On weak convergence of probability measures, channel capacity andcode error probabilities

The relationship between weak convergence of channel probability measures, channel capacity, and error probability of block codes is examined for memoryless channels with general input and output alphabets. It is shown that channel capacity is a lower semi-continuous function and that every block code with maximal probability of error δ for a nominal channel for any ϵ>0 can be modified such that the modification has a probability of error less than δ+ϵ for all channels in a sufficiently small neighborhood of the nominal channel     

Bit-to-bit error dependence in slotted DS/SSMA packet systems withrandom signature sequences

A technique is developed to find an accurate approximation to the probability of data bit error and the probability of packet success in a direct-sequence spread-spectrum multiple-access (DS/SSMA) packet radio system with random signature sequences. An improved Gaussian approximation to the probability of data bit error is performed. Packet performance is analyzed by using the theory of moment spaces to gain insight into the effect of bit-to-bit error dependence caused by interfering signal relative delays and phases which are assumed constant over the duration of a desired packet. Numerical results show that if no error control exists in the desired packet or if block error control is used when multiple-access interference is high, the error dependence increases the average probability of packet success beyond that predicted by models which use independent bit errors. However, when block error control is used and the multiple-access interference is low, the bit error dependencies cause a reduction in packet error performance     

A method to analyze performance of digital connections

A comprehensive performance analysis method that models, at bit level, the error performance of individual links in an end-to-end connection is presented. The link model accounts for the burst-error behaviour of each individual link. A method to concatenate several individual links and extract a model for the end-to-end connection is given. This resulting end-to-end model can be used to calculate performance measures such as bit error rate and block error rate for any given block size. A procedure to compute the probability distribution of errors within a specific block is also developed. Finally, a method to compute the probability distribution of blocks having a certain error rate over a given period of time is presented. The utility and power of the model are illustrated with the help of an example connection     

Markov characterization of digital fading mobile VHF channels

We apply techniques for the modeling of error sequences on digital communication channels to results of experiments undertaken on mobile VHF channels. The experiments were carried out using four different modulation schemes at some of the different standardized signaling rates. The modulation schemes used were: FSK @ 300 baud, DPSK @ 1200 baud, QPSK @ 1200 baud, and 8-ary PSK @ 1600 baud, and in each case, subcarrier modulation was used. The experiments were undertaken for urban as well as freeway environments. Fritchman-partitioned Markov chain models were derived throughout, and from the models, block error probability distributions were derived. These block error probability distributions or P(m,n) give the probability that a block of n bits will contain exactly m errors. We present P(⩾m,n) for 7-, 15-, 31-, 63-, 127-, and 255-b blocks, for the above-mentioned modulation schemes, in the mobile VHF environments mentioned. P(⩾m,n) denotes the probability that at least m errors will occur in a block of n bits. Furthermore, the P(⩾m,n) information presented here, should give some indication of the performance to be expected from block error-correcting schemes     

On the computation of the probability of post-decoding error events for block codes (Corresp.)

The computation of the probability of certain post-decoding error events is formulated for block codes whose weight distributions are known. A bounded-distance decoder for group codes is assumed. Numerical examples included show post-decoding character error rates for the Golay code and for some Reed-Solomon codes and the distribution of post-decoding errors for the same Reed-Solomon codes. An extension to errors and erasures decoding is also given.     

A method for convergence analysis of iterative probabilisticdecoding

A novel analytical approach to performance evaluation of soft-decoding algorithms for binary linear block codes based on probabilistic iterative error correction is presented. A convergence condition establishing the critical noise rate below which the expected bit-error probability tends to zero is theoretically derived. It explains the capability of iterative probabilistic decoding of binary linear block codes with sparse parity-check matrices to correct, with probability close to one, error patterns with the number of errors (far) beyond half the code minimum distance. Systematic experiments conducted on truncated simplex codes seem to agree well with the convergence condition. The method may also be interesting for the theoretical analysis of the so-called turbo codes     

On the error probability of coded frequency-hopped spread-spectrummultiple-access systems

A simple, exact calculation is presented of the probability distribution of the number of hits in a block of n symbols in a frequency-hopped, spread-spectrum, multiple-access communication system. While the sequence of hits is not Markovian, there is an underlying Markovian structure that allows the probability distribution of the number of hits to be calculated in a recursive fashion. Knowing the probability distribution of the number of hits makes it possible to calculate the probability of error for a system employing error correcting codes for several different types of receivers, including receivers with both errors and erasures. The numerical results show that both the approximation obtained by assuming the actual sequence of hits is Markovian and the approximation obtained by assuming the hits are independent are very good. When the number of frequency slots is not too small (less than five), calculations show that assuming the independence of hits gives an error probability accurate to within 1% of the actual error probability. Assuming the hits are Markovian gives error probabilities which are accurate to within 0.001%     

A channel model for characterization of the error data recoveredfrom compact discs

The data recovered from compact discs are contaminated with errors that exhibit a bursty behavior. A channel model is developed to characterize the error bursts and good-data gaps that occur in actual data recovered from compact discs. The probability of having m erroneous symbols in a data block of length n symbols is calculated using the channel model. This probability can be used to evaluate the performance of the error-correction code (ECC) used in the compact disc system     

Codes based on inaccurate source probabilities

Information theory obtains efficient codes by encoding messages in large blocks. The code design requires block probabilities that are often hard to measure accurately. This paper studies the effect of inaccuracies in the block probabilities and gives coding procedures that anticipate some of the worst errors. For an efficient code, the mean numberdof digits per letter must be kept small. In some cases the expected value ofdcan be related to the size of the sample on which probability estimates are based. To underestimate badly the probability of a common letter or block is usually a serious error. To ensure against this possibility, some coding procedures are given that avoid extremely long codewords. These codes provide a worthwhile insurance but are still very efficient if the probability estimates happen to be correct.     

Noise reduction of VQ encoded images through anti-gray coding

Noise reduction of VQ encoded images is achieved through the proposed anti-gray coding (AGC) and noise detection and correction scheme. In AGC, binary indices are assigned to the codevector in such a way that the 1-b neighbors of a code vector are as far apart as possible. To detect the channel errors, we first classify an image into uniform and edge regions. Then we propose a mask to detect the channel errors based on the image classification (uniform or edge region) and the characteristics of AGC. We also mathematically derive a criterion for error detection based on the image classification. Once error indices are detected, the recovered indices can be easily chosen from a “candidate set” by minimizing the gray-level transition across the block boundaries in a VQ encoded image. Simulation results show that the proposed technique provides detection results with smaller than 0.1% probability of error and more than 86.3% probability of detection at a random bit error rate of 0.1%, while the undetected errors are invisible. In addition, the proposed detection and correction techniques improve the image quality (compared with that encoded by AGC) by 3.9 dB     

Evaluation of the performance of error-correcting codes on aGilbert channel

Presents a combinatorial analysis to derive a closed-form expression for the number of transmission errors that occur in a block transmitted through a Gilbert channel. This expression simplifies the computations needed to investigate the tradeoffs among the decoding error probability, degree of interleaving, and the error-correction ability of a code. The authors illustrate how a designer may apply the method to determine different combinations of the degree of interleaving and error correction ability to achieve a specified decoding error rate     

Error statistics in data transmission over fading channels

We investigate the behavior of block errors which arise in data transmission on fading channels. Our approach takes into account the details of the specific coding/modulation scheme and tracks the fading process symbol by symbol. It is shown that a Markov approximation for the block error process (possibly degenerating into an identically distributed (i.i.d.) process for sufficiently fast fading) is a good model for a broad range of parameters. Also, it is observed that the relationship between the marginal error rate and the transition probability is largely insensitive to parameters such as block length, degree of forward error correction and modulation format, and depends essentially on an appropriately normalized version of the Doppler frequency. This relationship can therefore be computed in the simple case of a threshold model and then used more generally as an accurate approximation. This observation leads to a unified approach for the channel modeling, and to a simplified performance analysis of upper layer protocols     

Erasure and Error-Correction Decoding Algorithm for Spread-Spectrum Systems with Partial-Time Interference

An erasure-control decoding algorithm is discussed for frequency-hopped spread-spectrum systems operating in the presence of pulsed or partial-time interference, and some properties of the resulting error probability which are useful in designing effective decoding algorithms are derived. Based on numerical results, specific decoding algorithms which correct both erasures and errors are then presented for use with block codes employing hard decisions.     

Effects of correlated fading on frequency-hop communications with Reed-Solomon coding

The transmission of frequency-hop spread-spectrum signals over a frequency-selective fading channel results in correlation between the amplitudes of signals that occupy different frequency slots. This correlation produces dependent errors among symbols transmitted at different frequencies. For a system that employs block coding, this dependence results in dependent errors, even if the code symbols are interleaved over the dwell intervals. Using an appropriate mathematical model for wideband frequency-selective fading channel, we present analytical results on the code-word error probability for a simple coding scheme and simulation results for more complex codes.     

Analysis of Rake reception with multiple symbol weight estimation for antipodal signaling

Consider a Rake receiver for coherent binary antipodal signaling with: 1) a delayed received signal configuration; 2) weight estimation by matched filtering using the reference signal along with the decisions of the previous M symbol intervals; and 3) predetection maximal-ratio combining (MRC). The weight estimation errors here are not independent of the additive noise, and do not fit into the Gaussian weighting error model for MRC. Here we analyze the error performance of the receiver by obtaining the conditional symbol error probability, conditioned on past decisions, from the characteristic function of the decision variable, and getting the unconditional error probability (UEP) for a block of M consecutive symbols using a Markov model of the decision process. The channel is Rayleigh fading with independent and identically distributed branch gains. Results show that the error performance of the Gaussian distributed weighting error model is a bound for that of multiple symbol weight estimation by matched filtering, and the steady state UEP decreases with increase of M, but the amount of decrease reduces as M increases.     

Convolutionally interleaved PSK and DPSK trellis codes forshadowed, fast fading mobile satellite communication channels

Using a model from the literature, the performance of convolutionally interleaved phase-shift-keying (PSK) and differential phase-shift-keying (DPSK) trellis codes for digital speech transmission over shadowed mobile satellite communication channels is determined by computer simulation. First the characteristics of fading channels are examined and analyzed in terms of the probability distributions of amplitude, phase, and burst errors. A statistical method, using a histogram approach, is utilized along with the simulations of fading channels to generate these probability distributions. A test for channel burst error behavior is presented. A periodic convolutional interleaver/deinterleaver to be used with trellis coding to combat slow fading in digital, shadowed mobile satellite channels is designed. This interleaver ha less than half the time delay for the same bit error performance than a block interleaver. The results show that the periodic convolutional interleaver provides considerable improvement in the error and time delay performance of mobile satellite communication channels for up to average shadowing conditions as compared to other techniques     

Average- and Median-Based Smoothing Techniques for Improving Digital Speech Quality in the Presence of Transmission Errors

In digital speech communication, transmission errors generally introduce impulsive distortions in the received speech waveform. Smoothing of this waveform results at once in a squelching of the distortion component, and in an undesirable smearing of the speech. However, our experience with practical differential PCM (DPCM) codes (with adaptive quantizers and first-order predictors) has shown that if the error probability is fairly significant (for example, 0.025), the noise attenuation is perceptually desirable in spite of the attendant speech-muffling. Our observations are based on computer simulations and informal listening tests. The smoothing can be performed either on the received DPCM word (prediction error signal) or the reconstructed speech amplitude. (The two signals are identical in nondifferential PCM.) Smoothing algorithms can be either linear (based, for example, on running averages) or nonlinear (based, for example, on running medians). Studies with 3-bit quantizers indicate that with independently occurring transmission errors, smoothing of the prediction error signal is perceptually desirable, although the benefits decrease as a function of the predictor coefficient a, with the maximum advantage showing up fora = 0(PCM). Running averages and running medians seem to work equally well, and suggested block lengths for their computations are three or five samples. Results with clustered transmission errors show a higher advantage due to smoothing and a preference of linear methods and longer block lengths.     

On computing undetected error probabilities on the Gilbert channel

Two methods for computing the probability of undetected error on the Gilbert (1960) channel are examined. First, using a method proposed by Kittel (1978), we study some standard cyclic redundancy codes and compare the results with those on the binary symmetric channel. Then we consider a general method of approximate code evaluation, proposed by Elliott, which involves P(m, n), the probability of m errors in a block of length n bits. A nonrecursive technique for computing P(m, n) on the Gilbert channel is described     

Probability of Block Error for Very Slow Rayleigh Fading in Gaussian Noise

Expressions for probability of block error,P_{f}(M,N), the probability of more thanMerrors in a block ofNdigits, are derived for binary signaling over a channel with very slow nonselective Reyleigh fading and additive Gaussian noise. The analysis is applicable to noncoherent FSK, coherent FSK, and coherent PSK signaling. An asymptotic expression for the block error probability, suitable for high signal-to-noise ratio, is also derived. Computations ofP_{f}(0,N)are presented for a broad range of error probabilities and block sizes, and the characteristics of block error probabilities for the fading and nonfading cases are compared.     

The effect of alpha-particle-induced soft errors on memory systems with error correction

Using Poisson statistics, a model for the survival probability of integrated memory circuits having both hard and soft error bit failure mechanisms is developed. Calculations are made over a range of soft error generation rates and erasure intervals for both single and double error correction. It is shown that even if the soft errors are erased effectively instantaneously, there is still an impact on the probability of system survival which is a function of soft error generation rate, and that in the case of instantaneous erasure of soft errors, a system with N bit error correction will have a probability of survival at least as good as the same system with N-1 bit error correction, no matter how high the soft error generation rate.