In this study the authors derive a formula to compute the optimal signal constellation, bit error probabilities and decision thresholds for M-ASK (amplitude shift keying) with non-equal symbol probabilities. The resulting signal constellation is not equally spaced. The authors demonstrate the importance of assigning lower energy signals to symbols with higher probabilities. The authors show that the optimal system has a lower error probability than a conventional system, that is, a system with source coding, equiprobable symbols and equal space constellation. In several cases presented in this study the error probability of the optimal system is significantly lower than in the conventional system. The authors show that even with equal symbol probabilities the equal space constellation is only asymptotically optimal, not optimal for M≥4. However in the several cases that the authors have studied the difference in the error probabilities between ASK with optimal constellation and equal space constellation is insignificant in practice. The authors further extend our results in M-ASK to M-QASK (quadrature ASK) and similar improvements can be obtained.

We derive an equation for the bit-error probability (BEP) of Gaussian minimum-shift keying with limiter discriminator detection in Nakagami-m fading channels including selection and switch-and-stay combining. We compute the BEP for various selections of system parameters.

Optimal signal energies are derived for optimal binary digital communication systems with arbitrary signal probabilities and correlation with both coherent and noncoherent detection. The resulting bit-error probability (BEP) is computed and compared with the BEP of the same systems with equal signal energies. One of the conclusions is that for the coherent system with nonnegative correlation, and for the noncoherent system with arbitrary correlation, the optimal signals are on-off keying (OOK), i.e., the signal with probability p/spl les/0.5 has energy E/p, while the second signal has zero energy, where E is the average signal energy. The proposed system is also better than a system with source coding and equiprobable signals.

The effect of imperfect timing is analyzed in equal gain combining (EGC) and maximal ratio combining (MRC) techniques over Rayleigh and Nakagami-m fading channels with binary phase-shift keying modulation. In the case of EGC, the bit-error probability is derived, while in the case of MRC, error rate bounds are presented. Theoretical results are justified by computer simulation. Numerical results demonstrate that both EGC and MRC are fairly sensitive to timing errors, and comparatively, MRC is more sensitive.

We analyze the performance of a multiamplitude and multidimensional digital communication system with a simple detector in the presence of additive white Gaussian noise (AWGN). We compute the power and spectral efficiency of this scheme and compare them with those of conventional schemes with three different pulses.

In this paper, we analyze the performance (power and bandwidth efficiency) of a scheme with multidimensional signals, and both optimal detector and nonoptimal but simple detector, and show that they are inferior to known schemes with narrowband Nyquist signals and optimal detectors.

This paper investigates how a bandlimiting transmitter filter and the shape of the frequency pulse effect the bit-error probability of frequency shift keying with differential phase detection in satellite mobile channel. Numerical results are presented for the case when the transmitter and receiver filters are Butterworth filters of order N/sub T/=4 and N/sub R/=3, respectively, and the frequency shaping pulse is rectangular or raised cosine. It is shown that in all cases of practical interest, continuous phase modulation (with raised cosine pulse shaping) gives a lower bit-error probability and requires less bandwidth than frequency-shift keying (with rectangular pulse shaping) when all other parameters (number of symbols, Rician factor of the channel, order of filters, etc.) are the same.>

The author comments on the paper by Korn (IEEE Trans. on Commun., vol.38, no.11, p.1980-6, 1990). It is pointed out that a statement in parentheses following equation (52) in the original paper should be removed. Some new conclusions are drawn to replace the earlier erroneous ones.>

In M-ary, Gray coded digital communication the bit error probability is usually approximated by the symbol error probability divided by the number of bits in a symbol. This approximation is known to be excellent for phase shift keying with coherent detection at high-signal-to-noise ratios and Gaussian channel. Here it is illustrated that this approximation may also be good for M-ary frequency shift keying with limiter-discriminator-integrator detection on Gaussian, Rayleigh, and Rician channels.>

In M-ary, Gray coded DPSK, and FSK with differential phase detection the bit error probability is usually approximated by the symbol error probability divided by the number of bits in a symbol. This approximation is known to be excellent for PSK with large signal-to-noise ratios in a Gaussian channel. In other cases this approximation may be questionable. Here we compute the exact values of bit error probability for DPSK and FSK in Gaussian, Rayleigh, and Rician channels. We also compute the relative error involved in this approximation. The conclusion is that except for very low values of signal-to-noise ratio the approximation is good for all channels.>

The author derives a formula for error probability of partial-response continuous-phase modulation with differential phase detector and limiter discriminator detector in a multipath Rayleigh fading channel, taking into account frequency-selective fading, cochannel interference, Doppler frequency shift, and additive Gaussian noise while the receiver rejects a specified amount of adjacent channel interference. A formula for the error floor is also presented. Numerical results are presented for Gaussian minimum shift keying with a premodulation normalized filter bandwidth of 0.25. Under mild channel conditions and low energy-to-noise ratios, the best detector is an optimized two-bit differential detection; otherwise the best detector is the limiter discriminator detector.>

It is shown how to derive formulas for the error probability for M-ary differential phase shift keying with differential phase detection (DPD) and M-ary frequency shift keying with DPD, limiter-discriminator detection and limiter-discriminator-integrator detection in the satellite mobile channel (SMC) with narrowband receiver filter if such formulas are available for the Gaussian channel. The modification of the formulas involves only a redefinition of the noise power and autocorrelation function. Since the SMC contains as special cases the land mobile (Rayleigh) channel and the Gaussian channel, the derived formulas are valid for these channels as well. In fact the formula for the land mobile channel is in many cases reduced to a closed form, which does not contain an integral. The author computes the error probability for the four systems, and compares their performance assuming a third-order butterworth filter and M=2,4,8 symbols.>

Formulas are derived to find the error probability of partial-response frequency-shift keying with limiter discriminator detection, with and without decision feedback, for a satellite-mobile channel. The special cases of a Gaussian channel and a land-mobile channel are discussed. The formulas are applied to Gaussian minimum shift keying, and the error probability is computed as a function of energy-to-noise ratio, Gaussian filter bandwidth, Doppler frequency, maximum Doppler frequency, the ratio of powers in the specular and diffuse signal components, and the time delay between specular diffuse components.>

In a satellite mobile channel (SMC) and land mobile channel (LMC) because of fading and nonlinear power amplifiers, constant envelope modulation and noncoherent detection methods may outperform other schemes. It is shown how to compute the error floor for four noncoherent digital communication systems in satellite and land mobile channels. Differential phase shift keying (DPSK) with differential phase detection (DPD) or frequency shift keying (FSK) with DPD, limiter discriminator integrator detection. (LDID), or limiter discriminator detection (LDD) are studied. The error floor is the residual error probability when SNR is infinity, i.e. the error probability in the system is limited by the error floor. The error floor is computed as a function of Doppler frequency, modulation index, and ratio of powers in the specular and diffuse signal components for DPSK-DPD, FSK-DPD, FSK-LDID and FSD-LDD systems.>

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.>

A formula is derived for the error probability of partial-response continuous-phase modulation (which contains Gaussian minimum shift keying (GMSK) as a special case) with N-b (N=1, 2) differential phase detection (DPD) for the satellite mobile channel, which contains as special cases the Gaussian and Rayleigh channels. In the satellite mobile channel, the input signal is the sum of a direct component, a diffuse component, and white Gaussian noise. The receiver is either with or without decision feedback, and the decision region is optimized for 2-b DPD to minimize the error probability. The error probability for GMSK is computed as a function of signal-to-noise ratio and other system or channel parameters (Doppler frequency, Gaussian filter bandwidth, ratio of powers in the direct and diffuse signal components, etc.). It is shown that decision feedback is more effective for GMSK with narrow bandwidth. The 2-b DPD is superior to the 1-b DPD for low Doppler frequencies and signal-to-noise ratios. For practical vehicle velocities and bit rates, a 2-b DPD with decision feedback outperforms all other analyzed schemes when the signal-to-noise ratio is low.>

A class of digital phase-modulated signals called DCPM (digital complex phase-modulated) signals is introduced. In this class, the imaginary part of the phase is obtained from the real part by a linear time-invariant transformation. A method for computing the autocorrelation function (ACF) of DCPM is presented. It is shown that the ACF does not always exist. The power spectral density of DCPM is computed for Hilbert, identity, and zero transforms. It is also shown that DCPM with the Hilbert transform has a suppressed sideband and can yield significant bandwidth savings over zero transform schemes.>

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.>

We derive an equation for the error probability ofM-ary frequency shift keying with a limiter-discriminator-integrator detector in the presence of narrow-band filters in the transmitter and receiver. We present numerical results for the case of quaternary symbols, wide-band transmitter filter, and Butterworth filter in the receiver with 2, 3, and 4 poles. We optimize the sampling time and threshold setting for various values of filter bandwidth. We show the sensitivity of the error probability to errors in sampling time and threshold setting.

In this paper we present formulas for the computation of error probability in the presence of quadrature-channel or adjacentchannel interference in addition to intersymbol interference in a minimum shift keying system. The filters in the receiver and transmitter are arbitrary but with a finite number of poles. The effect of phase jitter in the main channel, phase and symbol timing misalignment in the interfering channels, and sampling time jitter is taken into account. The probability of error is averaged over the phase and symbol timing misalignment. Numerical results are presented for Butterworth filters in the receiver and transmitter with two, three, and four poles. Curves of error probability as a function of various variables (signal-to-noise ratio, bandwidth of receiver and transmitter filters, number of poles, channel frequency separation, phase jitter, sampling time, and symbol timing and phase misalignment) are presented. The method of this paper can readily be applied to other filters; hence, it can be used in the design and prediction of the performance of digital communication systems.

In this paper we study the effect of intersymbol and quadrature channel interference on the average error probability of 16-ary offset quadrature amplitude modulation with rectangular or sinusoidal shaping functions. We present numerical results assuming that the transmitter and receiver filters are identical Butterworth filters of orderN = 1,2,..., 6,and the decision rule is either fixed or adaptive.

In this paper, we derive a simple expression for the degradation in signal-to-noise ratio caused by intersymbol and adjacent channel interference in MSK systems for which the total response is nonnegative and the response is such that only two intersymbol and one adjacent channel terms are dominant. An MSK system with Gaussian filters in transmitter and receiver satisfy this condition. The resulting expression is simple because we simultaneously maximize the signal value from the main channel (by selecting the sampling time) and the degradation caused by the signals in the interfering channels. For such systems we derive the relation between the optimal filter bandwidth and channel frequency separation. Numerical results are presented for the case of Gaussian filters in both transmitter and receiver or receiver only and either one or two adjacent interfering channels.

The bandwidth in which 99 percent of the power is contained is presented for ASK, QASK, OQASK, PSK, FSK, and MSK. An example is given in which the bandwidth of FSK is one eighth the bandwidth of ASK and of the modulating signal. Error probability is presented for the various types of modulation with various detectors, detector filters and Butterworth receiver filters of orderN = 4or\infty. Comparisons are made between the performance and bandwidth efficiency of the various systems.

A generalized minimum shift keying (GMSK) signal is defined, and its equivalence to a modified offset quadrature shift keying signal is shown. A simple formula for the spectrum of a GMSK signal is presented and the spectrum and out-of-band power are computed for two examples.

Formulas for probability of error (Pe) and probability of erasure (Po) for a binary communication system with partial response signals (duobinary, dicode, modified duobinary) and either decision feedback or precoding are presented.The input data are binary,but the output data are ternary where the third symbol represents erasure.The system with decision feedback is not improved(penotreduced) by the introduction of erasure,while a significant improvement occurs in the system with precoding.

Minimum Shift Keying (MSK) is a method of modulation which can be viewed as a special case of Frequency Shift Keying or a special form of Offset Quadrature Phase Shift Keying; therefore, it can be detected either by a discriminator or with a pair of matched filters. In the latter case, the system is optimal, provided there are no bandlimiting filters in the transmitter and receiver. These filters introduce intersymbol interference and hence degrade the performance of the system. In this paper we investigate analytically the effect of a bandlimiting filter in the receiver on the error probability of the system, presenting numerical results when that filter is of the Butterworth type. It is shown that a very simple suboptimal detector outperforms the matched filter detector so long as2W/R_{b} < 1.3, where Rbis the bit rate andWis the single sided 3 dB bandwidth of the bandlimiting filter.

A binary bandpass communications system is considered, in which the ratio of bandwidth to intermediate frequency (IF) is not necessarily small. Closed-form expressions are given in the case when the receiver (IF) filter hasNdistinct poles. Explicit results, namely probabilities of error, are computed when the filter is a bandpass Butterworth filter. The effect ofk = f_{0}/f_{r}(ratio of half filter bandwidth to IF),f_{0}T(product of half filter bandwidth and bit duration),N, phase and time jitter on probability of error, is investigated.

Upper and lower bounds to the probability of error in a binary communication system with intersymbol interference and independent noise are computed. The symbols may be dependent or independent. The bounds are tight and practically coincide for a proper choice of certain parameters.

An upper and lower bound to the probability of error is presented for a digital communication system with dependent symbols affected by additive noise and intersymbol interference. Explicitly considered are two systems in which independent binary symbols are encoded into ternary dependent symbols, i.e., a bipolar code and a dicode. The bounds practically coincide under a proper choice of certain integers; hence the true value of probability of error can be computed as a function of signal to noise ratio.

The exact value of the probability of error is computed for binary communication systems with sampling detector or integrating-and-dumping detector, with nonreturn-to-zero or split-phase signals and single-pole filters in the receiver and transmitter. The effect of nonoptimal sampling time on the probability of error is presented.

Upper and lower bounds to probability of error in binary communication systems with intersymbol interference are presented. Closed-form formulas are given for systems with filters having a finite number of lumped elements. Explicit results are given when the filters are Butterworth filters of orderN = 1, 2,..., 10. Four systems are considered: with a sample detector (SD), with an integrate-integrated-and-dump detector (IDD), with a filter in receiver only, and with filters in receiver and transmitter. The transmitted signal is assumed to be of the nonreturn-nonreturn-to-zero (NRZ) form.

The probability of error is computed in four binary communication systems withNth order (N = 1, 2,...,10) Butterworth filters and split-phase signals. The systems have either a sampling detector or an integrating-and-dumping detector; the filter is either in the receiver only or both in the receiver and transmitter. A lower bound to the probability of error is also computed for modified binary communication systems in which prior to the decision the contribution of the past transmitted symbols is eliminated. In all systems the sampling time is optimized and the effect of nonoptimal sampling time on probability of error is demonstrated.

Three methods are described for the representation of continuous functions as lattice polynomials in terms of a set of generators. They are two-stage-parallel, three-stage-parallel, and series- parallel. Each representation is in direct correspondence with an analog function generator.

The sequential and nonsequential feedback systems for a binary source, first proposed by Turin, are considered. It is shown that in both cases the optimal signals belong to the family\pm A (y)[1 + \exp (\pm y)]^{-1}, whereA(y)is arbitrary. In the sequential scheme the bound on transmission rate is achieved for anyP_{e}by all members of this family, and in the nonsequential scheme for smallP_{e}only. The optimal signals within this family, i.e., signals for which\alpha =peak power/average power is minimal, result ifA(y) = \pm \sqrt{\alpha P_{av}} (1 -b e^{-|y|}). In the sequential case, the optimal scheme is compared with two suboptimal schemes, and the advantages of the first are demonstrated. Finally, the problem of sequential detection with time-varying thresholds is presented, and its equivalence to a problem with fixed thresholds is proved.

It is shown that the lower bound on zero-error capacityC_{0}as presented by Shannon [1] and Gallager [2] is\ln \alpha, where\alphais the maximum number of channel input symbols, no two of which have a common output symbol.

Optimal values for circuit parameters of nonlinear interpolating (NLI) half and full squarers and sine-wave generators are calculated. An exponential characteristic v = Cißis assumed for the nonlinear resistors. It is shown that optimal ß results in maximum fractional static errors of ±0.08 percent, ±0.32 percent, and ±0.17 percent for the respective types of function, if the simplest circuit configurations are used. These errors are materially reduced in more complicated NLI's.