For a bandwidth-limited, power-limited communication channel corrupted by additive white Gaussian noise, Shannon (1949) showed that, at any information rate below the channel capacity, an arbitrarily low error probability can be obtained when the transmitted signal is selected from a set a of M independent, white-Gaussian-noise-like waveforms having the prescribed power and bandwidth. He utilized a space whose points represent transmitted or received signals. Since for each possible received signal there is a most likely transmitted signal, this space is divided into decision regions {R/sub t/}, one associated with each member s/sub t/ (i=1, 2,..., M) of set s/sub t/ This article investigates some of the geometrical properties of these decision regions: their shape, size and number of faces, distances from s/sub t/ to R/sub t/'s nearest face or edges of various sorts, etc., thus providing insight into how random coding succeeds in achieving arbitrarily good performance.

Cramer and Leadbetter's (1967) result for the distribution of the local maxima of stationary Gaussian noise is studied and plotted. Its derivation is used for finding the distributions of the local maxima and minima of the envelope of narrowband Gaussian noise. These distributions, too, are studied and plotted, including the limiting cases of very wide and narrow noise spectra.

Like low excursions of the envelope of narrowband noise, those of the sum of a signal plus narrowband noise are hyperbolic in shape if the signal-to-noise ratio r is not high. When r/spl Gt/1, low excursions of the envelope are hyperbolic if the signal is frequency-modulated but have a quartic shape with one or two minima if it is not. This article illuminates various aspects of FM clicks.

Low minima of the envelope of narrowband noise are shown to be hyperbolic in shape. They are accompanied by large, short impulses in the instantaneous frequency of the noise. The statistics of these phenomena are described, and they are related to Rice's formula for the rate of zero-crossings of Gaussian noise.

There are just three regular polytopes (the hypercube, cross polytope and the simplex) in Euclidean (n>4) space. We calculate their dimensions, including the distance from the centroid to the periphery in a random direction-that of a white Gaussian-noise vector. As n/spl rarr//spl infin/, this distance becomes very predictable. It differs from the distance near which almost all of the volume and surface of the polytope lie.

It is shown that, when the input signal-to-interference ratio (SIR) is small, a biased nonlinearity that has a dead zone below some threshold value can provide a large enhancement in output SIR and signal-to-interference-plus-intermodulation ratio (SIIMR) if the threshold is set equal or close to the amplitude of the strong interference. In the absence of noise, the output SIR and SIIMIR are virtually independent of the input SIR, however low the latter may be. It Is shown that, for a dead-zone limiter, the output SIIMR in this case is -4.93 dB regardless of the input SIR. Under strong interference, any noise present in the input reduces the SIR and SIIMR improvement, but a biased nonlinearity can still provide an output SIR and SIIMR superior to the input SIR, i.e. an output SIIMR 6.35 dB below the input signal-to-noise ratio (SNR) instead of 6.02 dB below the input SIR as is the case with hard limiting.>

Novel forms are derived relatively simply for the spectrum of the square of a synchronous random pulse train. The lines in the discrete part of the spectrum are useful for bit and carrier synchronization in digital communication systems, whereas the continuous part of the spectrum limits the signal-to-noise ratio available in any given bandwidth from the spectral lines. The application of the formulas derived and some conclusions are discussed.>

Work on the parabolic shape of the topmost part of high noise peaks is extended to show that these peaks are all approximately proportional to the autocorrelation function, R( tau ), of the noise. It is shown that the large excursions of the envelope of narrowband Gaussian noise also tend to have the shape of the envelope of its autocorrelation function.>

For pt.I see ibid., vol.34, no.6, p.1396-1400 (1988). Approximate expressions are obtained for the distribution of the change in phase of a narrowband Gaussian noise plus a sinusoid during a long time interval. The expressions are based on S.O. Rice's results concerning the expected number of clicks in an FM receiver during the interval and on work stemming from his expression for the correlation function of the instantaneous frequency of narrowband Gaussian noise as well as his work on the statistical properties of a signal plus random noise.>

On the basis ofn\geq 2observations, confidence limits of the formX̄ \pm tS/ \sqrt{n}are constructed for the location (e.g., the median) of any distribution of known form with unknown location and dispersion (scale), whereX̄andSare the sample mean and "unbiased" standard deviation. Particular attention is given to the values oftneeded for the Cauchy and uniform distributions. The lattertsuffices for any (unknown) symmetric unimodal distribution ift \geq n - 1. A table compares these values oftforn=2,3,4, and5with those for the normal case, which are derived here very simply and are identical with those found by "Student." We are also able to include the case of a single observation(n=1), where confidence intervals of various forms are made just wide enough for the least favorable dispersion. They, therefore, include the true location with at least but, in general, not exactly the desired probability; these intervals involve a predetermined value that plays a role reminiscent of but quite different from that of the prior distribution that would enter into a Bayesian analysis. In addition, upper confidence limits for the dispersion are constructed forn \geq 1.

Nearly every digit (positive-integer partial quotient) of the simple continued fraction representing a random number carries3.4325bits of information about the number. This is reduced by 0.0081 bit if the preceding digit is already known and by a further0.0007bit if the second preceding digit is also known, Any finite sequence of digits is equally likely to occur forward and backward. Hence, the digit (source output symbol) following any digit likewise gives0.0081bit of information about the latter, and all succeeding (or preceding) digits together give0.0088bit despite the unbounded range of the statistical dependence among digits.

Despite the presence of AM-to-AM conversion, AM-to-PM conversion, and a strong interfering signal, a weak signal emerges undistorted from a memoryless nonlinear bandpass amplifier, suffering only a constant phase shift. A recent result of Chang and Shimbo concerning this situation is here clarified with the aid of a simpler, more direct derivation, which also yields the third-order intermodulation.

Expressions are found for the effect of an error in the delay of the preceding signal, which provides the reference phase for the decoding of the present signal in differential phase-shift-keying reception. The signal-to-noise ratio is allowed to be different for the two signals that are compared by the receiver's phase detector. The results are applicable to both binary and quaternary DPSK. In addition, an approximation is obtained for the error probability when the two Signal-to-noise ratios are equal to the same large value.

When an input consisting of a strong signal and a weak signal is handled digitally, the resulting quantization introduces not only noise and distortion of the stronger signal (which have been thoroughly studied elsewhere) but also intermodulation between the two. Third-order intermodulation is the principal source of spurious inband signals, and it is important, therefore, to determine its strength in comparison with that of the true weak-signal output, which is of the same order of magnitude. The assumption that the weaker signal's amplitudeais very much smaller than the quantization step size permits a relatively simple analysis by means of two different approaches. Both of them lead to a finite series for the amplitudec_{2}aof the intermodulation. This series is easily evaluated numerically, but simple approximations obtained by truncation of Euler's summation formula show the behavior of c2more clearly as a function of the amplitude of the stronger input and will be found adequate for most applications.

On the assumption that each input signal represents only a small part of the total input power, the output signals and output noise spectrum are found by extending the Barrett-Lampard diagonal expansion to the case of the quadrivariate distribution of input amplitude and phase at two different times. This expansion yields a decomposition of the nonlinearity into a summation of generalized Laguerre polynomials with uncorrelated outputs, whose spectra are simply autoconvolutions of the input spectrum. A slight generalization of the notion of memoryless nonlinearity allows this result to he applied very simply to devices, such as limiters and TWT amplifiers, that cause AM-to-PM conversion.

For the sum of two sinusoids of incommensurable frequencies, the number of zero-crossings per secondZis calculated. By averaging this result over a Rayleigh distribution for the amplitude of one of the sinusoids, Rice's formula is obtained for the numberzof zero-crossings per second of a sinusoid plus Ganssian noise of any spectrum. Simple approximations are obtained forzin the high-signal-to-noise-ratio and narrow-band cases. In addition, alternative derivations ofZandzare presented that further illuminate the significance of the foregoing results.

An analysis of the spectra of digital signals is presented that includes Lundquist's and Glance's results for PSK signals as special cases while at the same time being much simpler than previous analyses. The signal is assumed to be modulated in phase and/or amplitude by a sequence of pulses that may have different shapes for different symbols. In addition, the pulse representing a given symbol may be affected by neighboring symbols, so that intersymbol interference in the modulating process is permitted. The results, nevertheless, take a simple form, and they are applied to two binary cases.

The spectrum of a time-division-multiplexed (TDM) pulse-position-modulated (PPM) signal consists of three parts-the continuous part, which is simply the sum of the continuous parts of the spectra of the separate channels; lines on harmonics of the maximum overall pulse-repetition frequency (PRF), which are also independent of any permutation among the channels; and the lines on the other harmonics of the single-channel PRF. Simple expressions are found for all three parts under the assumption thatN - m - nchannels are empty,nare unmodulated, and the remainingmcontain independent, identically uniformly distributed pulse positions. A complicated expression is found for the variances of the lines in the last part of the spectrum under a random choice of the modulated and unmodulated channels, and a simple approximation is presented for it.

The signal-to-noise-ratio (SNR) performance of a pulse-position-modulation (PPM) receiver is well understood when the input SNR exceeds the threshold, but below it the output SNR deteriorates on account of false pulses due to noise. The calculation of their effect, which determines the threshold, has been considered a difficult problem. However, the use of Rice's 1948 result concerning the rate of occurrence of noise peaks permits a very simple analysis of this effect. Thus, a formula is obtained for the output SNR as a function of the input SNR that is valid somewhat below as well as near and above the threshold. From it the threshold is easily determined, and it is found to be higher than has sometimes been indicated. The paper includes a new, simpler derivation of Rice's result as well as a resolution of the demodulator output into its various com ponents which should also prove useful in other applications.

A general method is presented for determining not merely the output spectrum resulting from a nonlinear transformation of signals and noise but also the signal × signal distortion and intermodulation components, including their phases and their precise forms as functions of time. Recent work by Jain is thereby illuminated and generalized.

This paper considers the problem of the detection of a binary phase-modulated carrier which has been transmitted along with noise through a hard-limiting repeater, corrupted by additional noise, and demodulated by cross correlation and sampling at the receiver. Three equivalent expressions are obtained for the error probability of the receiver output. Two of these expressions take the form of an infinite series involving either confluent hypergeometric functions or modified Bessel functions. A third form allows representation of the error probability in terms of Rice'sIefunction. In the special case where the signal-to-noise ratios\rho_1^2and\rho_2^2at the limiter and receiver inputs are equal, the error probability is simply \begin{equation} P_e = \frac{1}{2} \exp (-p_1^2). \end{equation} Asymptotic expressions for the error probability for large and small signal-to-noise ratios are also derived. The error probability is found to be smaller than that of a linear system for all practical values of limiter and receiver input signal-to-noise ratios. The optimum repeater non-linearity is investigated and is shown to be a limiter at such signal-to-noise ratios.

The use of explicit forms for Walsh functions removes much of the confusion surrounding these interesting functions and permits simple proofs of their properties. Thus, for example, their period is far easier to determine than Alexandridis found, but their Fourier spectra are more complex than Schreiber's approximation suggests. For wave-form analysis, certain Walsh functions--the regular symmetric square waves--are more useful than are the others.

From the voltage-response characteristic of a memoryless nonlinearity the output amplitude in any harmonic zone is easily found as a function of the amplitude of a narrow-band input, but no general method has been known for inverting this (Chebyshev) transformation. The inversion is of interest because the best detector, bandpass non-linearity, or harmonic generator for various purposes, e.g., maximization of the output signal-to-noise ratio, is most readily described in terms of its amplitude-response characteristic for the desired harmonic zone. A simple integration over the amplitude-response characteristic (or describing function) is found to yield the required voltage-response curve.

The problem of distinguishing between two narrow-band signals in the presence of additive Gaussian noise when the forms of both signals are known except for phase is basically four dimensional, as the lack of phase information turns signal vectors into signal circles. The geometry of signal vectors and of the planes of signal circles is shown to account for the optimality of matched-filter detection. In the case of equal-energy equiprobable signals, it leads to the conclusion that a large variety of different pairs Of filters will perform exactly like a pair of matched filters. One such pair has orthogonal impulse responses, and its performance is, therefore, relatively easy to determine.

The signal\timessignal, noise\timesnoise, and signal\timesnoise output components of a nonlinearity are defined in a manner agreeing with intuition. The definitions are shown to accord with previous usage, for which they now provide a clear meaning. The case of a limiter is used as an illustrative example. The approach is applicable not only to fixed memoryless transformations but also to nonlinearities with memory that depend explicitly on time, such as frequency or phase modulation, demodulation, and bandpass nonlinearities.

No single measureM(X;y)of the amount of information that a specific valueyof a random variableYgives about another random variableXhas all of the desirable properties possessed by Shannon's measureI(X;Y) = E\{M(X;y)\}of the average mutual information ofXandY. It is shown that one of these properties (additivity) determines one particular form forM(X;y), while others (non-negativity or coordinate independance) determine a different form. The latter, which is the more useful and accepted information measure, is thus seen to be unique.

Using his characteristic-function approach, Rice (1945) obtained a double series for the autocorrelation function of a sinusoidal signal and Gaussian noise after passage through a memoryless nonlinearity. It is shown here that the output of the nonlinearity can be expressed as the sum of uncorrelated terms whose auto-correlation functions are the terms of Rice's double series. Such a decomposition of the output is shown to be generally possible if and only if the bivariate probability density functions of the input signal and the input noise can both be expressed in the diagonal form studied by Barrett and Lampard (1955), though not necessarily involving polynomials, as they can in the sinusoidal and Gaussian cases. In addition, a more direct and meaningful equation is found for the coefficients in Rice's double series.

The entropy power of a band-limited random process is the power of white Gaussian noise having the same entropy rate. Shannon's convolution inequality for entropy power states that the entropy power of the sum of two independent random processes is at least the sum of their entropy powers. This paper presents an improved version of Stam's proof of this inequality, which is obtained by mathematical induction from the one-dimensional case.

This paper presents a simple method for analyzing the effect of a nonlinearity sandwiched between band-pass filters, applying it to the case of weak signals when both filters pass the same band of frequencies, and obtaining completely general expressions for the output signal and noise when the output filter passes a harmonic of the input band.

Ifx(t)andy(t)are statistically independent stationary Gaussian random processes, each having correlation function\psi(\tau), mean squared value\delta^{2} = \psi(0), and spectral density\Psi(f), thenu(t) = x(t) \cos 2\pi Ft - y(t) \sin 2 \pi Ftis a stationary Gaussian random process with correlation function\psi(\tau) \cos 2 \pi F \tauand with spectral density\frac{1}{2} \Psi(f-F) + \frac{1}{2} \Psi(f + F), symmetric aboutFfor largeF. From this representation ofu(t)it is shown that the variance of the number of zeros ofu(t)in the interval(0, T)is, for integral2FT,\mbox{var} Z=\frac{1}{4}-\frac{1}{\pi^{2}}\arcsin^{2}\frac{\psi(T)}{\sigma^{2}}+\frac{2}{\pi^{2}}\int_{0}^{T} \frac{(T- \tau)\psi'^{2}(\tau)}{\sigma^{4}-\psi^{2}(\tau)}d \tau + O(1/F). This result complements that of Steinberg, {\em et al.}, giving varZfor wide-band Gaussian noise. The limit of(var Z)/TasT \rightarrow \inftyis evaluated for several spectra, and expressions are found for the variance of the number of zeros of the sum of the foregoing narrowband noise plus a sinusoid of frequencyF. From these results the low-frequency output spectral density of an FM receiver is obtained. Below the threshold the output signal-to-noise ratio is found to be\frac{\pi^{2}(1-\exp-A^{2}/2\sigma^{2})^{2}D^{2}_{rms}}{W \int_{0}^{\infty} \frac{\psi"^{2}(\tau)}{\sigma^{4} - \psi^{2}(\tau)} \exp - \frac{A^{2}}{\sigma^{2}- \psi(\tau)}d \tau}, whereA^{2}/2\sigma^{2}is the input signal-to-noise ratio,D_{rms}is the rms frequency deviation, assumed small enough not to affect the output noise, andWis the output bandwidth, assumed small compared to the input bandwidth. By the addition of the well known "triangular" noise, this expression is made valid through and above the threshold, thus unifying various results of Rice. The quieting of a wide-band FM receiver by a signal is also considered.

IfCis the greatest rate at which information can be reliably conveyed through a channel of bandwidthWthat accel)ts signals of average power no greater thanP, adding to them white Gaussian noise of average powerNand interference of average power no greater thanJwhich may depend on the signal being transmitted and on all of the other signals in the code book, then \begin{equation} C \leq W \log P/2J \qquad \qquad for P \geq 2J \end{equation} and \begin{equation} C = 0 \qquad \qquad \qquad for P \leq 2J. \end{equation} If \begin{equation} C_1= \begin{cases} W \log P^2 /4J(P- J)& for P \geq 2J\\ 0& for P \leq 2J \end{cases} \end{equation} and \begin{equation} C_2= \begin{cases} W \log P/(J + N)& for P \geq (J + N)^2 /J\\ W \log (1 + (\sqrt{P} - \sqrt{J})^2 /N)& for J \leq P \leq (J+N)^2/J\\ 0& for P \leq J. \end{cases} \end{equation} then, whenC_1 \leq C_2, \begin{equation} C_1 \leq C \leq C_2. \end{equation} WhenC_l \geq C_2, the maximum rate of reliable communication becomesC = C_2, which is attained by the use of randomly-selected white Gaussian noise-like signals.

Bounds are derived and plotted for the capacity of a bend-limited communication channel that accepts signals of specified average power as input, adding to them white gaussian noise and interference of specified average power which may depend on the signal being sent and on all of the other signals in the code book. The upper and lower bounds often coincide, and the exact value of the capacity is then determined.

The application of Minnick and Ashenhurst's technique of p-fold multiple-coincidence magnetic storage in an n x n array of cores is shown to require finding (p-2) orthogonal Latin squares of order n. The value of this technique lies in the reduction it effects in the disturbance of unselected cores. A method is suggested for reducing this disturbance to a level lower than that obtained by Minnick and Ashenhurst, by the application of reverse currents to the unselected interrogating wires during interrogation. The disturbance of unselected cores can be reduced to zero if p=n+1. In this case, (n+1) cores can be added to the store at the expense of a single additional interrogating wire. The resulting array of cores and interrogating wires is closely related to the finite projective geometry of order n.

Haying determined a cost in energy, time, or money of transmitting each of a set of symbols (e.g., dot and dash, pulses of various amplitudes, etc.), one may inquire as to the nature of a code using these symbols which will transmit a given amount of information at the least cost, or will transmit information at a given rate for the least cost per unit time. Such a code must use each symbol with a relative frequency given by a negative exponential of a linear combination of its cost and duration in the case of a noiseless communication channel. If noise is present, the cost of each symbol is effectively increased by a multiple of its "prevarication," i.e., the entropy of the received symbol when only the given symbol is transmitted.