Performance analysis of RS-coded M-ary FSK forfrequency-hopping spread spectrum mobile radios

Decoding performance of Reed-Solomon (RS) coded M-ary FSK with noncoherent detection in a frequency-hopping spread spectrum mobile radio channel is theoretically analyzed. Exact formulas and an approximate one for evaluating word error rates (WERs) of error correction and error-and-erasure correction schemes on decoding the RS codes are derived. It is shown that with K symbol erasure and C symbol error detection, RS coded M-ary FSK achieves the equivalent diversity order of (K+1)(C+1)     

A concatenated coded modulation scheme for error control

A concatenated coded modulation scheme is presented for error control in data communications. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary phase-shift keying (PSK) modulation. Error performance of the scheme is analyzed for an additive white Gaussian noise (AWGN) channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner-code and a relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The scheme is particularly effective for high-speed satellite communications for large file transfer where high reliability is required. A simple method is also presented for constructing block codes for M-ary PSK modulation. Soome short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft-decision Viterbi decoding algorithm. Furthermore, some of these codes are phase invariant under multiples of 45° rotation     

Multilevel trellis MPSK modulation codes for the Rayleigh fadingchannel

The multilevel coding technique is used for constructing multilevel trellis M-ary phase-shift-keying (MPSK) modulation codes for the Rayleigh fading channel. In the construction of a code, all the factors which affect the code performance and its decoding complexity are considered. The error performance of some of these codes based on both one-stage optimum decoding and multistage suboptimum decoding has been simulated. The simulation results show that these codes achieve good error performance with small decoding complexity     

Identities and approximations for the weight distribution of q-ary codes

An explicit formula is derived that enumerates the complete weight distribution of an (n, k, d) linear code using a partially known weight distribution. An approximation formula for the weight distribution of q-ary linear (n, k , d) codes is also derived. It is shown that, for a given q-ary linear (n, k, d) code, the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q=q ^-(n-k) as u becomes large. The error term is a decreasing function of the minimum weight of the dual. The results are also valid for nonlinear (n, M, d) codes with the minimum weight of the dual replaced by the dual distance     

On linear structure and phase rotation invariant properties ofblock M-PSK modulation codes

Two important structural properties of block M(=2^' )-ary PSK modulation codes, linear structure and phase symmetry, are investigated. An M-ary modulation code is first represented as a code with symbols from the integer group S_M-PSK=(0,1,2,---,M-1) under modulo-M addition. Then the linear structure of block M-PSK modulation codes over S_M-PSK with respect to modulo- M vector addition is defined, and conditions are derived under which a block M-PSK modulation code is linear. Once the linear structure is developed, the phase symmetry of block M-PSK modulation codes is studied. In particular, a necessary and sufficient condition for a block M-PSK modulation code that is linear as a binary code to be invariant under 2^h/180°M phase rotation, for 1⩽h⩽l is derived. Finally, a list of short 8-PSK and 16-PSK modulation codes is given, together with their linear structure and the smallest phase rotation for which a code is invariant     

Throughput analysis of a slotted frequency-hop multiple-accessnetwork

A link throughput analysis is presented for a slotted frequency-hop multiple-access (FHMA) packet radio network (PRN) operating in the presence of background noise, partial-band noise jamming, and partial-band tone jamming. The PRN consists of an arbitrary number of transceivers arranged in a paired-off topology. Forward error-correction coding is used for packet protection. M-ary FSK modulation is used with hard-decision decoding. Expressions are derived for the link throughput in terms of the channel cutoff rate and capacity. The dependency of the optimal processing gain, code rate, and jamming fraction on the population size, traffic intensity, bit energy to background noise ratio, and bit energy to jammer noise ratio is examined in detail. It is shown that a properly designed (optimized) PRN using random-access FHMA offers a significantly larger heavy-load throughput than a random-access frequency-division multiple-access PRN     

Generalized cutoff rate of time- and frequency-selective fadingchannels

The generalized cutoff rate of time- and frequency-selective fading channels is evaluated for M-ary frequency-shift keying (MFSK) and M-ary differential phase-shift keying (MDPSK) modulation with soft decoding. The optimal signaling rate and code rate for dispersive channels are evaluated. The guard time effect, is used in multipath spread channels, is evaluated for frequency-selective channels, and the optimal combination of signaling rate, code rate, and guard time is presented. Special attention is given to CCIR (International Radio Consultative Committee) HF channel models     

Error-correcting codes and nonlinear diversity combining againstworst case partial-band noise jamming of frequency-hopping MFSK systems

Error probability analyses are performed for a coded M-ary frequency-shift keying system (MFSK) using L hops per M-ary word frequency-hopping spread-spectrum waveforms transmitted over a partial-band Gaussian noise jamming channel. The bit error probabilities are obtained for a square-law adaptive gain control receiver with forward-error-control coding under conditions of worst-case partial-band noise jamming. Both thermal noise and jamming noise are included in the analyses. Performance curves are obtained for both block codes and convolutional codes with both binary and M-ary channel modulations. The results show that thermal noise cannot be neglected in the analysis if correct determinations of the optimum order of diversity and the worst-case jamming fraction are to be obtained. It is shown that the combination of nonlinear combining, M -ary modulation, and forward-error-control coding is effective against worst-case partial-band noise jamming     

On the performance of spectrally efficient trellis coded FMmodulation employing noncoherent FM demodulation

The authors analyze the problem of noncoherent FM demodulation of trellis-coded continuous-phase M-ary FSK (frequency-shift keying). The FM demodulation process is divided into two parts, the first being the actual noncoherent FM demodulation and the second being trellis decoding of the data. Upper bounds on the bit error rate as well as the 99% energy bandwidth are determined for the codes under consideration. In particular, the authors consider the trellis codes with rates 1/2 and 2/3 and symmetric and asymmetric signal constellations. Upper bounds on the probability of error are obtained for the symmetric and the optimum asymmetric cases. The optimum asymmetry is one which minimizes the bit error probability. The performance of this system is compared to that of the standard continuous phase modulation techniques employing noncoherent detection     

Spread-spectrum multiple-access performance of orthogonal codes forindoor radio communications

Direct-sequence spread spectrum, with its inherent resistance to multipath, is a promising technique for indoor wireless communication. To allow multiple users within a limited bandwidth, code division multiple access (CDMA) is needed. The bandwidth efficiency of M-ary CDMA systems in fading multipath indoor radio channels is analyzed. It is shown that M-ary signaling improves the bandwidth efficiency significantly when compared to binary signaling     

Pseudo-analog speech transmission in mobile radio communicationsystems

A low-complexity pseudo-analog speech transmission scheme is proposed for portable communications. It uses a speech coder based on adaptive differential pulse code modulation (ADPCM) in combination with a multilevel digital modulation technique such as M-ary DPSK or M-ary FSK and features low quantization noise, bandwidth efficiency, and robustness to transmission errors. A nonsymmetric M -ary DPSK scheme called skewed M-ary DPSK is proposed to enhance the noisy channel performance. Comparison to conventional analog FM and a digital speech transmission scheme using adaptive predictive coding and forward error correction (FEC) based on convolutional coding shows that the pseudo-analog system has the best objective signal-to-noise ratio performance under most channel conditions. Informal subjective evaluations rate the digital system superior to the pseudo-analog scheme for bad channels and conversely for good channels. It is concluded that the pseudo-analog system can be designed with low delay and high speech quality for good channels with high spectral efficiency     

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     

An improvement to generalized-minimum-distance decoding

A novel acceptance criterion that is less stringent than previous criteria is developed. The criterion accepts the codeword that is closest to the received vector for many cases where previous criteria fail to accept any codeword. As a result, the performance of generalized minimum distance (GMD) decoding is better if the new criterion is used. For M-ary signaling, the weights used in GMD decoding are generalized to permit each of the possible M symbol values to have a different weight     

Signal sets matched to groups

Recently, linear codes over Z_M (the ring of integers mod M) have been presented that are matched to M -ary phase modulation. The general problem of matching signal sets to generalized linear algebraic codes is addressed based on these codes. A definition is given for the notion of matching. It is shown that any signal set in N-dimensional Euclidean space that is matched to an abstract group is essentially what D. Slepian (1968) called a group code for the Gaussian channel. If the group is commutative, this further implies that any such signal set is equivalent to coded phase modulation with linear codes over Z_M. Some further results on such signal sets are presented, and the signal sets matched to noncommutative groups and the linear codes over such groups are discussed     

On the decoding of algebraic-geometric codes

A decoding algorithm for algebraic-geometric codes arising from arbitrary algebraic curves is presented. This algorithm corrects any number of errors up to [(d-g-1)/2], where d is the designed distance of the code and g is the genus of the curve. The complexity of decoding equals σ(n³) where n is the length of the code. Also presented is a modification of this algorithm, which in the case of elliptic and hyperelliptic curves is able to correct [(d-1)/2] errors. It is shown that for some codes based on plane curves the modified decoding algorithm corrects approximately d/2-g/4 errors. Asymptotically good q-ary codes with a polynomial construction and a polynomial decoding algorithm (for q⩾361 on some segment their parameters are better than the Gilbert-Varshamov bound) are obtained. A family of asymptotically good binary codes with polynomial construction and polynomial decoding is also obtained, whose parameters are better than the Blokh-Zyablov bound on the whole interval 0<σ<1/2     

Decoding algebraic-geometric codes up to the designed minimumdistance

A simple decoding procedure for algebraic-geometric codes C _Ω(D,G) is presented. This decoding procedure is a generalization of Peterson's decoding procedure for the BCH codes. It can be used to correct any [(d*-1)/2] or fewer errors with complexity O(n³), where d * is the designed minimum distance of the algebraic-geometric code and n is the codelength     

On repeated-root cyclic codes

A parity-check matrix for a q-ary repeated-root cyclic code is derived using the Hasse derivative. Then the minimum distance of a q-ary repeated-root cyclic code is expressed in terms of the minimum distance of a certain simple-root cyclic code. With the help of this result, several binary repeated-root cyclic codes of lengths up to n=62 are shown to contain the largest known number of codewords for their given length and minimum distance. The relative minimum distance d_min/n of q-ary repeated-root cyclic codes of rate r⩾R is proven to tend to zero as the largest multiplicity of a root of the generator g(x) increases to infinity. It is further shown that repeated-root cycle codes cannot be asymptotically better than simple-root cyclic codes     

Performance of fast frequency hopped noncoherent MESK witha fixed hop rate under worst case jamming

The performance of fast-frequency-hopped M-ary frequency-shift keying with a fixed hop rate is evaluated, utilizing the Chernoff union bound method. The performance criterion used is a throughput measure i.e., an information rate sustained by a system for a given bit error rate, normalized by the hop rate. Both uncoded and coded systems are considered. It is shown using the cutoff rate argument that coding can provide a few dB gain in throughput. This is confirmed by the performance evaluation of various convolutional and block codes. Both partial-band noise jamming and multitone jamming with one tone per M -ary band are considered. Jamming parameters are assumed to be the worst case against the coding channel. Determination of the optimum M is also addressed     

M-ary orthogonal signaling in the presence of Doppler

The performance degradation of an M-ary orthogonal keying (MOK) system due to relative motion between its transmitter and receiver can be minimized by selecting good signaling sets. A formulation for evaluating good signaling sets, or code sets, is developed for Walsh function data modulations. A union bound performance measure which closely approximates the exact probability of a demodulation error, is devised to allow for convenient evaluation of Walsh function codesets. The best Walsh function codesets for codevectors up to length 10 and for M=2, 4, 8 and 16, found by exhaustive search, are presented, along with their performance. In addition, codesets based on error-correcting codes are presented, along with a performance bound expressed in terms of the code's minimum distance     

Bounds for abnormal binary codes with covering radius one

The normality of binary codes is studied. The minimum cardinality of a binary code of length n with covering radius R is denoted by K(n,R). It is assumed that C is an (n,M)R code, that is, a binary code of length n with M codewords and covering radius R. It is shown that if C is an (n,M)1 code, then it is easy to find a normal (n ,M)1 code by changing C in a suitable way, and that all the optimal (n,M)1 codes (i.e. those for which M=K(n,1)) are normal and their every coordinate is acceptable. It is shown that if C is an abnormal (n,M) code, then n⩾9, and an abnormal (9118)1 code which is the smallest abnormal code known at present, is constructed. Lower bounds on the minimum cardinality of a binary abnormal code of length n with covering radius 1 are derived, and it is shown that if an (n,M)1 code is abnormal, then M⩾96