This paper presents new forms for an error probability associated, with the phase angle between two vectors perturbed by Gaussian noise. An interesting consequence of the underlying generic form is a new expression for the symbol error rate in MDPSK that is very similar to its counterpart for MPSK. The generic error probability is further shown to contain as special cases MPSK, MDPSK, CPFSK, and digital FM; and to give a new expression for the Marcum Q-function. The generic error probability also simplifies the error probability expressions in other situations such as nonorthogonal signaling, maximum likelihood differential detection of DPSK with block-by-block detection, and Gray coding of MDPSK.

The performance of convolutionally encoded narrow-band digital FM with Viterbi decoding was considered in some detail by Simon (1983) for a noncoherent limiter/discriminator (L/D) with integrate and dump (I&D) bit detection. Employing a new threshold receiver which averages the output of the I&D detector with the output of a sample and hold (S&H) detector, a 3-dB improvement over Simon's results for the bit error probability with FM clicks is shown to be achievable. At low error rates, the performance of this new receiver is, moreover, comparable to that obtained when the clicks are exactly removed by Simon's hypothetical "genie".

The family of distributions introduced by G.H. Hansmann in 1934 is considered. Apparently, it has gone unnoticed for over fifty years that there is a major error in Hansmann's work and that his solutions are incomplete in some cases. It is shown how to complete them and the necessity of doing so is demonstrated by means of two examples.>

The bit-error-rate performance of narrowband digital FM with limiter-discriminator detection is considered for the cases of integrate and dump postdetection filtering with partial-bit integration, and sample-and-hold bit detection at the discriminator output. Error rate curves are presented for Gaussian, six-element Butterworth, and two-stage synchronously tuned IF filters. The calculations illustrate just how much more E/sub b//N/sub 0/ is required as the partial-bit integration time goes from 100% down to the limiting case of sample and hold. The results show that it is important to have a well-designed IF filter especially if the entire bit time is not available for detection.>

A method is developed for calculating the probability density functionp(y)of the output of anRCfilter when the input is a particular kind of random telegraph wave. The method makes use of a computer to determine the numerical values of the coefficients in two series solutions, one of which contains a logarithm, of a fourth-order linear differential equation. The constants of integration are determined from the boundary conditions by a procedure that involves a summation of the series. Representative values and curves ofp(y)are presented, and the generality of the computer method is discussed briefly,

The problem of calculating the probability density function of the output of anRCfilter driven by a binary random process with intervals generated by an equilibrium renewal process is studied. New integral equations, closely related to McFadden's original integral equations, are derived and solved by a matrix approximation method and by iteration. Transformations of the integral equations into differential equations are investigated and a new closed-form solution is obtained in one special case. Some numerical results that compare the matrix and iteration solutions with both exact solutions and approximate solutions based upon the Fokker-Planck equation are presented.

The symbol error probabilityP_{E}(M)forM-ary DPSK is shown to be bounded in terms of a recent asymptotic approximationP_{asym}(M)by the inequalitiesP_{asym}(M) < P_{E}(M) < 1.03P_{asym}(M);M \geq 4, E_{b}/N_{0} \geq 1whereE_{b}/N_{0}is the bit energy-to-noise spectral density ratio. Aside from the wide range of validity and the closeness of the lower and upper bounds, this result is striking in light of the often held view that such asymptotic approximations are primarily of value only in the limitE_{b}/N_{0} \rightarrow \infty; thus, one of the goals of this note is to demonstrate that asymptotic methods can lead to extremely good error rate approximations in lieu of the more traditional and more widely used bounding techniques. The results are also noted to be applicable in other similar situations which commonly occur.

The bit error rate performance of binary DPSK with an arbitrary offset of the phase change vector is derived in the presence of noise correlation and power imbalance, and the special cases of conventional DPSK (0 phase change offset) and symmetric DPSK (\pi/2phase change offset) are compared. Precise conditions are given for optimum offsets, and it is shown that, from a practical standpoint, conventional DPSK gives the optimum performance of any offset scheme, and outperforms symmetric DPSK by an order of magnitude and more in error rate when the noise correlation becomes significant. It is also found that performance improvement over conventional DPSK can be realized, provided the noise correlation is known, by making the bit decision regions in phase space unequal, thereby demonstrating that the MAP estimator for DPSK without correlation is no longer optimum when the noise is correlated.

The error rate performance ofM-ary DPSK systems is investigated in the presence of phase change offset, noise correlation, and power imbalance. Both the terrestrial channel and the hard-limiting satellite repeater channel are considered. In the satellite repeater case, the noise correlations and power imbalances are assumed to be present on both the up- and downlinks. The emphasis is on exact results, or asymptotic approximations obtained therefrom, as opposed to error rate bounds. Most of the results are given in terms of integrals readily evaluated by quadrature. Performance curves showing the effects of the various impairments are presented. Some specific conclusions which follow from the results are that 1) the phase change offset is relatively unimportant forM \geq 4, 2) theM-ary case is less sensitive to imbalance than is the binary case, and 3) the satellite link performance depends strongly upon the quality of the uplink.

This note is an addendum to the recent paper "Distribution of the Phase Angle Between Two Vectors Perturbed by Gaussian Noise" [2]. A new derivation of the key equations in Roberts' characteristic function method for calculating the probability distribution of the modulo 2π phase difference between two narrow-band waveforms is given.

New results are given for the bit error probability for binary DPSK modulation over both terrestrial and satellite channels taking into account noise correlation and power imbalance. The new expressions are remarkably simpler than their previously published counterparts. For the terrestrial channel, the results are in closed form in terms of tabulated functions and, for the hard-limiting satellite channel, both in terms of integrals readily evaluated by quadrature and in terms of infinite series. Using the results, it is shown that the error rate does not depend on the noise correlation (down-link in the satellite case) so long as the a priori symbol probabilities are equal, thereby solving an "open problem" posed by Lee, French, and Hong. Also, the satellite case in which each of the up-link and down-link channels contains nonzero noise correlations is briefly treated.

The probability distribution of the phase angle between two vectors perturbed by correlated Gaussian noises is studied in detail. Definite integral expressions are derived for the distribution function, and its asymptotic behavior for large signal-to-noise is found for "small," "near\pi/2," and "large" angles. The results are applied to obtain new formulas for the symbol error rate in MDPSK, to calculate the distribution of instantaneous frequency, to study the error rate in digital FM with partial-bit integration in the postdetection filter, and to obtain a simplified expresion for the error rate in DPSK with a phase error in the reference signal. In the degenerate case in which one of the vectors is noise free, the results lead to the symbol error rate in MPSK.

The theory of error rates for narrow-band digital FM with limiter-discriminator detection and integrate and dump postdetection filtering is considered. The goal is that of simplifying the existing theory, which has evolved primarily in the frequency domain, and requires numerical integration and differentiation of Fourier series expansions, numerical convolution, and multibit pattern averaging. A fresh approach in the time domain, within certain ranges of frequency deviation ratio, and time-bandwidth product, leads to a theory which 1) gives Simple closed form solutions for the relevant system parameters, 2) allows for arbitrary IF filtering, 3) does not require numerical integration nor differentiation of Fourier series or computer convolutions, and 4) provides greater insight into the underlying error mechanisms. The subsequent, less involved BER calculations using the new approach compare favorably with previously published results. An unexpected bonus, resulting from the analysis, is that nonlinear IF filter phase characteristics can, in many cases, be shown to be unimportant in affecting the bit error rate.

Simulation results are presented for the bit error rate performance of binary and quaternary noncoherent CPFSK when used in conjunction withK=7,R=1/2convolutional encoding and Viterbi decoding. Although coding does not appear to offer any significant advantage in the binary case, the results indicate that the quaternary CPFSK modulation with three-symbol demodulation is a good match to the coding, with performance to within a decibel of coherent QPSK at an error rate of 10-5. Also, simplified optimum demodulator structures are presented, in the case of multisymbol processing, which require onlyNcorrelator branches in the case ofN-ary CPFSK regardless of the number of symbols being used by the demodulator. This is in contrast to conventional receivers which require Nncorrelator branches withn-symbol processing.

The broad-band noise performance of a spread spectrum system with frequency hopping and sequentially balanced modulation was analyzed in Part I of this paper. Here, the analysis is extended to include operation In jamming environments and in specular multipath. Bit-error probability and ranging accuracy curves are presented in each case.

A spread spectrum system incorporating sequentially balanced modulation, fast frequency hopping, phase-comparison tone ranging, and pseudoorthogonal block coding of BPSK data is analyzed from the standpoints of bit error performance and ranging accuracy. A tight upper bound analysis incorporates the effect of the limiter present in the system and accounts for nonzero inner products of the signaling alphabet code words. Somewhat simpler approximate expressions are obtained by a novel application of the central limit theorem. Both the upper bound and the approximate analyses are compared with the results of Monte Carlo simulations and experimental data.

The problem of determining the first-order probability density of a filtered version of hard-limitedRCnoise is considered. An integral equation for the density is derived by using Siegert's results on zero-crossing distributions forRCnoise. We also show how the integral equation can be obtained from a two-dimensional Fokker-Planck equation. The exact solution of the integral equation has not been found, but some general information is available. An iterative scheme for determining the moments has been found, and some information has been obtained regarding the behavior of the density as the variable approaches the ends of its range. One result of this portion of the investigation is a disproof of au expression for the density conjectured by Pawula and Tsai. Finally, curves for the density are obtained by solving the integral equation numerically.

Letx(t)be the Ornstein-Uhlenbeck process andy(t)the result of low-passRCfiltering of sgnx(t). This paper considers the problem of determining the first-order probability density function ofy(t). The approach is to apply the\nuth-order Fokker-Planck-Kolmogorov type equations. Based upon an assumption as to the linearity of a coefficient of the resulting differential equation, a closed-form solution is obtained forp(y). The result agrees with the previous work of Doyle, McFadden and Marx who solved the special case when the bandwidth of theRCfilter is twice the bandwidth of the input noise. The result also agrees, to within experimental error, with a Monte Carlo simulation over four orders of magnitude of variation of the ratio of the bandwidths of theRCfilter and the input process.

A method proposed by Pickard for estimating the center frequency of the power spectrum of a Gaussian process is analyzed in terms of the mean and variance of the estimator. Through limiting and differentiation, this method consists essentially of counting zero-crossings with positive and negative weights so that our analysis of the center-frequency estimator sheds some light on the zero-crossing problem. A simple expression for the variance of the estimator is obtained by a novel application of Price's theorem for zero-memory nonlinear devices with Gaussian inputs. The effects of nonideal limiting and, hence, nonideal counting upon the mean and variance of the estimator are examined in detail. The mathematical techniques employed are felt to have potential usefulness for related problems in communication theory.

In this paper, the classical Fokker-Planck-Kolmogorov equations are generalized to hold for conditional probability density functions of arbitrary random processes. Conditions are derived under which the generalized equations are of finite order both for one-dimensional and for vector random processes. An extension of the generalized equations which overcomes degeneracy occurring in the steady-state case is also presented.

Price's theorem for zero-memory nonlinear devices with Gaussian inputs is modified to make the theorem more amenable to the analysis of functions of more than two Gaussian variates. In the general case a single ordinary differential equation for the output correlation function is derived. Several examples illustrate the applicability of the modified version.