Duality for the group-coset probability ratio (Corresp.)

From the MacWilliams and Assmus-Mattson identifies, it is shown that Sullivan's lower bound on the ratio of the probability of a group and a coset has a dual relation with the upper bound.     

On the probability of error for decision-feedback equalizers (Corresp.)

Methods for calculating or bounding the probability of error in a decision-feedback equalizer when the noise component at the input of the quantizer is a sequence of independent and identically distributed random variables are briefly reviewed. A new upper bound on the probability of error of a decision-feedback equalizer is then derived. Unlike the methods reviewed, this bound is valid even when the noise component at the input of the quantizer is a serially dependent stationary stochastic process.     

Bounds on the probability of error for convolutional codes (Corresp.)

For the single error correcting convolutional codes introduced by Wyner and Ash, it is shown that if sufficiently few errors occur in an appropriate neighborhood of a block, the probability of correctly decoding that block is independent of errors outside that neighborhood. This fact is used to derive bounds on the bit error probability and the mean time to first error.     

On the entropy of continuous probability distributions (Corresp.)

A table is given of differential entropies for various continuous probability distributions. The formulas, some of which are new, are of use in the calculation of rate-distortion functions and in some statistical applications.     

On the estimation of asymptotic error probability (Corresp.)

In many actual learning problems, a sequence of decision functions is generated, and one has to estimate the limit of the error probabilities associated with these decision functions. This correspondence proposes a simple algorithm for the finite hypothesis testing problem. The procedure works in parallel with the iterative estimation of the decision function and utilizes in this way the same labeled samples for training and testing. A mild condition on the behavior of the probability of error of the sequence of decision rules is shown to imply strong convergence of a sequence of estimates of the probability of error.     

Threshold extension in weighted PCM with arbitrary probability distribution and nonuniform quantization (Corresp.)

The threshold in the output signal-to-noise ratio (SNR) of weighted pulse-code modulation (PCM) is determined for an arbitrary probability distribution of the input signal and a logarithmic quantizer. Numerical results are given which show a reduction of 5.37 dB in the threshold, relative to classical PCM, for a nonuniform probability distribution and a logarithmic quantizer.     

A probability model for simple closed random curves (Corresp.)

A Gibbsian probability measure is defined on the set of possible shapes of simple (i.e., not self-intersecting) and closed curves on a discretized lattice. Expressions have been derived for such quantifies as the entropy, average perimeter length, and area in terms of the model parameters, and an iterative procedure has been devised for generating sample objects according to this distribution.     

Simplified relationships for the probability distribution of the phase of a sine wave in narrow-band normal noise (Corresp.)

A new form of the probability distribution of the phase of a sine wave in narrow-band normal noise is derived. It leads to several different simple expressions for the probability distribution which render the previously required numerical integration of the probability density function unnecessary.     

Asymptotic properties of delay-time-weighted probability of error (Corresp.)

Asymptotic properties of expected distortion are studied for the delay-time-weighted probability of error distortion measured_n(x,tilde{x}) = n^{-1} sum_{t=0}^{n-1} f(t + n)[l - delta(x_t,tilde{x}_t)],, wherex = (x_0,x_1,cdots,x_{n-1})andtilde{x} = (tilde{x}_0,tilde{x}_1,cdots,tilde{x}_{n-1})are source and reproducing vectors, respectively, anddelta (cdot, cdot)is the Kronecker delta. With reasonable block coding and transmission constraintsx_tis reproduced astilde{x}_twith a delay oft + ntime units. It is shown that if the channel capacity is greater than the source entropyC > H(X), then there exists a sequence of block lengthncodes such thatE[d_n(X,tilde{X})] rigjhtarrow 0asn rightarrow inftyeven iff(t) rightarrow inftyat an exponential rate. However, iff(t)grows at too fast an exponential rate, thenE[d_n(X,tilde{X})] rightarrow inftyasn rightarrow infty. Also, ifC < H(X)andf(t) rightarrow inftythenE[d_n(X,tilde{X})] rightarrow inftyasn rightarrow inftyno matter how slowlyf(t)grows.     

On zero error probability of binary decisions (Corresp.)

In view of a certain randomized decision scheme, we simplify Rényi's result that concerns the conditions that a changing situation for binary decision must satisfy in order that the probability of decision errors tend to zero. We give its possible interpretation for a communication channel that has a tendency to become useless with time.     

On the decoder error probability for Reed - Solomon codes (Corresp.)

Upper bounds On the decoder error probability for Reed-Solomon codes are derived. By definition, "decoder error" occurs when the decoder finds a codeword other than the transitted codeword; this is in contrast to "decoder failure," which occurs when the decoder fails to find any codeword at all. These results imply, for example, that for aterror-correcting Reed-Solomon code of lengthq - 1over GF(q), if more thanterrors occur, the probability of decoder error is less than1/t!.     

Digital-detection probability of fluctuating pulsed radar signals (Corresp.)

This correspondence describes the digital-detection (binary integration, double threshold detection) properties for a fluctuating radar echo, assuming the Rice distribution for signal plus noise. Calculation results make clear the transition of detection properties from the nonfluctuating case to the extreme fluctuating case.     

An approximation for the distribution of shot noise (Corresp.)

If a shot noise distribution is conditioned on the number of Poisson events before it is expanded in Edgeworth's series, the resulting approximation may be more accurate than the usual direct expansion. We investigate the error of such an approximation, both asymptotically and for a particular example, and give an application to laser communication. We also give a general recurrence relation for determining the successive terms of Edgeworth's series.     

Concerning a bound on undetected error probability (Corresp.)

In the past, it has generally been assumed that the probability of undetected error for an(n,k)block code, used solely for error detection on a binary symmetric channel, is upperbounded by2^{-(n-k)}. In this correspondence, it is shown that Hamming codes do indeed obey this bound, but that the bound is violated by some more general codes. Examples of linear, cyclic, and Bose-Chaudhuri-Hocquenghem (BCH) codes which do not obey the bound are given.