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Optimality for Coherent Systems When Dimensionality Is not Specified: Regular Simplex Coding
Abstract
The variational approach to the signal-design problem described in the previous chapter was developed and exploited by Balakrishnan in obtaining the first known results on the optimality of the regular simplex signal structure when no restrictions are placed on the dimensionality of the signal space. We now develop these results for their own interests, as well as their applicability to the general problem with constraints on the dimensionality D. As shown in the previous chapter, the class of α for which D(α) = 0 contains all of the optimal α. Also, we concluded that the maximum value necessary for D is M − 1. Therefore, even in this case of no bandwidth restriction, only a finite bandwidth is required for the optimal signal set. This again does not violate Shannon’s channel-capacity theorem, since M and T can vary there but are fixed and finite here.
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Six parameters of importance in many communication systems are: (a) the rate at which digital information is transmitted; (b) the bandwidth of the system; (c) the signal power of the transmitted signals; (d) the noise power of disturbances in transmission; (e) the error probability in digits recovered at the receiver output; (f) the length of time that the transmitter and receiver can store their inputs. These six parameters cannot assume arbitrary values: certain sets of values cannot be realized. In a series of curves, this paper describes the boundary between compatible and incompatible sets of parameter values. In the model studied, it is assumed that the disturbance is additive Gaussian noise with constant power density spectrum in the transmission band.
We prove here the long conjectured fact that the regular simplex is the code of minimal error probability for transmission over the infinite-band Gaussian channel. The code is actually optimal for a rather wide class of assumed channel noises. We also establish the optimality of several other codes for the band-limited Gaussian channel.
In coherent PCM communication systems where the noise is additive and Gaussian (such as in space communication systems) two classes of nonredundant coding are of special interest. One is Regular Simplex Coding which yields the lowest possible error rate for fixed SNR. The other is Bi-orthogonal Coding because of its many desirable features from the practical point of view. In this paper the error rates per word and per bit are computed for both of these classes. The communication efficiency-a quantity of major importance in space communications-is also calculated for fixed error rate as a function of code size.
The basic problem of transmitting information through a noisy channel in a finite time with the least error is considered. The transmitted signals are either peak or average power limited, and the interference is presumed to be additive white gaussian noise. It is shown that the optimum signals are sequences of binary waveforms resembling a (2^m -l,m) Slepian group code, the generators of which can be obtained from a modified Reed Code. An analysis is also made of near-optimum codes which allow some inequality in the distance between code words in signal space. The advantage of these near-optimum codes is that for a small sacrifice in error probability, either the information rate can be substantially increased, or the bandwidth of the system greatly reduced. The details of these sytems are given and the results show that a many-fold improvement is obtained. An instrumentation of the detector is also presented to show how the code groups at the receiver can be effectively stored as an arrangement of resistors.
Formulas for the error probabilities of equicorrelated M -ary signals under optimum phase-coherent and phase-incoherent reception are derived in the form of previously untabulated single and double integrals. These integrals are amenable to computer evaluation for arbitrary M . Two modes of reception are considered. In the first, one of M equal energy equiprobable signals is known to be transmitted during a symbol interval of T seconds through a nonfading channel with additive white Gaussian noise. The receiver is assumed to be synchronized in time and frequency with the incoming signal, and reception is on a per-symbol basis. Furthermore, the cross-correlation coefficients between all the signals are equal. The probability of correct decision in both phase-coherent and phase-incoherent reception is derived exactly, as a function of the signal-to-noise ratio, the common cross-correlation coefficient, and the size of the signal set M . In the second mode of reception, the only difference is that a threshold is incorporated in the receiver. The probability of false detection and the probability of detection and correct decision are derived exactly for both phase-coherent and phase-incoherent reception as a function of the threshold level, the signal-to-noise ratio, the common cross-correlation coefficient, and the size of the signal set M . The method of reduction of multiple integrals presented here can be generalized, and may find application in other statistical studies in which the Gaussian density form is encountered under an integral.
M equally probable symbols may be encoded as continuous waveforms whose sample values are the coordinates of the vertices of a regular simplex in N dimensions. These vertices or code points, therefore, are equally spaced, and the resulting waveforms are adapted particularly to low signal-to-noise communication systems. An upper bound for the information rate in an additive Gaussian noise channel, based on the use of regular-simplex coded waveforms, has been calculated. The results indicate that for signal-to-noise energy ratios less than approximately unity and for error probabilities ranging from 10^{-2} to l0^{-8} , this rate is a sizable percentage of the ideal rate which has been derived by Shannon and Tuller.
Signal design theory is concerned with the problem of determining transmitter signal waveforms such that the probability of correct reception (or some other appropriate measure of communication efficiency) is maximized. The selection of signals must be performed under specified constraints of signal power and bandwidth and channel noise disturbance. In this paper three cases are treated for the coherent Gaussian channel: begin{enumerate} item white noise, equal signal energies, unequal message probabilities, item white noise, average signal power bounded, equal message probabilities, and item colored noise, average signal power bounded, equal message probabilities. end{enumerate} In each problem, necessary general conditions for signal optimality are derived, and specific solutions obtained for small and large signal-to-noise ratios (SNR's). Complete solutions are indicated for the case of three messages. It is shown that the regular simplex codes are always global solutions at large SNR. Other results are also presented.
First Page of the Article
A class of codes and decoders is described for transmitting digital information by means of bandlimited signals in the presence of additive white Gaussiau noise. The system, called permutation modulation, has many desirable features. Each code word requires the same energy for transmission. The receiver, which is maximum likelihood, is algebraic in nature, relatively easy to instrument, and does not require local generation of the possible sent messages. The probability of incorrect decoding is the same for each sent message. Certain of the permutation modulation codes are more efficient (in a sense described precisely) than any other known digital modulation scheme. PCM, ppm, orthogonal and biorthogonal codes are included as special cases of permutation modulation.
Advances in Communication
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- AV Balakrishnan
Regular Polyhedron Codes
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