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     

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     

Performance of FH/BFSK with generalized fading in worst casepartial-band Gaussian interference

For frequency-hopped (noncoherent) binary frequency shift keying (FH/BFSK) on a worst-case partial-band Gaussian interference channel, the bit error probability results are well known for the extreme cases where the signal is either nonfading or Rayleigh fading. In this work, the region between these extremes is filled in by considering the general Nakagami-m fading model. The worst-case partial-band Gaussian interference results are given by a one-parameter family which for m→∞ gives the Viterbi-Jacobs nonfading result, and for m=1 gives the Rayleigh fading result. In the latter case, a broadband interference strategy is optimal. Thus, the Nakagami- m results provide a smooth one-parameter bridge between the Viterbi-Jacobs channel and the Rayleigh fading channel. The results show that the worst-case interference fraction ρ increases as the fading variance increases, up to Rayleigh fading. Any fading less severe than Rayleigh, however slight the departure from Rayleigh, requires a partial-band strategy for sufficiently large E_b/N_I     

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     

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     

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     

Rayleigh fading compensation for QAM in land mobile radiocommunications

A pilot symbol-aided Rayleigh fading compensation is investigated for M-ary quadrature amplitude modulation (QAM) to achieve highly spectrally efficient land mobile communication systems. The optimum parameters for fading compensation, bit error rate (BER) performance against E_b/N₀ (energy per bit to the noise power spectrum density), adjacent channel interference, and cochannel interference for 16-QAM, 64-QAM, and 256-QAM, and the spectral efficiencies for these modulation schemes in Rayleigh fading environments are investigated by computer simulation. To further verify the effect of pilot symbol-aided fading compensation from a a practical point of view, a 16-QAM modem is implemented, laboratory experiments are executed, and the impact of the dynamic range limitation due to the resolution of the analog-to-digital (A/D) converters is evaluated, along with the imperfection of the analog circuits. It is demonstrated by computer simulation and laboratory experiments that the pilot symbol-aided fading compensation can sufficiently compensate for fast varying Rayleigh fading, and 16-QAM gives the highest spectral efficiency in the case of cellular systems     

Differential detection of π/4-shifted-DQPSK for digital cellularradio

The detection of π/4-shifted-DQPSK modulation using a tangent-type differential detector with an integrated symbol timing and carrier frequency offset correction algorithm is discussed. π/4-shifted-DQPSK modulation has been proposed for use in a high-capacity, TDMA-based digital cellular system being developed in the US; differential detection could potentially allow the production of low-complexity mobile units. Results obtained using the proposed IS-54 TDMA frame structure for base to mobile transmissions are presented. Theoretical and simulation bit-error-rate (BER) results are presented for static and Rayleigh fading channels. BER results are provided as a function of E_b/N₀ and C/I, where the interferer is a second π/4-shifted-DQPSK signal. Additional results are provided which show the BER sensitivity to Doppler frequency shifts, time delay spread, and carrier frequency offsets     

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     

The cutoff rate for on-off keying

It is well known that on-off keying (OOK), i.e. the transmission of a pulse for a data one (mark) and transmission of no signal for a data zero (space), exhibits a 3 dB inferiority to antipodal signalling, in terms of the signal-to-noise ratio required to achieve a specified bit error probability. When the channel is characterized by the cutoff rate R₀, this 3 dB inferiority persists, in the sense that the required E_b/N₀ to operate at a given value of R₀ is 3 dB greater for the OOK channel than for the antipodal channel. However, it is shown that, if the definition of cutoff rate is generalized in an appropriate way to reflect the fact that signal energy is expended only in the transmission of marks, part of this penalty is recovered. In fact, it is shown that, in the limit of low R₀, there is no penalty in signal-to-noise ratio. Even at rates of realistic interest, the penalty is significantly less than the expected 3 dB     

Multi-H phase-coded modulations with asymmetric modulationindexes

Multi-H phase-coded modulation (MHPM) is a bandwidth-efficient modulation scheme which offers substantial coding gain over conventional digital modulation schemes. MHPM with asymmetric modulation indices corresponding to the bipolar data +1 and -1 is considered, and numerical results for the minimum Euclidean distances are provided. It is shown that performance improvements on the error probability over conventional MHPM are gained with essentially the same bandwidth and a very slight modification in implementation. The upper bounds on the error probabilities as functions of observation intervals and received E_b/N₀ are also investigated in detail. It is concluded that the concept of asymmetric modulation indices for MHPM is attractive for bandwidth and power-efficient modulation     

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     

Repeated convolutional codes for high-error-rate channels

An error-correction scheme for an M-ary symmetric channel (MSC) characterized by a large error probability p_e is considered. The value of p_e can be near, but smaller than, 1-1/M, for which the channel capacity is zero, such as may occur in a jamming environment. The coding scheme consists of an outer convolutional code and an inner repetition code of length m that is used for each convolutional code symbol. At the receiving end, the m inner code symbols are used to form a soft-decision metric, which is passed to a soft-decision decoder for the convolutional code. The effect of finite quantization and methods to generate binary metrics for M>2 are investigated. Monte Carlo simulation results are presented. For the binary symmetric channel (BSC), it is shown that the overall code rate is larger than 0.6R₀, where R₀ is the cutoff rate of the channel. New union bounds on the bit error probability for systems with a binary convolutional code on 4-ary and 8-ary orthogonal channels are presented. For a BSC and a large m, a method is presented for BER approximation based on the central limit theorem     

Connectivity properties of a packet radio network model

A model of a packet radio network in which transmitters with range R are distributed according to a two-dimensional Poisson point process with density D is examined. To ensure network connectivity, it is shown that πR²D, the expected number of nearest neighbors of a transmitter, must grow logarithmically with the area of the network. For an infinite area there exists an infinite connected component with nonzero probability if π R²D>N₀, for some critical value N₀. It is shown that 2.195<N ₀<10.526     

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     

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     

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     

Error probabilities of fast frequency-hopped MFSK withnoise-normalization combining in a fading channel with partial-bandinterference

An error probability analysis performed for an M-ary orthogonal frequency-shift keying (MFSK) communication system employing fast frequency-hopped (FFH) spread-spectrum waveforms transmitted over a frequency-nonselective, slowly Rician fading channel with partial band interference is discussed. Diversity is obtained using multiple hops per data bit. Noise-normalization combining is employed by the system receiver to minimize partial-band interference effects. The partial-band interference is modeled as a Gaussian process. Thermal noise is also included in the analysis. Forward error correction coding is applied using convolutional codes and Reed-Solomon codes. Diversity is found to dramatically reduce the degradation of the noise-normalization receiver caused by partial-band interference regardless of the strength of the direct signal component. Diversity offers significant performance improvement when channel fading is strong, and performance improvement is obtained for high modulation orders (M>2). Receiver performance is improved when diversity, higher modulation orders, and coding are combined     

Refinements to the theory of error rates for narrow-band digital FM

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_b/N₀ 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     

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