
Original assignee: The Trustees Of Columbia University In The City Of New York
Priority date: 2005-03-18
Filing date: 2006-03-20
Publication date: 2006-09-28
Also published as: WO2006102178A3

Abstract

Time encoding machines with multiplicative coupling, and optional feedforward or feedback are provided. The time encoding machines, which are based on common oscillator circuits, are input/output equivalent to either an integrate-and- fire neuron with a variable threshold sequence or to an asynchronous sigma-delta modulator with a variable threshold sequence. An input bandlimited signal, which is time-encoded using the time encoding machines, can be perfectly recovered from the zero crossings of the modulated signal and the threshold sequence.

Description

SYSTEM AND METHOD FOR TIME ENCODING ANALOG SIGNALS

SPECIFICATION

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of United States provisional application Serial No. 60/663,276 filed March 18, 2005 and United States provisional application Serial No. 60/730,517 filed October 26, 2005, both of which applications are hereby incorporated by reference herein in their entireties.

FIELD OF THE INVENTION

The invention relates to electronic circuits and techniques for signal processing. In particular, the invention relates to analog-to-digital conversion and later digital-to-analog reconstruction of electrical signals.

BACKGROUND OF THE INVENTION

Conversion of analog signals (e.g., voice and video signals) into digital form is desirable for computer processing and transmission. Conversely, analog recovery of the digitized signal is desirable for replay.

Analog-to Digital (A/D) conversion of signals may be performed using suitable converter circuits. Analog signals have traditionally been encoded in the amplitude domain using common synchronous A/D converters, which involve regular sampling. However, with increasing miniaturization of electronic circuits and the corresponding use of low operating voltages, it may be beneficial for accurate reproduction to encode analog signals in the time domain.

Time encoding is a mechanism for representing the information contained in a continuous time, bandlimited, analog signal as the zero-crossings of a binary signal. Time encoding of a bandlimited function u = u(t), t e R, is a representation of u as a sequence of strictly increasing times (t_k), k e Z, where ϊl and Z denote the set of real numbers and integers, respectively. Alternatively, the output of the encoder is a binary signal z = z(t), ( e l, with zeros at times t_k, k e Z.

Circuit implementations of the time encoding mechanism may be referred to as Time Encoding Machines (TEMs). An early TEM based on the asynchronous sigma-delta modulator (ASDM) circuit is described by Roza et al. U.S. Patent No. 6,087,968 ("Roza"). In Roza's TEM, the ASDM generates an asynchronous duty cycle modulated square wave output z(t), which is representative of the input signal x(t). The output of the ASDM is sampled synchronously by a sampling circuit and processed through a decimating circuit to obtain a time encoded sequence version of the input signal. Roza's TEM introduces non-linearities in the output, which can impede accurate reconstruction of the analog signal.

Other circuit implementations of TEMs are described in Lazar International patent publication No. WO2004112298 and Lazar et al. International patent publication No. WO2004039021 (collectively "Lazar"). In particular, in addition to ASDM-based TEMs, Lazar describes TEMs that are based on filter banks and integrate-and-fire neuron circuits. Lazar also describes non-linear time-decoding algorithms that make a perfect recovery of a time encoded signal possible under certain encoding conditions. Circuit implementations of the time decoding algorithms may be referred to herein as time decoding machines (TDM).

Thus, known TEMs described in the literature are based on specialized circuits such as the ASDM or the integrate-and-fire neuron circuits. For more widely exploiting time encoding in signal processing, it would be advantageous to have TEM implementations that are based on other circuit structures. Consideration is now being given to developing TEM implementations that are based on circuit structures that are or will be commonly available in modern very or ultra large scale integrated circuit devices. Attention is directed in particular to developing TEMs that use simple garden variety circuits.

SUMMARY OF THE INVENTION

The present invention provides a class of time encoding machines that are based on oscillator circuits. The TEMs exhibit multiplicative coupling, and may optionally also include feedforward and feedback circuits. The oscillator circuits may be any common oscillator circuit subject only to the requirement that number of zeros of the waveform it generates satisfies on the average a Nyquist-type rate condition. Suitable oscillators may include, for example, the harmonic oscillator, the Hodgkin- Huxley neuron, the Van der Pol oscillator, and the Lorentz chaotic attractor. In the language of communication theory multiplicative coupling describes a process of non- linear modulation, or from a mathematical standpoint the oscillator waveform undergoes a time change.

The TEMs generalize a number of modulation schemes arising in communications and neuroscience, and provide a rich class of circuits for implementing novel A/D converters and non-linear modulation schemes for sensor networks. The inventive TEMs are designed for use in conjunction with time- decoding algorithms that make a perfect recovery or recovery with arbitrary error of the bandlimited input signals possible.

BRIEF DESCRIPTION OF THE DRAWINGS

Further objects, features and advantages of the invention will become apparent from the following detailed description taken in conjunction with the accompanying figures showing illustrative embodiments of the invention, in which:

FIG. l is a block diagram of a time encoding machine with multiplicative coupling, in accordance with the principles of the present invention.

FIG. 2 is a block diagram of an integrate-and-fire neuron circuit, which is input/output equivalent to the time encoding machine of FIG. I₅ in accordance with the principles of the present invention.

FIG. 3 is a block diagram representation of a recovery algorithm, in accordance with the principles of the present invention.

FIG. 4 is a block diagram of a time encoding machine with multiplicative coupling and feedforward, in accordance with the principles of the present invention.

FIG. 5 is a block diagram of a time encoding machine with multiplicative coupling and feedback, in accordance with the principles of the present invention.

FIG. 6 is an illustration of signal recovery and the recovery error associated with use of the recovery algorithm of FIG. 4, in accordance with the principles of the present invention FIGS. 7a and 7b are illustrations of the phase plane of Hodgkin-Huxley neuron oscillators in the Rinzel and the Modulated Rinzel Approximation, respectively. FIGS. 8a and 8b are illustrations of the spikes generated by Hodgkin- Huxley neuron oscillators in the Rinzel and the Modulated Rinzel Approximation, respectively.

Throughout the figures, the same reference numerals and characters, unless otherwise stated, are used to denote like features, elements, components or portions of the illustrated embodiments. Moreover, while the subject invention will now be described in detail with reference to the figures, it is done so in connection with the illustrative embodiments. It is intended that changes and modifications can be made to the described embodiments without departing from the true scope and spirit of the subject invention as defined by the appended claims.

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a general class of time encoding machines based on simple oscillator circuits. The inventive TEMs are configured to exhibit multiplicative coupling, and further may be configured to include other common circuits, for example, feedforward or feedback circuits.

FIG. 1 shows an exemplary TEM 100 with multiplicative coupling. TEM 100 includes an oscillator 110, which is modulated by an input bandlimited signal u(t), and a zero crossings detector 120. Oscillator 100 may be, for example, a common oscillator circuit that generates a stable limit cycle and provides an output signal y(t). (See FIGS. 7a-8b). The input signal u(t) with an offset b, is multiplicatively coupled to oscillator 110. Output y(t) of oscillator 110 is fed to zero crossings detector 120, which generates a time sequence of the zeros (i.e., a set of time events, called trigger times) of the oscillator waveform. In the presence of constant unit input, the output of oscillator 110 is described by a set of state space equations of the form f-^r(*^{λ (1)} where x 6 R " and f e I " are column vectors and f : R" -> E" is a continuous function. Under the assumption that for an arbitrary initial condition x(0) = X₀ , the set of differential equations (1) has an unique solution. The zeros of x_\, the first coordinate of x, denoted by (δ_/t), k e Z, are also called trigger times. For bandlimited input signals, u = u(t), t e R is a bounded continuous function on Ii with \u\ ≤ c; u models the input signal to the TEM, With multiplicative coupling, the output of the oscillator 110 is given by dy /dt = {b + u(t))f{y\ (2)

where y e R" is a column vector and b > c is a constant. Thus, b + u(t) > O₅ for all t, t e R. The zero crossings detector 120 detects the zeros

These zeros are denoted by (/_/f), k e IL

Intuitively, the essential dynamics of the set of equations with additive input are described by a limit cycle that is signal dependent. For the initial condition y(0) = x(0), the solution to equation (2) is given by y = x(bt + £ u(s)ds), (3)

for all t, t e B ₊, where x = x(t), t e E ₊, is the solution to (1) starting at

X₀.

It will be noted that the solution to equation (2) is derived from the solution to equation (1) via the time change t -» bt + J u(s)ds. In this light the

condition b + u(f) > 0 is very natural since it ensures that the changed time remains increasing.

The defining building blocks of TEM 100 have individually been employed in signal processing technology in a number of modulation schemes. For example, oscillator 110 described by equation (2) has been employed in common frequency modulation schemes. Similarly, zero crossing detector 120 has been employed in irregular sampling schemes. Therefore, it is expected that physical implementations of TEM 100 will be easily accomplished with existing circuit design and layout tools with few modifications. The operation of TEM 100 with multiplicative coupling is further understood by noting its input/output (I/O) circuit equivalence with a TEM based on an integrate-and-fire neuron having a variable threshold. TEMs based on an integrate-and-fire neuron having a fixed threshold are described, for example, in Lazar International patent publication No. WO2004112298 and Lazar et al. International patent publication No. WO2004039021, which are hereby incorporated by reference herein in their entireties. The variable threshold sequence of TEM 100 equivalent is given by the difference between the consecutive zeros of the waveform generated by oscillator 1 10 for unit input.

FIG. 2 shows a block diagram of an integrate-and-fire neuron TEM circuit having a variable threshold, which is input/output equivalent to TEM 100. The equivalence of the two TEM circuits is described in mathematical terms below.

The observable output of oscillator 110 can be exactly one of the coordinates of x. (e.g., coordinate x_\). The zeros of x_\(t) are denoted by (δ_/(), k e 7L Therefore, X₁ (^) = O, (4)

for all k e Z. Further as mentioned above, the trigger times (t_k), k e Z, are the zeros

^Of MO-

Since (δ*)₅ k e Z, are the set of zeros of x_\, and the zeros of y>_\ are given by (t_k), k e Z, equation

implies that

bt_k + £ u(s)ds = δ_k . (6)

Thus, the set of trigger times (t_k ), k e Z, and the set of zeros (δ_k), k e Z, verify the set of recursive equations ^ u{s)ds = δ_M - δ_k - b{t_M -t_k), (7)

or all k, k ^ ^"L,

Equation (5) defines the time transform ("t-transform"), which maps the amplitude information of (u(ή), t e IR, into the time sequence (t_k), k e Z. See e.g., A. A. Lazar and L. T. Toth, ''Perfect Recovery and Sensitivity Analysis of Time Encoded Bandlimited Signal," IEEE Transactions on Circuits and Systems-I, Vol. 51, No. 10, October 2004, pp. 2060-2073 ("Lazar and Toth"). Thus, encoding information with TEM 100 with multiplicative coupling is equivalent with encoding information with an integrate-and-fire neuron with variable threshold (δ^+i — δ^), k e Z. See e.g., A.A. Lazar, "Time Encoding with an Integrate-and-Fire Neuron with a Refractory Period," Neurocomputing, Vol. 58-60, pp. 53-58, 2005. Both TEM 100 and an integrate-and-fire neuron lead to the same trigger time sequence (fø) for all h, k e ^"L.

Formally, under the reasonable assumption that the variable threshold sequence of an integrate-and-fire neuron is identical to the difference between the consecutive zeros of the oscillator waveform generated for unit input, TEM 100 with multiplicative coupling (FIG. 1) and the integrate-and-fire neuron circuit 200 with variable threshold (FIG. 2) are input/output equivalent (e.g., when the integrator reset value is zero).

TEM 100 has the same advantageous characteristics as equivalent integrate-and-fire neuron circuit that the input bandlimited signal u(t) can be perfectly recovered from the zero crossings of the modulated signal and the threshold sequence. The recovery process is understood upon noticing that a linear function between two consecutive trigger times provides via the t-transforrn, an estimate of the integral of u(t) on the same interval. For a finite energy signal u(t), this estimate used in conjunction with the bandlimited and boundedness assumption on the same, enables a perfect reconstruction of the original signal even though the trigger times are irregular. In order to achieve perfect reconstruction, the distance between two consecutive trigger times should, at least on the on the average, be smaller than the distance between uniformly spaced samples in classical Shannon sampling theory. The mathematical methodology for signal recovery is based on computing the inverse of the t-transform under appropriate conditions. This methodology is the same or similar to recovery methodology described, for example, Lazar et al. International patent publication No. WO2004039021.

An exemplary "recovery algorithm" 300 based on the inverse t- transform is defined as: u(t) = ∑ c_kg(t - s_k ) (8) k e Z where s^ = (t_k + t_k+j)/2; the supremum of the distance between two consecutive zeros in the unitary input case is denoted by δ, i.e., δ = supj_teZ (δ_/t+i — δ^); vectors g = [gft-s/J], q = [δ_k+\ - δ_k -b(t_k - t_k+i)], and G = [ J*'^+l g(s - s_k )ds]\ and c

=G⁺q, where G⁺ is the pseudoinverse of G.

FIG. 3 shows a simple block diagram representation of recovery algorithm 300. The trigger time sequence and the binary signal z(t) have the same information content. Further, it is understood that the binary signal z(t) is related to the input signal u(t) by a non-linear function (i.e., a t-transform). By using the inverse of the non-linear relationship (i.e., the inverse t-transform) in a decoding algorithm, the input signal, u(t), can be recovered from knowledge of the zero crossings in z(t). If the zero crossings in z(t) are known precisely, then perfect recovery of the input signal is possible. If there is error attributable to the measurement of the zero crossings, the error in the recovery of u(t) is equivalent to the recovery of a signal using traditional amplitude sampling.

In accordance with the present invention, TEM 100 with multiplicative coupling may be extended to include further common circuits for additional signal processing that may be desirable or necessary in signal processing applications. For example, TEM 100 with multiplicative coupling may be extended to include feedforward circuits or feedback circuits, which have beneficial signal denoising characteristics. The t-transform for both the feedforward and feedback TEM schemes can easily be derived using the chain rule for derivatives. In both cases, suitable recovery algorithms (e.g., algorithm 300) may be applied to perfectly recover the input signal u(t).

FIG. 4 shows a TEM 400 in which TEM 100 with multiplicative coupling (FIG. 1) is coupled to a feedforward circuit 410. For simplicity, only the equivalent integrate- and-fire neuron representation 200 of TEM 100 is shown in FIG. 4. TEM 100 accepts a version of the bandlimited signal u = u(t), which is processed by feedforward circuit 410, as its input. This processed input to TEM 100 on the time interval [t_k, t_k+i] is given by v(t) J ' v(-?)ds, where t ≥ t_k . TEM 400 with multiplicative coupling and feedforward is described for all k, k e Z, by the equation

\_ι ^tM v(s) [] v(σ)dσds = δ_M -δ_k . (9) Thus, TEM 400 as shown in FIG. 4 is I/O equivalent to an integrate- and-fire neuron with variable threshold sequence [2(δ^+_/. - δ* )] ^υ2, k e Z.

In the general case, the t- transform describing the operation of TEM 400 may be described by h( j^ v(_S)ds ) = δ_k___] - δ_k (10)

where h is an arbitrary function on R. If equation 10 has a solution of the form ( J *^+l v(^)ds ) = h^~\δ_k=λ - δ_k), the original bandwidth limited signal can be h recovered from

( { %(s)ds = h^~\δ_M - δ_k) + b{t_M - t_k) . (1 1)

Thus, in this general case TEM 400 with multiplicative coupling and feedforward is I/O equivalent with an integrate-and-fire neuron with variable threshold. The variable threshold sequence of the neuron can be explicitly derived from the zeros of the oscillator's waveform for unit input.

FIG. 5 shows an exemplary TEM 500 in which TEM 100 with multiplicative coupling (FIG. 1) is coupled to a feedback circuit 510. TEM 500 derives its feedback 510 from the output of the zero crossings building block 120. Feedback circuit 510 may be designed to implement any suitable feedback scheme. For example, for feedback 510 may be implemented so that the input is composed as z{\) u(t) +b, where z(t) = sgn(yi(t)) for all for all t, t e R. TEM 500 with feedback can be shown to be input/output equivalent to an ASDM-based TEM, which is described by A.A. Lazar and L. T. Toth, "Perfect Recovery and Sensitivity Analysis of Time Encoded Bandlimited Signal," IEEE Transactions on Circuits and Systems-I, Vol. 51, No. 10, October 2004, pp. 2060- 2073. For example, for the input composed as z(t)u(t) +b, and assuming the convention that z(t) =1 for all t, t e [to, ti], the t-transform of TEM 500 is given by

J^ u(s)ds = (-I)Vw - δ_k + b{t_M - t_k)} (12)

Equation 12 is the same as the t-transform of an asynchronous sigma- delta modulator but with variable thresholds. See, e.g., Lazar and Toth for the fixed threshold case. Thus, TEM 500 with multiplicative coupling and feedback is I/O equivalent to an asynchronous sigma-delta modulator with variable thresholds. Further, algorithm 300 shown in FIG. 3 may be used for recovery of bandlimited input signals, which are encoded by TEM 500.

With renewed reference to FIG. 1, it will be understood that TEM 100 with multiplicative coupling may be build using any of a wide variety of common oscillators as oscillator 110. The only requirement on oscillator 110 is that Nyquist- type rate condition rs < 1 remain valid. Suitable oscillators may include, for example, the harmonic oscillator, the Hodgkin-Huxley neuron, the Van der Pol oscillator, and the Lorentz chaotic attractor. The choice of a particular oscillator type may be based on considerations of the non-linear modulation scheme desired. In an exemplary implementation, oscillator 110 may be a Van der Pol relaxation oscillator. The Van der Pol relaxation oscillator, is described by the set of equations: dx,

= x, (13) dt

The signal time encoding and recovery properties of TEM 100 were tested with oscillator parameters α = 500 and β = 0.5, for which the non-linear system has a periodic attractor. For a test input signal u = 0.2 sin (2π20t) + 0.3 sin (2π30t) + 0.5 sin (2π40t), TEM 100 was initialized at (1, 0) and the results evaluated on the time interval [0, 187.5] ms. FIG. 6 shows the recovered signal and the recovery error over this time interval. The recovered signal and the original are virtually indistinguishable. It is seen form FIG. 6 that a recovery error below 5 10^" can easily be achieved.

In another exemplary implementation, oscillator 110 in TEM 100 may be a Hodgkin-Huxley neuron oscillator. FIGS. 7a and 7b show phase plane properties of a Hodgkin-Huxley neuron oscillator without multiplicative coupling and with multiplicative coupling, respectively. In particular, FIGS. 7a and 7b show the phase plane in the two-dimensional Rinzel and modulated Rinzel approximation to the Hodgkin- Huxley equations, which are easier to visualize. FIG. 7a, which relates to an oscillator without multiplicative coupling shows an oscillator limit cycle. The corresponding spike train is shown in FIG. 8(a). FIG. 7b shows the oscillator limit cycle in the case of multiplicative coupling when the input signal is zero and the multiplicative constant b has an arbitrary value. The corresponding spike train is shown in FIG. 8(b).

In accordance with the present invention, the aforementioned TEMs and recovery algorithms may be implemented in any suitable media. The suitable media may include, without limitation, firmware, microcontrollers, microprocessors, integrated circuits, ASICS, computer readable media, and any other available media. It will be understood, further, that the foregoing is only illustrative of the principles of the invention, and that various modifications can be made by those skilled in the art, without departing from the scope and spirit of the invention, which is limited only by the claims that follow. For example, the inventive TEMs may use modulation schemes between transmitters and receivers that are different than those described herein. The oscillators used may derive not only from the electronics industry but may include nano-scale oscillators known from computational cell biology.

Claims

WHAT IS CLAIMED IS:

1. A time encoding machine (TEM) for encoding a bandlimited input signal u(t), the TEM comprising:

an oscillator circuit which is multiplicatively coupled to the input signal; and

a zero crossing detector circuit coupled to the output of the oscillator circuit,

wherein the input signal undergoes a process of non-linear modulation.

2. The TEM of claim 1 , further comprising a bias b, so that b +u(t) > 0 for all operating times t.

3. The TEM of claim 1, wherein the zero crossing detector circuit generates a sequence of zeros of the oscillator waveform.

4. The TEM of claim 1 , wherein the oscillator circuit comprises one of a harmonic oscillator, a Hodgkin-Huxley neuron, a Van der Pol oscillator, and a Lorentz chaotic attractor.

5. The TEM of claim 1, wherein the circuit parameters and operating parameters of the TEM are selected to optimize recovery of the input signal u(t) from its encoded version..

6. The TEM of claim 1, comprising further circuits to additionally process a signal internal to the TEM, namely, any one of the oscillator input signal, oscillator output signal, zero crossings detector input signal, zero crossings detector output signal.

7. The TEM of claim 6, wherein the further circuit comprises a feedforward circuit.

8. The TEM of claim 7, wherein the feedforward circuit is configured to denoise the oscillator input signal.

9. The TEM of claim 7, wherein the circuit parameters and operating parameters of the TEM are selected to optimize recovery of the input signal u(t) from its encoded version.

10. The TEM of claim 6, wherein the further circuit comprises a feedback circuit.

11. The TEM of claim 10, wherein the feedback circuit is configured so that the output of the zero crossings detector is returned to the TEM input.

12. The TEM of claim 10, wherein the circuit parameters and operating parameters of the TEM are selected to optimize recovery of the input signal u(t) from its encoded version.

13. A method for time encoding an input signal u(t), the method comprising:

using the input signal to non-linearly modulate an oscillator circuit; and

using a zeros crossing detector circuit to generate time sequence of the zeros of the oscillator waveform.

14. The method of claim 13, wherein using the input signal to non-linearly modulate an oscillator circuit comprises multiplicative coupling the input signal to the oscillator circuit.

15. The method of claim 13 further comprising applying a bias b to the input signal u(t) so that b +u(t) > 0 for all operating times t.

16. The method of claim 13 further comprising selecting circuit parameters and operating parameters to optimize recovery of the input signal u(t) from its encoded version.

17. The method of claim 13 further comprising denoising the input signal by coupling a feedforward circuit to the oscillator circuit input.

18. The method of claim 17 further comprising selecting circuit parameters and operating parameters of the oscillator circuit and the zeros crossing detector circuit to optimize recovery of the input signal u(t) from its encoded version.

19. The method of claim 13 further comprising coupling the output of the zeros crossing detector circuit to the input of the oscillator circuit using a feedback circuit.

20. The method of claim 19 further comprising selecting circuit parameters and operating parameters of the oscillator circuit and the zeros crossing detector circuit to optimize recovery of the input signal u(t) from its encoded version.