Spectral shaping and digital synthesis of an M-ary timeseries

The use of pulse shaping to control transmitted spectral density precisely is examined. A digital filter architecture is described that not only mitigates the traditional problems of lengthy development intervals and cost manufacturing methods, but offers the additional features of intrinsically coding high-speed binary (M=2) data into M-ary symbols while ensuring highly reproducible, baud-normalized, transmitter pulse shaping. A conceptual basis for the digital synthesis method is first described, including a functional circuit appropriate to the simplest filter realizations. Spectral effects internal to the filter are considered, and a simple method to obtain desired transmitted spectra is outlined. It is shown that even relatively short pulses used in high-level modulation systems lead to impractical memory storage demands; however, the simple expedient of segmenting the finite impulse function greatly reduced the individual memory requirements, though it necessitates intermediate adding operations. Experimental examples illustrate the design methodology for quaternary (M=4) data signals in a Nyquist communication channel and serve as points of reference for addressing performance and design flexibility     

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)     

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     

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     

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     

M-ary frequency shift keying withlimiter-discriminator-integrator detector in satellite mobile channelwith narrowband receiver filter

An expression is derived for the error probability of M-ary frequency shift keying with a limiter-discriminator-integrator detector and a narrowband receiver filter in the satellite mobile channel. This channel contains, as special cases, the Gaussian and Rayleigh (land mobile) channels. The error probability is computed as a function of various system parameters for M=2, 4, 8 symbols and a third-order Butterworth receiver filter     

Phase noise effects on M-ary PSK trellis codes

The performance of M-ary phase-shift keying (PSK) coded systems in the presence of added white Gaussian noise (AWGN), with a noisy carrier reference in the receiver, is examined. The results are obtained by combining the Chernoff bound technique with the generating function approach. The bit error probability analysis is carried out for 8- and 16-PSK with code rates of 2/3 and 3/4, respectively. The numerical results obtained demonstrate the error performance degradation due to a Tikhonov phase noise process with parameter α     

Limiter-discriminator detection of a coherent optical heterodyneM-ary CPFSK receiver

The performance of a coherent optical M-ary continuous-phase frequency-shift-keying (CPFSK) receiver using limiter-discriminator (L-D) detection is investigated. It is shown that L-D detection of CPFSK optical signals offers the best performance for a large normalized IF beat spectral linewidth, ΔνT. When the modulation index is unity, the receiver is immune to laser phase noise and can produce (M/4) exp (-SNR) symbol error probability, which may be considered as the upper bound if the optimal modulation index is used (SNR is the signal-to-noise ratio per symbol). Optimum modulation indexes are 0.8 and 1 at ΔνT=1% and ΔνT=2%, respectively, for M=4, 8, and 16     

Probability of error of M-ary PSK and DPSK on a Rayleighfading channel

M-ary phase-shift keying and differential phase-shift keying (DPSK) on a slow fading Rayleigh channel without diversity is investigated. Expressions for the distribution of the phase angle between a vector with Rayleigh amplitude distribution and a noiseless reference, and between two vectors both with Rayleigh amplitude distribution perturbed by Gaussian noise are obtained     

Performance of convolutionally coded optical M-PPM systemswith imperfect slot synchronization

Performance of convolutionally coded M-ary pulse position modulation (M-PPM) systems in the presence of slot synchronization errors is evaluated for the shot-noise-limited photon-counting receiver and the avalanche photodetection receiver. Both hard and soft (δ-max) demodulation results are given, and two soft-decision metrics are investigated     

Coherent detection of M-ary phase-shift keying in thesatellite mobile channel with tone calibration

A formula is derived for the error probability of M-ary phase-shift keying (PSK) in the satellite mobile channel when the signal is accompanied by a pilot tone which is used in the receiver for coherent detection of the signal. The author computes the error probability as a function of the signal-to-noise ratio, the ratio of powers in the specular and diffused signal component, the ratio of bandwidth in the pilot tone and signal extraction filters, and the numbers of symbols, i.e. M, while optimizing the ratio of powers in the pilot tone and signal. He computes results when the diffused component is not delayed with respect to the specular component, as well as for the opposite case, i.e. when the symbol in the diffused component is independent of the symbol in the main component     

The performance of reference-based M-ary PSK with trelliscoded modulation in Rayleigh fading

The performance of trellis-coded M-ary phase shift keying (MPSK) is analyzed for three types of phase detection system; differential detection, pilot-symbol-aided detection and pilot-tone-aided detection. A near-exact solution to the probability of event error in Rayleigh fading is derived, and two extremely tight upper bounds generated. They improve the Chernoff upper bound by at least a factor of four at medium and high signal-to-noise ratios (SNR)     

Baseband pulse shaping to improve M-ary FSK in satellitemobile systems

Baseband pulse shaping is proposed to improve the performance of M-ary FSK with limiter discriminator integrator detection in a satellite mobile channel. Numerical results obtained for binary, quaternary, and octonary signaling with a Gaussian and a third-order Butterworth IF filter indicate that pulse shaped signals perform significantly better than frequency shift keying (FSK) signals. The error rate floor in the presence of a Doppler shift is also analyzed. Numerical results indicate that the error rate floor of pulse-shaped signals is significantly lower than that of FSK signals     

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     

Error probability of M-ary FSK with differential phasedetection in satellite mobile channel

Formulas are derived for the error probability of M-ary frequency shift keying (FSK) with differential phase detection in a satellite mobile channel. The received signal in this channel is composed of a specular signal, a diffuse signal, and white Gaussian noise; hence, the composite signal is fading and has a Rician envelope. The error probability is shown to depend on the following system parameters: (1) the signal-to-noise ratio; (2) the ratio of powers in the specular and diffuse signal components; (3) the normalized frequency deviation; (4) the normalized Doppler frequency; (5) the maximum normalized Doppler frequency; (6) the correlation function of the diffuse component, which depends on the normalized Doppler frequency and the type of the antenna; (7) the number of symbols; and (8) the normalized time delay between the specular and diffuse component (t _d/T) where 1/T is the symbol rate. Except for T_d/T, all normalized parameters are the ratios of the parameter value and symbol rate. The Doppler frequency depends on the velocity of the vehicle and the carrier frequency. The error probability is computed as a function of the various parameters. The bit error probability is plotted as a function of signal-to-noise ratio per bit and other system parameters     

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     

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×N Booth encoded multiplier generatorusing optimized Wallace trees

The architecture of a design method for an M-bit by N -bit Booth encoded parallel multiplier generator are discussed. An algorithm for reducing the delay inside the branches of the Wallace tree section is explained. The final step of adding two N±M-1-bit numbers is done by an optimal carry select adder stage. The algorithm for optimal partitioning of the N ±M-1-bit adder is also presented     

Rotationally invariant trellis-coded modulations withmultidimensional M-PSK

Using a multidimensional approach, the author discovers a large family of rotationally invariant trellis-coded M-PSK (M-ary shift keying) schemes, M⩾8, with nominal coding gains ranging from 3 to 5 dB and with bandwidth requirements the same as, or even less than, those of uncoded M/2-PSK schemes at the same information bit rate. The rotationally invariant schemes have performance and complexities comparable to the best known nonrotationally-invariant trellis-coded two-dimensional M-PSK schemes. Computer simulation results for these schemes, assuming an additive white-Gaussian-noise (AWGN) channel, are reported