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     

Optimization of signal sets for partial-response channels. I.Numerical techniques

Given a linear, time-invariant, discrete-time channel, the problem of constructing N input signals of finite length K that maximize minimum l₂ distance between pairs of outputs is considered. Two constraints on the input signals are considered: a power constraint on each of the N inputs (hard constraint) and an average power constraint over the entire set of inputs (soft constraint). The hard constraint, problem is equivalent to packing N points in an ellipsoid in min(K,N-1) dimensions to maximize the minimum Euclidean distance between pairs of points. Gradient-based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems. Two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves Shannon capacity, assuming additive white Gaussian noise     

Focused codes for channels with skewed errors

Consider a channel with inputs and outputs in the field F _q(q>2). It is said that the channel is skewed on a set B⊂F_q* if the additive noise generated by the channel is likely to lie in B, i.e. B is a set of common errors. The concern is the construction of focused codes that are appropriate for such channels. It is said that a code is (t₁,t₂)-focused on B if it can correct up to t₁+t₂ errors provided at most t₁ of those errors lie outside of B; the strategy is to offer different levels of protection against common and uncommon errors and so provide novel tradeoffs between performance and rate. Techniques for constructing focused codes and bounds on their rates are described     

Performance limits of coded multilevel DPSK in cellular mobileradio

Performance limits of coded multilevel differential PSK (MDPSK) in multipath Rayleigh fading channels are described. The simple Gaussian metric is assumed for reasons for practicality even though it is not the maximum likelihood. The channel cutoff rate, R₀ of MDPSK is analyzed based on the metric. Account is taken of additive white Gaussian noise (AWGN), cochannel interference, and multipath channel delay spread. For the analysis of the spectrum efficiency of a cellular mobile radio system employing coded MDPSK, its service area is defined as the area in which, with a bit rate of R information bit/symbol (R⩽R₀), reliable communications are possible. Three optimal information bit rates are determined from the channel cutoff rate to minimize the required average signal energy per information bit-to-noise power spectral density ratio (E_b/N₀) to maximize the tolerable r.m.s. delay spread τ_r.m.s. and to maximize the spectrum efficiency     

The bandwidth performance of a two-element adaptive array withtapped delay-line processing

The bandwidth performance of a two-element adaptive array with a tapped delay line behind each element is examined. It is shown how the number of taps and the delay between taps affect the bandwidth performance of the array. An array with two weights and one delay behind each element is found to yield optimal performance (equal to that obtained with continuous-wave interference) for any value of intertap delay between zero and T₉₀/B, where T ₉₀ is a quarter-wavelength delay time and B is the fractional signal bandwidth. Delays less that T₉₀ yield optimal performance but result in large array weights. Delays larger than T₉₀/B yield suboptimal signal-to-interference-plus-noise ratio when each element has only two weights. For delays between T₉₀/B and 4T₉₀/B , the performance is suboptimal with only two taps but approaches the optimal if more taps are added to each element. Delays larger than T₉₀/B result in suboptimal performance regardless of the number of taps used     

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     

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     

Differential detection in quadrature-quadrature phase shift keying(Q2PSK) systems

A generalized quadrature-quadrature phase shift keying (Q² PSK) signaling format is considered for differential encoding and differential detection. Performance in the presence of additive white Gaussian noise (AWGN) is analyzed. Symbol error rate is found to be approximately twice the symbol error rate in a quaternary DPSK system operating at the same E_b/N₀. However, the bandwidth efficiency of differential Q²PSK is substantially higher than that of quaternary DPSK. In differential detection, it is quite reasonable to suspect that errors tend to occur in pairs. It is shown that when the error is due to AWGN, the ratio of double error rate to single error rate can be very high, and the ratio may approach zero at high signal-to-noise ratio. In an attempt to improve the error rate performance, differential detection through maximum-likelihood decoding based on multiple or N symbol observations is considered     

On the performance of an ATM switch with multichannel transmissiongroups

A routing architecture applying the concept of multichannel transmission groups (MCTGs) for ATM systems is proposed. A queuing analysis of an internally nonblocking ATM switch employing this MCTG concept with partially shared output buffers is presented. The analysis is based on the discrete-time DA///D/c /B queuing model. Both bulk input traffic bulk-size distribution (A) and deterministic traffic (D₁ +. . .+D_N) are considered. The impact of switch speedup on the performance is also taken into account. It is shown that the MCTG architecture yields better performance in terms of delay and cell loss probability than its single channel counterpart. It is also found that the switch speedup required to closely approximate the optimal performance obtained by having the switch fabric run N times as fast as the input and output channels, where N is the size of the switch, is rather small compared to N. This makes the practical realization of the proposed switch architecture feasible     

Performance of 16-state SQAM in a nonlinearly amplifiedmultichannel interference environment

The performance of 16-state SQAM (superposed-QAM) system, in a nonlinearly amplified multichannel environment, in the presence of additive white Gaussian noise (AWGN), intersymbol interference (ISI), and adjacent channel interference (ACI), is experimentally analyzed via computer simulation. Various channel conditions, such as channel spacings between the main and adjacent channels, and fade depth on the desired main channel, are examined. Baseband signal waveshape of 16-SQAM as well as the receive low-pass filter bandwidth are optimized to achieve a good P(e) performance. The result shows that 16-SQAM system is more efficient than MAMSK (multiamplitude MSK) in the frequency utilization     

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     

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     

Burst error statistics of simulated Viterbi decoded BPSK on fadingand scintillating channels

The author presents and analyzes burst error statistics of a soft-decision Viterbi decoder when the transmitted signal is encoded with the 313 (3, 1/2) or 31123 (5, 1/2) convolutional codes, modulated via coherent binary phase-shift keying (BPSK) for the additive white Gaussian noise (AWGN) channel, and subjected to slow and nonselective scintillation/fading modeled by the Nakagami-m distribution. These statistics were generated by Monte-Carlo simulations, and presented in terms of burst error length average and quantile (90 and 99%) statistics versus SNR (E_b/N₀) parameterized by the fading intensity parameter m. The results indicate how Viterbi decoder burst error statistics vary with the fading/scintillation intensity m for Nakagami-m channels, and, consequently, provide information important to the design of interleaved or noninterleaved concatenated coding schemes for such channel environments     

Truncation length for Viterbi decoding

A bound is derived and analyzed for the bit error rate (BER) of a Viterbi decoder with survivor truncation. Estimates of the SNR (signal-to-noise ratio) loss on the AWGN (additive white Gaussian noise) channel due to truncation are obtained for convolutional codes. Larger truncation lengths are required than the smallest value that does not effectively decrease the code's free distance, especially at low E _b/N₀     

Multiuser signal detection using sequential decoding

The application of sequential decoding to the detection of data transmitted over the additive white Gaussian noise channel by K asynchronous transmitters using direct-sequence spread-spectrum multiple access (DS/SSMA) is considered. A modification of R.M. Fano's (1963) sequential-decoding metric, allowing the messages from a given user to be safely decoded if its E_b/N₀ exceeds -1.6 dB, is presented. Computer simulation is used to evaluate the performance of a sequential decoder that uses this metric in conjunction with the stack algorithm. In many circumstances, the sequential decoder achieves results comparable to those obtained using the much more complicated optimal receiver     

Rate-compatible punctured convolutional codes (RCPC codes) andtheir applications

The concept of punctured convolutional codes is extended by punctuating a low-rate 1/N code periodically with period P to obtain a family of codes with rate P/(P+l), where l can be varied between 1 and (N-1)P. A rate-compatibility restriction on the puncturing tables ensures that all code bits of high rate codes are used by the lower-rate codes. This allows transmission of incremental redundancy in ARQ/FEC (automatic repeat request/forward error correction) schemes and continuous rate variation to change from low to high error protection within a data frame. Families of RCPC codes with rates between 8/9 and 1/4 are given for memories M from 3 to 6 (8 to 64 trellis states) together with the relevant distance spectra. These codes are almost as good as the best known general convolutional codes of the respective rates. It is shown that the same Viterbi decoder can be used for all RCPC codes of the same M. the application of RCPC codes to hybrid ARQ/FEC schemes is discussed for Gaussian and Rayleigh fading channels using channel-state information to optimise throughput     

Multiple differential detection of continuous phase modulationsignals

A sequence estimation algorithm for the differential detection of the continuous phase modulation (CPM) signals, yielding significant gains in BER performance and with considerable resistivity to fading, is introduced. These advantages, along with the reduced hardware complexity, low cost, and fast synchronization which characterize the differential detector, make the proposed receiver useful for land mobile radio and mobile-satellite communications. The new receiver is based on multiple differential detection. The multiple differential detection strategy provides the decoder with more information regarding the transmitted data and applies a noise decorrelation process on the received signal, useful to the sequence estimation. The algorithm is derived in a general form, and can be applied on any CPM scheme, with any degree of complexity. The authors have evaluated the receiver for two of the most popular CPM schemes, the tamed frequency modulation (TFM) and Gaussian minimum-shift keying (GMSK) (with B₁ T=0.25), in the presence of additive white Gaussian noise (AWGN) and Rician fading. The BER performance evaluation results indicated significant gains and considerable reduction of error floors. In AWGN improvements close to 9 dB have been verified     

Quantization loss in convolutional decoding

The loss in quantizing coded symbols in the additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) or quadrature phase-shift keying (QPSK) modulation is discussed. A quantization scheme and branch metric calculation method are presented. For the uniformly quantized AWGN channel, cutoff rate is used to determine the step size and the smallest number of quantization bits needed for a given bit-signal-to-noise ratio (E_b/N₀) loss. A nine-level quantizer is presented, along with 3-b branch metrics for a rate-1/2 code, which causes an E_b/N₀ loss of only 0.14 dB. These results also apply to soft-decision decoding of block codes. A tight upper bound is derived for the range of path metrics in a Viterbi decoder. The calculations are verified by simulations of several convolutional codes, including the memory-14, rate-1/4 or -1/6 codes used by the big Viterbi decoders at JPL     

Comments on `GMSK with differential phase detection in thesatellite mobile channel'

The error probability results shown by I. Korn (see ibid., vol.38, no.11, p.1980-6, 1990) indicate that the error floor is higher for systems with decision feedback (DF). It was concluded that DF gives a lower error probability only for smaller values of the normalized bandwidth B_tT of the premodulation Gaussian filter, higher values of the ratio of powers in the direct and diffuse signal components K, and a lower range of signal-to-noise ratio. It is shown that this conclusion is not correct by theoretically analyzing the case of the land mobile channel where K=0 (or -∞ dB) and deriving a simple closed-form expression for the error probability for 1 bit differential detection with DF. It is shown that DF reduces the error probability for all values of B_tT and signal-to-noise ratios. The formula derived can be easily evaluated not only for Gaussian minimum shift keying (GMSK) but for all partial-response continuous-phase-modulation (PRCPM) signals     

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