Asymptotic performance of M-ary orthogonal modulation ingeneralized fading channels

The asymptotic (M→∞) probability of symbol error P_e,_m for M-ary orthogonal modulation in a Nakagami-m fading channel is given by the incomplete gamma function P(m, mx) where x=In 2/(E_b/N₀) and E_b is the average energy per bit. For large signal-to-noise ratio this leads to a channel where the probability of symbol error varies as the inverse mth power of E_b/N₀. These channels exist for all m⩾1/2. The special case of m=1 corresponds to Rayleigh fading, an inverse linear channel     

Secret key agreement by public discussion from common information

The problem of generating a shared secret key S by two parties knowing dependent random variables X and Y, respectively, but not sharing a secret key initially, is considered. An enemy who knows the random variable Z, jointly distributed with X and Y according to some probability distribution P_XYZ, can also receive all messages exchanged by the two parties over a public channel. The goal of a protocol is that the enemy obtains at most a negligible amount of information about S. Upper bounds on H(S) as a function of P_XYZ are presented. Lower bounds on the rate H (S)/N (as N→∞) are derived for the case in which X=[X₁, . . ., X _N], Y=[Y₁, . . ., Y_N] and Z=[Z₁, . . ., Z_N] result from N independent executions of a random experiment generating X_i, Y_i and Z_i for i=1, . . ., N. It is shown that such a secret key agreement is possible for a scenario in which all three parties receive the output of a binary symmetric source over independent binary symmetric channels, even when the enemy's channel is superior to the other two channels     

Asymptotic performance of M-ary orthogonal signals inworst case partial-band interference and Rayleigh fading

It is shown that for worst-case partial-band jamming, the error probability performance (for fixed E_b/N_I) becomes worse with increasing M for (M>16). The asymptotic probability-of-error is not zero for any E_b/N _I(>ln 2), but decreases inverse linearly with respect to it. In the fading case, the error-probability performance (for fixed E_b/N₀) improves with M for noncoherent detection, but worsens with M for coherent detection. For large E_b/N₀ the performance of the Rayleigh fading channel asymptotically approaches the same limit as the worst case partial-band jammed channel. However, for values of M at least up to 4096, the partial-band jammed channel does better. While it is unlikely that an M-ary orthogonal signal set with M>1024 will be used in a practical situation, these results suggest an important theoretical problem; namely, what signal set achieves reliable communication     

Error performance of a digitally processed binary ESK frequencyhopping receiver in the presence of large Doppler

Analysis is made of the effects of Doppler on the error rate performance of a low data rate binary FSK frequency hopping receiver, employing a discrete Fourier transform (DFT) technique for baseband detection. Bit detection decision is made by locating the maximum of the DFT outputs which, in the frequency domain, are assumed to be separated by 1/T where T is the bit period. Both the worst case and average error performances are obtained and presented as a function of E_b/N₀ for various values of M where E_b/N₀ is the signal bit energy-to-noise density ratio and M is the degree of freedom associated with the Doppler uncertainty window. The E _b/N₀ degradation as a function of M is also presented     

The calculation of the probability of detection and the generalizedMarcum Q-function

A highly reliable, accurate, and efficient method of calculating the probability of detection, P_N(X,Y ), for N incoherently integrated samples, where X is the constant received signal-to-noise ratio of a single pulse and Y is the normalized threshold level, is presented. The useful range of parameters easily exceeds most needs. On a VAX/11 computer with double precision calculations, better than 13-place absolute accuracy is normally achieved. There is a gradual loss of accuracy with increasing parameter values. For example, for N=10⁹, and with both NX and Y near 10⁷, the accuracy can drop to ten places. The function P_N(X,Y ) can be equated to the generalized Marcum Q-function, Q_m(α,β). The corresponding limits on α and β are roughly 4500 for the 13-place accuracy and 60000 for ultimate (INTEGER×4) limit     

Upper bounds on capacity for a constrained Gaussian channel

A low-pass and a bandpass additive white Gaussian noise channel with a peak-power constraint imposed on otherwise arbitrary input signals are considered. Upper bounds on the capacity of such channels are derived. They are strictly less than the capacity of the channel when the peak-power constrain is removed and replaced by the average-power constraint, for which the Gaussian inputs are optimum. This provides the answer to an often-posed question: peak-power limiting in the case of bandlimited channels does reduce capacity, whereas in infinite bandwidth channels it does not, as is well known. For an ideal low-pass filter of bandwidth B, the upper bound is Blog 0.934P/(N₀B) for P/( N₀B)≫1, where P is the peak power of the input signal and N₀/2 is the double-sided power spectral density of the additive white Gaussian noise     

Optimal probability-of-error thresholds and performance for twoversions of the sign detector

Expressions are obtained for specifying the optimal error probability (minimum P_e) thresholds λ₀₁ and λ₀₂ for the traditional and modified sign detectors, respectively. These thresholds are shown to depend on the parameters p, P₁, and M where: M is the number of observations z_i used in the test statistic; P₁=P(H₁ ) is the prior probability for hypothesis H₁ that signal s₁ is present and 1-P₁ =P(H₀) corresponds to the hypothesis H₀ that signal s₀ is present; and p=Pr{z_i⩾0|H₁} with s₀=0 for the traditional sign detector and p=Pr{z_i⩾λ|H₁ }=Pr{z_i<λ|H₀} with λ =(s₀+s₁)/2 for the modified sign detector. The expressions for λ₀₁ and λ₀₂, are given explicitly, and shown to be independent of P₁ for sufficiently large M. Optimal P_e versus M performance curves, corresponding to both versions of the sign detector, are obtained for a representative range of values for p and P₁     

A linear lightwave Benes network

The authors consider linear lightwave networks with a single waveband that have N inputs, each with a transmitter, and N outputs, each with a receiver, interconnected by optical links, broadcast stars, and wavelength-independent 2×2 switches. The transmitters and receivers can tune to C different wavelengths. The authors describe a rearrangeably nonblocking network that is a modification of the Benes network and uses transmitters that are fixed tuned and switches with two states. The network uses [1+o(1)] N/log₂(N/C) switches, which is shown to be nearly the minimum number. It is also shown that, if C =o(log N), then a rearrangeably nonblocking network requires [1+o(1)]Nlog₂N switches even if the switches have more than two states     

Combined tone and noise jamming against coded FH/MFSK ECCM radios

The authors conjecture that a proper combination of partial-band tone jamming (PBTJ) and full-band noise jamming (FBNJ) under a given total jamming power constraint may be more effective than PBTJ alone, not only for the case with a low (E_s/N_J), but also for the case with high E_s/N_J, since the FBNJ can corrupt the jamming state information (JSI). Assuming this combination of PBTJ and FBNJ jamming, they consider three cases of receiver processing-the hard decision (HD) metric without JSI, the HD metric with perfect JSI, and the maximum-likelihood (ML) metric using Viterbi's ratio threshold (VRT) to generate a 1-b symbol decision quality indicator. System performance is evaluated in terms of the Chernoff bound on the probability of symbol error. From extensive numerical analysis the authors conclude that, for the case of the HD metric without JSI, PBTJ-only jamming is the worst form of jamming, as expected, since the receiver does not use JSI at all; for the other cases, a combination of PBTJ and FBNJ is the worst, with the worst ratio of PBTJ power to FBNJ power a function of the values of M and E_s/N_J     

Diffusion and noise in GaAs material and devices

The variation of the diffusion coefficient D(E) versus the electric field strength E is determined at 300 K in n-type GaAs (N_D=3×10^-17 cm^-3 ), using pulsed high-frequency noise measurements. D(E) is found to increase slightly at low field, then to decrease down to one tenth of its ohmic value near the threshold field. Long (⩾4 μm) real n⁺-n-n⁺ Gunn diodes, with an arbitrary doping profile, can be modeled. Comparisons are made, and excellent agreement is found, between experimental and theoretical characteristics of two real diodes, with notch and with gradual doping profiles. The doping profile N_D(x ) is shown to have a considerable influence on the diode behavior, in regard to the electric field profile as well as the noise characteristics. Using the impedance field method, the noise current is modeled and found to by very sensitive in the D(E) variation law, in particular in the range of 2.5-4 kV/cm. The agreement between the experimental noise and the computed noise of real diodes is quite satisfactory when using the D(E) determined     

Wavelength dependence of 1/f noise in the light output oflaser diodes: an experimental study

The optical power emitted by a monomode GaAlAs laser is filtered with a monochromator. The 1/f noise in the filtered emission is found to be directly dependent on the noncoherent emission, such as S_pαP_nc^m. Here s_p is the spectral density of the 1/f fluctuations, P_nc is the average noncoherent power, m=3/2 under spontaneous emission, and m=4 in the superradiation and laser regions. Study of the 1/f noise in the optical power in a band centered at the laser wavelength and with variable bandwidth shows three operating regions. (1) LED region (at low currents): the fluctuations with a 1/f spectrum are uncorrelated in wavelength. (2) Superradiation region (at currents close to the threshold): the fluctuations are correlated. (3) Laser region: the 1/f noise apparently is dominated by noncoherent emission within a small optical band around the laser wavelength     

Statistical properties of timing jitter due to data echo in digitalmodem receivers

Consideration is given to the influence of the noise and data sequences present in the received data signal on a nondecision-aided timing recovery scheme in digital modem receivers, It is known that white noise is not particularly disturbing for timing recovery, whereas data signals such as local echo (or residual echo) introduce a bias, called jitter, into the sampling recovered phase. It is shown that when the disturbing data signal has power P less than the power S of the useful signal (whose timing must be recovered), the jitter is sinusoidal with amplitude proportional to the ratio P/S. In the opposite situation, the bias increases indefinitely with time     

Resolution function of nonsinusoidal radar signals. I.Range-velocity resolution with rectangular pulses

A generalization of a previously published ambiguity function that applies to radar known as large-relative-bandwidth radar, carrier-free radar, impulse radar, or nonsinusoidal radar is discussed. This radar has attracted attention because of its ability to penetrate absorbing materials used in the stealth technology. Another good application is the detection of moving targets with a small radar cross section by a look-down radar, which calls for a thumbtack ambiguity function. Since a small radar cross section in this application is typically due to the small size of the target that is coated with absorbing material, the antistealth feature of the nonsinusoidal radar is implicitly being used. The principle is presented of a resolution function (tentatively called the range-velocity or the range-Doppler resolution function) based on processing a nonsinusoidal signal consisting of N characters with a time separation T_D and each character consisting of a sequence of L binary pulses of duration T . It is shown that range-velocity resolution functions approaching the ideal thumbtack function are easy to obtain. The blind speeds of the pulse-Doppler radar with sinusoidal carrier do not inherently occur, and all velocities are observed as true velocities rather than as velocities modulo the first blind speed (velocity ambiguity)     

Combined effects of phase sweeping transmitter diversity andchannel coding

A novel transmitter diversity scheme that generates forced fading to improve the performance of channel coding is proposed and investigated. Since the required phase sweeping frequency is much smaller than the transmission bit rate, bandwidth expansion is negligible. A sinusoidal phase sweeping function ΔΘ sin(2πf_Ht) is employed in laboratory experiments using 32 kbit/s quarternary differential phase shift keying (QDPSK) with differential detection and BCH(23,12) code. It is shown that ΔΘ=200° and f_H=67 Hz can be used when m (interleaving depth)=10 b. Hence, excessively long interleaving is not required by the use of transmitter diversity. Under very slow Rayleigh fading (f_D=1 Hz), a measured improvement of 4.8 dB is obtained at a word error rate of 10^-2 without receiver diversity. Applications include paging systems that require very simple receivers     

Self-avoiding random loops

A random loop, or polygon, is a simple random walk whose trajectory is a simple Jordan curve. The study of random loops is extended in two ways. First, the probability P_n(x,y) that a random n-step loop contains a point (x,y) in the interior of the loop is studied, and (1/2, 1/2) is shown to be (1/2)-(1/ n). It is plausible that P_n(x,y) tends toward 1/2 for all ( x,y), but this is not proved even for (x,y)=(3/2,1/2) A way is offered to simulate random n-step self-avoiding loops. Numerical evidence obtained with this simulation procedure suggests that the probability P_n (3/2,1/2)≈(1/2)-(c/n), for some fixed c      

More on the decoder error probability for Reed-Solomon codes

A combinatorial technique similar to the principle of inclusion and exclusion is used to obtain an exact formula for P_E (u), the decoder error probability for Reed-Solomon codes. The P_E(u) for the (255, 223) Reed-Solomon code used by NASA and for the (31, 15) Reed-Solomon code (JTIDS code) are calculated using the exact formula and are observed to approach the Qs of the codes rapidly as u gets large. An upper bound for the expression |P_E(u)/ Q-1| is derived and shown to decrease nearly exponentially as u increases     

Polarization insensitive homodyne detection with all opticalprocessing based on the photorefractive effect

An optical homodyne PSK detection scheme based on the photorefractive effect in BaTiO₃ has been investigated. The system uses polarization locking to implement a polarization insensitive receiver using all optical processing and only one detector. A theoretical analysis of the detection process is presented and verified experimentally. The system is shown to have a sensitivity of -52 dBm with S/N=2. The rate at which the system responds to changes in the incoming signal state of polarization is also investigated     

Measurement of the characteristics of the optical-microwavedouble-resonance effect of the 87Rb D1line for application as an atomic frequency standard

The OMDR (optical-microwave double resonance) spectrum of ⁸⁷ Rb with the aim of using a frequency-stabilized GaAs semiconductor laser instead of an Rb lamp as a pumping source in a gas-cell-type Rb frequency standard. Natural isotope ⁸⁷Rb was sealed in a glass cell with buffer gases (Ar/N₂=1.2, total pressure=39 torr). The double resonance signal in the 5P_1/2(F=2)←5S_1/2( F=1) transition appearing at the resonance to the F=2←1 hyperfine transition of the 5S_1/2 state was detected. The optimum operational cell temperature was 56°C. The peak-to-peak frequency width of the atomic hyperfine resonance discriminator used to stabilize the microwave frequency shifts induced by detuning of the laser frequency, changes in the laser and microwave powers, and temperature drift of the cell were investigated     

Bounds on maximum throughput for digital communications withfinite-precision and amplitude constraints

The problem of finding the maximum achievable data rate over a linear time-invariant channel is considered under constraints different from those typically assumed. The limiting factor is taken to be the accuracy with which the receiver can measure the channel output. More precisely, the following problem is considered. Given a channel with known impulse response h(t), a transmitter with an output amplitude constraint, and a receiver that can distinguish between two signals only if they are separated in amplitude at some time t ₀ by at least some small positive constant d, what is the maximum number of messages, N_max, that can be transmitted in a given time interval [0,T]? Lower bounds on N_max can be easily computed by constructing a particular set of inputs to the channel. The main result is an upper bound on N_max for arbitrary h(t). The upper bound depends on the spread of h(t), which is the maximum range of values the channel output may take at some time t₀>0 given that the output takes on a particular value α at time t=0. Numerical results are shown for different impulse responses, including two simulated telephone subscriber loop impulse responses     

Automatic decision threshold level control in direct-sequencespread-spectrum systems

An algorithm for automatic decision threshold-level control in direct-sequence spread-spectrum systems is presented and analyzed. Using this algorithm, instantaneous setting of the threshold is possible. An additional loop called a threshold loop is used to improve the system performance. This loop is based on a constant false alarm rate (CFAR) criterion. The purpose of the algorithm is to set a decision threshold in the system which will provide a small probability of false alarm (P₃→0) and a large probability of signal detection (P_d→1) at the same time. The analysis has shown that the threshold probability distribution function (PDF) is narrower (equaling better performance) in the case of the CFAR algorithm, but this comes at the cost of a considerably larger integration time for the system (the system time response is slower). As a compromise, the instantaneous algorithm is modified so that k successive samples of the signal are used to determine the value of the threshold. This permits better control of the shape and position of the threshold PDF with respect to the P₃ and P _d curves. At the same time, the time response of the system is good and can be easily controlled