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     

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 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     

A note on D-ary Huffman codes

An upper bound on the redundancy of D-ary Huffman codes in terms of the probability p₁ of the most likely source letter is provided. For large values of p₁, the bound improves the one given by R.G. Gallager (1978). Additionally, some results known for the binary case (D=2) are extended to arbitrary D-ary Huffman codes. As a consequence, a tight lower bound that corrects a bound recently proposed by J.D. Golic and M.M. Obradovic (1987) is derived     

A Taylor synthesis for principal solution patterns ofmultiplicative antenna systems

A multiplicative (cross-correlation) receiving antenna system with a linear aperture can have a power pattern P₀(u ) (the so-called principal-solution power pattern) whose spatial frequency transfer function (SFTF) is uniform over the entire spatial frequency (SF) bandwidth. A modified principal solution system which retains the uniform SFTF except for smooth transitions at the ends of the SF passband is described. The transitions are due to a change in the original pattern P₀(u), which suffers from high sidelobes, to a Taylor (1955) synthesis pattern P_T (u) which involves a slowly varying envelope pattern. All of the slowly varying envelope sidelobes of P_T(u ) are set at the same appropriate low level, e.g. -30 dB. The aperture weighting distributions are free of singularities, unlike those for P₀(u), and can be sampled to provide the current weightings for a linear multiplicative array     

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     

Exponential-type error probabilities for multiterminal hypothesistesting

Multiterminal hypothesis testing is considered, subject to the exponential-type constraint α_n⩽exp(-nr) on the error probability of the first kind. The problem is to determine the minimum β*_n of the error probability of the second kind under the given constraint at limited rates R₁ and R₂ for observing the respective pairs of variables. Good lower bounds on the power exponent for β*_n are presented by invoking basic properties of r-divergent sequences. In particular, the one-bit and the zero-rate compression cases are considered. The power exponent for the former and a lower bound for the latter are established     

Statistical inference under multiterminal rate restrictions: adifferential geometric approach

A statistical inference problem for a two-terminal information source emitting mutually correlated signals X and Y is treated. Let sequences Xⁿ and Yⁿ of n independent observations be encoded independently of each other into message sets M_X and M_Y at rates R₁ and R ₂ per letter, respectively. This compression causes a loss of the statistical information available for testing hypotheses concerning X and Y. The loss of statistical information is evaluated as a function of the amounts R₁ and R ₂ of the Shannon information. A complete solution is given in the case of asymptotically complete data compression, R₁, R₂→0 as n→∞. It is shown that the differential geometry of the manifold of all probability distributions plays a fundamental role in this type of multiterminal problem connecting Shannon information and statistical information. A brief introduction to the dually coupled e-affine and m-affine connections together with e -flatness and m-flatness is given     

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     

Capacity of the Gaussian arbitrarily varying channel

The Gaussian arbitrarily varying channel with input constraint Γ and state constraint Λ admits input sequences x=(x₁,---,X_n) of real numbers with Σx_i²⩽nΓ and state sequences s=(S₁,---,s_n ) of real numbers with Σs_i²⩽nΛ; the output sequence x+s+V, where V=(V₁,---,V_n) is a sequence of independent and identically distributed Gaussian random variables with mean 0 and variance σ². It is proved that the capacity of this arbitrarily varying channel for deterministic codes and the average probability of error criterion equals 1/2 log (1+Γ/(Λ+σ²)) if Λ<Γ and is 0 otherwise     

Observation of a large number of new laser lines from12CD3F gas optically pumped with a continuouslytunable high-pressure pulsed CO2 laser

An observation is reported of a large number of new laser lines from ¹²CD₃F gas optically pumped with a continuously tunable high-pressure pulsed CO₂ laser. Making use of the coincidence of the 10 μm P and R branches of CO₂ with the v₃ and v₆ vibrational-rotational absorption bands of ¹²CD₃F, 180 laser lines were found in the wavenumber range between 8 and 55 cm^-1, all of them yet unknown; these lines are studied for characteristic properties of laser action. All laser lines are assigned as pure rotational transitions in the vibrational excited or ground states     

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     

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      

Dependence of the frequency shift of the optical-microwave doubleresonance signal in the Cs D2 line on the pumpingfrequency and power of a GaAs semiconductor laser

The OMDR (optical-microwave double resonance) effect in the Cs D₂ line was studied for realizing a gas-cell-type Cs atomic frequency standard. A glass cell containing Cs with buffer gases (Ar/N₂=1.26, total pressure=39 torr) was placed in a TE₀₁₂ mode microwave cavity at a temperature of 45°C and was pumped using a GaAs semiconductor laser frequency locked to an external interferometer tuned to the 6P_3/2 (F=2,3,4)←6 S_1/2(F=3) transition. The OMDR signal appearing at the resonance to the F=4←3 hyperfine transition of the 6S_1/2 state shifted with detuning of the laser frequency and with change of the laser and microwave powers. The dependence of the shift on these variables around an optimum operating condition was obtained as, Δν_MW[Hz]=-(0.31±0.02) {1+(0.44±0.15) (ΔP_L/P_L)} Δν_L [MHz]-10(ΔV_MW/V _MW)     

A chemically driven visible laser transition using fastnear-resonant energy transfer

Evidence is obtained which demonstrates the potential for developing purely chemical visible lasers based on rapid near-resonant energy transfer from metastable excited triplet states of germanium and silicon monoxide (a³Σ⁺,b ³Π) to select metal atoms. In this study, the Group IIIA metal atoms were chosen as the energy receptors for the energy transfer-pump sequence. Excited triplet states were generated from the Ge-O₃, Ge-N₂, Si-N₂O, and Si-NO₂ reactions; the bulk of the experimental results was obtained with a germanium-based system. The energy stored in the long-lived triplet states is transferred to pump X²P_1/2 thallium, indium, and gallium atoms to their lowest lying ² S_1/2 states. The authors observe a system of temporal behavior which suggests the creation of a population inversion producing a gain condition and forming the basis for full cavity oscillation on the Tl 7²S_1/2-6P _3/2 transition at 535 nm     

Tight bounds on the redundancy of Huffman codes

A method for deriving optimal upper bounds on the redundancy of binary Huffman codes in terms of the probability p₁ of the most likely source letter is presented. This method will be used to compute bounds for all p₁⩾1/127, which were previously known only for a few special cases. Furthermore, the known optimal lower bound for binary Huffman codes is generalized to arbitrary code alphabets and some upper bounds for D-ary Huffman codes, 2⩽D<∞, are given, which are the tightest possible for all p₁⩾1/2     

Bayes estimation of reliability under a random environment governedby a Dirichlet prior

The reliability function of a component whose lifetime is exponentially distributed with a known parameter λ>0 is R (t|λ)=exp (-λt). If an environmental effect multiplies the parameter by a positive factor η, then the reliability function becomes R(t|η,λ)=exp(-ηλt). The authors assume that η itself is random, and its uncertainty is described by a Dirichlet process prior D(α) with parameter α=MG₀, where M>O represents an intensity of assurance in the prior guess, G₀, of the (unknown) distribution of η. Under squared error loss, the Bayes estimator of R(t|η,λ) is derived both for the no-sample problem and for a sample of size n. Using Monte Carlo simulation, the effects of n, M, G₀ on the estimator are studied. These examples show that: (a) large values of n lead to estimates where the data outweigh the prior, and (b) large values of M increase the contribution of the prior to the estimates. These simulation results support intuitive ideas about the effect of environment and lifetime parameters on reliability     

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     

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