This paper presents a comparative analysis of the performance of three UWB (ultra wideband) DSSS (direct sequence spread spectrum)/Rake modems used to communicate with mobile platforms over new mobile outdoor and indoor dense multipath UWB channel models developed in a companion paper. The Rake modems analyzed differ in the techniques used to estimate the channel and are called "decision-directed" "parallel- probe" and "serial-probe" modems. The performance analysis includes the effects of time-selective distortion and noise on the estimated tap weights wherein the time constant of the tap weight measurements and the number of Rake taps are optimized. Numerical results are presented for 2 GHz bandwidth transmitted signals, a 7.5 GHz carrier frequency, and channel BPSK symbol rates of .15, 1.5, and 15 Megasymbols/sec. The well known rate 1/2, constraint length 7 convolutional code using 3 bit soft decision Viterbi decoding is assumed. One moving and one stationary platform are postulated, with a velocity of 100 ft/sec for the outdoor channel and 5 ft/sec for the indoor channel. Required values of Es/Noto achieve 10-6decoded error rate are computed as a function of multipath spread.

The WSSUS channel model has provided a useful tool for modeling random time variant linear channels in the four decades since its introduction. However, with the advent of UWB (Ultra Wideband) communications, it is necessary to replace this model which is valid for Narrowband (NB) channels. This paper develops such a replacement. The development is based upon the tapped delay line model of a linear time- variant channel, which is valid for any ratio of bandwidth to carrier frequency. For simplicity, the analysis assumes a dense multipath environment such as may be experienced in outdoor and indoor urban environments. The thrust of this study is to derive an UWB channel model for dense multipath that will allow a meaningful comparative modem performance evaluation rather than to develop a new precise characterization of UWB channels. Such a model should be useful in preparation of UWB communication standards for mobile links.

This paper presents analytical and numerical results on the sample size required to achieve a specified root mean square (RMS) error in estimating the error rate for flat fading channels having complex Gaussian statistics. The analysis shows that for the large sample sizes normally used in estimating error rates, k, the required sample size normalized to the required sample size for independent symbol fading, can be expressed in the form k=1+d/spl beta/ where d is the symbol rate normalized to the Doppler spread of the channel. For a given modem, /spl beta/ is a function of the error probability and the order of diversity. It is shown that if the Doppler spread measure used is proportional to the RMS Doppler spread, P will be relatively insensitive to the shape of the Doppler power spectrum. Numerical results are presented for Lth order diversity reception of binary phase shift keying (PSK), differential PSK, and frequency shift keying (FSK) signals and for five different Doppler power spectra. Ideal maximal ratio combining is assumed for the PSK modem, and square law combining is assumed for the DPSK and FSK modems.

This paper compares the effectiveness of linear and decision feedback equalizers in counteracting the effect of frequency-selective distortion in microwave relay links. A quadratic frequency-selective model of the channel is employed and particular attention is given to channels with a null in the passband. Both staggered QPSK and QPR modems are studied. Performance is studied for error probability and mean squared error. The general conclusion is reached that a decision feedback equalizer is required for robust performance.

This paper is concerned with characterizing the link between an airplane and a satellite. Attention is focused on the effect of indirect paths scattered from the surface of the earth. Applicable propagation-theoretic and system function-theoretic work is reviewed and integrated. Some new and some known expressions for channel correlation functions are presented for the "steepest descent" channel model.

An exact analysis is carried out on the effects of noise and surface scatter multipath on the one-way ranging errors of a singlesideband tone ranging modem used in an airplane-satellite link. With appropriate redefinition of parameters the results are shown to apply approximately to double-sideband tone ranging modems at high signal/ noise and signal/multipath ratios. The theoretical results are applied to obtain numerical results on ranging error for typical system geometries and sea states and for a representative hemispherical-coverage aircraft antenna. Multipath is shown to cause larger ranging errors than previously indicated. The need for additional channel measurements is pointed out.

This paper concerns the problems of the measurability and measurement of random time-variant linear channels. With regard to measurability, a new, less stringent channel measurability criterion is proposed to supersede the BL product introduced by Kailath. This criterion involves the area of occupancy of the Doppler-delay spread function (or its dual). By using time and bandwidth constraints on the input and output of a channel, channel measurement is reduced to the measurement of a discrete set of finite parameters. Optimal measurement techniques are described and their performances determined for both known and unknown channel correlation functions.

This paper derives upper and lower bounds on binary error probability as a function of signal-to-noise ratio for several digital data systems operating over a complex Gaussian fading channel. Bounds of varying degrees of tightness are obtained by placing certain physically meaningful constraints on the allowable detected noise probability distributions. Detailed derivations are given for bounds corresponding to constraints on the "crest-factor" of the detected noise and on the ratio of peak noise power to average signal power. The calculations include the effects of diversity corn-billing and are applicable to frequency-shift keying (FSK), binary and quaternary phase-shift keying (PSK) using a pilot tone as reference, and binary and quaternary PSK using the previous signaling element as a phase reference. It is an interesting result of this paper that for moderate noise crest factors the upper and lower bounds can be rather close. In the particular case of nondiversity operation, for example, the lower bound actually becomes approximately equal to the upper bound at SNR's of30dB or greater for noises with a crest factor as high as20dB.

This paper determines the relationship between the rms bandwidth of the envelope of a narrow-band Gaussian process and the rms bandwidth of the process itself for a general nonlinear (no-memory) envelope detector. The interesting result is shown that the output rms bandwidth depends only on the input rms bandwidth and the nonlinear device and not on the shape of the input spectrum. As an illustrative example, it is shown that the rms bandwidth of the output of a\nuth-law envelope detector is\sqrt{\nu/2}times the rms bandwidth of the input process. Some attention is given to the determination of higher moments of the output spectrum in terms of moments of the input spectrum.

This paper develops a concept of duality, called time-frequency duality, which is applicable to a class of networks called communication-signal-processing networks. Such networks consist of an interconnection of basic elements such as filters, mixers, delay lines, etc. The usefulness of time-frequency duality stems from the fact that two situations which may be quite distinct physically can have identical behavior patterns except for an interchange of the roles played by time and frequency. As a result, the solution to a detection or estimation problem may be found directly from the solution of the dual problem, if known, merely by replacing variables and quantities by their duals. This application of time-frequency duality is illustrated in the paper by the problem of measuring the transfer function of a scatter medium by means of an optimal gating operation prior to spectrum analysis. Another benefit to be obtained from time-frequency duality is the generation of new ideas for communication signal processing techniques. We illustrate this type of benefit by constructing the dual of the Kineplex communication system. A third benefit is the additional insight gained into a communication problem by the ability to look at it in another way. This benefit is illustrated by the characterization of time-variant linear channels. It is demonstrated that such channels may be characterized in an interesting symmetrical manner in time and frequency variables by defining dual system functions.

This paper considers the properties of the tap gain functions in a tapped delay line model of the orbital dipole channel. Specifically, auto-correlation and cross-correlation functions are computed for these tap functions in terms of assumed statistics for the motion of illuminated dipoles. It is demonstrated that when a certain inequality involving the transmitted signal bandwidth is satisfied, these correlation functions become identical to those which would arise from the corresponding tap gain functions of a suitably defined channel consisting of a continuum of motionless scintillating uncorrelated scatterers. In effect, this inequality sets an upper bound to the bandwidth of the transmitted signal, beyond which the orbital dipole channel may not be modeled as such a continuum of uncorrelated scatterers. Simplified expressions for tap gain correlation functions are derived for the case of large signal bandwidths, including the subcase in which the above mentioned inequality is satisfied. In the latter case the tap gain functions are shown to become mutually uncorrelated for all time shifts.

In this paper, an analysis is made of a communication system in which the information-bearing signal phase modulates a Gaussian noise carrier. The effect of additive Gaussian noise and linear filtering on the first-order statistics of the receiver output noise and on the character of the output signal are determined. It is shown that with regard to determining the distortion of the output signal, the system may be replaced by a single linear filter whose input is the modulated signal impressed on a sinusoidal rather than noise carrier. In this way, conventional FM techniques may be used for the determination of signal distortion.

A narrow-band process is conveniently characterized in terms of a complex envelope whose magnitude is the envelope, and whose angle is the phase variation of the actual narrow-band process. When the narrow-band process is normally distributed, the complex envelope has the properties of a complex normally distributed process. This paper investigates the approach to the complex normally distributed form of the complex envelope of the output of a narrow-band filter when the input is wide-band non-Gaussian noise of a certain class, and the bandwidth of the narrow-band filter approaches zero. The non-Gaussian input consists of a train of pulses having identical waveshapes, but random amplitudes and phases. While the derivations assume statistical independence between pulses, it is shown that the results are valid for a certain interesting class of dependent pulses. The Central Limit Theorem is proved in the multidimensional case for the output process.

This paper is concerned with certain aspects of the problem of discriminating between two Gaussian processes. The emphasis is on determining approximate optimum detector structures which avoid some of the mathematical difficulties inherent in the evaluation of the exact optimum detector structure. To this end, an approach termed the "inverse operator" approach is presented which leads to approximate detector structures via the Neumann series expansion of linear operator theory. These approximate detectors are found by using a finite number of terms in an "optimun detector" expansion which results from the use of the above Neumann series expansion. A sufficient condition for the rapid convergence of the optimum detector expansion is found to be that the eigenvalues of a certain operator have magnitudes much less than unity. An upper bound is derived for the error incurred at the detector output by the use of a finite number of terms in the optimum detector expansion. Error probabilities are calculated for the case in which the detector outputs may be assumed approximately normally distributed. From an output "signal-to-noise" ratio point of view, it is shown that the performance of the optimum detector and approximate detectors will differ negligibly if the above eigenvalues have squared magnitudes much less than unity. Some upper bounds are derived for the largest eigenvalue (magnitude).

This paper is concerned with the effect of hard limiting on the signal detectability of a system consisting of a limiter, narrow-band filter, and envelope detector in cascade. The input to the system is a pulsed IF signal immersed in noise whose power spectrum is uniform over a band of widthWcycles. Assuming that the noise bandwidthWis much larger than the bandwidth of the narrow-band filter, the probability distribution of the output of the filter will approach Gaussian. A bivariate Edgeworth series approximation is necessary to handle the narrow-band-filter output since the "in-phase" and "quadrature" components of the narrow-band-filter output are statistically dependent random variables. An expression is derived for the probability of incorrect dismissal that requires the numerical evaluation of single integrals only. From the same bivariate Edgeworth series, an expression is derived for the probability-density function of the output of the envelope detector for the zero-input-signal case. Subsequent integration leads to the probability of false alarm.

The methods of inverse probability have been used by Woodward and others to obtain lower bounds on variances of radar parameter estimators. Previous results on the lower bounds of variances of delay and Doppler estimators have assumed that the reflecting object travels with a constant line of sight velocity and does not cause scintillation in the radar return. Using the inverse probability approach, this paper derives expressions for the minimum variances of estimators of the delay, Doppler, and Doppler rate of a radar return assumed to consist of a long train of pulses with independent scintillation from pulse to pulse.

Two channels carry noise waveforms,N_0 (t) + N_1 ( t)andN_0 (t) + N_2 (t), whereN_0 (t)is a common narrow-band Gaussian noise andN_1 (t)andN_2 (t)are independent narrow-band Gaussian noises associated with each channel. The outputs of each channel are sent through detectors whose outputs,F(x, y), are identical homogeneous functions of the components,xandy, of their inputsN, whereN(t) = x(t) \cos \omega_0 t + y(t) \sin \omega_0 t. LetR_{12}(\tau)be the normalized cross-correlation function of the two detector outputs. It is shown that to determineR_{12}(\tau)it suffices to know the normalized auto-correlation functionR_0(\tau)of the output of a single such detector when the input isN_0(t); i.e., ifR_0(\tau) = G(\sigma_0^2, p(\tau))wherep(\tau)and\sigma_0are the normalized auto-correlation function and rms of either component ofN_0, then it is shown thatR_{12}(\tau) = G(\sigma_0^2, Z_{\rho}(\tau))whereZ = [\{1 + (\sigma_1^2/\sigma_0^2)\} \{(1 + (\sigma_2^2/\sigma_0^2)\}]^{-\frac{1}{2}}.