This book focuses on performance evaluation techniques for wireless systems in fading. It consists of five parts. The first part introduces system performance measures, types of fading encountered by wireless systems, and the basic types of communication (coherent, partially coherent, and noncoherent). The second part focuses on mathematical tools that are useful in performance evaluation. A comprehensive and detailed performance evaluation for different receptions, several channel models, and the performance measures outlined earlier is addressed in the third part of the book. The fourth part covers uncoded multiuser communications, while the fifth and last part of the book deals with coded digital communication in fading. A rich set of reference is provided. The book would be useful as a reference but not as a college textbook.

This letter addresses the problem of energy detection of an unknown signal over a multipath channel. It starts with the no-diversity case, and presents some alternative closed-form expressions for the probability of detection to those recently reported in the literature. Detection capability is boosted by implementing both square-law combining and square-law selection diversity schemes

We consider multiple-bit differential detection (MBDD) of differentially encoded shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient and popular constant-envelope modulation. We propose two MBDD schemes that are based on a recent continuous phase modulation interpretation of SOQPSK. We show that the performance of these MBDD schemes approaches that of coherent detection (CD) as the multiple-bit observation N window increases. The first scheme uses a detection window that spans the full-bit observation window (F-MBDD), and is shown to require very large values of N to approach the performance of CD. This presents a practical problem since the complexity of MBDD grows exponentially with N. The second scheme is an improved version (I-MBDD) with a detection window that is shortened to N-2 bit intervals. Although the complexity of I-MBDD also increases exponentially with N, it represents a significant improvement since only modest values of N are needed for high performance. These performance characteristics are identified via a detailed performance analysis, which provides asymptotic formulas for the bit error probability that are confirmed with computer simulations. The analysis is also used to find the symmetric frequency pulse shapes with the best and worst error performance. Finally, we develop a simplified and practical decision feedback differential detection algorithm that achieves near-optimal performance with complexity that grows only linearly with N.

We consider multiple-bit differential detection (MBDD) of differentially encoded shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient and popular constant-envelope modulation. We propose two MBDD schemes that are based on a recent continuous phase modulation (CPM) interpretation of SOQPSK. We show that the performance of these MBDD schemes approaches that of coherent detection as the multiple-bit observation window N increases. The first scheme uses a detection window that spans the full N-bit observation window (F-MBDD) and is shown to require very large values of N to approach the performance of coherent detection. This presents a practical problem since the complexity of MBDD grows exponentially with N. The second scheme is an improved version (I-MBDD) with a detection window that is shortened to N - 2 bit intervals and requires more modest values of N for high performance. These performance characteristics are identified via a detailed performance analysis, which provides asymptotic formulas for the probability of bit error that are confirmed with computer simulations.

We investigate the differences between allowable differential encoding strategies and their associated bit-error probability performances for quadrature phase-shift keying (QPSK) and offset QPSK modulations when the carrier demodulation reference signals are supplied by the optimum (motivated by maximum a posteriori estimation of carrier phase) carrier-tracking loop suitable for that modulation. In particular, we show that in the presence of carrier-synchronization phase ambiguity but an otherwise ideal loop, both the symbol and bit-error probabilities in the presence of differential encoding are identical for the two modulations. On the other hand, when in addition the phase error introduced by the loop's finite signal-to-noise ratio is taken into account, it is shown that the two differentially encoded modulations behave differently, and their performances are no longer equivalent. A similar statement has previously been demonstrated for the same modulations when the phase ambiguity was assumed to have been perfectly resolved by means other than differential encoding.

The present paper comments on the paper by Zogas and Karagiannidis (IEEE Trans. Commun., vol.53, no.11, p.1790-4, 2005 November)

In this letter, we bring to the reader's attention some interesting properties of certain line codes that appear to not have been mentioned in the literature, that, in some cases, can be used to enhance their performance when used for digital communication with particular application to radio frequency identification. In addition, we correct a previously reported result pertaining to the performance of such codes.

Starting with the maximum a posteriori (MAP) estimation approach, this paper derives the optimum (in the MAP estimation sense) means for performing symbol timing recovery in the absence of carrier phase information (i.e., prior to carrier phase recovery). Specifically, we examine the necessary modification of a well-known form of coherent symbol synchronizer, namely, the data transition tracking loop (DTTL) to allow its operation in the absence of carrier phase information, i.e., as a so-called noncoherent symbol sync loop. By employing such a noncoherent scheme, one can eliminate the need for iteration between the carrier and symbol sync functions, as typically takes place in receivers that more commonly perform carrier tracking and acquisition prior to symbol timing. The performance of both the linear and nonlinear versions of this noncoherent DTTL is obtained by a combination of analysis and simulation, and compared with that of the corresponding coherent DDTLs.

Starting with the maximum a posteriori (MAP)estimation approach, this paper derives the optimum (in the MAP estimation sense) means for performing symbol-timing recovery in the absence of carrier-phase information (i.e., prior to carrier-phase recovery). Specifically, we examine the necessary modification of a well-known form of coherent symbol synchronizer,namely, the data transition tracking loop (DTTL), to allow its operation in the absence of carrier-phase information, i.e., as a so-called noncoherent symbol sync loop. By employing such a noncoherent scheme, one can eliminate the need for iteration between the carrier and symbol sync functions,as typically takes place in receivers that more commonly perform carrier tracking and acquisition prior to symbol timing. The performance of both the linear and nonlinear versions of this noncoherent DTTL is obtained by a combination of analysis and simulation, and compared with that of the corresponding coherent DTTLs.

This paper proposes and analyzes an alternative form of switch and examine combining, namely one that waits for a channel coherence time if all the available diversity paths fail to meet a predetermined minimum quality requirement. This scanning through the available diversity paths followed by waiting process is repeated indefinitely until a path with an acceptable quality is found. The performance of the resulting "scan and wait" combining (SWC) scheme is studied in terms of its average probability of error and ergodic capacity. The paper looks also into (i) the delay associated with SWC by deriving the statistics of the waiting time needed before satisfactory transmission occurs and (ii) the underlying complexity by studying the statistics of the number of diversity paths estimated per channel access. Selected numerical examples show that, for a fixed average number of channel estimates per channel access, SWC outperforms traditional diversity combining schemes at the expense of a negligible time delay.

The literature is relatively sparse in performance analysis of diversity combining schemes over Weibull fading channels, despite the fact that the Weibull distribution is often found to be suitably fit for empirical fading channel measurements. In this paper, we capitalize on some interesting results due to Lieblein on the order statistics of Weibull random variables to derive exact closed-form expressions for the combined average signal-to-noise ratio (SNR) as well as amount of fading (AF) at a generalized selection combining (GSC) output over Weibull fading channels. We also use some simple AF-based mappings between the fading parameters of the Weibull distribution and those of the Nakagami, Rice, and Hoyt distributions to obtain the approximate but accurate average SNR and AF of GSC over these types of channels. The mathematical equations are validated and illustrated by some numerical examples for scenarios of practical interest. Copyright © 2006 John Wiley & Sons, Ltd.

We have demonstrated that the performance (as measured by its variance) of a popular SNR estimator can be improved by increasing the number of observables per symbol (accomplished by partitioning the symbol interval into a larger (than two) even number of subdivisions) but still processing them in successive pairs. We have shown that the amount of improvement is a function of the true SNR being estimated and continues to increase monotonically with increasing SNR. Despite its ad hoc nature, the modified estimator tracks the Cramer-Rao bound on the variance of SNR estimators with known or unknown data over the entire range of SNR values, a property not previously achievable with the traditional version of this estimator.

The tracking performance of a simplified (no transition detector in the in-phase arm and no delay in the quadrature arm) data transition tracking loop (SDTTL) suggested in the conference literature (Chiu and Lee, 1987) a number of years ago for symbol synchronization of an NRZ data stream is presented. It is shown that, contrary to what was previously stated in Chiu and Lee, 1987, the SDTTL and the traditional DTTL have performances that are quite different, the latter always being the more superior. In fact, asymptotically for large symbol SNR, the mean-squared timing error of the SDTTL is 3 dB worse than that of the DTTL which considerably negates its implementation simplification. Even at low SNR, where neither scheme would be motivated by maximum a posteriori (MAP) estimation of symbol sync considerations, the DTTL still outperforms the SDTTL.

In this paper, we study the optimality of bit detection for coherent M-ary phase-shift keying (PSK) and M-ary quadrature amplitude modulation (QAM), and noncoherent M-ary frequency-shift keying (FSK) signal sets. For M-PSK and M-QAM signal constellations, we employ Gray mapping, consider 8-PSK and 16-QAM signal sets as representative of the general results, and derive the log-likelihood ratio (LLR) for each bit forming the symbol. Using the LLRs, we derive the average bit-error probability (BEP) for the individual bits, and show that the decision regions and the corresponding average BEP for the case of M-PSK coincide with those obtained with the optimal symbol-based detector, whereas, in general, this is not the case for both M-QAM and M-FSK.

A method by which a popular signal-to-noise ratio (SNR) estimator can be reconfigured to yield improved performance is proposed. The reconfiguration consists of partitioning the symbol interval into a larger (but even) number of subdivisions than the two that have been traditionally suggested for this estimator but still processing the observables in successive pairs. The optimum number of subdivisions is shown to depend on the SNR range in which the true SNR lies; however, we also show that these SNR regions can be significantly widened with very little loss in performance. Most important is the fact that with this reconfiguration, the SNR estimator tracks the Cramer-Rao bound on the variance of the estimator over the entire range of SNR values.

A modification of the Alamouti code originally proposed for RF wireless applications is described that allows it to be applied in scenarios such as free-space optical communication with direct detection where unipolar modulations like pulse-position modulation and on-off keying are traditionally used to convey the information. The modification of the code and associated decision metric is such as to maintain all of the desirable properties of the original scheme.

A robust SNR estimator is proposed and its performance analyzed for a variety of cases dealing with its operation in receivers where little or no information is known regarding other system parameters such as carrier phase and frequency, order of the modulation, and data symbol knowledge.

The moment-generating function (MGF)-based method, which has previously been applied with great success in evaluating the error probability performance of more traditional space-time block and trellis codes, is equally useful when applied to systems employing super-quasi-orthogonal space-time trellis codes. Its utility in exactly evaluating the pairwise error probability has been demonstrated for both fast and slow Rayleigh fading channels with the results obtainable either in closed form or as a single integral with finite limits and an integrand composed solely of trigonometric functions.

A new metric is proposed for multiple-symbol differential-detection of phase noise-impaired binary differential phase-shift-keying (DPSK) signals. The performance of the proposed metric is evaluated in terms of the overall bit error rate (BER) using analytical means. It is assumed that the observation interval extends over three consecutive symbol intervals and that the received signal is corrupted by oscillator phase noise.

We consider the problem of soft-detection for communication systems employing double differential modulation and forward error correction codes, for example, a convolutional code. We propose a soft-detector (based on the multiple symbol heuristic detector) with low computational complexity. Simulation results are provided to show the superior performance of the new soft-detector.

In this work, we analyze the pairwise error probability (PEP) performance of space-time codes in the presence of transmit fading correlation. Unlike receive correlation, the effect of transmit correlation depends on the specific space-time code used. We show that at low SNR, depending on the code, transmit correlation can either improve or degrade the PEP performance. By analyzing the worst case scenario for asymptotically high or low SNR, we show that, to guarantee robust performance for arbitrary transmit correlation, the minimum eigenvalue of the codeword pair difference matrix should be maximized among all codeword pairs. Numerical and simulation results are given to corroborate the theoretical analysis.

We propose a generalized selection combining (GSC) scheme for binary signaling in which a subset of diversity branches providing the largest magnitude of log-likelihood ratio (LLR) are selected and combined. It is shown that the bit-error probability with maximum ratio combining (MRC) or square-law combining of L branches is identical to that with LLR-based GSC of L/2 branches. We also propose a simple, but suboptimal, GSC based on a noncoherent envelope detection and discuss its potential advantages over the conventional signal-to-noise-ratio-based GSC and MRC.

We show that military standard (MIL-STD) shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient constant-envelope modulation, can be represented in the form of a cross-correlated trellis-coded quadrature modulation. Similarly, we show that offset QPSK (OQPSK) can be decomposed into a "degraded" trellis encoder and a memoryless mapper. Based on the representations of OQPSK and MIL-STD SOQPSK as trellis-coded modulations (TCMs), we investigate the potential coding gains achievable from the application of simple outer codes to form a concatenated coding structure with iterative decoding. For MIL-STD SOQPSK, we describe the optimum receiver corresponding to its TCM form and then propose a simplified receiver. The bit-error rate (BER) performances of both receivers for uncoded and coded MIL-STD SOQPSK are simulated and compared with that of OQPSK and Feher-patented QPSK (FQPSK). The asymptotic BER performance of MIL-STD SOQPSK is also analyzed and compared with that of OQPSK and FQPSK. Simulation results show that, compared with their uncoded systems, there are significant coding gains for both OQPSK and MIL-STD SOQPSK, obtained by applying iterative decoding to either the parallel concatenated coding scheme or the serial one, even when very simple outer codes are used.

Two algorithms for double-differential detection of multiple phase-shift keying modulation are proposed, based on a least-squares criterion and a generalized-likelihood ratio test, respectively. While both algorithms take advantage of the performance gain obtained by observing the received signal over an observation interval longer than that required for symbol-by-symbol detection, the former has the important advantage of reduced implementation complexity, whereas the latter offers better performance.

The performance and tradeoffs of multipulse position modulation (MPPM) are investigated and compared with that of the traditional (single) pulse position modulation (PPM) scheme typically employed on the optical direct-detection channel. While the primary motivation for the consideration of the problem is to provide performance improvement for deep-space optical communications where narrow high-peak-power transmitted pulses offer significant advantages in terms of detection probabilities and background suppression capabilities at the receiver, the results obtained are sufficiently generic as to apply to other applications.

We derive the exact pairwise error probability (PEP) for space-time coding over quasi-static Rayleigh fading channels in the presence of spatial fading correlation. We show that receive correlation always degrades the PEP for all signal-to-noise ratios (SNRs). We quantify the effect of receive correlation by employing the notion of "majorization". We show that the stronger the receive correlation, the worse the PEP for all SNRs. We show that at low SNR, depending on the codes, transmit correlation can either improve or degrade the PEP performance. We show that to guarantee robust performance for arbitrary transmit correlation, the minimum eigenvalue of the codeword pair difference matrix should be maximized among all codeword pairs.

In this letter, based on the exact pairwise-error probability, we derive the union bound on the symbol-error probability (SEP) of the differential unitary space-time (DUST) modulation employing group codes. Instead of using the rank-and-determinant or Euclidean distance criteria, we optimize the cyclic group codes such that the union bound on the SEP is minimized for a predetermined scenario, taking into account the number of transmit and receive antennas and the operating signal-to-noise ratio (SNR). Our simulation results show that for a wide range of SNRs, the codes with the minimum union bound for a particular SNR outperform the codes designed based on rank-and-determinant or Euclidean distance criteria.

Abstract We investigate the performance of non‐coherent frequency shift keying (NCFSK) with multi‐branch post‐detection switched diversity systems. After an appropriate statistical description of the square‐law detector output, we derive generic formulas for the symbol/bit error probability of NCFSK for both switch and stay combining (SSC) and switch and examine combining (SEC) schemes. We show that while the SSC scheme does not benefit from more than two diversity paths when the paths are identically distributed, the error performance of SEC improves as the number of antenna paths increases. We consider two detection schemes for NCFSK, one for M‐ary FSK and another limited to only BFSK. We show that the second scheme leads to lower error probability performance when both are used to detect binary signals. The mathematical formalism is illustrated with a number of selected numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.

We extend the method based on the moment generating function (MGF), previously applied to analyzing the performance of orthogonal space-time block and trellis codes, to super-orthogonal codes that combine set-partitioning with a super set of orthogonal space-time block codes in such a way as to provide full diversity with increased rate and improved coding gain. It is shown that the maximum-likelihood (ML) metric and expressions for the pairwise error probability (PEP), previously developed for the Alamouti code combined with multidimensional trellis-coded modulation (TCM), can be readily extended to the super-orthogonal case.

For asymptotically high SNR, the design criteria of differential unitary space-time (DUST) coding are based on the rank-and-determinant criterion (Hughes, B.L., 2000; Hochwald, B.M. and Sweldens, W., 2000). For a large number of receive antennas or low SNR scenarios, the Euclidean distance criterion is appropriate for differential unitary space-time codes (Wang, J. et al., 2002). In practical systems, with a reasonable number of transmit and receive antennas, the SNR is neither asymptotically high nor asymptotically low. Therefore it is interesting to find what is a good code design criterion at medium range SNR. Based on the exact pairwise error probability, we derive the union bound on the symbol error probability (SEP) of the DUST modulation. Instead of using the rank-and-determinant or Euclidean distance criteria, we optimize the codes such that the union bound on the SEP is minimized for a predetermined scenario taking into account the number of transmit and receive antennas and the operating SNR. Our simulation results show that for a wide range of SNRs, the codes with the minimum union bound for a particular SNR outperform the codes designed based on rank-and-determinant or Euclidean distance criteria.

This paper presents another look at the problem of energy detection of unknown signals over different fading channels. We start with the no diversity case and present some alternative closed-form expressions for the probability of detection (P/sub d/) to those recently reported in [V.I. Kostylev, May 2002]. We then investigate the system performance when different diversity schemes are employed. It is shown that there is not much improvement in the probability of detection when either the probability of false alarm (P/sub f/) exceeds 0/1 or the average signal-to-noise ratio (SNR) exceeds 20 dB. In addition, receiver operating characteristic (ROC) curves comparing the performance of equal-gain combining (EGC), selection combining (SC), and switch and stay combining (SSC) are presented. As an example, EGC introduces a gain of two orders of magnitude from the probability of miss perspective compared to the no diversity case while both SC and SSC introduce a gain of about one order of magnitude.

We propose a generalized selection combining (GSC) scheme for binary signaling in which M diversity branches providing the largest magnitude of log-likelihood ratio (LLR) are selected and combined. The bit error probability provided by LLR-based GSC serves as a lower bound on the bit error probability provided by any GSC techniques. We also propose a suboptimal GSC based on a noncoherent envelope detection. We derive the bit error probability with LLR-based and envelope-based GSC techniques and examine their power gains over the conventional SNR-based GSC technique. We show that the bit error probability with maximum ratio combining or square-law combining of L branches is identical to that with LLR-based GSC of L/2 branches.

Analogous to multiple symbol differential detection of quadrature phase-shift-keying (QPSK), a multiple bit differential detection scheme is described for offset QPSK that also exhibits continuous improvement in performance with increasing observation interval. Being derived from maximum-likelihood (ML) considerations, the proposed scheme is purported to be the most power efficient scheme for such a modulation and detection method. Extension of the result to shaped offset QPSK is also possible.

We derive the performance of the optimum noncoherent FSK diversity combining receiver and compare it with that which implements the more traditional noncoherent postdetection EGC scheme to reveal the degree of suboptimality of the latter.

This paper develops, analyzes, and optimizes a simpler form of dual-branch switch-and-stay combining (SSC), namely, one that relies on the output signal plus noise rather than the instantaneous signal-to-noise ratio (SNR) to trigger the switching between the diversity branches. Analysis supported by numerical results show that the newly proposed postdetection SSC scheme outperforms predetection SSC and this performance gain increases as the channel conditions improve in terms of average SNR and/or severity of fading. In brief, when simplicity of implementation is of primary concern, as is the case, for example, in mobile units, the proposed scheme offers an attractive low-complexity solution to mitigate the deleterious effects of multipath fading.

Analogous to multiple-symbol differential detection of quadrature phase-shift keying (QPSK), a multiple-bit differential detection scheme is described for offset QPSK that also exhibits continuous improvement in performance with increasing observation interval. Being derived from maximum-likelihood considerations together with the representation of differentially encoded offset QPSK as a continuous phase modulation, the proposed scheme is purported to be the most power-efficient scheme for such a modulation and detection method.

We derive the performance of the optimum noncoherent frequency-shift keying diversity combining receiver and compare it with that which implements the more traditional noncoherent postdetection equal gain combining scheme to reveal the degree of suboptimality of the latter.

The paper extends the notion of postdetection switched combining introduced by the authors to M-ary orthogonal modulation (Alouini and Simon, ibid., p.1591-1602, 2003) and proceeds to analyze and evaluate its average bit-error rate performance for the dual-branch case. The particular variation chosen for the switching strategy uses a different model than that previously used for the binary case and although it results in a slightly poorer performance, it nevertheless outperforms conventional (predetection) switched combining for all values of M. Results are obtained for a variety of popular channel models including Rayleigh, Rician, and Nakagami-m fading. Because of its relative simplicity of implementation, the proposed scheme once again offers a very attractive low-complexity solution for mitigating the deleterious effects of multipath fading.

We present new exponential bounds for the Gaussian Q function (one- and two-dimensional) and its inverse, and for M-ary phase-shift-keying (MPSK), M-ary differential phase-shift-keying (MDPSK) error probabilities over additive white Gaussian noise channels. More precisely, the new bounds are in the form of the sum of exponential functions that, in the limit, approach the exact value. Then, a quite accurate and simple approximate expression given by the sum of two exponential functions is reported. The results are applied to the general problem of evaluating the average error probability in fading channels. Some examples of applications are also presented for the computation of the pairwise error probability of space-time codes and the average error probability of MPSK and MDPSK in fading channels.

This letter evaluates some new integrals whose integrand involves the generalized Marcum Q function, both of whose arguments are linear functions of the integration variable. The resulting closed-form expressions obtained for these integrals are then applied to the error probability evaluation of noncoherent and differentially coherent detection with postdetection equal gain combining over Rayleigh and Nakagami fading channels.

A great deal of attention has been devoted in the literature to studying the bit error rate (BER) performance of diversity combining systems in the presence of Rayleigh, Rice, and Nagakami-m fading. By comparison, the literature is relatively sparse in comparable analyses over log-normal channels which typically characterize shadowing from indoor obstacles and moving human bodies. One reason for this disparity stems from the difficulty in evaluating the exact average BER when log-normal variates are involved, using, for example, the moment-generating function (MGF) approach, due to the inability of expressing the MGF itself in a simple closed form. Since it is possible to evaluate the marginal and joint statistical moments as well as the cumulative distribution function (CDF) associated with a log-normal distribution in closed form, we rather focus here on other performance measures, namely, average combined output signal-to-noise ratio, amount of fading, and outage probability. The first two performance measures depend only on the moments, whereas the outage probability depends solely on the cdf. Closed-form expressions (in terms of known functions), single-integral representations, or upper and lower bounds are obtained for these measures corresponding to maximal-ratio combining, selection combining, and switch-and-stay combining schemes, allowing for the possibility of correlation between the two branches. Numerical evaluations of these expressions illustrating the performances of each individual diversity type as well as comparisons among them are also presented.

The Nuttall (1972) Q function is introduced and, for the case where the sum of its two order indices is odd, a closed-form expression is derived for it in terms of the Marcum (1950) Q function and modified Bessel functions of the first kind. Alternatively, a finite-range integral expression is also given for it, which has the identical form to that of the generalized Marcum Q function. Finally, digital communication performance applications involving the Nuttall Q function are presented to justify the value of the results.

We derive a compact expression for the moment-generating function (MGF) of the output of a generalized selection combiner for independent but nonidentically distributed Rayleigh fading paths which is then used to derive a single integral expression for the symbol error probability of, for example, M-PSK. Because of the simple product form of the MGF and its ability to be expressed as a partial fraction expansion, we are then able to invert this function to derive the probability density function and cumulative distribution function of the combiner output from which the outage probability can then be evaluated in closed form. An example, of the latter is given for an exponentially decaying power delay profile.

For a general class of constant-energy trellis-coded modulation (TCM) schemes with 2/sup /spl nu// states, necessary and sufficient conditions to guarantee that a maximum-likelihood sequence estimator (MSLE) can decode each symbol with a fixed delay of /spl nu/ symbols are derived. Additive white Gaussian noise (AWGN) is assumed. Minimum shift keying (MSK) is a special case that belongs to the family of modulation schemes with /spl nu/=1. It is shown that when these conditions are met, the minimum squared Euclidean distance is upper-bounded by 4E/sub s/, where E/sub s/ is the signal's energy per interval. Necessary and sufficient conditions to achieve the upper bound are given and it is shown that these conditions are met if and only if the TCM scheme can be implemented as pulse amplitude modulation (PAM) using a pulse that extends over /spl nu/+1 symbols. Signals that achieve this upper bound and maximize the power within a given bandwidth are found. The bandwidth efficiency of such schemes is significantly higher than that of MSK.

Abstract A few years back, a generic analysis approach referred to as the MGF method was introduced for the purpose of simplifying the evaluation of the performance of digital communication over fading channels. A particular case of high interest that fell under the umbrella of the methodology was the evaluation of the average error probability performance of systems that employ receive diversity to combat multipath as well as the possible inclusion of error‐correcting coding/decoding for improving power efficiency. In this paper, we extend the methodology to space‐time coded systems. Illustrative examples are explicitly given for space‐time trellis coding. The emphasis of the paper is on the performance analysis tools and their application, as opposed to the design of the space‐time codes themselves. Copyright © 2002 John Wiley & Sons, Ltd.

We analyze the average bit error rate and outage probability performance of two types of conditional diversity combining schemes in which each branch signal-to-noise ratio is tested against a fixed predetermined threshold and applied to the combiner only if its value exceeds this threshold. The two schemes, respectively referred to as absolute threshold generalized selection combining and normalized threshold generalized selection combining differ from one another only in the manner in which the threshold is chosen. Nevertheless, when operating over a generalized fading channel, the two schemes have a markedly different behavior as reflected in the numerical results presented. The analytical method taken to obtain these results follows the moment generating function approach used previously by the authors for analyzing the performance of more conventional diversity combining schemes.

A great deal of attention has been devoted in the literature to studying the bit error rate (BER) performance of diversity combining systems in the presence of Rayleigh, Rice, and Nakagami-m fading. By comparison the literature is relatively sparse in comparable analyses over log-normal channels which typically characterize shadowing from indoor obstacles and moving human bodies. One reason for this disparity stems from the difficulty in evaluating the exact average BER when log-normal variates are involved using, for example, the moment generating function (MGF) approach due to the inability of expressing the MGF itself in a simple closed form. Since however, it is possible to evaluate the marginal and joint statistical moments of a log-normal distribution in closed form, we rather focus here on two other performance measures, namely, average combined output signal-to-noise ratio and amount of fading both of which depend only on these moments. Closed-form expressions are obtained for these two measures corresponding to maximal-ratio combining (MRC), selection combining (SC), and switch and stay combining (SSC) schemes allowing for the possibility of correlation between the two branches.

Based on a representation of FQPSK as a trellis-coded modulation, this paper investigates the potential improvement in power efficiency obtained from the application of simple (small number of states) outer codes to form a concatenated coding arrangement with iterative decoding. Several possible configurations for the concatenation are suggested and specific numerical results are presented for one of these in order to demonstrate the large coding gains that are achievable even when using a reduced complexity FQPSK receiver. The end result of these investigations is a system which has application in scenarios requiring a high degree of both power and bandwidth efficiency.

Based on a representation of FQPSK as a trellis-coded modulation, this paper investigates the potential improvement in power efficiency obtained from the application of simple (small number of states) outer codes to form a concatenated coding arrangement with iterative decoding. Several possible configurations for the concatenation are suggested and specific numerical results are presented for one of these in order to demonstrate the large coding gains that are achievable even when using a reduced complexity FQPSK receiver. The end result of these investigations is a system which has application in scenarios requiring a high degree of both power and bandwidth efficiency.

The literature is relatively sparse in performance analyses of diversity combining schemes over Weibull fading channels, despite the fact that the Weibull distribution is often found to be a suitable fit to empirical fading channel measurements. We capitalize on some interesting results due to Lieblein on the order statistics of Weibull random variables, to derive exact closed-form expressions for the combined average signal-to-noise ratio (SNR) as well as amount of fading (AF) at a generalized selection combining (GSC) output over Weibull fading channels. We also use some simple AF-based mappings between the fading parameters of the Weibull distribution and those of the Nakagami, Rice, and Hoyt distributions to obtain the approximate but accurate average SNR and AF of GSC over these type of channels. The mathematical equations are validated and illustrated by some numerical examples for scenarios of practical interest.

This paper presents new exact closed-form expressions for the outage probability of cellular systems (without diversity reception) in which the desired user follows a Rician distribution whereas the interferers are subject to correlated Rayleigh fading. It then capitalizes on a numerical technique by D. Raphaeli (see IEEE Trans. Inf. Theory, vol.42, p.1002-7, 1996) dealing with the accurate evaluation of the cumulative distribution function of quadratic forms in complex Gaussian random variables to solve for the outage probability in the more general case of arbitrary fading statistics, average fading powers, diversity branches correlation, and/or correlation between the interferers. In some scenarios, the numerical approach reduces to finite-sum closed-form expressions.

Building on the developments in the performance analysis of generalized selection combining (GSC), this paper examines the optimum spreading bandwidth for a fixed-complexity GSC diversity receiver operating over independent identically distributed Rayleigh paths. For this purpose, the study considers three performance criteria: (1) average combined signal-to-noise ratio (SNR) at the GSC output; (2) average bit error probability (BEP); and (3) outage probability of the instantaneous combined SNR at the GSC output. For the average BEP criterion, results are presented for both coherent and noncoherent combining. For the average combined SNR and some instances of the average BEP optimization problem, an accurate approximate estimate of this optimum bandwidth in the form of a solution of a transcendental equation is provided. In other cases, where the optimization is not easily tractable in an analytic fashion, a numeric-search procedure is used to find this optimum bandwidth for different performance criteria and system parameters of interest. Finally, simplified rule-of-thumb-type formulas are also presented as a good reference for picking the optimum spreading bandwidth given a set of system parameters and a particular performance criterion of interest.

In this letter we first consider the maximum-likelihood sequence estimator for multiple symbol differential detection (MSDD) over the slow fading diversity channel. Since this optimum decision metric results in a complex receiver implementation whose average bit-error probability (BEP) performance is difficult (if not impossible) to obtain analytically, we then focus our attention on evaluating the average BEP for MSDD with diversity reception in the form of postdetection equal-gain combining (EGC) giving emphasis to its ability to bridge the gap between EGC of conventional differentially detected M-PSK and maximal-ratio combining of coherently detected M-PSK with differential encoding.

An expression for the cumulative distribution function evaluated at zero of the difference of two chi-square variates with different number of degrees of freedom is derived and applied to the outage probability computation of cellular mobile radio systems in fading channels. In particular, a generic result is developed for this probability which takes on several forms, the simplest of which, is a single integral with finite limits and an integrand composed of elementary (exponential and trigonometric) functions. The results are applicable to cellular systems that are subject to independent identically distributed (i.i.d.) interfering signals and that employ maximal-ratio combining reception over i.i.d. diversity paths. Various fading channel models are assumed for the desired signal and interferers. For each desired signal/interferer combination, the outage probability is expressed in closed form in terms of a set of parameters that characterize the particular scenario. A tabulation of these various scenarios and their associated parameters for channels of practical interest is included.

Using a recently developed moment generating function-based approach for the performance evaluation of digital communications over fading channels, we present a unified approach for the exact performance analysis of binary direct-sequence code division multiple access (DS-CDMA) systems operating over generalized frequency-selective fading channels. The results are applicable to single carrier systems employing RAKE reception as well as to multicarrier DS-CDMA systems with frequency diversity. Aside from simplifying previous results both analytically and computationally, the proposed approach also gives a solution for many situations which heretofore defied a simple form. Copyright © 2001 John Wiley & Sons, Ltd.

We first consider the maximum-likelihood sequence estimation (MLSE) for multiple symbol differential detection (MSDD) over the slow fading diversity channel. Since this optimum decision metric results in a complex receiver implementation whose average bit error probability (BEP) performance is difficult (if not impossible) to obtain analytically, we then focus our attention on evaluating the average BEP for MSDD with diversity reception in the form of postdetection equal-gain combining (EGC) giving emphasis to its ability to bridge the gap between EGC of conventional differentially detected M-PSK and maximal-ratio combining (MRC) of coherently detected M-PSK with differential encoding.

The average bit error rate (BER) performance of M-ary orthogonal noncoherent frequency-shift-keying (FSK) over frequency-selective Nakagami-m fading channels with an exponentially decaying power delay profile is studied. Exact analytical results when suboptimum square-law combining type of reception is used are first presented. Metrics and performance of maximum likelihood (ML) type of receivers are then discussed. The performance results of optimum ML receivers are obtained via extensive Monte-Carlo computer simulations, and are compared with the the analytical average BER results obtained for the suboptimum square-law combining receivers. Numerical and simulation results confirm that the optimum ML receivers outperform square law receivers for strictly positive values of the power decay factor and do not suffer from the well-known "noncoherent combining loss" which affects the performance of square law receivers at low signal-to-noise ratio values. In addition, these results show that suboptimum receivers perform actually quite well with respect to their optimum counterpart in general but the performance difference between them becomes more significant as the Nakagami (1960) fading parameter and/or the number of diversity paths increases.

Using the notion of the "spacing" between ordered exponential random variables, a performance analysis of the generalized selection combining (GSC) diversity scheme over Rayleigh fading channels is presented and compared with that of the conventional maximal-ratio combining and selection combining schemes. Starting with the moment generating function (MGF) of the GSC output signal-to-noise ratio (SNR), we derive closed-form expressions for the average combined SNR, outage probability, and average error probability of a wide variety of modulation schemes operating over independently, identically distributed (i.i.d.) diversity paths. Because of their simple form, these expressions readily allow numerical evaluation for cases of practical interest. The results are also extended to the case of non-i.i.d. diversity paths.

Outage probability is an important performance measure of communication systems operating over fading channels. Relying on a simple and accurate algorithm for the numerical inversion of the Laplace transforms of cumulative distribution functions, we develop a moment generating function-based numerical technique for the outage probability evaluation of maximal-ratio combining (MRC) and postdetection equal-gain combining (EGC) in generalized fading channels for which the fading in each diversity path need not be independent, identically distributed, nor even distributed according to the same family of distributions. The method is then extended to coherent EGC but only for the case of Nakagami-m fading channels. The mathematical formalism is illustrated by applying the method to some selected numerical examples of interest showing the impact of the power delay profile and the fading correlation on the outage probability of MRC and EGC systems.

We derive expressions for the exact bit-error probability (BEP) for the detection of coherent binary phase-shift keying signals of the optimum combiner employing space diversity when both the desired signal and a Gaussian cochannel interferer are subject to flat Rayleigh fading. Two different methods are employed to reach two different, but numerically identical, expressions. With the direct method, the conditional BEP is averaged over the fading of both signal and interference, With the moment generating function based method, expressions are derived from an alternative representation of the Gaussian Q-function.

Strict upper and lower bounds of exponential-type are derived for the generalized (mth order) Marcum Q-function which enable simple evaluation of a tight upper bound on the average bit-error probability performance of a wide class of noncoherent and differentially coherent communication systems operating over generalized fading channels. For the case of frequency selective fading with arbitrary statistics per independent fading path, the resulting upper hound on performance is expressed in the form of a product of moment generating functions of the instantaneous power random variables that characterize these paths.

Using a simple finite integral representation for the bivariate Rayleigh (1889) cumulative distribution function previously discovered by the authors, we present expressions for the outage probability and average error probability performances of a dual selective diversity system with correlated slow Rayleigh fading either in closed form (in particular for binary differential phase-shift keying) or in terms of a single integral with finite limits and an integrand composed of elementary (exponential and trigonometric) functions. Because of their simple form, these expressions readily allow numerical evaluation for cases of practical interest. The results are also extended to the case of slow Nakagami-m fading using an alternate representation of the generalized Marcum (1950) Q-function.

Presented here is a unified approach to evaluating the error-rate performance of digital communication systems operating over a generalized fading channel. What enables the unification is the recognition of the desirable form for alternate representations of the Gaussian and Marcum Q-functions that are characteristic of error-probability expressions for coherent, differentially coherent, and noncoherent forms of detection. It is shown that in the largest majority of cases, these error-rate expressions can be put in the form of a single integral with finite limits and an integrand composed of elementary functions, thus readily enabling numerical evaluation.

This paper introduces an improved nonlinear parallel interference cancellation scheme for code-division multiple access (CDMA) that significantly reduces the degrading effect on the desired user of interference from the other users that share the channel. The implementation complexity of the scheme is linear in the number of users and operates on the fact that parallel processing simultaneously removes from each user a part of the interference produced by the remaining users accessing the channel the amount being proportional to their reliability. The parallel processing can be done in multiple stages. The proposed scheme uses tentative decision devices at the multiple stages to produce the most reliably estimated received data for generation and cancellation of user interference. Simulation results are given for a multitude of different situations, in particular, those cases for which the analysis is too complex.

We present a unified approach to determine the exact bit error rate (BER) of noncoherent and differentially coherent modulations with single and multichannel reception over additive white Gaussian noise and generalized fading channels. The multichannel reception results assume independent fading in the channels and are applicable to systems that employ post-detection equal gain combining. Our approach relies on an alternate form of the Marcum Q-function and leads to expressions of the BER involving a single finite-range integral which can be readily evaluated numerically. Aside from unifying the past results, the new approach also allows for a more general solution to the problem in that it includes many situations that in the past defied a simple solution. The best example of this occurs for multichannel reception where the fading on each channel need not be identically distributed nor even distributed according to the same family of distributions.

Using an alternate form of the Gaussian probability integral discovered a number of years ago, it is shown that the solution to a number of previously considered communication problems can be simplified and, in some cases, made more accurate (i.e. exact rather than bounded). These problems include the evaluation of: (1) the bit-error probability of uncoded phase shift keying (PSK) with Costas loop tracking; (2) word-error probability of antipodal modulation in the presence of fading; (3) bit-error probability of coded M-ary PSK (MPSK) over the memoryless fading channel with given channel-state information; (4) conditional symbol-error probability of MPSK in the presence of carrier synchronization error; and (5) the average error probability for the binary additive white Gaussian noise (AWGN) intersymbol interference channel. Also obtained is a generalization of this new alternate form to the case of a two-dimensional Gaussian probability integral with arbitrary correlation which can be used to evaluate the symbol-error probability of MPSK with I-Q unbalance.

Exact and approximate statistical models (analytical and simulation), based on average- and maximum-likelihood ratio tests, are studied and compared for the purpose of establishing an accurate assessment of the performance of noncoherent SFH M-FSK intercept receivers. Both continuous and discontinuous phase M-FSK cases are considered with important differences between the two.>

Practical M-FSK systems experience a combination of time and frequency offsets (errors). This paper assesses the deleterious effect of these offsets, first individually and then combined, on the average bit error probability performance of the system. Exact expressions for these various error probability performances are derived and evaluated numerically for system parameters of interest. Also presented are upper bounds on average symbol error probability for the case of frequency error alone which are useful in assessing the absolute and relative performance of the system. Both continuous and discontinuous phase M-FSK cases are considered when timing error is present, the latter being much less robust to this type of offset.>

The optimum hop timing estimator, based on classical likelihood-ratio (LR) theory, is derived for noncoherent slow and fast frequency-hopped M-FSK intercept receivers. Such receivers have no a priori knowledge of the hopping code and thus the solution to this estimation problem differs considerably from the more commonly considered case of the friendly receiver. The implementation and performance of the LR hop timing structures are presented and compared with that of other previously documented suboptimum schemes. In particular, when compared to the multiple-hop autocorrelation (MHAC) approach and the "ping-pong" approach, the average-likelihood ratio (ALR) and maximum-likelihood ratio (MLR) optimum estimators offer an improvement in performance of several dB in input SNR. Robustness issues for the optimal estimator are examined and alternative configurations are proposed that offer enhanced insensitivity to unknown frequency offsets.>

Exact and approximate statistical models (analytical and simulation), based on average-and maximum-likelihood ratio tests, are studies and compared for the purpose of establishing an accurate assessment of the performance of coherent SFH M-FSK intercept receivers.>

This paper derives metrics for maximum-likelihood differential detection of uncoded and trellis coded MPSK and QAM transmitted over Rayleigh and Rician fading channels. Receiver structures based on these metrics are proposed and their error probability performance analyzed and/or simulated. The results represent a generalization of the notion of multiple symbol differential detection, previously introduced by the authors for MPSK over an AWGN, to the fading channel and other modulations. For the coded cases, ideal interleaving/deinterleaving is assumed and furthermore the presence or absence of channel state information. An interesting side result is that for a constant envelope modulation transmitted over a fading channel with unknown but rapidly-varying phase error (the other extreme to the slowly-varying phase error case normally assumed for differential detection), under certain practical assumptions, it is shown that the optimum receiver is of the limiter-discriminator type.>

Optimum detectors have previously been derived for fast frequency-hopped (FFH) signals with M-ary frequency-shift-keyed (MFSK) data modulation received in additive white Gaussian noise (FFH here implies that a single MFSK tone is transmitted per hop), The present paper extends that work to the more analytically complex category of slow frequency-hopped (SFH) signals with multiple MFSK tones per hop. A special subset of the SFH/MFSK format that receives particular attention in the paper is the case of continuous-phase modulation (CPM) for which the carrier phase is assumed to be constant over the entire hop. A fundamental conclusion is that SFW/CPM modulation is advantageous not only to the communicator but also to a sophisticated noncoherent detector. By applying techniques developed in the paper to exploit the continuous-phase characteristic, an intercept receiver of reasonable complexity will perform appreciably better than traditional channelized detectors.>

This paper presents a simple closed form expression for the conditional (on the timing error) bit error probability of an otherwise ideal DPSK receiver. Numerical results are given for average bit error probability obtained by averaging the conditional error probability over a Tikhonov PDF which is a typical characterization of the timing error in practical bit synchronizers.>

The authors show that by using the known (or estimated) value of carrier tracking loop SNR in the decision metric, it is possible to improve the error probability performance of a partially coherent MPSK system relative to that corresponding to the commonly used ideal coherent decision rule. Using a maximum-likelihood approach, an optimum decision metric is derived and shown to take the form of a weighted sum of the ideal coherent decision metric (i.e., correlation) and the noncoherent decision metric which is optimum for differential detection of MPSK. The performance of a receiver based on this optimum decision rule is derived and shown to provide continued improvement with increasing length of observation interval (data symbol sequence length). Unfortunately increasing the observation length does not eliminate the error floor associated with the finite loop SNR. Nevertheless, in the limit of infinite observation length, the average error probability performance approaches the algebraic sum of the error floor and the performance of ideal coherent detection, i.e., at any error probability above the error floor, there is no degradation due to the partial coherence. While the above is strictly speaking only true in the limit of infinite observation length, it is shown that one can virtually achieve this limiting behavior with practical size observation lengths. Furthermore, the performance is quite insensitive to mismatch between the estimate of loop SNR (e.g., obtained from measurement) fed to the decision metric relative and its true value.>

The authors examine maximum-likelihood block detection of uncoded full response continuous phase modulation (CPM) over an additive white Gaussian noise (AWGN) channel. Both the maximum-likelihood metrics and the bit error probability performances of the associated detection algorithms are considered. The special and popular case of minimum-shift-keying (MSK) corresponding to h=0.5 and constant amplitude frequency pulse is treated separately. The many new receiver structures that result from this investigation can be compared to the traditional ones that have been used in the past both from the standpoint of simplicity of implementation and optimality of performance.>

A variety of schemes for performing differential detection in environments characterized by frequency offset are discussed. All of the schemes involve encoding the input phase information as a second-order difference and performing an analogous second-order differential detection at the receiver. Because of the back-to-back differential detection operations at the receiver, The performance of most of the schemes is considerably degraded relative to that of first-order differential detection schemes. However, the latter are quite sensitive to frequency offset and, in many instances, cannot be used at all. It is demonstrated that via a simple enhancement of using a 2 T s (instead of T s) delay in the second stage of the encoder and first stage of the decoder, the performance degradation can be significantly reduced. This result is significant in view of the fact that it comes without any penalty in implementation complexity.>

Several modulation and coding advances supported by NASA are summarized. To support long-constraint-length convolutional code, a VLSI maximum-likelihood decoder, utilizing parallel processing techniques, which is being developed to decode convolutional codes of constraint length 15 and a code rate as low as 1/6 is discussed. A VLSI high-speed 8-b Reed-Solomon decoder which is being developed for advanced tracking and data relay satellite (ATDRS) applications is discussed. A 300-Mb/s modem with continuous phase modulation (CPM) and codings which is being developed for ATDRS is discussed. Trellis-coded modulation (TCM) techniques are discussed for satellite-based mobile communication applications.>

A differential detection technique for MPSK (multiple-phase shift keying), which uses a multiple-symbol observation interval, is presented, and its performance is analyzed and simulated. The technique makes use of maximum-likelihood sequence estimation of the transmitted phases rather than symbol-by-symbol detection as in the conventional differential detection. Thus, the performance of this multiple-symbol detection scheme fills the gap between conventional (two-symbol observation) differentially coherent detection of MPSK and ideal coherent of MPSK with differential encoding. The amount of improvement gained over conventional differential detection depends on the number of phases M and the number of additional symbol intervals added to the observation. What is particularly interesting is that substantial performance improvement can be obtained for only one or two additional symbol intervals of observation. The analysis and simulation results presented are for uncoded MPSK.>

The use of interleaving/deinterleaving in trellis-coded modulation systems to reduce the SNR loss due to imperfect carrier demodulation references is demonstrated. Both the discrete carrier (phase-locked loop) and the suppressed carrier (Costas loop) cases are considered, and the differences between the two are clearly demonstrated by numerical results. The special case of convolutional codes is also treated and illustrated with an example of practical interest.>

The idea of using a multiple (more than two) symbol observation interval to improve error probability performance is applied to differential detection of trellis-coded multiple phase-shift keying (MPSK) over an additive white Gaussian noise (AWGN) channels. An equivalent Euclidean distance measure per trellis branch is determined for this detection scheme. This is used to define an augmented (larger multiplicity) trellis code whose distance measure is the conventional squared Euclidean distance typical of conventional trellis-coded modulation on the AWGN. Such an augmented multiple trellis code is a convenient mathematical tool for simplifying the analysis. Results are obtained by a combination of analysis and computer simulation. It is shown that only a slight increase (e.g. one symbol) in the length of the observation interval will provide a significant improvement in bit error probability performance.>

An open-loop technique is presented for estimating and correcting Doppler frequency shift in an M-ary differential phase-shift-keyed (MDPSK) receiver. The novelty of the scheme is based on the observation that whereas the change in phase of the received signal over a full symbol contains the sum of the data (phase) and the Doppler-induced phase shift, the same change in phase over half a symbol (within a given symbol interval) contains only the Doppler-induced phase shift. Thus, by proper processing, the latter can be estimated and removed from the former. Analytical and simulation results are given for the variance of the above estimator, and the error probability performance of the MDPSK receiver is evaluated in the presence of the Doppler correction. Next, the practical considerations associated with the application of this technique on bandlimited Nyquist channels are discussed and incorporated into the final design. It is shown that the receiver can, in the absence of timing jitter, be designed to allow combined Doppler correction and data detection with no penalty due to intersymbol interference (ISI). The effects of ISI due to timing jitter are assessed by computer simulation.>

The authors demonstrate a trellis coded modulation technique referred to as multiple trellis coded modulation (MTCM) wherein more than one channel symbol per trellis branch is transmitted. They have found simple two-state trellis codes for symmetric MPSK multiple phase-shift keying and AM modulations that can achieve 3-dB gain over uncoded modulation at very high signal-to-noise ratios without bandwidth expansion or reduction in information bit rate. The gain of these codes with respect to previously reported two-state trellis codes is between 1 and 2 dB at very high signal-to-noise ratios, depending on the number of bits per Hertz transmitted. These gains are achieved for those of the equivalent conventional trellis codes with the same number of states in the trellis diagram. The authors note that additional computations per branch are needed for the multiple trellis coding scheme. The concept can be extended to a higher number of states and other types of modulations.>

It has been well established that the appropriate criterion for optimum trellis-coded modulation design on the additive white Gaussian noise channel is maximization of the free Euclidean distance. It is shown that when the trellis-coded modulation is used on a Rician fading channel with interleaving/deinterleaving, the design of the code of optimum performance is guided by other factors, in particular, the length of the shortest error-event path, and the product of branch distances (possibly normalized by the Euclidean distance of the path) along the path. Although maximum free distance (d/sub free/) is still an important consideration, it plays a less significant role the more severe the fading is on the channel. These considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over Rician fading channels. If no interleaving/deinterleaving is used, then once again the design of the trellis code is guided by maximizing d/sub free/.>

A previous work on criteria for designing trellis-coded MPSK modulation to achieve minimum error probability performance on the Rician fading channel (see ibid., vol.36, no.9, p.1004-1012, Sep. 1988) is extended. It is demonstrated that allowing for multiple symbols per trellis branch, i.e., multiple trellis-coded modulation (MTCM), provides an additional degree of freedom for designing a code to meet the optimization on the fading channel. Diversities larger than those achievable with conventional trellis codes having the same number of trellis states are now attainable, it is under these conditions that MTCM achieves its full potential.>

The combination of trellis coding and MPSK signaling with the addition of asymmetry to the signal set is discussed with regard to its suitabllity as a modulation/coding scheme for the fading mobile satellite channel. For MPSK, introducing nonuniformity (asymmetry) into the spacing between signal points in the constellation buys a further improvement in performance over that achievable with trellis coded symmetric MPSK, all this without increasing the average or peak power, or changing the bandwidth constraints imposed on the system. Whereas previous contributions have considered the performance of trellis coded modulation transmitted over an additive white Gaussian noise (AWGN) channel, the emphasis in this paper is on the performance of trellis coded MPSK in the fading environment. The results will be obtained by using a combination of analysis and simulation. It will be assumed that the effect of the fading on the phase of the received signal is fully compensated for either by tracking it with some form of phase-locked loop or with pilot tone calibration techniques. Thus, our results will only reflect the degradation due to the effect of the fading on the amplitude of the received signal. Also, we shall consider only the case where interleaving/deinterleaving is employed to further combat the fading. This allows for considerable simplification of the analysis and is of great practical interest. Finally, the impact of the availability of channel state information on average bit error probability performance is assessed.

Traditionally symmetric signal constellations, i.e., those with uniformly spaced signal points, have been used for both uncoded and coded systems. Although symmetric signal constellations are optimum with no coding, the same is not necessarily true for coded systems. This paper shows that by designing the signal constellations to be asymmetric, one can, in many instances, obtain a performance gain over the traditional symmetric constellations combined With trellis coding. In particular, we consider the joint design ofn/(n + 1)trellis codes and asymmetric2^{n+1}-point signal constellations, which has no bandwidth expansion relative to an uncoded 2n-point symmetric signal set. The asymptotic performance gains due to coding and asymmetry are evaluated in terms of the minimum free Euclidean distance dfreeof the trellis. A comparison of the maximum value of this performance measure to the minimum distance dmin, of the uncoded system is an indication of the maxiamm reduction in requiredE_{b}/N_{0}that can be achieved for arbitrarily small system bit error rates. Bit error probability analysis is carried out for general cases. A few examples are given to show the performance gain due to the asymmetry of the signal set. It is to be emphasized that the introduction of asymmetry into the signal set does not affect the bandwidth or power requirements of the system; hence, the abovementioned improvements in performance come at little or no cost. Asymmetric signal sets in coded systems first appear in the work of Divsalar and Yuen [1], [2]. Here we expand upon these results by considering various types of asymmetric signal sets combined with the optimum (in the sense of maximum dfree) trellis code having 2, 4, 8, and 16 states. The numerical results obtained will clearly demonstrate the tradeoff between the additional savings in requiredE_{b}/N_{0}and the additional complexity (more trellis states) needed to achieve it.

This paper evaluates the double symbol error rate (average probability of two consecutive symbol errors) in differentially detected narrow-band FM. Numerical results are presented for the special case of MSK with a Gaussian IF receive filter. It is shown that, not unlike similar results previously obtained for the single error probability of such systems, large inaccuracies in predicted performance can occur when intersymbol interference is ignored.

A recursive solution is presented for the probability density function of the sum ofNindependent, random phase vectors. The recursion parameter isN, the number of vectors in the sum. This approach allows one to rapidly compute a complete set of these density functions for values ofN = 2, 3, ..., N_{max}, where Nmaxtypically corresponds to the total number of system users in a multiuser FHMA/MFSK application, or one plus the total number of jamming tones in an FH/MFSK spread-spectrum application. Such evaluations are necessary for exact calculation of the average error probability performance of such systems.

A simple circular state-diagram method has been recently proposed [1], [2] for the modeling, analysis, and optimization of straight serial search code acquisition in spread-spectrum systems. Herein, the approach is generalized to arbitrary serial strategies, such as theZ-search expanding window search, etc., and arbitrary a priori probability distributions for the code phase uncertainty. The basic idea is the construction of equivalent circular state diagrams, wherefrom the transform domain description, i.e., the generating or characteristic function of the stochastic acquisition process is derived. This general method is readily systematized, circumvents the need for complicated time-domain combinatorial arguments, and allows past results [3], [4] to be obtained as particular examples. In principle, the generating function can be used to calculate any moment of the acquisition process; here, however, due to complexity considerations, we restrict our attention to the first moment and its associated performance measure, the mean acquisition time. Based on the derived analytical expressions for the mean acquisition time, an expanding-window single-dwell search strategy has been optimized in terms of the required number of partial windows, parameterized by the prior distribution of the phase uncertainty.

The performance of uncoded narrow-band FM with limiter/discriminator (L/D) detection over the additive white Gaussian noise channel has received widespread attention [1]-[14] over the years both from simulation and analytical standpoints. By comparison, little or nothing has appeared in the literature for the performance of this modulation/demodulation technique when used together with coding/decoding. The purpose of this paper is to partially fill this void by considering the potential gain offered by the combination of convolutionally encoded narrow-band FM with L/D detection and Viterbi decoding. Both hard and soft decision decoding cases will be considered with some rather surprising differences between the performances of the two. In particular, it will be shown that a large departure exists from the 2 dB advantage traditionally associated with soft decision decoding which is explained in terms of the mismatch between the coding channel and the decoding metric peculiar to this modulation/demodulation technique.

The error probability performance of differential detection of narrow-band FM is determined and compared with the analogous results for limiter-discriminator detection of the same modulation. It is shown that over a large class of benign and hostile environments, e.g., Gaussian IF filter,1 \leq BT \leq 3, h \leq 1, AWGN, partial-band noise jamming, the differential detector offers no theoretical performance advantage over the limiter-discriminator receiver with integrate-and-dump postdetection filtering.

This paper examines by analytical methods the power spectral densities of digital modulations (in particular, staggered and unstaggered quadrature modulations) passed through band-limited nonlinear channels. Previously observed (by computer simulation or hardware measurement) behavior of such spectra with regard to the suppression or restoration of its sidelobes after passing through the nonlinearity is verified analytically. Several examples corresponding to specific quadrature modulations and filter-nonlinearity combinations are presented as illustrations of the general results.

The performance of differentially coherent detection of frequency-hopped QASK in the presence of partial-band noise and partial-band multitone jamming is presented. In each case, the worst case jamming strategy is determined which consists of specifying the worst case partial-band fraction and the corresponding maximum average error probability. The results obtained are compared with those ofM-ary FH-DPSK operating in the same jamming environment.

Using a geometric approach, the performance ofM-ary FH-DPSK in the presence of partial-band multitone jamming is evaluated. The optimal jamming strategy is determined as a function of the number of signaling levelsMand the ensuing results are used to determine worst case bit error probability performance as a function of this same parameter. It is demonstrated that, forM=2m, the best performance is obtained forM=4.

In this paper the concept of differentially coherent detection of differentially phase-encoded QASK signals (DQASK) is described, and a receiver structure proposed. The remainder of the paper analyzes the performance of DQASK in the presence of a Gaussian noise environment with comparisons made to coherent detection of QASK.

This paper generalizes and corrects previous results on the tracking performance of two-channel Costas-type carrier synchronization loops. In particular, the mean-square phase jitter performance of these loops is evaluated when the data formats on the two channels are arbitrary. Particular attention is given to the case where the loops contain active arm filters with or without hard limiters following them. For each case, it is demonstrated that selection of the channel gain ratio, as motivated by MAP estimation theory, does not guarantee optimum loop tracking performance. In fact, it is shown that, in some instances, a conventional single-channel Costas loop will outperform the two-channel version with the MAP choice of gain ratio. Instead, it is suggested that the gain ratio be chosen to directly minimize the mean-square phase tracking jitter which is equivalent to minimizing the loop's "squaring loss." When this is done, it is shown that, in some cases, one can obtain significant improvement in this performance measure. In all cases, however, the two-channel loop will now outperform the single-channel version.

This paper considers the performance of QPR in the presence of the identical jamming postulated in Part I. Although a QPR system employs a single sample detector in its receiver, while QASK (or QPSK) requires a matched-filter type of receiver., it is shown that the coherent detection performances of the two in the presence of the intentional jammer have definite similarities.

This paper examines the performance of coherent QPSK and QASK systems combined with FH or FH/PN spread spectrum techniques in the presence of partial-band multitone or noise jamming. The worst-case jammer and worst-case performance are determined as functions of the signal-to-background noise ratio (SNR) and signal-to-jammer power ratio (SJR). Asymptotic results for high SNR are shown to have a linear dependence between the jammer's optimal power allocation and the system error probability performance.

The power spectral density of the output symbol sequence of a convolutional encoder is computed for two different input symbol stream source models, namely, an NRZ signaling format and a first-order Markov source. In the former, the two signaling states of the binary waveform are not necessarily assumed to occur with equal probability. The effects of alternate symbol inversion on this spectrum are also considered. The mathematical results are illustrated with many examples corresponding to optimal performance codes. It is demonstrated that only for the case of a purely random input source (e.g., NRZ data with equiprobable symbols) and a particular class of codes is the output spectrum identical to the input spectrum except for a frequency scaling (expansion) by the reciprocal of the code rate. In all other cases, the output spectrum is sufficiently changed relative to the input spectrum that the commonly quoted statement "a convolutional encoder produces a bandwidth expansion by a factor equal to the reciprocal of the code rate" must be exercised with care.

A technique is presented for generating phase plane plots on a digital computer which circumvents the difficulties associated with more traditional methods of numerical solving nonlinear differential equations. In particular, the nonlinear differential equation of operation is formulated as a pair of coupled differential equations with phase error and phase error derivative as dependent variables and arc length as the independent variable, which is uniformly distributed along the phase plane trajectory. The sinusoidal phase-locked loop and polarity-type Costas loop are used as illustrations of the proposed method of solution.

Inverting alternate symbols of the encoder output of a convolutionally coded system provides sufficient density of symbol transitions to guarantee adequate symbol synchronizer performance, a guarantee otherwise lacking. Although alternate symbol inversion may increase or decrease the average transition density, depending on the data source model, it produces a maximum number of contiguous symbols without transition for a particular class of convolutional codes, independent of the data source model. Further, this maximum is sufficiently small to guarantee acceptable symbol synchronizer performance for typical applications. Subsequent inversion of alternate detected symbols permits proper decoding.

This paper investigates the tracking performance of a Costas loop in the presence of single and multiple in-band pulsed RFI signals and receiver additive noise. It is shown that the loop's tracking performance is very sensitive to the pulse durations of the RFI signals, as well as their repetition rates and peak powers relative to those of the desired signal. The presence of in-band RFI can affect the loop's tracking performance in basically three ways. 1) The loop tracks the composite phase of the RFI instead of the phase of the desired signal when the effective RFI power is greater than that of the signal. 2) When the effective RFI power is not sufficiently large to capture the loop, there still will exist static phase error biases which cause the loop to lock at a phase between that of the desired signal and the composite RFI. 3) There is an increase in the rms phase jitter over the nominal design. These latter two effects degrade the bit error probability performance of the receiver. More significantly, if the nominal loop bandwidth is not designed properly to account for the increase in jitter phenomenon, frequent cycle slipping may result.

The potential false lock problem associated with the sampled data type of Costas loop implementation is addressed in this paper. The term "alias lock" is used here to differentiate this type of false lock behavior from the data sideband false lock behavior of analog Costas loops [1]-[8]. It is shown that the sampled-data version of the conventional Costas loop, sampled at a rate1/T_{s}, can alias lock at frequencies that are multiples of1/2T_{s}away from the carrier frequency. It is also shown that the alias lock problem of the sampled data version of the Costas loop with hard-limited in-phase channel is further Compounded by the potential occurrence of false lock frequencies at rational multiples of1/2T_{s}, away from the carrier. The false lockS-curves ofI-Qloops and decision-directedI-Qloops are investigated in detail, with and without additive noise. Close agreement between theory and earlier experimental results is also demonstrated.

Starting with MAP estimation theory as a basis for optimally estimating carrier phase of BPSK and QPSK modulations, it is shown in this paper that the closed loop phase trackers, which are motivated by this approach, are indeed closed loop optimum in the minimum mean-square phase tracking jitter sense. The corresponding squaring loss performance of these so-called MAP estimation loops is compared with that of more practical implementations wherein the hyperbolic tangent nonlinearity is approximated by simpler functions.

Quadriphase receivers, like biphase receivers, have the ability to false lock on a sideband of the data modulation. The theory associated with this phenomenon for receivers of binary phase-shift-keying (BPSK), using Costas loop demodulation, has recently been documented in the literature. This paper considers the corresponding theory for receivers of balanced quadriphase-shift-keying (QPSK) employing a quadriphase Costas loop (or equivalent fourth-power loop) for demodulation. Specific closed form expressions for false lock performance are developed and numerically evaluated for the particular case of single pole arm filters and an NRZ data format for each of the two statistically independent quadrature modulations.

The ability of a Costas loop to false lock on a data sideband is a problem which must be dealt with in the design of suppressedcarrier receivers which employ such loops for carrier reconstruction. For conventional Costas loops wherein the error signal is formed from the product of two analog signals, the false lock problem has recently been investigated by the author and others and is now well understood. For implementation reasons associated with the reduction of dc offsets, it is often desirable to hard limit the output of the in-phase channel and replace the analog multiplier which forms the above product with a chopper-type device. The false lock behavior of such a Costas loop with hard-limited in-phase channel is quite different from that of the conventional Costas loop and is the subject of investigation in this paper. Results are also presented for a modified version of the Costas loop wherein the quadrature arm filter is removed. This configuration has recently been suggested as a means of improving the false lock performance of the Costas loop with hard-limited in-phase channel.

The calculation of the optimum performance of suppressed carrier receivers with Costas loop tracking is directly related to evaluating the loop's so-called squaring loss. Recent work by the author and others presented specific numerical results for this loss when the input data were biphase-L(Manchester coded) and the Costas loop arm filters were of then-pole Butterworth type. These results were largely obtained by numerical integration on a digital computer. This paper presents a partial fraction expansion technique for arriving at closed form expressions for squaring loss for Costas loops with arbitary arm filters and NRZ as well as Manchester coded data. Specific closed form results are given for one and two pole Butterworth filters as examples.

It is becoming increasingly popular in the design of suppressed carrier receivers, which employ Costas loops for earrier reconstruction, to hard-limit the output of the in-phase channel. Doing so allows replacement of the analog multiplier, which forms the loop error signal, with a chopper-type device which typically exhibits much less dc offset. The false lock behavior of such a hard-limited loop was recently investigated and shown to be quite different from that of the conventional Costas loop without the hard limiter. This paper presents the companion, analysis of the tracking performance of the hard-limited loop and assesses the penalty, if indeed it is a penalty rather than an improvement, in this performance relative to the conventional Costas loop with an analog third multiplier. In particular, for the case ofRCarm filters and NRZ data, the squaring loss (or equivalently the linear loop tracking jitter) is evaluated and illustrated as a function of the ratio of arm filter bandwidth to data rate and data signal-to-noise ratio. Superimposed on these numerical results will be the corresponding ones for the conventional Costas loop. As a finale, the equivalence in operation of the Costas loop with hard-limited in-phase channel and a baseband modulation carrier reconstruction loop referred to as a demod/ remod loop is discussed.

The problem of deriving "optimum" receiver structures for synchronization of BPSK, QPSK, and unbalanced QPSK (UQPSK) modulations has been addressed by many authors starting with the wellknown maximum a posteriori (MAP) estimation technique as motivation. In general, the closed loop structures which result from application of this theory are impractical because of the difficulty of implementing the hyperbolic tangent nonlinearity which is suggested by using the gradient of the likelihood function as an error control signal in the loop. To arrive at practical realizations, one ordinarily approximates this nonlinearity with simpler, more easily implementable functions. This paper reexamines some of these approximations and the optimum structures which result. Using a slightly more sophisticated approximation of the nonlinearity, new and interesting structures are presented which overcome some of the deficiencies posed by MAP receiver structures for QPSK and UQPSK suggested by authors in the past. The equivalence of the QPSK structure with the well-known quadriphase Costas loop is discussed. Furthermore, it is argued that the new UQPSK structure allows carrier reconstruction at all ratios of data rates and powers in the two channels even in the limit of balanced QPSK.

A previous Costas loop study by the author demonstrated that for biphase modulation considerable improvement in tracking performance could be obtained by employing active arm filters of the integrate-and-dump type as opposed to passive arm filters. The purpose of this part, then, is to demonstrate that a similar performance improvement can be obtained for an unbalanced QPSK modulation.

A simple technique for calculating the error probability performance and associated noisy reference loss of practical unbalanced QPSK receivers is presented. The approach is based on expanding the error probability conditioned on the loop phase error φ in a power series in φ and then, keeping only the first few terms of this series, averaging this conditional error probability over the probability density function of φ. Doing so results in an expression for the average error probability which is in the form of a leading term representing the ideal (perfect synchronization references) performance plus a term proportional to the mean-squared crosstalk. Thus, the additional error probability due to noisy synchronization references occurs as an additive term proportional to the mean-squared phase jitter\sigma_{\phi}^{2}directly associated with the receiver's tracking loop. Similar arguments are advanced to give closed-form results for the noisy reference loss itself.

This brief correspondence notes the equivalence between the optimumM-ary DPSK receiver given recently by Park and another form of this receiver suggested earlier in the literature but not referenced in Park's paper.

When a Costas loop is operated in a PM mode, such as on the Space Shuttle OrbiterS-band uplink during the time when both data and a ranging subcarrier are linearly modulated on the carrier, and if the data modulation index is less than\pi/2, the loop must reconstruct its carrier reference from an input signal whose carrier component is not completely suppressed. This paper establishes a theory for the performance of Costas loops in the presence of residual carrier as a function of the key system parameters and applies this theory toward predicting the performance degradation in the PM mode (relative to PSK operation) of the ShuttleS-band uplink. The corresponding performance of the ranging channel is also given as a function of these same system parameters.

Unbalanced quadriphase-shift-keying (QPSK) is an attractive means for transmitting two digital data streams which in general have different average powers, data rates, and data formats. Previous analyses of the tracking performance of Costas loop demodulators of unbalanced QPSK have accounted only for the filtering effect produced by the loop's two arm filters on the equivalent additive noise perturbing the loop. When the bandwidth of these filters is selected on the order of the data rate, as is typical of optimum Costas loop design, the filtering degradations of the data modulations themselves and the crossmodulation noise produced by their multiplication in the loop often cannot be neglected. The purpose of this paper is to incorporate these additional filtering effects into the analysis. Many of the results obtained herein are in the form of closed-form expressions which can easily be evaluated numerically for design and performance prediction purposes.

This paper systematically analyzes the signal-signal-to-noise ratio degradations which can potentially occur due to data asymmetry in digital transmission systems. Suitable asymmetry models are developed and error probability performance for various types of data detectors (integrate-and-dump filter, filter-sample detector, and gated-integrate-and-dump filter) is derived. Although this work was done to resolve problems being encountered in the ShuttleKu-band return link design, specifically for the 50 Mbit/s convolutionally encoded channel (NRZ format), generalizations are made which provide results for other cases of interest (otherKu-band return link channels, or other systems entirely). This paper therefore considers Manchester data formats (in addition to NRZ) and uncoded transmission (in addition to convolutionally coded transmission). The effects of bandlimiting are also considered.

A classical problem in digital frequency-shifted keyed (FSK) demodulation is the evaluation of the bit error probability performance when an estimator-correlator that incorporates a phaselocked loop (PLL) is employed. Although some attention has been devoted to this problem in the past, an accurate account of the mechanism which produces decision errors has not yet been advanced. This paper examines a special case, viz., a first-order PLL preceded by a wideband IF filter, of the above problem using a new approach which is based upon the renewal Markov process theory and the Meyr distribution. In particular, the ad hoc approach of invoking the Gaussian assumption on the decision variable and patching it with a correction term based on Rice's click theory is not used. Rather, the effective noise is properly characterized by unfolding the renewal Markov process associated with the loop phase error. As a slight extension of the results, the performance of the above PLL detector operating on low data rate PSK is given and demonstrated to be approximately 3 dB superior to that of FSK reception. The theory and analysis presented herein apply to the special case where the bandwidth of the IF filter preceding the first-order PLL is required to be several times the data rate because of frequency uncertainties due to channel Doppler and oscillator instabilities, but the frequency deviation to data rate ratio may be chosen small (if desired) to optimize system error probability performance. In addition to presenting results for the case where the oscillator instabilities are assumed absent and channel Doppler is prefectly tuned out at the receiver oscillator, the effects of small residual Doppler on bit error probability performance is considered. In all cases tested, excellent agreement was obtained between theory and computer simulation results.

To optimize the threshold of a pseudonoise (PN) spread spectrum modem for use over an aircraft/satellite communications link at SHF, the effects of Doppler must be taken into account. Reconstitution of carrier phase by a Costas loop to coherently demodulate the PSK data and also the delay-lock error voltage has typically been the practice in PN modems intended for ground applications. To accommodate the platform dynamics, the Costas loop must have a relatively wide bandwidth, and this implies a significant threshold degradation. An alternate implementation employs a noncoherent carrier tracking loop which maintains frequency lock rather than phase lock. Now, the delay-lock error voltage is noncoherently demodulated. For the airborne application, analysis and simulations show this implementation will extend the receiver's tracking threshold significantly (up to 6 dB) for the worst case dynamics profile. An experimental project was undertaken to modify an existing ground PN modem (AN/USC-28, ADM version) for flight test. A software implementation of the digital tracking algorithms was selected where a HP-2100A minicomputer controls carrier frequency and PN code phase via digital phase shifters. The Costas demodulator for extracting PSK data resides entirely in software, and is completely segregated from PN tracking. In laboratory testing of the receiver with simulated dynamics and in actual flight tests, the demonstrated performance was found to approach closely the goals established by the analyses and simulations.

The optimum design and performance of two noncoherent PN tracking loop configurations, namely, the delay-locked loop and tau-dither loop, are described. In particular, the bandlimiting effects of the bandpass arm filters are considered by demonstrating that for a fixed data rate and data signal-to-noise ratio, there exists an optimum filter bandwidth in the sense of minimizing the loop's tracking jitter. Both the linear and nonlinear loop analyses are presented and the region of validity of the former relative to the latter is indicated. In addition, numerical results are given for several filter types. For example, assuming ideal bandpass arm filters, it is shown that the tau-dither loop requires approximately 1 dB more signal-to-noise ratio than the delay-locked loop for equal rms tracking jitters.

The performance of suppressed carrier receivers with Costas loop tracking is optimized by proper choice of loop arm filter bandwidth. In particular, it is shown that for a variety of passive arm filter types, there exists, for a given data rate and data signal-signal-to-noise ratio, an optimum filter bandwidth in the sense of minimizing the loop's squaring loss. For the linear theory case, this is equivalent to minimizing the loop's tracking jitter. When symbol synchronization is known, it is shown that by replacing the passive arm filters with active filters, i.e., integrate-and-dump circuits, one can achieve an improvement in carrier-to-noise ratio of as much as 4 to 6 dB depending on the passive arm filter type used for comparison and the value of data signal-to-noise ratio (SNR).

The implementation and tracking performance of symbol synchronizers for Manchester coded data is presented with motivation provided by maximum a posteriori (MAP) estimation theory. In particular, it is shown herein that for uncoded data symbols, with, however, arbitrary data transition probabilities, certain physically realizable closed-loop structures, which are readily implemented in practice, are suggested by the theory. The tracking performance of these loops is optimized and comparisons are made among the various configurations over a wide range of system parameters. Although not the major intent of the paper, the acquisition problem is briefly addressed.

In recent years, minimum-shift-keying (MSK) has gained increasing popularity as a modulation technique because of its desirable spectral properties. Quite often, the spectral concentration provided by MSK is not sufficient to meet requirements on out-of-band energy spillover. In these situations, one might apply additional input pulse shaping m such a way as to still maintain constant envelope signals. The properties of such MSK-type signals are the subject of this paper. Specific examples are included as illustrations of the theory both for the binary andM-ary cases.

This paper demonstrates that an offset quadrature amplitude-shift-keyed (OQASK) signal can be represented as an ncomponent version of a minimum-shift-keyed (MSK) signal. For example, the signal set obtained by summing two MSK signals which are 6 dB different in power and are formed by continuous phase, frequency modulating the same oscillator with two independent binary antipodal data streams, is spectrally equivalent to an OQASK signal set composed of 16 signals in which the symbol pulse is a half-cycle sinusoid rather, than the conventional rectangular form. Such generalizations as the above allow for potentially simpler implementation of spectrally shaped OQASK.

Much has been said in the literature with regard to the problem of establishing symbol synchronization in binary baseband digital communication systems. By comparison, the literature contains little information relating to the extraction of symbol sync from multilevel baseband data. With the recent interest in multilevel amplitude-shift keying (MASK) and quadrature amplitude-shift keying (QASK) as signaling techniques for multilevel digital communications systems, the problem of providing symbol synchronization in the receivers of such systems becomes paramount. This paper presents a technique for extracting symbol sync from an MASK or QASK signal which has been transmitted over an infinite bandwidth white Gaussian noise channel. The scheme is essentially a generalization of the data transition tracking loop (DTTL) which has heretofore been used in PSK systems. The performance of the loop is analyzed in terms of its mean-squared symbol sync jitter and its effects on the data detection process in MASK and QASK systems.

A carrier regeneration loop which generates highly coherent quadrature reference signals for quadrature amplitude-shift keying (QASK) demodulation is presented. The loop employs the principle of decision feedback and has a structure analogous to a decision feedback loop for quadriphase signals suggested earlier in the literature. The error probabihty performance of QASK is computed in the presence of the noisy carrier reference signals provided by the above loop. It is demonstrated that when the ratio of data rate to loop bandwidth is 50 or greater, then for all practical purposes, ideal QASK error probability performance is achieved.

In this paper, we focus our attention on the decisionfeedback approach to carrier synchronization of offset quadrature phase-shift-keyed (QPSK) and more generally offset quadrature amplitude-shift-keyed (QASK) systems. In particular, we present two decision-feedback loops for tracking offset QPSK and offset QASK, which are modifications of comparable loops previously given in the literature for carrier synchronization of QPSK and QASK respectively. The performance gains obtained using such synchronization techniques are discussed. In particular, we show that the use of offset-quadrature communications rather than conventional quadrature communications permits a doubled loop bandwidth at fixed data rate, signal-to-noise ratio and probability of error. However, the more efficient phase-tracking loop (decision-feedback as opposed to phase-locked loop) used reduces substantially the total signal-to-noise ratio required.

Selection of a particular signal set array for a bandwidthConstrained multiple phase-and-amplitude-shift-keyed (MPASK) communication system for a linear additive Gaussian noise channel requires consideration of factors such as average and/or peak power versus symbol error probability, signal amplitude dynamic range, simplicity of generation and detection, and number of bit errors per symbol error (Gray code properties). A simple technique is presented for generating and optimally detecting the honeycomb (hexagonal.) signal set, i.e., the signal set that has the tightest sphere-packing properties. The symbol and bit error probability performance of this set is compared to other two-dimensional signal sets that have been investigated in the literature, and is shown to be slightly superior from an average power standpoint. The paper concludes with a comparison of all of these signal sets from the standpoint of the factors listed above.

Digital communication networks used for the distribution of high-speed digital information are currently the subject of design studies for many civil and military applications. This paper presents results that are useful in such studies as well as in network planning. In particular, the paper is concerned with the problems of carrier synchronization and noisy reference detection of polyphase signals. Reconstruction of coherent references for the detection of polyphase [N-ary phase-shift-keyed (PSK)] signals is considered and analyzed for three carrier reconstruction loops, namely,Nth power (multiply-and-divide) loops, generalized Costas (I-Q) loops, and extensions of data-aided (modulation wipeoff) loops. General expressions for the error probability are developed when the reconstructed reference signals are noisy. These expressions are evaluated numerically for cases of practical interest and compared with differentially coherent reception.

This paper investigates the detail capabilities and performance characterization of systems that employL-orthogonal signaling techniques.L-orthogonal signals represent a unified set of signals wherein the polyphase and orthogonal (biorthogonal) signal sets are included as special cases. This fact is important since orthogonal (biorthogonal) and polyphase signaling sets represent opposing forces as far as tradeoffs between error probability, energy-to-noise ratio, and bandwidth requirements are concerned. Bounds on the performance of the optimum receiver and the performance of various suboptimum (practical) receiver structures are given. Coherent and differentially encoded signals are also pursued. Various comparisons and tradeoffs are made by means of numerical evaluation of the error-probability expressions.

Any digital communication system that employs coherent detection requires coherent reference signals for proper operation. This paper is concerned with the transmission and detection of differentially encoded multiple phase-shift-keyed (MPSK) signals and the ambiguity resolution problem that results from suppression of the transmitted carrier. In particular, the paper presents the analysis and performance of differentially encoded coherent MPSK systems that reconstruct coherent reference signals by means of generalized Costas orNth power loops. The performance of such systems is then compared with that of ideal reception of MPSK signals and differentially coherent detection of differentially encoded MPSK signals. Emphasis is placed upon the special cases of quadriphase and octaphase signaling.

In the design of phase-coherent receivers employing bandpass limiters, it is customary to specify system performance relative to its value at a fixed design point. For a given design point, it is well known that an optimum tradeoff can be found between the power allocated to the carrier and sideband signals. This paper describes an attempt to further improve the performance of such coherent carrier systems by optimizing the design point based upon a given practical optimization criterion. The single-channel system is treated in detail and a brief discussion is given on how to extend the optimization technique to a two-channel system.

We consider here the problem of selecting the lowpass sampling bandwidth for a digital mechanization of a matchedfilter bit-synchronizer combination. In particular, if digital data at a rateRbits/second plus Gaussian noise is passed through a filter of bandwidthKRand then sampled at the Nyquist rate, i.e.,2 KR, then how small can we chooseKwithout paying an appreciable penalty in the SNR performance of the data detector?

Many applications of the bandpass limiter involve either coherent or noncoherent demodulation following the limiter. In this paper we study the performance of a noncoherent frequency-shift keying (FSK) receiver when it is preceded by a bandpass limiter. In particular, we obtain expressions for signal suppression- factor, output signal-to-noise ratio (SNR), and error probability from which one can assess the degradation in performance of the receiver due to the presence of the limiter. Both narrow-band and wide-band cases are treated thus covering situations where no frequency uncertainty exists (i.e., known carrier frequency) as well as large-frequency uncertainties. Also discussed is the first-order signal plus noise probability density function (pdf) following noncoherent demodulation.

A simplified procedure for calculating the channel capacity of a cascade ofNidentical discrete memoryless nonsingular channels is presented. The result depends only upon theMeigenvalues andMeigenvectors of any one of the subchannel transition matrices. Thus, for smallMand largeN(the usual case of interest) the result represents a considerable saving in computation relative to the standard technique of finding the overall channel transition matrix. The procedure is illustrated for anNcascade of binary symmetric channels.

The design of time-limited binary signals imbedded in colored Gaussian noise resulting from passing white noise through a channel of known transfer function is investigated. It is shown that, for a class of channels whose magnitude characteristic is expressible as the ratio of polynomials with the numerator divisible by and of lower or the same degree as the denominator, the output noise kernel (correlation function) and its inverse will have Fourier transforms which are reciprocals of each other. The above assumption results in an expression for probability of error which is independent of the received waveform and depends solely on the transmitter signal-to-noise ratio. The additional freedom of design afforded by the above enables one to impose further restrictions in the form of energy concentration in the frequency domain on the received signal without deterioration of system performance. Some such bandwidth constraints and their associated optimum signal designs are considered in this paper.