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     

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     

Uncoded and coded performance of MFSK and DPSK in Nakagami fadingchannels

The author presents uncoded and coded performance results for noncoherent M-ary frequency-shift keying (MFSK) and differentially coherent binary phase-shift keying (DPSK) in a slow nonselective Nakagami-m (1960) fading channel. He gives simple expressions for the asymptotic slopes of probability of bit error for large signal-to-noise ratio and shows that the effective order of diversity compared to an uncoded Rayleigh channel is the product of two parameters, one for the channel and one for the code. He also compares the uncoded Nakagami-m results to those of the Rician channel in order to show performance differences between these two generalized fading channel models     

Multiple trellis coded frequency and phase modulation

Multiple trellis coded modulation of constant envelope frequency and phase modulated signal sets (MTCM/FPM) is investigated for performance on the additive white Gaussian noise (AWGN) channel and on the one-sided normal, Rayleigh and Rician fading channels. The Nakagami- m fading model is used as an alternative to the Rician fading model to calculate the error probability upper bound for trellis-coded schemes on the fading channel. The likeliness and the disparity between the upper bounds to the error probability for the two fading models are discussed. The design criteria for the one-sided normal fading channel, modeled by the Nakagami-m distribution, are observed to be the same as those for the Rayleigh-fading channel. For the MTCM/FPM schemes, it is demonstrated that the set partitioning designed to maximize symbol diversity (optimum for fading channels) is optimum for performance on the AWGN channel as well. The MTCM/FPM schemes demonstrate improved performance over MTCM/MPSK schemes and TCM/FPM schemes on the AWGN channel and the fading channel     

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     

On the performance of rate 1/2 convolutional codes with QPSK onRician fading channels

Consideration is given to the bit error probability performance of rate 1/2 convolutional codes in conjunction with quaternary phase shift keying (QPSK) modulation and maximum-likelihood Viterbi decoding on fully interleaved Rician fading channels. Applying the generating function union bounding approach, an asymptotically tight analytic upper bound on the bit error probability performance is developed under the assumption of using the Viterbi decoder with perfect fading amplitude measurement. Bit error probability performance of constraint length K=3-7 codes with QPSK is numerically evaluated using the developed bound. Tightness of the bound is examined by means of computer simulation. The influence of perfect amplitude measurement on the performance of the Viterbi decoder is observed. A performance comparison with rate 1/2 codes with binary phase shift keying (BPSK) is provided     

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     

ALTAIR VHF/UHF observations of multipath and backscatterenhancement

During the Defense Nuclear Agency (DNA) PEAK (propagation effects assessment-Kwajalein) experiment in August 1988, the ALTAIR VHF/UHF wide bandwidth radar was used to track spherical satellites in low-earth orbit. The purpose of the experiment was to obtain radar data during the most severe natural propagation disturbances. The PEAK experiment was successful, giving many measurements of strong scintillation as well as the first measurements of frequency-selective fading on propagating radar pulses. Experimental results are used to demonstrate an enhancement, due to scattering, in the average received power that is observed during severe scintillation. The observed statistics of the enhancement are compared to analytic calculations using the Nakagami-m distribution with very good agreement     

1/f noise in MODFETs at low drain bias

The 1/f noise in normally-on MODFETs biased at low drain voltages is investigated. The experimentally observed relative noise in the drain current S_I/I² versus the effective gate voltage V_G=V_GS-V_off shows three regions which are explained. The observed dependencies are S_I/I²∝V_G ^m with the exponents m=-1, -3, 0 with increasing values of V_G. The model explains m =-1 as the region where the resistance and the 1/f noise stem from the 2-D electron gas under the gate electrode; the region with m=0 at large V_G or V_GS≅0 is due to the dominant contribution of the series resistance. In the region at intermediate V_G , m=-3, the 1/f noise stems from the channel under the gate electrode, and the drain-source resistance is already dominated by the series resistance     

A linear bound for sliding-block decoder window size

The author gives an upper bound on the necessary length of a sliding-block decoder window for finite-state codes from arbitrary n -ary data into any constrained system Σ with capacity at least log(n) presented by a graph G with memory m and anticipation a. Specifically, it is shown that the ACH code construction algorithm can be used to construct a code with a sliding-block decoder at rate t:t and with window length m+a+2t, where t is upper-bounded by a linear function of the number of states of G. It is demonstrated that this is the best one can do in the sense that any general upper bound on the decoder window length for finite-state codes into systems Σ with finite memory must grow at least linearly with the number of states of the graph G presenting Σ     

Evaluation of the quantization error in denominator-separable 2-Drecursive filters

The evaluation of the quantization error in two-dimensional (2-D) digital filters involves the computation of the infinite square sum J=Σ^∞_m=φΣ ^∞_n=φy² (m, n). A simple method is presented for evaluating J based on partial fraction expansion and using the residue method provided the Z-transform Y(Z₁, Z₂) of the sequence y(m, n) having quadrant support is a causal bounded input, bounded output (BIBO) stable denominator-separable rational function. The value of J is expressed as a sum of simple integrals which can easily be evaluated. The simple integrals are tabulated for ready reference. The proposed method is suitable for analytical as well as numerical computation and can easily be programmed     

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     

A practical reliability metric for block codes used on binary-inputchannels

A reliability metric (RM) for a block code is defined to be a function that operates on both the decoder input (a block of channel output) and the decoder output (the codeword estimate) and produces a real number as a measure of the reliability of the decoder decision. The best RM has the disadvantage of depending on the codeword probabilities. Thus, the ideal RM is defined as the value that would be computed for equally likely codewords. The implementation of the ideal RM is too costly for most applications. The author proposes an easily implemented RM for binary-input memoryless channels (for state-observable channels with a freely evolving state such as some fading channels) when the codewords consist of n 2^m-ary symbols, each of which is transmitted serially by m uses of the channel. Simulation results for some BCH codes and some Reed-Solomon codes used in a simple ARQ system show that the proposed RM performs nearly as well as the ideal RM and much better than a previously proposed practical RM     

Limited search trellis decoding of convolutional codes

The least storage and node computation required by a breadth-first tree or trellis decoder that corrects t errors over the binary symmetric channels is calculated. Breadth-first decoders work with code paths of the same length, without backtracking. The Viterbi algorithm is an exhaustive trellis decoder of this type; other schemes look at a subset of the tree or trellis paths. For random tree codes, theorems about the asymptotic number of paths required and their depth are proved. For concrete convolutional codes, the worst case storage for t error sequences is measured. In both cases the optimal decoder storage has the same simple dependence on t. The M algorithm and algorithms proposed by G.J. Foschini (ibid., vol.IT-23, p.605-9, Sept. 1977) and by S.J. Simmons (PhD. diss., Queens Univ., Kingston, Ont., Canada) are optimal, or nearly so; they are all far more efficient than the Viterbi algorithm     

Delay and coding gain analysis of a truncated type-II hybrid ARQprotocol

Time delay of a truncated type-II hybrid ARQ protocol with one retransmission is analyzed by using queueing theory. The truncated protocol uses rate 1/2 error correcting codes. Parity bits for error correction are transmitted only when they are needed. It is shown that the truncated type-II hybrid protocol has a bounded delay and bounded queue length under typical communication traffic conditions. The error performance of the truncated protocol is further analyzed for a normalized Nakagami-m fading channel and compared with the performance of a conventional (untruncated) protocol. The results show that the truncated protocol provides coding gain improvement over the untruncated type-II hybrid ARQ protocol. Thus, it is suggested that the truncated protocol is best suited to power- and delay-limited mobile applications     

Fast decoding of codes from algebraic plane curves

Improvement to an earlier decoding algorithm for codes from algebraic geometry is presented. For codes from an arbitrary regular plane curve the authors correct up to d*/2-m² /8+m/4-9/8 errors, where d* is the designed distance of the code and m is the degree of the curve. The complexity of finding the error locator is O(n^7/3 ), where n is the length of the code. For codes from Hermitian curves the complexity of finding the error values, given the error locator, is O(n²), and the same complexity can be obtained in the general case if only d*/2-m²/2 errors are corrected     

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     

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     

A coding scheme for m-out-of-n codes

A scheme for the construction of m-out-of-n codes based on the arithmetic coding technique is described. For appropriate values of n, k, and m, the scheme can be used to construct an (n,k) block code in which all the codewords are of weight m. Such codes are useful, for example, in providing perfect error detection capability in asymmetric channels such as optical communication links and laser disks. The encoding and decoding algorithms of the scheme perform simple arithmetic operations recursively, thereby facilitating the construction of codes with relatively long block sizes. The scheme also allows the construction of optimal or nearly optimal m-out-of-n codes for a wide range of block sizes limited only by the arithmetic precision used     

Minimum distance of logarithmic and fractional partialm-sequences

Two results are presented concerning the partial periods (p-p's) of an m-sequence of period 2ⁿ-1. The first proves the existence of an m-sequence whose p-p's of length approximately (n+d log₂ n) have minimum distance between d and 2d for small d. The second result is of an asymptotic nature and proves that the normalized minimum distance of p-p's whose length is any fraction of the period of the m-sequence, approaches 1/2 as the period of m-sequence tends to infinity