We show that for a wide range of channels and code ensembles with pairwise-independent codewords, with probability tending to 1 with the code length, expurgating an arbitrarily small fraction of codewords from a randomly selected code results in a code attaining the expurgated exponent.

The statement of Theorem 1 in [1] should have read as follows.

This paper shows that the probability that the error exponent of a given code randomly generated from a pairwise-independent ensemble is smaller than a lower bound on the typical random-coding exponent tends to zero as the codeword length tends to infinity. This lower bound is known to be tight for i.i.d. ensembles over the binary symmetric channel and for constant-composition codes over memoryless channels. Our results recover both as special cases and remain valid for arbitrary alphabets, arbitrary channels—for example finite-state channels with memory—, and arbitrary pairwise-independent ensembles. We specialize our results to the i.i.d., constant-composition and cost-constrained ensembles over discrete memoryless channels and to ensembles over finite-state channels.

This paper shows that, for discrete memoryless channels, the error exponent of a randomly generated code with independent codewords converges in probability to its expectation—the typical error exponent. For high rates, the result follows from the fact that the random-coding error exponent and the sphere-packing error exponent coincide. For low rates, instead, the convergence is based on the fact that the union bound accurately characterizes the error probability. The paper also zooms into the behavior at asymptotically low rates, and shows that the normalized error exponent converges in distribution to the standard Gaussian or a Gaussian-like distribution. We also state several results on the convergence of the error probability and error exponent for generic ensembles and channels.

We derive a lower bound on the typical random-coding (TRC) exponent of pairwise-independent codeword ensembles used over a finite-state channel (FSC) at rates below capacity. Under some conditions, we also show that the probability of selecting a code from the ensemble with an error exponent larger than our lower bound tends to one as the codeword length tends to infinity. Our result, presented here for the FSC, also applies to compound channels.

We study the convergence in distribution of the error exponent of random codes, defined as the negative normalized logarithm of the probability of error, of both i.i.d. and constant-composition ensembles over discrete memoryless channels. For a constant number of messages, the distribution of the error exponent converges to that of the minimum of a set of independent normal random variables. For an increasing sub-exponential number of messages, the error exponent converges to a normal distribution, independent of the number of messages. As a byproduct, we provide a new method to prove the convergence to a normal distribution of an infinite number of random variables based on a modification of the Wasserstein metric.

For random-coding ensembles with pairwise-independent codewords, we show that the probability that the exponent of a given code from the ensemble being smaller than an upper bound on the typical random-coding exponent is vanishingly small. This upper bound is known to be tight for i.i.d. ensembles over the binary symmetric channel and for constant-composition codes over memoryless channels. Our result recovers these as special cases and remains valid for arbitrary alphabets and channel memory, as well as arbitrary ensembles with pairwise independent codewords.

This paper studies the error exponent of i.i.d. randomly generated codes used for transmission over discrete memoryless channels with maximum likelihood decoding. Specifically, this paper shows that the error exponent of a code, defined as the negative normalized logarithm of the probability of error, converges in probability to the typical error exponent. For high rates, the result is a consequence of the fact that the random-coding error exponent and the sphere-packing error exponent coincide. For low rates, instead, the proof of convergence is based on the fact that the union bound accurately characterizes the probability of error.

This paper proposes an efficient simulation method based on importance sampling to estimate the random-coding error probability of coded modulation. The technique is valid for complex-valued modulations over Gaussian channels, channels with memory, and naturally extends to fading channels. The simulation method is built on two nested importance samplers to respectively estimate the pairwise error probability and generate the channel input and output. The effect of the respective number of samples on the overall bias and variance of the estimate of the error probability is characterized. For a memoryless channel, the estimator is shown to be consistent and with a small variance, growing with the square root of the code length, rather than the exponential growth of a standard Monte Carlo estimator.

Saddlepoint approximations to the error probability are derived for multiple-cost-constrained random coding ensembles where codewords satisfy a set of constraints. Constant-composition inputs over a binary symmetric channel are studied as a particular case. For codewords with equiprobable empirical distribution, the analysis recovers the same error exponent and pre-exponential polynomial decay as the uniform i.i.d. ensemble and provides an explicit formula for the loss in prefactor (third-order term) incurred by the constant-composition ensemble.

This paper studies the random-coding exponent of joint source-channel coding for the multiple-access channel with correlated sources. For each user, by defining a threshold, the messages of each source are partitioned into two classes. The achievable exponent for correlated sources with two message-dependent input distributions for each user is determined and shown to be larger than that achieved using only one input distribution for each user. A system of equations is presented to determine the optimal thresholds maximizing the achievable exponent. The obtained exponent is compared with the one derived for the MAC with independent sources.

Saddlepoint approximations and expansions of the random coding union bound are derived for the i.i.d. random coding ensemble. Using the inverse Laplace transform of lattice and strongly non-lattice distributions, our results recover the random coding error exponent and refine the pre-exponential coefficient of the error probability. Explicit characterization of the terms are given for the binary symmetric channel and for the binary input AWGN channel.

Saddlepoint approximations to the pairwise error probability and to the random coding union bound are derived for the cost-constrained random coding ensemble. For the special case of the AWGN channel, an alternative expression to approximate the Shannon bound for optimal spherical codes is found.

In this paper, an achievable error exponent for the multiple-access channel with two independent sources is derived. For each user, the source messages are partitioned into two classes and codebooks are generated by drawing codewords from an input distribution depending on the class index of the source message. The partitioning thresholds that maximize the achievable exponent are given by the solution of a system of equations. We also derive both lower and upper bounds for the achievable exponent in terms of Gallager's source and channel functions. Finally, a numerical example shows that using the proposed ensemble gives a noticeable gain in terms of exponent with respect to independent identically distributed codebooks.

This paper provides an asymptotic expansion of the error probability, as the codeword length n goes to infinity, in quasi-static binary symmetric channels. After the leading term, namely the outage probability, the next two terms are found to be proportional to and respectively. Explicit characterizations of the respective coefficients are given. The resulting expansion gives an approximation to the random-coding union bound, accurate even at small codeword lengths.

This paper derives an achievable random-coding error exponent for joint source-channel coding over a multiple access channel with correlated sources. The codebooks are generated by drawing codewords from a multi-letter distribution that depends on the composition of the source message.

This paper studies the random-coding union (RCU) bound to the error probability in quasi-static fading channels. An asymptotic expansion and a normal approximation to the RCU bound suggest that the error probability converges to the outage probability as 1/n, where n is the codeword blocklength. We particularize our results for Rayleigh fading, and compare them with the conventional normal approximation.

This paper revisits the model order selection problem in the context of second-order spectrum sensing in cognitive radio. Taking advantage of the recent interest on the generalized likelihood ratio (GLR), the asymptotic performance of the minimum description length (MDL) rule under unknown noise variance is addressed. In particular, by exploiting the asymptotically Chi-squared distribution of the GLR, a complete characterization of the error probability is reported, instead of approximating only the missed-detection probability as done in the literature.

The frequency diversity exhibited by cyclostationary signals is exploited in this paper. A novel rank-1 frequency-domain representation of a digital waveform is proposed to address the generalized likelihood ratio (GLR) detection of a cyclo-stationary signal with unknown white noise. With the aim of avoiding the well-known sensitivity of cyclostationary-based detectors to frequency-selective fading channels, a parametric channel model based on the coherence bandwidth is adopted and incorporated in the GLR test. The proposed detector outperforms the classical spectral correlation magnitude detectors by exploiting the rank-1 structure of small spectral co-variance matrices.

This study addresses the optimal spectrum sensing detection based on the complete or partial side‐information on the signal and noise statistics. The use of the generalised‐likelihood ratio test (GLRT) involves maximum‐likelihood (ML) estimation of the nuisances. ML estimation of the unknowns is especially challenging for wideband cognitive radio because closed‐form solutions are often not available. Based on the equivalence between the wideband regime and the low‐signal‐to‐noise ratio regime, this study provides a general kernel framework for GLRT spectrum sensing. It is shown that any GLRT detector exclusively depends on the projection of the sample covariance matrix of the data onto a given underlying kernel that reflects the available side‐information in the problem. The kernels in several scenarios of interest are derived, including the widespread single and multi‐frequency channelisation cases. Theoretical interpretations and numerical results show the trade‐off between detection performance and the degree of side‐information on the most informative statistics for detection, that is, the modulation format and spectrum distribution of the primary users.

This paper addresses the problem of blind detection of a wide-sense stationary (WSS) signal over fading channels. We propose a test statistic which is optimal from a correlation-matching perspective that shows invariance with respect to the noise power and the channel gain. In the blind scenario, we derive the quadratic sphericity test (QST) which exploits the structure of the fading channel as a squared mean to arithmetic mean ratio of the eigenvalues of the autocorrelation matrix of the observations. We provide numerical results to assess the performance of the QST in several fading scenarios, as well as the benchmarking to other blind and non-blind detectors.

In this work, we show the existence of sampling walls in signal detection of nonuniformly sampling wideband signals in the presence of noise uncertainty. A sampling wall is the sampling rate below which the target error probabilities (namely the missed detection and false alarm probabilities) cannot be achieved at a given signal to noise ratio (SNR) regardless the number of acquired samples. Contrary to existing works, we address the signal detection as a binary hypotheses testing problem without having to reconstruct neither the signal nor the spectrum from the observations. Specifically, we adopt a Bernoulli distributed sampler as it exhibits good tradeoff properties between complexity and performance. We show that Bernoulli nonuniform sampling suffers from noise enhancement, which translates into a whitening effect in the correlation of the legacy signal. Therefore, in the presence of noise uncertainty, we derive explicit expressions for sampling walls as a function of the legacy signal occupation, the SNR and the noise uncertainty level. Finally, numerical results are further provided to assess the behavior of the sampling walls and signal detection performance.

The performance in signal detection is evaluated by the error (false-alarm and missed-detection) probabilities. However, calculating these probabilities is a difficult task in practice. This paper studies the asymptotic behavior of the energy-detector and the estimator-correlator by means of the Stein's lemma. The Stein's lemma is an information-theory result that provides the best achievable error exponent in the error probabilities when the number of observations goes to infinity. The derived closed-form expressions explain how detection performance is driven by the detector parameters and the second-order statistics of the problem. More specifically, it is shown that the error exponents depend on the signal-to-noise ratio (SNR) and the observation size. The prime focus is to establish a link between the required observation size for a fixed error probability as a function of the SNR. Numerical results show the tightness of the lemma.

The authors consider the robust Tomlinson–Harashima precoding (THP) for downlink multiuser multiple-input multiple-output orthogonal frequency-division multiplexing systems with quantised feedback. The authors discuss vector channel feedback strategies in the frequency and time domains, and develop a robust version of THP that takes into account of error statistics of the channel state information, that consists of the optimal feedforward filters, feedback filters and the receive filters. Feedback techniques are developed to exploit the spatial correlations in realistic 3GPP channel models by applying dimension reduction and scalar-quantisation. Extensive simulations results are provided to demonstrate the performance of the proposed robust THP design as well as the channel feedback scheme.

The spectrum analysis problem based on the periodogram and Capon estimates from a nonuniformly sampled signal is addressed. The nonuniform sampling process is cast as a linear projection matrix which encompasses several recent sampling strategies such as compressive sampling. A rank-1 correlation- matching approach is proposed to derive a general filter-bank framework that allows the formulation of the nonuniform periodogram and Capon estimates as particular cases. The performance of the novel periodogram and Capon estimates is analyzed in a cognitive radio scenario where the primary system is a multiband signal with sparse spectrum. Both theoretical and numerical results show that the denoising process required by correlation-matching plays an important role in spectral analysis based on nonuniform sampling.

In this paper, we consider a novel compressed correlation-matching (CCM) approach for spectrum sensing of wideband sparse signals. We derive a general closed-form estimate of the wideband sparse signal level from compressed observations, while providing physical interpretation of the problem. The formulation allows straightforward application to signal processing problems of interest, such as generalized likelihood ratio test (GLRT) spectrum sensing for wideband cognitive radio. Simulation results are reported to assess the behavior of the CCM method.

The problem of spectrum sensing in multi-frequency cognitive radio systems is addressed. We show that as the sensed bandwidth increases, the primary user detection is governed by a low signal-to-noise ratio (low-SNR) regime. By means of low-SNR approximations, we show that the optimal generalized likelihood ratio test (GLRT) only depends on the second order statistics of the observations and on a shaping kernel that highlights the relevant parameters required for detection. Furthermore, the ML estimates of the unknown model parameters are derived for multi-frequency systems, which allow closed-form expressions for the GLRT statistic. The detection performance and the kernel interpretation are supported with simulation results.

We consider the spectrum sensing problem in cognitive radio networks. We offer a framework for optimal joint detection and parameter estimation when the secondary users have only a small number of signal samples. We discuss the finite-sample optimality of the generalized likelihood ratio test (GLRT) and derive the corresponding GLRT spectrum sensing algorithms by exploiting the statistics of the received signal and the prior information on the channel, noise, as well as the data signal. An iterative GLRT sensing algorithm, and a simple non-iterative GLRT sensing algorithm are developed for slow and fast-fading channels, respectively, with the latter also serving as an approximate sensing method for slow-fading channels. The proposed techniques are also extended for spectrum sensing in orthogonal frequency-division multiple-access (OFDMA) systems and in multiple-input multiple-output (MIMO) systems. It is seen that the proposed simple non-iterative fast-fading GLRT sensing algorithm offers the best performance in all systems under considerations, including slow fading channels, fast fading channels, OFDMA systems, and MIMO systems, and it significantly outperforms several state-of-the-art spectrum sensing methods in these systems when there is noise uncertainty.