Variations on the Gallager bounds, connections, and applications

There has been renewed interest in deriving tight bounds on the error performance of specific codes and ensembles, based on their distance spectrum. We discuss many reported upper bounds on the maximum-likelihood (ML) decoding error probability and demonstrate the underlying connections that exist between them. In addressing the Gallager bounds and their variations, we focus on the Duman and Salehi (see IEEE Trans. Commun., vol.46, p.717-723, 1998)variation, which originates from the standard Gallager bound. A large class of efficient bounds (or their Chernoff versions) is demonstrated to be a special case of the generalized second version of the Duman and Salehi bounds. Implications and applications of these observations are pointed out, including the fully interleaved fading channel, resorting to either matched or mismatched decoding. The proposed approach can be generalized to geometrically uniform nonbinary codes, finite-state channels, bit interleaved coded modulation systems, and it can be also used for the derivation of upper bounds on the conditional decoding error probability.     

Chernoff bounds on pairwise error probabilities of space-time codes

We derive Chernoff bounds on pairwise error probabilities of coherent and noncoherent space-time signaling schemes. First, general Chernoff bound expressions are derived for a correlated Ricean fading channel and correlated additive Gaussian noise. Then, we specialize the obtained results to the cases of space-time-separable noise, white noise, and uncorrelated fading. We derive approximate Chernoff bounds for high and low signal-to-noise ratios (SNRs) and propose optimal signaling schemes. We also compute the optimal number of transmitter antennas for noncoherent signaling with unitary mutually orthogonal space-time codes.     

Error performance of MPSK trellis-coded modulation overnonindependent Rician fading channels

This paper presents new upper bounds on the pairwise error probability (PEP) of trellis-coded modulation (TCM) schemes over nonindependent Rician fading channels. Cases considered are coherent and pilot-tone-aided detection and differential detection of trellis-coded multilevel phase-shift keying (TC-MPSK) systems. The average bit-error probability P_b can be approximated by truncating the union bound. This method does not necessarily lead to an upper bound on P_b, and, hence, the approximation must be used with simulation results. In addition, for Rayleigh fading channels with an exponential autocovariance function, bounds resembling those for memoryless channels have been derived. The bounds are substantially more accurate than Chernoff bounds and hence allow for accurate estimation of system performance when the assumption of ideal interleaving is relaxed     

A new twist on the Marcum Q-function and its application

A new form of the Marcum (1950) Q-function is presented that has both computational and analytical advantages. The new form is particularly useful in simplifying and rendering more accurate the analysis of the error probability performance of uncoded and coded partially coherent, differentially coherent, and noncoherent communication systems in the presence of fading. It also enables simple upper and lower bounds to be found analogous to the Chernoff bound on the Gaussian Q-function     

Cauchy–Schwarz bound on the generalized Marcum Q‐function with applications

The Cauchy–Schwarz bounding technique is used to derive useful bounds on the generalized Marcum Q‐function and its complement. Three new exponential‐type bounds on Q_M (α, β) are derived, and these are found to be tight and useful for a number of applications of interest. One such example is the derivation of an upper bound on the average symbol error rate probability for noncoherent and differentially coherent communication systems over generalized fading channels. It is shown that these exponential‐type bounds are considerably tighter than the Chernoff bound (Rappaport SS. IEEE Trans. on Information Theory 1971; 17 : 497–498) counterpart. Numerical results also reveal that the tightness of one of the exponential‐type bounds is comparable to the bound obtained in Simon and Alouini (IEEE Trans. on Communications 2000: 359–366), while another is found to be superior than that obtained in Simon and Alouini over a wide range of arguments. Copyright © 2001 John Wiley & Sons, Ltd.     

Performance of trellis-coded CPM with iterative demodulation anddecoding

Interleaved trellis-coded systems with full response continuous-phase modulation (CPM) are considered. Upper bounds on the bit-error rate performance are derived for coherent detection on the additive white Gaussian noise and flat Rayleigh fading channels by considering the trellis code, interleaver, and CPM modulator as a serially concatenated convolutional code. A coherent receiver that performs iterative demodulation and decoding is shown to provide good bit error performance. Finally, a noncoherent iterative receiver is proposed and is shown to perform close to the coherent iterative receiver     

Bounds on the error performance of coding for nonindependentRician-fading channels

New upper bounds on the error performance of coded systems for Rician channels are presented. The fading channels need not be fully interleaved to obtain meaningful performance results. These bounds hold for coherent, differentially coherent and noncoherent demodulation of binary signals. They provide a useful analytical approach to the evaluation of the error performance of convolutional or block coding and they may be generalized to M-ary signals and trellis modulation. The approach allows for complex bounds using the fine structure of the code, for simpler bounds similar to those on memoryless channels and finally for a random coding bound using the cutoff rate of the channel. The analysis thus permits a step by step evaluation of coded error performances for Rician-fading channels     

Exponential-type bounds on the generalized Marcum Q-function withapplication to error probability analysis over fading channels

Strict upper and lower bounds of exponential-type are derived for the generalized (mth order) Marcum Q-function which enable simple evaluation of a tight upper bound on the average bit-error probability performance of a wide class of noncoherent and differentially coherent communication systems operating over generalized fading channels. For the case of frequency selective fading with arbitrary statistics per independent fading path, the resulting upper hound on performance is expressed in the form of a product of moment generating functions of the instantaneous power random variables that characterize these paths     

Performance analysis on LDPC-Coded systems over quasi-static (MIMO) fading channels

In this paper, we derive closed form upper bounds on the error probability of low-density parity-check (LDPC) coded modulation schemes operating on quasi-static fading channels. The bounds are obtained from the so-called Fano- Gallager?s tight bounding techniques, and can be readily calculated when the distance spectrum of the code is available. In deriving the bounds for multiple-input multiple-output (MIMO) systems, we assume the LDPC code is concatenated with the orthogonal space-time block code as an inner code. We obtain an equivalent single-input single-output (SISO) channel model for this concatenated coded-modulation system. The upper bounds derived here indicate good matches with simulation results of a complete transceiver system over Rayleigh and Rician MIMO fading channels in which the iterative detection and decoding algorithm is employed at the receiver.     

Performance of Space–Time Codes: Gallager Bounds and Weight Enumeration

Since the standard union bound for space-time codes may diverge in quasi-static fading channels, the limit-before-average (LBA) technique has been exploited to derive tight performance bounds. However, it suffers from the computational burden arising from a multidimensional integral. In this paper, efficient bounding techniques for space-time codes are developed in the framework of Gallager bounds. Two closed-form upper bounds, the ellipsoidal bound and the spherical bound, are proposed that come close to simulation results within a few tenths of a decibel. In addition, two novel methods of weight enumeration operating on a further reduced state diagram are presented, which, in conjunction with the bounding techniques, give a thorough treatment of performance bounds for space-time codes.     

New Gallager Bounds in Block-Fading Channels

In this paper, we propose a new upper bound on the error performance of binary linear codes over block-fading channels by employing Gallager's first- and second-bounding techniques. As the proposed bound is numerically intensive in its general form, we consider two special cases, namely, the spherical bound and the DS2-exponential bound, which are found to be tight in nonergodic and near-ergodic block-fading channels, respectively. The tightness of the proposed bounds is demonstrated for turbo codes. Many existing bounds for quasistatic or fully interleaved fading channels can be viewed as special cases of the proposed Gallager bound     

A union bound on the error probability of binary codes over block-fading channels

Block-fading is a popular channel model that approximates the behavior of different wireless communication systems. In this paper, a union bound on the error probability of binary-coded systems over block-fading channels is proposed. The bound is based on uniform interleaving of the coded sequence prior to transmission over the channel. The distribution of error bits over the fading blocks is computed. For a specific distribution pattern, the pairwise error probability is derived. Block-fading channels modeled as Rician and Nakagami distributions are studied. We consider coherent receivers with perfect and imperfect channel side information (SI) as well as noncoherent receivers employing square-law combining. Throughout the paper, imperfect SI is obtained using pilot-aided estimation. A lower bound on the performance of iterative receivers that perform joint decoding and channel estimation is obtained assuming the receiver knows the correct data and uses them as pilots. From this, the tradeoff between channel diversity and channel estimation is investigated and the optimal channel memory is approximated analytically. Furthermore, the optimal energy allocation for pilot signals is found for different channel memory lengths.     

General random coding bounds: AWGN channels to MIMO fading channels

Random coding bounds are obtained for multiple-input multiple-output (MIMO) fading channels. To derive the result in a compact and easy-to-evaluate form, a series of combinatorial codeword enumeration problems are solved for input-constrained MIMO fading channels. The bounds obtained in this paper are shown useful as performance prediction measures for MIMO systems which employ turbo-like block codes as the outer code to derive the space-time inner code. The error exponents for MIMO channels are also derived from the bounds, and then compared with the classical Gallager error exponents as well as the channel capacities. The random coding bounds associated with the maximum likelihood receiver exhibit good match with the extensive system simulation results obtained with a turbo-iterative receiver.     

Tight exponential upper bounds on the ML decoding error probability of block codes over fully interleaved fading channels

We derive tight exponential upper bounds on the decoding error probability of block codes which are operating over fully interleaved Rician fading channels, coherently detected and maximum-likelihood decoded. It is assumed that the fading samples are statistically independent and that perfect estimates of these samples are provided to the decoder. These upper bounds on the bit and block error probabilities are based on certain variations of the Gallager bounds. These bounds do not require integration in their final version and they are reasonably tight in a certain portion of the rate region exceeding the cutoff rate of the channel. By inserting interconnections between these bounds, we show that they are generalized versions of some reported bounds for the binary-input additive white Gaussian noise channel.     

Broad-band fading channels: signal burstiness and capacity

Medard and Gallager (2002) showed that very large bandwidths on certain fading channels cannot be effectively used by direct sequence or related spread-spectrum systems. This paper complements the work of Medard and Gallager. First, it is shown that a key information-theoretic inequality of Medard and Gallager can be directly derived using the theory of capacity per unit cost, for a certain fourth-order cost function, called fourthegy. This provides insight into the tightness of the bound. Secondly, the bound is explored for a wide-sense-stationary uncorrelated scattering (WSSUS) fading channel, which entails mathematically defining such a channel. In this context, the fourthegy can be expressed using the ambiguity function of the input signal. Finally, numerical data and conclusions are presented for direct-sequence type input signals     

Specular coherent and noncoherent optimal detection for unresolvedmultipath Ricean fading channels

This paper considers specular coherent and noncoherent optimal detection for unresolved multipath Ricean fading channels with known delays. The focus is on receiver structures and performance. Specular coherent detection employs the carrier phase of the Ricean specular component, while noncoherent detection does not. Therefore, a specular coherent detector must be augmented with a carrier phase estimator for the specular component. The structures considered are generalization of the well-known RAKE receiver to the unresolved multipath case. It is shown that both optimal structures perform a decorrelation operation before combining, which is essential to eliminating error floors under multipath unresolvability conditions. Furthermore, the noncoherent optimal receiver includes an inherent estimator for the specular component phasor. It is shown that the specular coherent and noncoherent structures converge at high SNR. This result is confirmed through analytical and numerical performance evaluation. Little performance gains can be obtained by the use of specular coherent detection for orthogonal frequency-shift keying and to a lesser extent for differential phase-shift keying over mixed mode Ricean/Rayleigh fading channels, making noncoherent demodulation attractive in these cases     

非频率选择性莱斯慢衰落信道中M元正交信号的接收

本文推导出了非频率选择性莱斯慢衰落信道中Ｍ元正交信号的相干接收机与非相干接收机的误比特率计算公式，并给出了一些数值结果。最后，对相干接收机和非相干接收机的性能进行了比较     

Performance Bounds for Nonbinary Linear Block Codes Over Memoryless Symmetric Channels

The performance of nonbinary linear block codes is studied in this paper via the derivation of new upper bounds on the block error probability under maximum-likelihood (ML) decoding. The transmission of these codes is assumed to take place over a memoryless and symmetric channel. The new bounds, which are based on the Gallager bounds and their variations, are applied to the Gallager ensembles of nonbinary and regular low-density parity-check (LDPC) codes. These upper bounds are also compared with sphere-packing lower bounds. This study indicates that the new upper bounds are useful for the performance evaluation of coded communication systems which incorporate nonbinary coding techniques.      

Improved-performance noncoherent weighted-coefficients diversity combiner for DPSK and NC-FSK signals in nonidentical Nakagami fading channels

It is well-known that noncoherent equal-gain combining (NC-EGC) is the simplest combining technique for noncoherent and differentially coherent communication systems. However, for nonidentical Nakagami-m channels (channels having nonuniform multipath intensity profile (MIP) and/or arbitrary non-integer fading parameters), the use of NC-EGC has three main disadvantages. First, its performance serves as a lossy upper bound to that of the optimum diversity combiner. Second, it results in complicated expressions for the system average error performance. Third, it incurs noncoherent combining loss (does not aid the use of diversity) at relatively low average signal-to-noise ratio (SNR). In this letter, we propose a modified version of the NC-EGC, which is a noncoherent combiner with weighting coefficients, to overcome the disadvantages of the conventional one. We show that this alternative combiner does provide improvements over the conventional N.C-EGC for all values of average SNRs, it does not incur any noncoherent combining loss, and it leads to a design of the receiver whose average error performance can be evaluated easily.     

Integral representation and bounds for Marcum Q-function

A new expression for the Marcum Q-function involving an integral over a fixed interval is given. Tight upper and lower bounds are then derived and applied to the performance evaluation of noncoherent and differentially coherent detection of digital modulation over Nakagami fading channels