Memoryless Gaussian Channel Research Articles

Dynamic signaling games with quadratic criteria under Nash and Stackelberg equilibria

This paper considers dynamic (multi-stage) signaling games involving an encoder and a decoder who have subjective models on the cost functions. We consider both Nash (simultaneous-move) and Stackelberg (leader–follower) equilibria of dynamic signaling games under quadratic criteria. For the multi-stage scalar cheap talk, we show that the final stage equilibrium is always quantized and under further conditions the equilibria for all time stages must be quantized. In contrast, the Stackelberg equilibria are always fully revealing. In the multi-stage signaling game where the transmission of a Gauss–Markov source over a memoryless Gaussian channel is considered, affine policies constitute an invariant subspace under best response maps for Nash equilibria; whereas the Stackelberg equilibria always admit linear policies for scalar sources but such policies may be non-linear for multi-dimensional sources. We obtain an explicit recursion for optimal linear encoding policies for multi-dimensional sources, and derive conditions under which Stackelberg equilibria are informative.

An Achievable Region for the Cognitive Interference Relay Channel

In this paper, the causal cognitive interference relay channel (CCIRC) is considered. The CCIRC is obtained from the classic cognitive interference channel by introducing a full-duplex relay that has causal and noisy access to the transmitted signals and aids the communication between the two users. For this channel model, a simple upper bound on the capacity is initially established. Then, we derive an achievable rate region that jointly applies cooperative Han–Kobayashi rate splitting, block Markov encoding, superposition coding, Marton’s binning encoding, and backward decoding. In particular, these bounds are derived for both discrete memoryless model and Gaussian channel. Further, both our theoretical and numerical results indicate that the derived achievable rate region encompasses all known available results in the literature.

On the Capacity of Cloud Radio Access Networks With Oblivious Relaying

We study the transmission over a network in which users send information to a remote destination through relay nodes that are connected to the destination via finite-capacity error-free links, i.e., a cloud radio access network. The relays are constrained to operate without knowledge of the users’ codebooks, i.e., they perform oblivious processing. The destination, or central processor, however, is informed about the users’ codebooks. We establish a single-letter characterization of the capacity region of this model for a class of discrete memoryless channels in which the outputs at the relay nodes are independent given the users’ inputs. We show that both relaying a-la Cover–El Gamal, i.e., compress-and-forward with joint decompression and decoding, and “noisy network coding” are optimal. The proof of the converse part establishes, and utilizes, connections with the Chief Executive Officer source coding problem under logarithmic loss distortion measure. Extensions to general discrete memoryless channels are also investigated. In this case, we establish the inner and outer bounds on the capacity region. For memoryless Gaussian channels within the studied class of channels, we characterize the capacity region when the users are constrained to time-share among Gaussian codebooks. Furthermore, we also discuss the suboptimality of separate decompression and decoding and the role of time sharing.

Joint Source–Channel Coding for Broadcasting Correlated Sources

This paper studies lossy transmission of a memoryless bivariate Gaussian source over a bandwidth-mismatched memoryless Gaussian broadcast channel with two receivers, where each receiver is interested in reconstructing only one source component. For both bandwidth expansion and compression regimes, novel hybrid digital/analog (HDA) coding schemes are proposed. With appropriate choice of parameters, our schemes are shown to specialize to separate source–channel coding studied by Gao and Tuncel, and is, therefore, superior to it in both bandwidth regimes. Our scheme for bandwidth expansion also outperforms the HDA coding scheme of Behroozi et al. On the other hand, if a proposed conjecture (supported by numerical observations) is indeed true, the same superiority follows for the bandwidth compression regime as well. Finally, when the bandwidth expansion/compression ratio approaches 1, both of our schemes become optimal as their performance approaches that of the bandwidth-matched scheme of Tian et al.

Joint Source-Channel Coding With Feedback

This paper quantifies the fundamental limits of variable-length transmission of a general (possibly analog) source over a memoryless channel with noiseless feedback, under a distortion constraint. We consider excess distortion, average distortion, and guaranteed distortion ( $d$ -semifaithful codes). In contrast to the asymptotic fundamental limit, a general conclusion is that allowing variable-length codes and feedback leads to a sizable improvement in the fundamental delay-distortion tradeoff. In addition, we investigate the minimum energy required to reproduce $k$ source samples with a given fidelity after transmission over a memoryless Gaussian channel, and we show that the required minimum energy is reduced with feedback and an average (rather than maximal) power constraint.

On the Capacity of Vector Gaussian Channels With Bounded Inputs

The capacity of a deterministic multiple-input multiple-output channel under the peak and average power constraints is investigated. For the identity channel matrix, the approach of Shamai et al. is generalized to the higher dimension settings to derive the necessary and sufficient conditions for the optimal input probability density function. This approach prevents the usage of the identity theorem of the holomorphic functions of several complex variables which seems to fail in the multi-dimensional scenarios. It is proved that the support of the capacity-achieving distribution is a finite set of hyper-spheres with mutual independent phases and amplitude in the spherical domain. Subsequently, it is shown that when the average power constraint is relaxed, if the number of antennas is large enough, the capacity has a closed-form solution and constant amplitude signaling at the peak power achieves it. Moreover, it will be observed that in a discrete-time memoryless Gaussian channel, the average power constrained capacity, which results from a Gaussian input distribution, can be closely obtained by an input where the support of its magnitude is a discrete finite set. Finally, we investigate some upper and lower bounds for the capacity of the non-identity channel matrix and evaluate their performance as a function of the condition number of the channel.

Cognitive interference channel: achievable rate region and power allocation

In this study, the authors consider a state-dependent two user interference channel. The two users sharing the spectrum are assumed to be cognitive and each user has a non-causal access to the signal from the other user. For this channel model, an achievable rate region is established for both discrete memoryless model and Gaussian channel. In particular, the achievable rate region is obtained by combining Han–Kobayashi rate splitting coding scheme, superposition coding, Gelfand–Pinsker coding scheme and zero-forcing dirty paper coding. Furthermore, the sum rate maximisation and the associated power allocation problem are studied, numerically and theoretically. The corresponding numerical examples show that the proposed combined coding scheme outperforms the existing schemes in the sense of achievable rate region. Moreover, the effectiveness of the optimal power allocation between the two cognitive nodes is also shown.

List Decoding—Random Coding Exponents and Expurgated Exponents

Some new results are derived concerning random coding error exponents and expurgated exponents for list decoding with a deterministic list-size L. Two asymptotic regimes are considered, the fixed list-size regime, where L is fixed independently of the block length n, and exponential list size, where L grows exponentially with n. We first derive a general upper bound on the list-decoding average error probability, which is suitable for both regimes. This bound leads to more specific bounds in the two regimes. In the fixed list-size regime, the bound is related to known bounds and we establish its exponential tightness. In the exponential list-size regime, we establish the achievability of the well-known sphere packing lower bound. An immediate byproduct of our analysis in both regimes is the universality of the maximum mutual information list decoder in the error exponent sense. Finally, we consider expurgated bounds at low rates, both using Gallager's approach and Csiszár-Körner-Marton approach, which is equivalent for the optimal input assignment. The latter expurgated bound, which involves the notion of multi-information, is also modified to apply to continuous alphabet channels, and in particular, to the Gaussian memoryless channel, where the expression of the expurgated bound becomes quite explicit.

Insufficiency of Linear-Feedback Schemes in Gaussian Broadcast Channels With Common Message

We consider the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(K\geq 2\) </tex-math></inline-formula> -user memoryless Gaussian broadcast channel (BC) with feedback and common message only. We show that linear-feedback schemes with a message point, in the spirit of Schalkwijk and Kailath’s scheme for point-to-point channels or Ozarow and Leung’s scheme for BCs with private messages, are strictly suboptimal for this setup. Even with perfect feedback, the largest rate achieved by these schemes is strictly smaller than capacity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(C\) </tex-math></inline-formula> (which is the same with and without feedback). In the extreme case where the number of receivers <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(K\to \infty \) </tex-math></inline-formula> , the largest rate achieved by linear-feedback schemes with a message point tends to 0. To contrast this negative result, we describe a scheme for rate-limited feedback that uses the feedback in an intermittent way, i.e., the receivers send feedback signals only in few channel uses. This scheme achieves all rates <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(R\) </tex-math></inline-formula> up to capacity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(C\) </tex-math></inline-formula> with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(L\) </tex-math></inline-formula> th order exponential decay of the probability of error if the feedback rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(R_{\text {fb}}\) </tex-math></inline-formula> is at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\((L-1)R\) </tex-math></inline-formula> for some positive integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(L\) </tex-math></inline-formula> .

Coding Schemes and Asymptotic Capacity for the Gaussian Broadcast and Interference Channels With Feedback

A coding scheme is proposed for the memoryless Gaussian broadcast channel with correlated noises and feedback. For all noise correlations other than ±1, the gap between the sum-rate that the scheme achieves and the full-cooperation bound vanishes as the signal-to-noise ratio tends to infinity. When the correlation coefficient is -1, the gains afforded by feedback are unbounded and the prelog is doubled. When the correlation coefficient is +1, we demonstrate a dichotomy that if the noise variances are equal, then feedback is useless, and otherwise, feedback affords unbounded rate gains and doubles the prelog. The unbounded feedback gains, however, require perfect (noiseless) feedback. When the feedback links are noisy, the feedback gains are bounded, unless the feedback noise decays to zero sufficiently fast with the signal-to-noise ratio. Extensions to more receivers are also discussed as is the memoryless Gaussian interference channel with feedback.

On optimal policies for control and estimation over a Gaussian relay channel

The problem of causal transmission of a memoryless Gaussian source over a two-hop memoryless Gaussian relay channel is considered. The source and the relay encoders have average transmit power constraints, and the performance criterion is mean-squared distortion. The main contribution of this paper is to show that unlike the case of a point-to-point scalar Gaussian channel, linear encoding schemes are not optimal over a two-hop relay channel in general, extending the sub-optimality results which are known for more than three hops. In some cases, simple three-level quantization policies employed at the source and at the relay can outperform the best linear policies. Further a lower bound on the distortion is derived and it is shown that the distortion bounds derived using cut-set arguments are not tight in general for sensor networks.

Secure Communication Over Parallel Relay Channel

We investigate the problem of secure communication over parallel relay channel in the presence of a passive eavesdropper. We consider a four terminal relay-eavesdropper channel which consists of multiple relay-eavesdropper channels as subchannels. For the discrete memoryless model, we establish outer and inner bounds on the rate-equivocation region. The inner bound allows mode selection at the relay. For each subchannel, secure transmission is obtained through one of two coding schemes at the relay: decoding-and-forwarding the source message or confusing the eavesdropper through noise injection. For the Gaussian memoryless channel, we establish lower and upper bounds on the perfect secrecy rate. Furthermore, we study a special case in which the relay does not hear the source and show that under certain conditions the lower and upper bounds coincide. The results established for the parallel Gaussian relay-eavesdropper channel are then applied to study the fading relay-eavesdropper channel. Analytical results are illustrated through some numerical examples.

Maximum Likelihood Based Measurement of Interference Level and Noise Power for Memoryless Gaussian Channel with Deterministic Interference

The model of memoryless Gaussian channel with deterministic interference are common in various engineering applications. In this letter we propose a non-data-aided algorithm for joint measurement of the interference level and the noise power. The algorithm applies an iterative approach based on the maximum likelihood criterion. The Cramer-Rao bound (CRB) is also derived as the performance benchmark. Simulation results show that the proposed approach has fast convergence speed and its measurement performance is almost identical to the CRB.

Minimum Energy to Send $k$ Bits Through the Gaussian Channel With and Without Feedback

The minimum achievable energy per bit over memoryless Gaussian channels has been previously addressed in the limit when the number of information bits goes to infinity, in which case it is known that the availability of noiseless feedback does not lower the minimum energy per bit, which is -1.59 dB below the noise level. This paper analyzes the behavior of the minimum energy per bit for memoryless Gaussian channels as a function of k, the number of information bits. It is demonstrated that in this nonasymptotic regime, noiseless feedback leads to significantly better energy efficiency. In particular, without feedback achieving energy per bit of -1.57 dB requires coding over at least k=106 information bits, while we construct a feedback scheme that transmits a single information bit with energy -1.59 dB and zero error. We also show that unless k is very small, approaching the minimal energy per bit does not require using the feedback link except to signal that transmission should stop.

Stabilization and Disturbance Attenuation Over a Gaussian Communication Channel

We consider the problem of stabilizing an unstable system driven by a Gaussian disturbance using a feedback signal transmitted over a memoryless Gaussian communication channel. By applying the concept of entropy power, we show that the mean square norm of the state vector must satisfy a lower bound that holds for any causal, measurable communication and control strategies that result in signals having well defined differential entropy. In addition, we show that use of nonlinear, time varying strategies does not allow stabilization over a channel with a lower signal-to-noise ratio than that achievable with linear time invariant state feedback. Finally, we show that for scalar systems the lower bound on the mean square norm of the state is tight, and achievable using linear time invariant communication and control.

On the optimal transmit strategy for the MIMO bidirectional broadcast channel

In this work the transmit covariance matrix optimization problem for the discrete memoryless MIMO Gaussian bidirectional broadcast channel is studied. A half-duplex relay node establishes bidirectional communication between two nodes using a decode-and-forward protocol. In the initial multiple access phase both nodes transmit their messages to the relay node. In the succeeding phase the relay broadcasts an optimal re-encoded message so that both nodes can decode the other's message using their own message as side information. The capacity region of the bidirectional broadcast channel is completely characterized by a weighted rate sum maximization problem, which can be solved by a simple iterative fixed point algorithm. If an efficient transmit covariance matrix is invariant with respect to the joint subspace spanned by the channels, then different combinations of the part transmitted on the orthogonal subspaces result in equivalent transmit strategies with different ranks. A closed-form procedure to obtain the optimal transmit covariance is derived for the case where the rank of the channels is equal to the number of antennas at the relay node and a full-rank transmission is optimal. It shows the complicated structure of the optimal eigenspace, which depends on the weights and the mean transmit power constraint. For parallel channels the optimal solution is completely characterized and discussed, which also solves the optimal power allocation problem for a single-antenna OFDM system.

Hybrid digital-analog coding with bandwidth compression for gaussian source-channel pairs

Three hybrid digital-analog (HDA) systems, denoted by HDA-I, HDA* and HDA-II, for the coding of a memoryless discrete-time Gaussian source over a discrete-time additive memoryless Gaussian channel under bandwidth compression are studied. The systems employ simple linear coding in their analog component and superimpose their analog and digital signals before channel transmission. Information-theoretic upper bounds on the asymptotically optimal mean squared error distortion of the systems are obtained under both matched and mismatched channel conditions. Allocation schemes for distributing the channel input power between the analog and the digital signals are also examined. It is shown that systems HDA* and HDA-II can asymptotically achieve the optimal Shannon-limit performance under matched channel conditions. Low-complexity and low-delay versions of systems HDA-I and HDA-II are next designed and implemented without the use of error correcting codes. The parameters of these HDA systems, which employ vector quantization in conjunction with binary phase-shift keying modulation in their digital part, are optimized via an iterative algorithm similar to the design algorithm for channel-optimized vector quantizers. Both systems have low complexity and low delay, and guarantee graceful performance improvements for high CSNRs. For memoryless Gaussian sources the designed HDA-II system is shown to be superior to the HDA-I designed system. When applied to a Gauss-Markov source under Karhunen-Loeve processing, the HDA-I system is shown to provide considerably better performance.

Secret Key Agreement by Soft-Decision of Signals in Gaussian Maurer's Model

We consider the problem of secret key agreement in Gaussian Maurer's Model. In Gaussian Maurer's model, legitimate receivers, Alice and Bob, and a wire-tapper, Eve, receive signals randomly generated by a satellite through three independent memoryless Gaussian channels respectively. Then Alice and Bob generate a common secret key from their received signals. In this model, we propose a protocol for generating a common secret key by using the result of soft-decision of Alice and Bob's received signals. Then, we calculate a lower bound on the secret key rate in our proposed protocol. As a result of comparison with the protocol that only uses hard-decision, we found that the higher rate is obtained by using our protocol.

Novel Low-Density Signature for Synchronous CDMA Systems Over AWGN Channel

Novel low-density signature (LDS) structure is proposed for transmission and detection of symbol-synchronous communication over memoryless Gaussian channel. Given N as the processing gain, under this new arrangement, users' symbols are spread over N chips but virtually only dv < N chips that contain nonzero-values. The spread symbol is then so uniquely interleaved as the sampled, at chip rate, received signal contains the contribution from only dc < K number of users, where K denotes the total number of users in the system. Furthermore, a near-optimum chip-level iterative soft-in-soft-out (SISO) multiuser decoding (MUD), which is based on message passing algorithm (MPA) technique, is proposed to approximate optimum detection by efficiently exploiting the LDS structure. Given beta = K/N as the system loading, our simulation suggested that the proposed system alongside the proposed detection technique, in AWGN channel, can achieve an overall performance that is close to single-user performance, even when the system has 200% loading, i.e., when beta = 2. Its robustness against near-far effect and its performance behavior that is very similar to optimum detection are demonstrated in this paper. In addition, the complexity required for detection is now exponential to dc instead of K as in conventional code division multiple access (CDMA) structure employing optimum multiuser detector.

The Capacity Region of Frequency-Selective Gaussian Interference Channels Under Strong Interference

This paper presents the capacity region of frequency-selective Gaussian interference channels under the condition of strong interference, assuming an average power constraint per user. First, a frequency-selective Gaussian interference channel is modeled as a set of independent parallel memoryless Gaussian interference channels. Using nonfrequency selective results, the capacity region of frequency-selective Gaussian interference channels under strong interference is expressed mathematically. Exploiting structures inherent in the problem, a dual problem is constructed for each independent memoryless channel, in which both mathematical and numerical analysis are performed. Furthermore, three suboptimal methods are compared to the capacity-achieving coding and power allocation scheme. Iterative waterfilling, a suboptimal scheme, provides close-to-optimum performance and has a distributed coding and power allocation scheme, which are attractive in practice.