In this paper, widely linear filtering (WLF) techniques are applied to the demodulation of both pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) systems with multiple receiver antennas and multiple cochannel interferers. In the proposed implementation, the existing correlations between the real (I) and imaginary (Q) parts of the complex-valued baseband received signal are exploited using a widely linear (WL) maximum likelihood (ML) receiver. Several bounds and approximations are developed to analyze the average symbol error rate (SER) performance, and to analyze the tradeoff between the diversity advantage and interference cancellation (IC) gain in a flat Rayleigh fading channel. Two new results are shown. First, we show that the IC ability of WL receivers is independent of the modulation type used by the desired signal but the gain depends mainly on the modulation type employed by the individual interferers and the total number of antennas. Secondly, assuming that the system uses a mixture of PAM and QAM signals, we show that a WL receiver with N antennas can reject any combination of M1PAM, M2QAM interferers satisfying the constraint: M1+ 2M21/2 - M2. In contrast, a conventional receiver whose performance is independent of the modulation characteristics of the individual interferers can reject only up to M1+ M2interferers, where M1+ M21- M2. When the system contains either PAM, or a mixture of PAM and QAM type CCI, it is shown that WL processing offers a significant performance advantage with a moderate increase in the receiver complexity.

In this paper, we study the effect of cochannel interference (CCI) on the performance of partially coherent BPSK and QPSK in uncorrelated L-branch equal-gain combining systems. We consider a generalized propagation model wherein the desired and interfering signals undergo Nakagami-m or Rician fading with different amounts of fading severity. Further, the interfering signals are assumed to be asynchronous symbol timing with the desired signal, so that the effect of cross-signal intersymbol interference (ISI) is taken into account. Using a convergent Fourier series method, we derive extensive analytical results for the average bit error probability and the SNR gain penalty caused by the interference signals for different signal to-interference ratio levels. The numerical results presented in this paper demonstrate the system performance under very realistic propagation and detection conditions including CCI, carrier phase error recovery, cross-signal ISI, generalized fading channels, and AWGN. Hence our results are expected to be of significant practical use for such scenarios.

In this paper, we introduce a generalized widely linear (WL) equalizer for quadrature amplitude modulation (QAM) systems with single/multiple antennas. In our proposed implementation, the WL receiver first separates the in-phase (I) and quadrature (Q) parts of the complex-valued baseband received signal and jointly filters the two branches for signal detection. Infinite length WL minimum mean-square error (WL-MMSE) linear, and WL decision-feedback-equalizer (WL-DFE) settings are derived and performance is analyzed in co-channel interference limited channels. It is shown that, in frequency selective Rayleigh fading channels, the interference cancellation (IC) gain depends mainly on the rank (r) of the interference correlation matrix (ICM) which is defined as the covariance of the vector-valued signal which consists of the real and imaginary parts of the noise-plus-interference signal collected at multiple antenna branches. Assuming that the DFE feedback path is error free, we show that a WL QAM receiver with N antennas exhibits full IC capability (that is complete interference removal) when the ICM is rank deficient i.e., when: r1pulse-amplitude-modulation (PAM) and M2QAM interferers satisfying the constraint: M1+ 2M2< 2N. Simulation results show that, in the presence of PAM-type interference, the gain of WL-DFE is reduced by decision feedback errors while the IC benefit of WL-MMSE is limited by the noise enhancement problem. Nevertheless, the proposed receivers are shown to be useful in cellular systems that employ a combination of PAM and QAM schemes.

This paper analyzes the average bit error probability (BEP) of the differential binary and quaternary phase-shift keying (DBPSK and DQPSK respectively) with multiple-input multiple-output (MIMO) systems employing postdetection equal gain combining (MIMO EGC) diversity reception over Rayleigh fading channels. Finite closed-form expressions for the average BEP of DBPSK and DQPSK are presented. Two approaches are introduced to analyze the error rate of DQPSK. The proposed structure for the differential phase-shift keying (DPSK) with MIMO EGC provides a reduced-complexity and low-cost receiver for MIMO systems compared to the coherent phase-shift keying system (PSK) with MIMO employing maximal ratio combining (MIMO MRC) diversity reception. Finally, a useful procedure for computing the associated Legendre functions of the second kind with half-odd-integer order and arbitrarily degree is presented.

This paper derives the average bit error probability (BEP) of differential quaternary phase shift keying (DQPSK) with postdetection equal gain combining (EGC) diversity reception over independent and arbitrarily correlated fading channels. First, using the associated Legendre functions, the average BEP of DQPSK is analyzed over independent Rayleigh, Nakagami-m, and Rician fading channels. Finite-series closed-form expressions for the average BEP of DQPSK over L-branch independent Rayleigh and Nakagami-m fading channels (for integer Lm) are presented. Besides, a finite-series closed-form expression is given for the average BEP of differential binary phase shift keying (DBPSK) with EGC over independent Rician fading channels. Second, an alternative approach is propounded to study the performance of DQPSK over arbitrarily correlated Nakagami-m and Rician fading channels. Relatively simple BEP expressions in terms of a finite sum of a finite-range integral are proposed. Moreover, the penalty in signal to noise ratio (SNR) due to arbitrarily correlated channel fading is also investigated. Finally, the accuracy of the results is verified by computer simulation.

In this paper, we analyze the performance of partially coherent BPSK systems in the presence of intersymbol interference. Generalized flat fading channels including Rayleigh, Rician, and Nakagami-m are investigated, and Tikhonov-distributed phase error processes are assumed. We consider diversity system with L-branch signals that being detected using post detection equal-gain combining (EGC) receiver. The diversity branches are assumed to experience slow, flat, and uncorrelated fading. Two performance criteria are considered: the average bit-error probability (BEP) and the signal-to-noise ratio (SNR) gain penalty due to the sampling time jitter. Quazi-analytical simulation results are also presented.

This paper analyzes the average bit error prob- ability (BEP) of the differential binary and quaternary phase shift keying (DBPSK and DQPSK respectively) with multiple input/multiple output (MIMO) systems employing postdetection equal gain combining (EGC) diversity reception (MIMO EGC) over Rayleigh fading channels. Finite closed form expressions for the average BEP of DBPSK and DQPSK are presented. The average BEP of DQPSK is obtained using the associated Legendre functions. The proposed structure for the differential phase shift keying (DPSK) with MIMO EGC provides a reduced- complexity and low-cost receiver for MIMO systems compared to the coherent phase shift keying system (PSK) employing maximum ratio combining (MRC) diversity reception (MIMO MRC). Finally, a useful procedure for computing the associated Legendre functions of the second kind with half integer order and arbitrarily degree is presented.

This paper derives the average bit error probability (BEP) of the differential quaternary phase shift keying (DQPSK) with postdetection equal gain combining (EGC) diversity reception over arbitrarily correlated Nakagami-m and Rician fading channels. Our main contribution lies in presenting relatively simple BEP expressions, the error expressions are given in terms of a finite sum of a finite-range integral. The penalty in signal to noise ratio (SNR) due to arbitrarily correlated channel fading is investigated. It is observed that the performance degradation due to the correlation among the diversity branches in Rician fading channels is severer than that in Nakagami-m fading channels as a result of the presence of the specular component.

A method for computing the exact average symbol error probability (SEP) of the rectangular M-ary quadrature amplitude modulation (MQAM) with maximum ratio combining (MRC) diversity over L independent and equal-correlated Nakagami-m fading channels with arbitrary fading index m is presented. Two models of MQAM receivers are analyzed. Closed-form expressions are provided for the average SEP in terms of the Appell and Gauss hypergeometric functions. Simplified error expressions are proposed when the product Lm is either integer or half integer. The distinction between the performance of the two MQAM schemes is discussed. Finally, the penalty in signal to noise ratio (SNR) due to equal-correlated Nakagami-m channel fading is investigated

In the forward link of a signal-to-interference ratio (SIR)-based fast power-controlled CDMA system, the fraction of base station (BS) power allocated to a mobile station (MS) is considered as a key factor affecting the system performance. By using our proposed macrodiversity non-orthogonality factor, we establish a unified analytical model to characterize the distribution of the fraction of BS power allocated to an MS in either the non-soft handoff mode or the soft handoff mode. By using that distribution and limiting the maximum fraction of BS power available to an MS, a closed-form expression of the capacity at a certain outage probability is obtained. The effects of system parameters, such as the available Rake fingers, the soft handoff threshold, the unbalance of the BS power, and the power control error, on the capacity are investigated from the viewpoint of the limitation on the fraction of BS power allocated to an MS. Simulation results show that soft handoff does not always improve capacity and the capacity gain may result depending on the choice of the system parameters.

A new method is developed to analyze the performance of partially coherent PSK systems in wireless channels with equal-gain combining diversity receiver. Two performance criteria are considered: the average bit error probability and the probability distribution of the combiner SNR (SNR reliability). Tikhonov-distributed phase error processes are assumed and generalized fading channels including Rayleigh, Rician, and Nakagami-m are investigated. We evaluate the detection loss suffered by the carrier recovery for different SNR reliability levels when BPSK and QPSK systems are used in wireless channels. The analysis is based on a convergent infinite series for the distribution of the sum of random variables. The convergence rate of the proposed series is investigated and the analytical results are presented along with providing results obtained by simulation.

In this paper, we analyze the bit error probability (BEP) of L-branch BPSK channels in the presence of cochannel interference (CCI) and carrier phase error. We consider a generalized propagation model in where the desired and interfering signals allow to follow Nakagami-m or Rician fading with different amounts of fading strength. Also, a general asynchronous CCI signals are analyzed so that the effect of cross-signal intersymbol interference (ISI) is taken into account. The diversity branches are independent, not necessarily identically distributed or having equal power levels, and they experience slow frequency-nonselective fades. Extensive error rate results are presented and several comparisons based on the interfering users' signals condition are established

Previous studies on the second statistics, that is, level crossing rates (LCR) and average fade durations (AFD), of maximum ratio combining for uncorrelated Nakagami channels assume identical diversity branches. This assumption rarely holds in the real systems. In this paper, we derive closed-form expressions of the LCR and AFD for a general case of independent diversity branches with unequal average signal powers. Our analysis reveals that unbalance in the diversity branches acts as if there were correlations among them. Simulation results demonstrate that larger LCR and AFD are resulted when the unbalance exists among the diversity branches. Furthermore, the effects of the decay rate, the fading severity, and the total number of unbalanced diversity branches on the LCR and AFD are discussed.

We introduce an analytical model to characterize the distribution of the base station (BS) power in the forward link of CDMA systems, which leads to a closed-form expression of the blocking probability. Real system parameters, such as the soft handoff threshold and the power control error, are taken into consideration. The knowledge of the distribution of the BS power can be used to design a fine call admission control in the CDMA systems.

Useful upper and lower bounds are presented for the average bit-error probability (BEP) of uncoded partially coherent BPSK and QPSK systems. For the first time, the bounds are derived taking into consideration wireless fading impairment. The bounds are shown to be tight and easy to evaluate for phase error noise parameters likely to be encountered in practice and for SNR that are needed in moderate fading environments. A precise relationship between the lower bounds and the average degradation caused by the carrier phase error are given. Also, the analysis of the upper bound establishes an expression for the irreducible errors due to imperfect phase recovery for a coherent QPSK system. The accuracy of both bounds is illustrated by comparing the bound results to those obtained by more complex exact series analysis.

The outage and handoff probabilities for the reverse link of wireless communication systems have been extensively studied (A.J. Viterbi et al., Oct. 1994) (Yu-Wen Chang and E. Geraniotis, 1996). But there are no similar studies on the forward link. We establish an analytical model to study the effects of handoff threshold and path loss margin on the outage. The effects of channel propagation statistics and signal correlation are also addressed. The simulation results show how the soft handoff region and outage probability can be adjusted by varying the path loss margin and handoff threshold. Furthermore, the path loss margin of the forward link is compared to that of the reverse link.

For uncorrelated Nakagami channels in wireless systems, the level crossing rates (LCKs) and average fade durations (AFDs) of maximum ratio combining have been derived. However, this assumption is usually violated in the real world. In this paper, we derive the LCRs and AFDs for correlated Nakagami channels, which have not been addressed in the literature. Simulation results demonstrate that the LCRs and AFDs of the correlation signals are larger than those of the uncorrelated signals at lower levels. Furthermore, the effects of fading severity of the channel and the number of diversity branches on the LCRs and AFDs are also studied.

Binary and quaternary -differential coherent phase-shift keying (DBPSK, DQPSK) are widely used modulations in digital wireless systems. However, a simple expression for the probability density function (pdf) of the decision statistic and the error rate performance of such systems under post-detection equal gain combining (KGC) and generalized fading channels are not currently available in the literature. This paper presents a discussion of these subjects in detail. Our analytic study is based on a convergent infinite series for the sum of independent random variables that has been derived using Fourier series approximation method. Generalized fading channels including Rayleigh, Ricean, and Nakagami-m are investigated. The channels are assumed to be independent, frequency-nonselective, and slowly varying. The accuracy of our results is verified through simulation.

In this paper we evaluate the performance of a bandwidth allocation algorithm used to offer packet data services in the cdma2000 reverse link. In order to reduce interference levels and increase capacity, the proposed approach avoids the use of reverse link signaling by taking advantage of the traffic characteristics of the user applications. Capacity is defined as the number of users supported while complying with the quality-of-service requirements of the applications being serviced. The QoS attributes are the packet delay and probability of outage. The performance of the proposed approach is evaluated using computer simulation. It is compared with schemes that employ the access channel (ACH) to transmit requests for bandwidth. Results show that the proposed approach complies with the assumed application's QoS requirements, and outperforms the ACH based scheme by nearly doubling the system capacity.

The orthogonality factor was used to simplify the expression of the E/sub b//N/sub 0/ of a mobile station communicating with a single base station in the forward link of CDMA systems. This paper extends the orthogonality factor to express the E/sub b//N/sub 0/ of a mobile station communicating with multiple base stations by using the channel-gain matched combination approach. A simplified expression of the E/sub b//N/sub 0/ is derived by averaging over fast fading and using average orthogonality factor. This new expression of E/sub b//N/sub 0/ simplifies the capacity and outage calculations of the CDMA systems in the forward link. The relationship between the average E/sub b//N/sub 0/ in practical CDMA systems and that obtained using the simplified model is discussed. The analysis shows that averaging over fast fading leads to a conservative evaluation of the CDMA system performance.

We explore an approach for bandwidth allocation for the centralized MAC protocol employed by DOCSIS (data over cable service interface specification) compliant cable networks. The proposed approach is based on the statistics of the source traffic models. It aims at achieving low excess bandwidth while still offering delays that are acceptable from the point of view of the application's QoS requirements. The access and total delays as well as the excess bandwidth are our performance metrics. Computer programs were used to compute the performance metrics for a system servicing two applications with different QoS requirements. The selected applications are an Internet video game, which is delay sensitive, and data transfer, which is delay tolerant. The performance metrics were computed for different offered loads. Results show our approach yields QoS complying delays while achieving high bandwidth utilization.

Characterizing the traffic generated by common user applications could prove useful in the design of mechanisms for medium access, admission control, etc. As an example, centralized algorithms that use knowledge of traffic patterns to schedule transmission of stations in access networks could be developed efficiently utilize network resources. Traffic characterization can be done in two steps. In the first step the applications that use network resources are identified, as well as the percentage of time they are active. The second step describes the traffic generation pattern of an active application. In this work we analyze network traces generated by four Internet game applications in order to characterize their traffic generation pattern. This characterization is achieved by obtaining a statistical description of different parameters associated with a traffic model. The model is shown to be robust, that is, the parameters of the model can be characterized using the same probability distributions independent of the application. It is shown that the gamma and uniform distributions can be used to characterize the random parameters of the model. Observed values of the parameter moments are presented.

The objective of this proper is to develop an analytical method of calculating the average power spectral density (PSD) and bit error rate (BER) of Gaussian minimum shift keying (GMSK) modulation scheme. GMSK modulation has been widely used in terrestrial and satellite communication systems. No method, other than simulation is currently available to calculate GMSK PSD. In this paper we express the PSD in a compact Hermitian form suitable for numerical computation. We also analyze the BER performance of a coherent I-Q receiver. The relationship between out of band radiation attenuation and BER degradation is clearly shown. The degradation caused by the transmit and receive filters is estimated. The formulas given in this paper can be extended to arbitrary pulse shapes of finite duration.

An analytical technique based on Gram-Charlier series expansion is presented for the computation of the error probability of equal gain combiner (EGC) with partially coherent fading signals. Detection loss caused by synchronization phase errors is computed for several bit error probability levels. It is shown that the carrier phase precision requirement can be reduced through signal combining.

CDMA is a very attractive technique for PCS because of its potentially large capacity and well-known ability to resolve multipath fading. The next step will be to enhance the performance of CDMA system efficiently and optimally. Two new frequency planning schemes are proposed. One uses a frequency reuse factor of 3 and the other uses a cell partitioning. To show the advantages of two new frequency planning schemes, the capacity of CDMA systems is compared. First of all, the capacity of the current CDMA system is analyzed with the consideration of power control. Then, the performance of the CDMA systems using new frequency planning schemes is obtained. The results show that the current frequency reuse factor of 1 is not always optimal, even though it is the simplest frequency planning to deploy. The new frequency planning schemes provide more capacity and are more adaptive in the cases of non-uniformly distributed traffic loads.

This paper presents the analysis and performance of the optimum diversity receiver in mobile radio environments with QAM data transmission, two-ray frequency-selective Rayleigh-fading channel, Gaussian noise and co-channel interference (CCI), while the symbol timing offset between the desired and co-channel interference is considered. With this non-zero random offset value, the co-channel interference is shown as a wide-sense stationary process (WSS). An optimum diversity receiver is proposed for the new WSS co-channel interference model. Linear and decision-feedback equalization techniques are used in the design of the optimum diversity receivers. By simulation techniques, we show that the degradation due to the WSS co-channel interference is smaller than that caused by the equivalent additive Gaussian noise or cyclostationary cochannel interference, especially in highly co-channel interference dominated channels and also in the system where a decision-feedback equalizer is employed.

Lower bounds on the error probability for BPSK and QPSK systems are presented when the detection process is corrupted by a noisy phase reference. These bounds are derived from the application of Jensen's inequality to convex functions and from the utilization of the moment space bounding technique. A set of improved lower bounds using Jensen's inequality is also derived. We provide explicit relationships between the derived bounds and the average power losses in the studied signals caused by the imperfect detection process. These averages have simple expressions and can be used in system designs to provide lower bounds on the detection loss.

Mobility of the wireless user creates uncertainty in demand and non-optimum use of radio resources. Current wireless networks are periodically re-configured manually to improve network performance. Dynamic channel assignment (DCA) methods can be used to reconfigure radio channels automatically. But many of these DCA techniques use "parochial" allocation schemes which consider only the received interference levels and do not consider the impact of the new interference generated by a particular channel assignment on other cells. These DCA algorithms typically allocate channels on an instantaneous need basis so there is little computation time to consider the impact of network wide interference. This paper will propose a new DCA approach which uses time intervals to estimate traffic trends. These traffic trends will be used to estimate radio requirements for each period. All investigated DCA algorithms will consider network wide interference impact and select radio configurations which minimize overall interference. This paper will examine the performance of two different DCA approaches, the gradient descent algorithm (GDA) and cell level radio tuning algorithm (CLRTA) methods. The different DCA algorithms will also be compared to static allocation algorithms. The GDA is shown to have superior performance to that of the CLRTA by approximately 2-4 dB depending on the number of available radio channels. The DCA approach is approximately 3-5 dB better than conventional fixed channel allocation (FCA) scheme.

This paper analyzes the performance of space diversity structures combined with finite-tap decision-feedback equalization technique in a mobile radio environment with QAM transmission and frequency-selective fading. Two optimum diversity structures, combining-diversity and selection-diversity, are considered in our analysis. By analytic techniques and simulation, we show that for the same number of taps the improvement is always greater with combining-diversity structure. The combining-diversity with a certain finite number of taps can outperform the selection-diversity with infinite taps. The numbers of taps to approach optimum performance (infinite-tap bound) in outage for different mobile environments are also determined. The decision-feedback equalizer not only provides better performance than the linear equalizer, but also needs a smaller number of taps to approach the optimum performance, especially in the combining diversity system or in the severe time-dispersion environment.

Differential phase-shift keying (DPSK) and coherent phase-shift keying (CPSK) are two of the techniques widely used in digital communications over channels such as satellite, terrestrial radio, and voiceband telephone. Because of the linear nature of the demodulation process in CPSK, the error performance of CPSK with various sources of impairments has been exhaustively treated, but even with additive Gaussian noise, only asymptotic approximations are available to determine the detection performance ofM-ary DPSK,M> 2. In this note, the error performance of nonbinary DPSK in additive Gaussian noise channels has been bounded in terms of tabulated functions and various approximations are then derived.

By using principles of analytic continuation, upper and lower bounds on the error probability of a canonical binary system corrupted by additive interference and independent zero-mean Gaussian noise are derived. The bounds, which are simple functions, require only the evaluation or bounding of the moment generating function of the interference. For large signal-to-noise ratios (SNR) and even for moderately large interference, the bounds are shown to be tight and useful.

An analytical study to predict multipath fading outages in terrestrial digital radio systems is described. The method, which is quite general, makes specific use of previously reported statistics on multipath fading. It is applied, in this study, to 6- and 11-GHz systems carrying 90 Mbits/s per RF channel. For purposes of comparing alternate approaches, three different modulations are considered, namely, (i) conventional QPSK at 45 Mbits/s, using two polarizations per channel; (ii) Offset-QPSK at 45 Mbits/s using two polarizations per channel; and (iii) 8-level PSK at 90 Mbits/s, using one polarization per channel. In all cases, a fourthorder Butterworth transmit filter is assumed which enforces FCC spectral emission requirements. The channel is treated as a two-path propagation medium, the receiver is assumed to be coherent, and various practical assumptions are invoked regarding the recovered carrier phase, timing phase, receive filter response, bit error rate objectives and fade margin. Results are given in terms of expected yearly fraction of multipath outage per repeater hop vs. hop length. We find that the choice of operating band has only a minor effect on the results; that OffsetQPSK systems suffer somewhat more multipath outage than conventional QPSK systems; that 8-level systems suffer more multipath outage than 4-level systems (although this difference may be offset by cross-polarization effects in the latter); and that all the approaches considered require repeater spacings of 15 mi or less in order to meet short-haul outage objectives over average terrain.

One of the most interesting challenges in the field of digital radio is to generate and transmit signals which meet the spectral requirements and which have high detection efficiencies. The spectral efficiency of PSK-type modulation with overlapping baseband pulses is known to be better than that with nonoverlapping pulses of the same form; but, to our knowledge, no analysis is presently available to determine their detection efficiency. In this paper a method has been given to evaluate the error probability performance of PSK-type modulation with baseband pulses of finite overlap and mutually independent samples. It is shown that the error probability can be computed with any desired accuracy. For BPSK and for some typical transmit and receive filters and for a transparent channel corrupted by additive Gaussian noise we compare the detection efficiencies when the baseband modulation pulses have zero and nonzero overlap and when the transmitted signal satisfies the same spectral requirement. The nonoverlapping rectangular signaling is shown to be superior to overlapping raised-cosine signaling, but the differential degradation is less than 0.8 dB if the twosided 99-percent bandwidth W99satisfies1.1/T \leq W_{99}\leq 1.6/T. The analytical results presented here for a binary system can be extended with minor modifications to anyM-ary PSK system,M > 2.

For phase-shift keying (PSK) systems a method has been presented to determine the filter parameters so that the degradation produced by the bandlimiting filters is tolerable and as many channels as possible can be accommodated in a given RF band. Thermal noise present in the system and intersymbol and interchannel interference are considered the major sources of impairment in the system, and these are the only sources of impairment considered in this paper. The method is here applied to a binary coherent PSK (CPSK) system when unequalized Butterworth and Chebyshev filters of different complexities are used as receiving filters, and suitable filter parameters have been determined.

We derive upper and lower bounds on the probability of error ofM-ary CPSK systems subject to intersymbol interference and additive Gaussian noise. The bounds are expressed in terms of the error probability obtained with a finite pulse train and some parameters associated with the residual pulse train. Methods are given to compute the probability of error with a finite number of interference terms and it is shown that the difference between the upper and the lower bounds is a monotone decreasing function of the number of pulses in the finite pulse train. The applicability of the method to compute the probability of error with any desired degree of accuracy is illustrated by examples for quaternary and octonary systems.

For a digital system with correlated digital symbols, we derive upper and lower bounds on the probability of error when the system is subject to intersymbol interference and additive Gaussian noise. The bounds are expressed in terms of the error probability obtained with a finite number of intersymbol interference terms and some parameters associated with the remainder terms. We also show that the difference between the upper and lower bounds can be made arbitrarily small and that the error probability can be computed to any desired degree of accuracy. We give some examples to illustrate the method.

Upper and lower bounds on the spectral density of a sinusoidal carrier frequency modulated by a band-limited Gaussian noise signal are derived and we show that these bounds are very close to each other if the ratio of the rms frequency deviation to the maximum modulating frequency is even moderately high. We then show that the results obtained from the quasi-static approximation, often used in such cases, are too small for largefand that this low-frequency approximation cannot be used in cases where the behavior on the tails is of importance.

In order to optimize the design of a coherent phase-shift-keyed (CPSK) system, it is necessary to estimate the amount of degradation produced by intersymbol interference. In this paper, an upper bound to the probability of error ofm-ary CPSK systems is given when an ideal CPSK signal is passed through a linear time-invariant noisy filter. This bound can be used to estimate the maximum amount of deterioration produced by intersymbol interference, and hence to choose the filter and channel parameters appropriately. It is assumed that allmsymbols have equal a priori probabilities and that the noise is Gaussian.