Israel Bar-David

Capacity and coding for the gilbert-elliott channels

The Gilbert-Elliott channel, a varying binary symmetric channel, with crossover probabilities determined by a binary-state Markov process, is treated. In general, such a channel has a memory that depends on the transition probabilities between the states. A method of calculating the capacity of this channel is introduced and applied to several examples, and the question of coding is addressed. In the conventional usage of varying channels, a code suitable for memoryless channels is used in conjunction with an interleaver, with the decoder considering the deinterleaved symbol stream as the output of a derived memoryless channel. The transmission rate is limited by the capacity of this memoryless channel, which is often considerably less than the capacity of the original channel. A decision-feedback decoding algorithm that completely recovers this capacity loss is introduced. It is shown that the performance of a system incorporating such an algorithm is determined by an equivalent genie-aided channel, the capacity of which equals that of the original channel. The calculated random coding exponent of the genie-aided channel indicates a considerable increase in the cutoff rate over that of the conventionally derived memoryless channel. >

The capacity of average and peak-power-limited quadrature Gaussian channels

The capacity C(/spl rho//sub a/, /spl rho//sub p/) of the discrete-time quadrature additive Gaussian channel (QAGC) with inputs subjected to (normalized) average and peak power constraints, /spl rho//sub a/ and /spl rho//sub p/ respectively, is considered. By generalizing Smiths results for the scalar average and peak-power-constrained Gaussian channel, it is shown that the capacity achieving distribution is discrete in amplitude (envelope), having a finite number of mass-points, with a uniformly distributed independent phase and it is geometrically described by concentric circles. It is shown that with peak power being solely the effective constraint, a constant envelope with uniformly distributed phase input is capacity achieving for /spl rho//sub p//spl les/7.8 (dB 4.8 (dB) per dimension). The capacity under a peak-power constraint is evaluated for a wide range of /spl rho//sub p/, by incorporating the theoretical observations into a nonlinear dynamic programming procedure. Closed-form expressions for the asymptotic (low and large /spl rho//sub a/ and /spl rho//sub p/) capacity and the corresponding capacity achieving distribution and for lower and upper bounds on the capacity C(/spl rho//sub a/, /spl rho//sub p/) are developed. The capacity C(/spl rho//sub a/, /spl rho//sub p/) provides an improved ultimate upper bound on the reliable information rates transmitted over the QAGC with any communication systems subjected to both average and peak-power limitations, when compared to the classical Shannon formula for the capacity of the QAGC which does not account for the peak-power constraint. This is in particular important for systems that operate with restrictive (close to 1) average-to-peak power ratio /spl rho//sub a///spl rho//sub p/ and at moderate power values. >

Information rates of photon-limited overlapping pulse position modulation channels

A direct-detection photon-limited optical communication channel that uses pulse position modulation (PPM) under a pulsewidth constraint is considered. Overlapping PPM (OPPM) allows multiple positions per pulsewidth, as well as fractional modulation indices (number of pulsewidths per frame) requiring more refined timing than that needed for conventional disjoint PPM (DJPPM). It is shown that even at moderate values of the expected photon count per pulse (Q) --such as needed for high data rates--OPPM outperforms on-off keying (OOK) in both capacity and cutoff rate, even though OOK is uniformly superior to DJPPM. Moreover, efficient use of OPPM is possible with equiprobable input symbols, whereas OOK requires inconvenient asymmetrical inputs to achieve capacity and high cutoff rate efficiencies (nats/ photon). At lower data rates, where capacity efficiency is the prime criterion, a significant advantage (- 20 percent) over DJPPM can be achieved up to efficiencies of about 0.7 nats/ photon. The M-ary photon-limited OPPM channel can be viewed as an ambiguity and erasure channel, in the sense that some channel outputs are ambiguous in only some input symbols and only if no photons are counted is there ambiguity in all input symbols. For large M ambiguities cause bursts of erasures of data symbols. Masseys interlaced encoding, as well as conventional encoding followed by interleaving, are adaptable to this bursty channel, and effect an increase in its cutoff rate comparable to the increase obtainable with DJPPM by the same techniques.

An implicit sampling theorem for bounded bandlimited functions

The following sampling theorem is proved: Let f ( t ) be a bounded band-limited function, possibly a sample of a nonstationary stochastic process, such that | f ( t )| B . Denote by w o the appropriately defined bandwidth of f ( t ). Let t k denote the set of instants for which f ( t ) = C cos 2 πwt , with C > B and w > w 0 . Then f (0) and t k determine uniquely f ( t ). Namely f ( t ) is represented, up to a multiplicative constant, by its sine-wave-crossings , i.e., by the set of its argument values at which it crosses a given sinusoid the amplitude and the frequency of which exceed, respectively, the bound on f ( t ) and the limit on its band. The reconstruction of f ( t ) from f (0) and t k is a noncausal operation. A practical feedback scheme that interpolates a causal estimate of f ( t ) from the set of its past sine-wave-crossings and from f (0) is introduced. The input to the circuit is a binary waveform: its phase changes occur at t k and its amplitude is linear in f (0).

On the Rice model of noise in FM receivers

The so-called click-and-Gaussian-noise model for the output of a limiter-discriminator receiver for frequency-modulated signals, first proposed by S.O. Rice in 1963, is reviewed. A short survey is presented of the subsequent research that it generated and of some practical applications that it motivated. A more detailed analysis of parameters encountered in the Rice model is carried out with emphasis on their pertinence to click detection. These results are applied to the understanding of the limitations and to the interpretation of the performance of several noise threshold extension techniques for analog modulations and of error reduction techniques for digital modulations that depend on click detection and elimination. >

On information transfer by envelope-constrained signals over the AWGN channel

Using T.E. Duncans theorem (1970) on the relation between mutual information and the mean-square error of the optimum causal estimator of a random signal in additive white Gaussian noise (AWGN), the maximum achievable information transfer over the AWGN channel is derived with the random telegraph wave input. The information transfer is bounded and symptotically determined for the Wiener phase-modulated process input at large signal-to-noise ratio (SNR). Both results are compared to the information transfer for the capacity-achieving Gauss-Markov input process. For both the Wiener phase-modulated and the Gauss-Markov processes the information transfer increases asymptotically as the square root of SNR, but for the random telegraph wave it increases only as its logarithm. >

Augmented APP (A 2 P 2 ) module for a posteriori probability calculation and channel parameter tracking

The augmented a posteriori probability (APP) module, denoted by A/sup 2/P/sup 2/, comprises two mutually supporting algorithms: (1) a soft-input soft-output (SISO) APP module, adjusted to output edge metric information and (2) a recursive estimator for the channel parameters that benefits from this information at each step of the recursion. The thus-estimated parameters appropriately transform, in turn, the channel output signals that feed the SISO module. When applied to decoding a parallel concatenated convolutional code (PCCC) transmitted by binary phase-shift keying through a channel with frequency offset (0.08/T/sub s/ Hz) and phase jitter (0.23-rad RMS), concentrated at 0.01/T/sub s/ Hz, (T/sub s/-symbol duration), the degradation compared to fully coherent reception is a small fraction of 1 dB, without use of a preamble.

Minimum-mean-square-error estimation of photon pulse delay (Corresp.)

The minimum-mean-square-error (rose) estimation of the delay of a coherent pulse of photons by a direct-detection receiver is shown to depend on the shape of the pulse envelope. Whereas with smooth envelopes the minimum mse decreases only as the expected energy in the pulse, with sharp-edged envelopes it decreases as its square, provided the optimum estimator is used. Occurrence of photoelectrons due to additive noise increases the rose by an additional term, proportional to the expected intensity of the noise. It approximately equals the photon-limited term at a signal-to-noise power ratio of 17 dB in the case of rectangular envelopes.

Upper bounds on capacity for a constrained Gaussian channel

A low-pass and a bandpass additive white Gaussian noise channel with a peak-power constraint imposed on otherwise arbitrary input signals are considered. Upper bounds on the capacity of such channels are derived. They are strictly less than the capacity of the channel when the peak-power constrain is removed and replaced by the average-power constraint, for which the Gaussian inputs are optimum. This provides the answer to an often-posed question: peak-power limiting in the case of bandlimited channels does reduce capacity, whereas in infinite bandwidth channels it does not, as is well known. For an ideal low-pass filter of bandwidth B, the upper bound is Blog 0.934P/(N/sub 0/B) for P/(N/sub 0/B)>>1, where P is the peak power of the input signal and N/sub 0//2 is the double-sided power spectral density of the additive white Gaussian noise. >

Forward collision resolution-a technique for random multiple-access to the adder channel

Consider M-Choose-T communications: T users or less, out of M potential users, are chosen at random to simultaneously transmit binary data over a common channel. A method for constructing codes that achieve error-free M-Choose-T communication over the noiseless adder channel (AC), at a nominal rate of 1/T bits per channel symbol per active user, is described and an efficient decoding procedure is presented. The use of such codes is referred to as forward collision resolution (FCR), as it enables correct decoding of collided messages without retransmissions. For any given T a code is available that yields a stable throughput arbitrarily close to 1 message/slot. Furthermore, if the occurrence of collisions is made known to the transmitters, such a throughput can be maintained for arbitrary T,T >