Discrete-time Gaussian Channel Research Articles

Private Key and Decoder Side Information for Secure and Private Source Coding.

We extend the problem of secure source coding by considering a remote source whose noisy measurements are correlated random variables used for secure source reconstruction. The main additions to the problem are as follows: (1) all terminals noncausally observe a noisy measurement of the remote source; (2) a private key is available to all legitimate terminals; (3) the public communication link between the encoder and decoder is rate-limited; and (4) the secrecy leakage to the eavesdropper is measured with respect to the encoder input, whereas the privacy leakage is measured with respect to the remote source. Exact rate regions are characterized for a lossy source coding problem with a private key, remote source, and decoder side information under security, privacy, communication, and distortion constraints. By replacing the distortion constraint with a reliability constraint, we obtain the exact rate region for the lossless case as well. Furthermore, the lossy rate region for scalar discrete-time Gaussian sources and measurement channels is established. An achievable lossy rate region that can be numerically computed is also provided for binary-input multiple additive discrete-time Gaussian noise measurement channels.

The Bit Error Performance and Information Transfer Rate of SPAD Array Optical Receivers

In this paper the photon counting characteristics, the information rate and the bit error performance of single-photon avalanche diode (SPAD) arrays are investigated. It is shown that for sufficiently large arrays, the photocount distribution is well approximated by a Gaussian distribution with dead-time-dependent mean and variance. Because of dead time, the SPAD array channel is subject to counting losses, part of which are due to inter-slot interference (ISI) distortions. Consequently, this channel has memory. The information rate of this channel is assessed. Two auxiliary discrete memoryless channels (DMCs) are proposed which provide upper and lower bounds on the SPAD array information rate. It is shown that in sufficiently large arrays, ISI is negligible and the bounds are tight. Under such conditions, the SPAD array channel is precisely modelled as a memoryless channel. A discrete-time Gaussian channel with input-dependent mean and variance is adopted and the properties of the capacity-achieving input distributions are studied. Using a numerical algorithm, the information rate and the capacity-achieving input distributions, subject to peak and average power constraints are obtained. Furthermore, the bit error performance of a SPAD-based system with on-off keying (OOK) is evaluated for various array sizes, dead times and background count levels.

Weak–Noise Modulation–Estimation of Vector Parameters

We address the problem of modulating a parameter onto a power–limited signal, transmitted over a discrete–time Gaussian channel and estimating this parameter at the receiver. Continuing an earlier work, where the optimal trade–off between the weak–noise estimation performance and the outage probability (threshold–effect breakdown) was studied for a single (scalar) parameter, here we extend the derivation of the weak–noise estimation performance to the case of a multi–dimensional vector parameter. This turns out to be a non–trivial extension, that provides a few insights, and it has some interesting implications, which are discussed in depth. Several modifications and extensions of the basic setup are also studied and discussed.

Tradeoffs Between Weak-Noise Estimation Performance and Outage Exponents in Nonlinear Modulation

We focus on the problem of modulating a parameter onto a power-limited signal transmitted over a discrete-time Gaussian channel and estimating this parameter at the receiver. Considering the well-known threshold effect in the non-linear modulation systems, our approach is the following: instead of deriving upper and lower bounds on the total estimation error, which weighs both weak-noise errors and anomalous errors beyond the threshold, we separate the two kinds of errors. In particular, we derive upper and lower bounds on the best achievable tradeoff between the exponential decay rate of the weak-noise expected error cost and the exponential decay rate of the probability of the anomalous error event, also referred to as the outage event . This outage event is left to be defined as a part of the communication system design problem. Our achievability scheme, which is based on lattice codes, meets the lower bound at the high signal-to-noise limit and for a certain range of tradeoffs between the weak-noise error cost and the outage exponent.

Relations Between Information and Estimation in Discrete-Time Lévy Channels

Fundamental relations between information and estimation have been established in the literature for the discrete-time Gaussian and Poisson channels. In this work, we demonstrate that such relations hold for a much larger class of observation models. We introduce the natural family of discrete-time L\'evy channels where the distribution of the output conditioned on the input is infinitely divisible. For L\'evy channels, we establish new representations relating the mutual information between the channel input and output to an optimal expected estimation loss, thereby unifying and considerably extending results from the Gaussian and Poisson settings. We demonstrate the richness of our results by working out two examples of L\'evy channels, namely the gamma channel and the negative binomial channel, with corresponding relations between information and estimation. Extensions to the setting of mismatched estimation are also presented.

Asymptotically equivalent sequences of matrices and capacity of a discrete-time Gaussian MIMO channel with memory

Using some recent results on asymptotically equivalent sequences of matrices, we present in this paper, a new derivation of the capacity formula given by Brandenburg and Wyner for a discrete-time Gaussian multiple-input-multiple-output channel with memory. In this paper, we tackle not only the case considered by them, where the number of channel inputs and the number of channel outputs are the same, but also when both numbers are different.

Discrete-time Gaussian interfere-relay channel

In practical wireless relay communication systems, nondestination nodes are assumed to be idle not receiving signals while the relay sends messages to a particular destination node, which results in reduced bandwidth efficiency. To improve the bandwidth efficiency, we relax the idle assumption of non-destination nodes and assume that non-destination nodes may receive signals from sources. We note that the message relayed to a particular node in such a system gives rise to interference to other nodes. To study such a more general relay system, we consider, in this paper, a relay system in which the relay first listens to the source, then routes the source message to the destination, and finally produces interference to the destination in sending messages for other systems. We obtain capacity upper and lower bounds and study the optimal method to deal with the interference as well as the optimal routing schemes. From analytic results obtained, we find the conditions on which the direct transmission provides higher transmission rate. Next, we find the conditions, by numerical evaluation of the theoretical results, on which it is better for the destination to cancel and decode the interference. Also we find the optimal source power allocation scheme that achieves the lower bound depending on various channel conditions. We believe that the results provided in this paper will provide useful insights to system designers in strategically choosing the optimal routing algorithms depending on the channel conditions.

Capacity of Gaussian Channels With Duty Cycle and Power Constraints

In many wireless communication systems, radios are subject to a duty cycle constraint, that is, a radio can only actively transmit signals over a fraction of the time. For example, it is desirable to have a small duty cycle in some low power systems; a half-duplex radio cannot keep transmitting if it wishes to receive useful signals; and a cognitive radio needs to listen and detect primary users frequently. This paper studies the capacity of point-to-point scalar discrete-time Gaussian channels subject to a duty cycle constraint as well as an average transmit power constraint. An idealized duty cycle constraint is first studied, which can be regarded as a requirement on the minimum fraction of nontransmissions or zero symbols in each codeword. Independent input with a unique discrete distribution is shown to achieve the channel capacity. In many situations, numerically optimized on-off signaling can achieve much higher rate than Gaussian signaling over a deterministic transmission schedule. This is in part because the positions of nontransmissions in a codeword can convey information. A more realistic duty cycle constraint is also studied, where the extra cost of transitions between transmissions and nontransmissions due to pulse shaping is accounted for. The capacity-achieving input is correlated over time and is hard to compute. A lower bound of the achievable rate as a function of the input distribution is shown to be maximized by a first-order Markov input process, whose stationary distribution is also discrete and can be computed efficiently. The results in this paper suggest that, under various duty cycle constraints, departing from the usual paradigm of intermittent packet transmissions may yield substantial gain.

At Low SNR, Asymmetric Quantizers are Better

We study the capacity of the discrete-time Gaussian channel when its output is quantized with a 1-bit quantizer. We focus on the low signal-to-noise ratio (SNR) regime, where communication at very low spectral efficiencies takes place. In this regime, a symmetric threshold quantizer is known to reduce channel capacity by a factor of 2/π, i.e., to cause an asymptotic power loss of approximately 2 dB. Here, it is shown that this power loss can be avoided by using asymmetric threshold quantizers and asymmetric signaling constellations. To avoid this power loss, flash-signaling input distributions are essential. Consequently, 1-bit output quantization of the Gaussian channel reduces spectral efficiency. Threshold quantizers are not only asymptotically optimal: at every fixed SNR, a threshold quantizer maximizes capacity among all 1-bit output quantizers. The picture changes on the Rayleigh-fading channel. In the noncoherent case, a 1-bit output quantizer causes an unavoidable low-SNR asymptotic power loss. In the coherent case, however, this power loss is avoidable provided that we allow the quantizer to depend on the fading level.

Bounds on the Information Rate for Sparse Channels with Long Memory and i.u.d. Inputs

In this paper we propose new bounds on the achievable information rate for discrete-time Gaussian channels with intersymbol interference (ISI) and independent and uniformly distributed (i.u.d.) channel input symbols drawn from finite-order modulation alphabets. Specifically, we are interested in developing new bounds on the achievable rates for sparse channels with long memory. We obtain a lower bound which can be achieved by practical receivers, based on MMSE channel shortening and suboptimal symbol detection for a reduced-state channel. An upper bound is given in the form of a semi-analytical solution derived using basic information theoretic inequalities, by a grouping of the channel taps into several clusters resulting in a newly defined single-input multiple-output (SIMO) channel. We show that the so obtained time-dispersive SIMO channel can be represented by an equivalent single-input single-output (SISO) channel with a significantly shorter channel memory. The reduced computational complexity allows the use of the BCJR algorithm for the newly defined channel. The proposed bounds are illustrated through several sparse channel examples and i.u.d. input symbols, showing that the upper bound significantly outperforms existing bounds. Performance of our lower bound strongly depends on the channel structure, showing best results for minimum-phase and maximum-phase systems.

Communication by Energy Modulation: The Additive Exponential Noise Channel

Communication across an additive exponential noise (AEN) channel is studied. The AEN channel is derived from a continuous-time Gaussian channel by assuming decoherence between the signal and noise components. Decoherence renders impossible the use of the complex amplitude and information is transmitted by modulating the signal energy. The AEN channel is shown to perform closely - in terms of channel, information rate, and error probability of binary codes, including the use of bit-interleaved coded modulation - to an equivalent discrete-time Gaussian channel of identical signal-to-noise ratio. Constellations are designed for low signal-to-noise ratios, where the minimum energy per bit is not universally attained by all modulations, and for high signal-to-noise ratios, where an equiprobable nonuniform constellation 0.76 dB away from capacity is described. The multiple-access AEN channel is also considered: its capacity region coincides with that of the Gaussian equivalent; feedback does not, however, enlarge the capacity region.

Refinements of the half-bit and factor-of-two bounds for capacity in Gaussian channel with feedback

We consider the upper bounds of the finite blocklength capacity C/sub n,FB/(P) of the discrete time Gaussian channel with feedback. We also let C/sub n/(p) be the nonfeedback capacity. We prove the relations C/sub n/(P)/spl les/C/sub n,FB/(P)/spl les/C/sub n/(/spl alpha/P)+ 1/2 ln(1+1//spl alpha/) and C/sub n/(P)/spl les/C/sub n,FB/(P)/spl les/(1+1//spl alpha/)C/sub n/(/spl alpha/P) for any P>0 and any /spl alpha/>0, which induce the half-bit and factor-of-two bounds given by Cover and Pombra (1989) in the special case of /spl alpha/=1.

On the capacity of the discrete-time Gaussian channel with delayed feedback

Gives an upper bound on the finite block length capacity of the discrete-time nonstationary Gaussian channel with delayed feedback. With the aid of minimization of a quadratic form, it is proved that the L time-delayed feedback capacity C/sub nFB//sup L/(P) and the nonfeedback capacity C/sub n/(P) satisfy C/sub n/(P)/spl les/C/sub nFB//sup L/(P)/spl les/C/sub n/(P*) where P* is concretely given. The authors give a sufficient condition for C/sub nFB//sup L/(P) to be increased by L time-delayed feedback. Finally the authors also give a necessary and sufficient condition for C/sub nFB//sup 2/(P) of a class of Gaussian channel to be increased by 2 time-delayed feedback. >

An upper bound to the capacity of discrete time Gaussian channel with feedback. II

We give an upper bound on the finite block length capacity of a discrete time nonstationary Gaussian channel with feedback. With the aid of minimization of a quadratic form, it is proved that the feedback capacity C/sub n,FB/(P) and the nonfeedback capacity C/sub n/(P) satisfy C/sub n/(P)/spl les/C/sub n,FB/(P)/spl les/C/sub n/(P/sup */) where P/sup */ is concretely given. >

Necessary and sufficient condition for capacity of the discrete time Gaussian channel to be increased by feedback

The problem of whether the capacity of a discrete-time Gaussian channel is increased by feedback or not is considered. It is well known that the capacity of a white Gaussian channel under an average power constraint is not changed by feedback. The necessary condition under which the capacity of a nonwhite Gaussian channel with blockwise white noise is increased by feedback is given. >

Exponential error bounds for random codes on Gaussian arbitrarily varying channels

The main objective is to develop exponential bounds to the best error probability achievable with random coding on the Gaussian arbitrarily varying channel (GAVC) in the one case where a (strong) capacity exists (i.e., with peak time-averaged power constraints on both the transmitter and interference). The GAVC models a channel corrupted by thermal noise and by an unknown interfering signal of bounded power. The upper and lower bounds to the best error probability achievable on this channel with random coding are presented. The asymptotic exponents of these bounds agree in a range of rates near capacity. The exponents are universally larger than the corresponding exponents for the discrete-time Gaussian channel with the same capacity. It is further shown that the decoder can be taken to be the minimum Euclidean distance rule at all rates less than capacity. >

Coding capacity for a class of additive channels

Coding capacity is considered for a class of additive dimension-limited channels. The channels may be with or without memory, stationary or nonstationary. The constraint is partially given in terms of an increasing family of finite-dimensional subspaces. A general expression for the capacity is obtained, which depends on the family of subspaces and the relation between the noise covariance and the covariance giving the energy-frequency constraint on the transmitted signal. This result holds for all classical discrete-time Gaussian channels and for continuous-time Gaussian channels with fixed time of transmission, so long as a peak energy constraint is used on the codewords. The expression also provides upper bounds for a class of non-Gaussian channels. Several results are obtained that aid in calculation of capacity for specific applications. For this class of channels, it is shown that coding capacity is equal to information capacity. Error bounds are given for Gaussian channels.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Information rates for a discrete-time Gaussian channel with intersymbol interference and stationary inputs

Bounds are presented on I/sub i.i.d./-the achievable information rate for a discrete Gaussian Channel with intersymbol interference (ISI) present and i.i.d. channel input symbols governed by an arbitrary predetermined distribution p/sub x/(x). The lower and upper bounds on I/sub i.i.d./ and I are formulated. The bounds on I/sub i.i.d./ are calculated for independent equiprobably binary channel symbols and for causal channels with ISI memory of degree one and two. The bounds on I/sub i.i.d./ are compared to the approximated (by Monte Carlo methods) known value of I/sub i.i.d./ and their tightness is considered. An application of the new lower bound on I/sub i.i.d./ yields an improvement on previously reported lower bounds for the capacity of the continuous-time strictly bandlimited (or bandpass) Gaussian channel with either peak or simultaneously peak power and bandlimiting constraints imposed on the channel's input waveform.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The capacity region of the symbol-asynchronous Gaussian multiple-access channel

An equivalent discrete-time Gaussian channel parametrized by the signal cross-correlations is derived to obtain an equivalent channel model with discrete-time outputs. The main feature introduced by the lack of symbol synchronism is that the channel has memory. This is due to the overlap of each symbol transmitted by a user with two consecutive symbols transmitted by the other user. It is shown that if the transmitters are assigned the same waveform, symbol asynchronism has no effect on the two-user capacity region of the white Gaussian channel which is equal to the Cover-Syner pentagon, whereas if the assigned waveforms are different (e.g., code division multiple access), the symbol-asynchronous capacity region is no longer a pentagon. An alternative representation of the capacity region which results in a particularly compact characterization of the fundamental limits of the multiple-access channel in the region of signal-to-noise ratios is also considered. >

Capacity of discrete time Gaussian channel with and without feedback, II

The problem considered in this paper is whether the capacity of a discrete time Gaussian channel is increased by feedback or not. It is well known that the capacity of a white Gaussian channel under the average power constraint is not changed by feedback. We give some conditions under which the capacity of a discrete time Gaussian channel is increased by feedback. It is also shown that there exists a non-white Gaussian channel whose capacity is not increased by feedback.