In the evolving landscape of wireless communications for 5G, 6G, and beyond, the deployment of unmanned aerial vehicles (UAVs) has emerged as a groundbreaking strategy to expand coverage areas due to their flexibility and ease of deployment. Simultaneously, reflecting intelligent surfaces (RISs) have introduced a transformative paradigm aimed at improving key performance metrics, such as average sum-rate and energy efficiency (EE). The seamless integration of advanced technologies, including UAVs, RIS, and nonorthogonal multiple access (NOMA), presents a promising avenue for significantly boosting the performance and efficiency of next-generation communication systems. This study investigates EE maximization for two scenarios in a NOMA-enabled mmWave network: 1) multi-UAV-mounted base stations (BSs) and 2) multi-UAV-mounted distributed RIS. In both cases, each UAV serves a NOMA cluster with imperfect successive interference cancellation (SIC), capturing the impact of hardware impairments in real-world NOMA systems. For each scenario, an optimization problem is formulated to maximize EE by jointly optimizing the beamforming matrix, phase shift matrix, NOMA power allocation, and UAV 3-D placement. The nonconvex problems are tackled using both model-based and model-free deep reinforcement learning (DRL) algorithms under constraints, such as minimum Quality of Service (QoS), beamforming and phase shift limits, and UAV trajectory constraints. The simulation results demonstrate that the proposed DRL algorithms significantly enhance spectral efficiency (SE) and EE, showcasing their suitability for 6G communication systems. Furthermore, a comparative analysis with orthogonal multiple access (OMA) and spatial-division multiple access (SDMA) confirms that NOMA outperforms both techniques, achieving substantial gains in efficiency and performance.

Identification of modulation and space‐time block code (STBC) is an important task of receivers in applications such as military, civilian, and commercial communications. Here, we consider multiple‐input multiple‐output (MIMO) systems. We propose two methods for STBC identification when the modulation is known. We also introduce a method for joint identification of code and modulation. Additionally, we present an enhanced zero‐forcing (ZF) equaliser to improve the separation between the features of different classes. Higher order cumulants are used as the statistical features. In the first method of STBC identification, after the proposed equalisation, received data samples are segmented, and then using the mean and Frobenius norm of the covariance matrix of extracted cumulants as threshold values, the STBCs are detected. This method requires the knowledge of the threshold values for each modulation type and noise power. In the second method, the knowledge of noise power and the type of modulation are not needed. After the proposed ZF equalisation, feature vector based on the cumulants is calculated, and then support vector machine (SVM) classifier is used to classify different STBCs. In the third method, joint detection of STBC and modulation type is performed using the proposed ZF equaliser and mapping the received samples. This method deals with the theoretical values calculated from the cumulants of four STBCs and four modulations, where there is no need to know the noise power. The results indicate that the proposed methods perform well even at low signal‐to‐noise ratios (SNRs).

Sleep has great effect on physical health and quality of life. Electroencephalogram (EEG) signal is used in studying sleep process and recently, time–frequency transforms are increasingly utilised in EEG signal analysis. This study proposes an efficient method for sleep stages classification based on a time–frequency transform, namely Stockwell transform. In the introduced method, at first, the Stockwell transform is used to map each 30 s epoch of EEG signal into the time–frequency domains, which results in a complex‐valued matrix. Then, the frequency domain is divided into different non‐overlapping segments, leading to several matrices. After that, entropy features are extracted from the obtained matrices. In order to determine the sleep stage of each epoch, the computed features are applied to classifier. Support vector machine, weighted K ‐nearest neighbour, and ensemble bagged tree classifiers are considered. The Pz–Oz and Fpz–Cz channels of EEG signal from Sleep‐EDF data set and C3–A2 channel from ISRUC‐Sleep data set are used in this research. The results indicate that the proposed method outperforms the recently introduced methods.

Static bandwidth allocation for variable bit rate (VBR) video traffic forfeits the available bandwidth. Prediction of the next frame size is thus useful in dynamic bandwidth allocation. It has been shown that VBR video traces are long-range dependent, which makes one-frame-ahead prediction insufficient for dynamic bandwidth allocation. Several studies have been conducted based on the linear autoregressive (AR) model to address VBR traffic prediction. In this paper, we propose the use of a nonlinear model from the AR family called logistic smooth transition autoregressive (LSTAR) to predict VBR video traffic. Furthermore, we introduce adaptive algorithms, including least mean square (LMS), normalized LMS (NLMS), kernel LMS (KLMS), and normalized KLMS (NKLMS), to obtain the parameters of the LSTAR model used in long-range VBR traffic prediction. In the proposed model, we do not separate traffic of different frame types and use only one predictor, which results in lower computational complexity. The performance of the proposed predictor for different prediction steps was evaluated and compared with recently introduced predictors. The results indicate that the proposed nonlinear LSTAR-based predictor yields better results than the optimum linear AR predictor, i.e., Wiener-Hopf and others.

This study presents an efficient handoff algorithm for heterogeneous networks comprising macrocells, microcells, picocells, and femtocells. The proposed algorithm is based on call admission control (CAC) for selecting target base station (BS) from a list of neighbouring candidate BSs, as well as using the technique for order of preference by similarity to ideal solution (TOPSIS) as decision method. The introduced algorithm takes into account multiple criteria including measured received signal strength (RSS) and signal‐to‐interference‐plus noise ratio (SINR), predicted RSS and SINR, and number of free resource blocks of neighbouring BSs. With the use of TOPSIS, neighbouring BSs are ordered based on their priority determined using the five mentioned criteria. Selection of target BS from ordered list is then performed using CAC. Also, coverage expansion is used to retain the connection when there is not any BS for handoff. In addition, in order to predict the RSS and SINR samples, the logistic smooth transition autoregressive model is used. Performance of the proposed algorithm is evaluated in terms of ping‐pong rate, outage probability, and throughput. The results indicate efficiency of the proposed algorithm in reducing number of unnecessary handoffs while significantly increasing throughput and decreasing connection dropping when compared with conventional algorithms.

The aim of this study is to improve the performance of two‐tier macro/femtocell networks using a power control approach. In wireless networks, power control plays an important role in improving a number of performance parameters such as co‐channel interference and outage probability reduction, throughput increasing, and power saving. This study explores the evolution of centralised power control algorithm based on femtocell base station (FBS) clustering and predicted signal‐to‐interference‐plus‐noise ratio (SINR) of users. To reduce the computational complexity of centralised algorithm, dense deployed femtocells are considered in different clusters. In this case, femtocells inside one cluster make considerable interference to each other, while the interferences from femtocells of other clusters are negligible. Moreover, because of the non‐linearity of SINR samples, non‐linear logistic smooth transition autoregressive (LSTAR) model is used to model the SINR data, and then the next SINR samples are predicted from the previous samples. According to the clustered FBSs and predicted SINR, the proposed power control scheme is applied to femtocell network in the downlink. The results demonstrate that the introduced method improves the outage probability and throughput and outperforms previous methods significantly.

This study presents an efficient fractional frequency reuse scheme for macrocell network which uses six directional antennas to cover the outer region and analyses the performance. Cross‐tier interference between macrocell and femtocell networks leading to throughput and outage probability degradation is a major issue in two‐tier networks when femtocells are deployed in co‐channel with macrocells. In this study, the received interference from the macrocell network to femtocell network is analysed. Then, based on the degree of received interference, an algorithm for orthogonal resource allocation to femtocells with low cross‐tier interference is presented. According to the introduced scheme, macrocell uses all resource blocks along with a portion of the spectrum allocated to femtocell while both networks operate simultaneously. Closed‐form expressions for downlink throughput and outage probability of femtocells are also attained as well as the throughput of macrocell network. The results demonstrate that the proposed hierarchical cellular network achieves higher throughput and lower outage probability for femtocell when compared to other hierarchical macro–femto cellular networks.

Cognitive radio and cooperative networks are increasingly regarded as revolutionary technologies that utilise the radio spectrum efficiently. In this study, spectrally encoded (spread‐time) code division multiple access (SE‐CDMA) scheme is proposed as a flexible and adaptable technique for physical layer of cognitive relay networks. All primary and secondary users transmit their signals to a common base station (BS) and then the BS selects one of the cognitive secondary users based on feedback information to relay the primary user's (PU's) data. To study the efficiency of the system, two scenarios are considered. In the first scenario, the secondary users transmit their information using the SE‐CDMA technique and the PU is considered narrowband. In the second scenario, both primary and secondary users utilise the SE‐CDMA technique. The performance of the system is evaluated in terms of the error probability of the PU, the outage probability of the primary and secondary users and comparing theoretical results with simulations. Moreover, the performance of the proposed method is compared with orthogonal frequency division multiple access‐based cognitive relaying scheme. The results indicate the efficiency of the SE‐CDMA technique.

In this study, the problem of variable bit rate (VBR) video traffic prediction is addressed. VBR traffic prediction is necessary in dynamic bandwidth allocation for multimedia quality of service control strategies. Autoregressive (AR) models have been widely used in VBR traffic prediction where the least mean square (LMS)‐based methods were utilised for parameter estimation. However, they are ineffective when the traffic is dynamic in nature. In this study, using the Brock, Dechert, and Scheinkman (BDS) test, it is shown that the video traffic is non‐linear. Kernel is an efficient tool to convert non‐linear data into linear one in a higher‐dimensional space. The kernel LMS (KLMS) method is proposed to forecast the next frame sizes of I, B and P frames as well as the next group‐of‐pictures (GOP) size of video traffic. Extensive simulations were performed on different video traces where different performance metrics were considered. KLMS results were very close to those of the Wiener–Hopf optimum solution and better than the results of commonly used normalised LMS and other algorithms such as the least mean kurtosis (LMK), wavelet LMK, adaptive network fuzzy inference system (ANFIS) and neural networks.

In this study, a blind steganalysis approach which accurately discloses low rate data hiding schemes is proposed. The absolute values of neighbouring joint density (absNJ) are used for feature extraction. In this way, the intra‐ and inter‐block situations are employed providing a variety of different features. Aside from the absolute values of discrete cosine transform (DCT) coefficients, the differential DCT coefficients are also exploited to extract the features. Moreover, the absNJ features will be extended and used over differential DCT coefficients. It is shown that applying the P th power of the DCT coefficients instead of their first power gains more discriminative features. Then, using the ensemble classifier, the cover image is discriminated from the stego one. Experimental results indicate the efficiency of the new scheme in comparison with the previously presented schemes.

The aim of this paper is to present a non‐linear statistical model to fit and forecast the signal‐to‐interference plus noise ratio (SINR) in two‐tier heterogeneous cellular networks which consist of macrocells and femtocells. Since in these networks the number and locations of femtocell base stations (FBS) are variable, SINR forecasting can be useful in some areas such as power control and handover management. So far, linear autoregressive (AR) models have commonly been used in forecasting the received signal strength (rss) in macrocellular networks. However, AR modelling results in high mean square error (MSE) when data are non‐linear. This paper focuses on SINR which takes into account signal strength, interference and noise effects. Moreover, macro‐femto cellular network is considered. The F ‐test results show that the SINR data are non‐linear, leading to use non‐linear models instead of AR model. A non‐linear logistic smooth threshold AR (LSTAR) model is utilised to model and forecast the SINR data. Kolmogorov–Smirnov (K‐S) test demonstrates that LSTAR provides good fitness to the SINR samples. The results indicate that LSTAR model achieves much better performance in modelling and forecasting of SINR data than the AR model.

In this study, an efficient handover algorithm based on the received signal strength (RSS) prediction is presented for two‐tier macro–femtocell networks in which, because of the fading effects of channel and short coverage range of femtocells, ping‐pong handovers may take place. In the proposed approach, first each mobile station (MS) uses the recursive least square algorithm for predicting the RSS from the candidate base stations (BSs) including both femtocell and macrocell BSs. Then, according to the predicted RSS values, several future values of signal‐to‐interference plus noise ratio (SINR) are calculated. Afterwards, the candidate list of BSs is pruned according to the estimated future SINR values and the predicted RSS of each BS. Finally, the target BS which yields the highest throughput, is opted for handover. Through extensive simulations, the effects of speed of MSs and the density of femtocell BSs on the outage probability (OP), throughput and the ping‐pong rate of MSs are studied. The results show that the proposed handover algorithm outperforms the previous ones and improves the throughput of MS while it reduces the OP and the number of ping‐pong handovers.

A new method for wideband spectrum sensing in cognitive radio networks is proposed. Since the problem of estimating the number of occupied channels can be considered as the problem of estimating the number of signal sources in array signal processing, so the model used for direction of arrival (DOA) estimation is utilised here for spectrum‐sensing modelling. In the proposed algorithm, wideband spectrum is divided into subchannels, where each subchannel resembles a sensor in array processing for DOA estimation. Furthermore, the detection problem in practical situations is complicated because noise is most likely non‐Gaussian and non‐stationary unlike the assumption of previously presented algorithms. Therefore, in the proposed algorithm, a generalised autoregressive conditional heteroscedasticity model is used to model the additive noise. The number and locations of the occupied subchannels will be jointly estimated using the maximum likelihood approach. The introduced method is blind as there is no need for initial information about the primary users’ signals and noise variance. The results indicate the efficiency of the proposed method.

Recent studies have revealed that spectrum congestion is due to the inefficient usage of spectrum against the availability. Cognitive radio is viewed as an approach for improving the utilisation of radio spectrum. The spectrally encoded (spread‐time) code‐division multiple access (SE‐CDMA) technique (which is regarded as the dual of spread spectrum CDMA) is considered and its performance in cognitive radio networks is studied. In cognitive network spectrum, overlay and underlay techniques are employed to enable the primary and secondary users to coexist while improving the overall spectrum efficiency. It is shown that SE‐CDMA provides considerable flexibility to design overlay and underlay waveforms that are scenario‐dependent. The performance of SE‐CDMA method is evaluated in additive white Gaussian noise and fading channels in terms of both primary and secondary users. Moreover, its performance is compared with cognitive radio using soft‐decision spectrally modulated spectrally encoded waveforms. The results indicate the efficiency of SE‐CDMA.

In this study, a new space time block code (STBC) is proposed for long term evolution-advanced (LTE-A) system that is equipped with three time slots and two antennas. The introduced STBC has the following characteristics: i) it achieves rate one and full diversity, ii) its maximum likelihood (ML) decoding requires a joint detection of three real symbols, iii) the minimum determinant value (MDV) does not vanish by increasing signal constellation size, and iv) bit error rate (BER) results show that the proposed STBC outperforms previously presented schemes for LTE-A at high SNRs.

Eye state analysis (open or closed) is an important step in fatigue detection. In this study, an efficient algorithm for eye state detection is proposed. At first, a new face detection method is presented for noisy images that finds the face area in the input image well. Then, novel algorithms for detection of eye region and eye state are introduced. The performance of the proposed method is evaluated on four different databases namely FERET, Aberdeen, IMM and CVL which contain more than 5700 images with different descents, positions, light conditions and glasses. The results show that the new method achieves more accuracy rate than the previously presented algorithms, while it does not need training data and is also computationally efficient.

The aim of this study is to take advantage of both shape and texture properties of image to improve the performance of image indexing and retrieval algorithm. Further, a framework for partitioning image into non‐overlapping tiles of different sizes, which results in higher retrieval efficiency, is presented. In the new approach, the image is divided into different regions (tiles). Then, the energy and standard deviation of Hartley transform coefficients of each tile, which serve as the local descriptors of texture, are extracted as sub‐features. Next, invariant moments of edge image are used to record the shape features. The shape features and combination of sub‐features of texture provide a robust feature set for image retrieval. The most similar highest priority (MSHP) principle is used for matching of textural features and Canberra distance is utilised for shape features matching. The retrieved image is the image which has less MSHP and Canberra distance from the query image. The proposed method is evaluated on three different image sets, which contain about 17 000 images. The experimental results indicate that the proposed method achieves higher retrieval accuracy than several previously presented schemes, whereas the computational complexity and processing time of the new method are less than those of other approaches.

This paper addresses the estimation of the integer part of the carrier frequency offset (CFO) in orthogonal frequency division multiplexing (OFDM) systems. A novel method is presented based on the maximum likelihood (ML) approach that jointly estimates the channel impulse response (CIR) and the integer CFO. The new scheme estimates the CIR by using pilots and obtains the estimate of the integer CFO by taking advantage of the effect of the CFO on the CIR and data symbols. The performance of the new method is evaluated in Rayleigh fading channels. The results demonstrate that the proposed scheme significantly outperforms the previous methods.

The exact symbol and bit error probabilities of linearly modulated signals in frequency non-selective Rician fading channels with Lth diversity branches are derived, where coherent detection with maximum ratio combining (MRC) is used at the receiver. For performance evaluation, the multiplicative distortion is combined with the additive Gaussian noise and form an additive noise is formed. This method allows computing the exact symbol and bit error probabilities of M-ary phase shift keying (M-PSK), M-ary quadrature amplitude modulation (M-QAM), M-ary amplitude modulation (M-AM) and M-ary amplitude modulation phase modulation (M-AMPM) for any arbitrary bit mappings. The results contain several versions of one simple integral. The error probabilities are also obtained for Rayleigh channel, which is a special case of Rice channel. Further, the results for Nakagami-m channel with the integer parameter are found. The results quite match with the previous ones while having simple forms. Also, for some unsolved cases, the exact error probabilities are presented. Simulations are carried out to verify the analytical evaluations.

In this paper, we propose a hybrid algorithm including Genetic Algorithm (GA), Ant Colony Optimization (ACO), and Simulated Annealing (SA) metaheuristics for increasing the contrast of images. In this way, the contrast enhancement is obtained by globally transformation of the input intensities. ACO is used to generate the transfer functions which map the input intensities to the output intensities. SA as a local search method is utilized to modify the transfer functions generated by ACO. GA has the responsibility of evolutionary process of ants' characteristics. The results indicate that the new method performs better than the previously presented methods from the subjective and objective viewpoints.

In this paper, we consider the internally coded time-hopping spread-time multiple access scheme introduced in. We evaluate the performance of the correlator receiver followed by the Viterbi algorithm in fading channel. We analyze the effect of multiple access interference (MAI) and narrowband interference (NBI). We also provide simulation results and further evaluate the performance of the system for the ultra wide band (UWB) application with the UWB channel model specified by the IEEE 802.15.4a group. For comparison, we consider the conventional ST-CDMA system. The results indicate that our method has much better performance while it has the same bandwidth and spectral efficiency as the conventional ST-CDMA system.

A new internally coded scheme that combines frequency hopping (FH) and multicarrier (MC) code division multiple access (CDMA) techniques using a super-orthogonal encoder is proposed. In this method, the total bandwidth is partitioned into Ns1 disjoint bands and each band is also segmented into Ns2 subbands. On the basis of a super-orthogonal encoder output and a pseudorandom sequence, one of the Ns1 bands is selected. The data bit is then transmitted in the band in the form of the multicarrier FH (MC-FH) CDMA scheme, that is, Ns2 carriers are transmitted in the Ns2 subbands of the selected band. At the receiver, both hard and soft detectors are used. The performance of the proposed method in the additive white Gaussian noise (AWGN) and fading channels is evaluated. The results indicate that the proposed method outperforms the previously presented uncoded and coded MC-FH-CDMA systems, where the data are transmitted over the whole bandwidth, keeping the same bandwidth (spectral efficiency). Further, in the new method, the carriers hop in part of the total bandwidth, and hence coherent detection is more feasible.

A new time-hopping ultrawideband (TH–UWB) CDMA scheme for indoor wireless communications is presented. In the proposed method, the duration of each bit is divided into Ns1 frames, each one containing Ns2 subframes. Two pseudorandom sequences are assigned to each user. During each bit interval, based on the output of a super-orthogonal encoder and the user's first dedicated pseudorandom sequence, the transmitter selects one of the Ns1 frames and then transmits Ns2 narrow pulses in that frame, one in each of the Ns2 subframes. The location of the pulse in each subframe is determined by the user's second dedicated PN sequence. Four different detection techniques are considered at the receiver front end, namely thresholded hard decision, strict hard decision, soft decision and chip-based decision. Their performances are analysed and the results are compared with those of the previously introduced coded and uncoded TH–UWB systems. The results indicate that the proposed scheme has the best performance without requiring any extra bandwidth. It is also shown that the chip-based decoding technique works better in moderate and high SNRs while the soft decision method has better performance in low SNRs.

We study and examine the effect of fading channels on the performance of spread-time/code division multiple access (ST/CDMA). We consider two different models for fading channel and obtain the average signal to interference plus noise ratio (SIR) for ST/CDMA in both cases. We also obtain the SIR for spread-spectrum/CDMA (SS/CDMA) and compare the results with that of ST/CDMA. Depending upon the transmitted chip pulse shape, and in the absence of thermal noise, spread-time (ST) performance is equal to or greater than spread spectrum (SS). For example ST/CDMA has 0.62 dB higher SIR value compared to SS/CDMA with root raised cosine pulse shape with rolloff factor 0.22 used in W-CDMA. The spread-time technique was first proposed by Crespo et al. (1991, 1995), and US Patents no. 5177768, 5175744, 5175743, 5173923, and is considered as the time-frequency dual of spread spectrum technique.

The conventional single user detector in DS-CDMA (direct sequence code division multiple access) systems involves multiple access interference and the near-far effect which cause the limitation of capacity. On the other hand, the computational complexity of the optimum multiuser detector grows exponentially with the number of users. There has been a lot of interest in suboptimal multiuser detectors with less complexity and reasonable performance. In this paper we apply the classic and new modified genetic algorithms for multiuser detection of DS/CDMA signals. It is shown that the classic genetic algorithm (GA) reaches an error floor at high signal to noise ratios while the proposed method has higher performance than the classic one and is comparable to the optimum single user detector with much less complexity than the optimum multiuser detector.

We compute the outputs of matched filters for spread-time CDMA with and without the effect of window which is used for truncating the resulting pulse. We obtain the signal to interference plus noise ratio (SIR) in both cases and compare the results. We examine the effect of different multiuser detectors used for spread spectrum (SS) in spread-time (ST). It is shown that ST and SS have the same performance while ST has the ability of matching the transmitted spectrum to the channel. Furthermore we consider the effect of synchronization error in the absence of any window. It is shown that synchronization error in timing causes infinite ISI terms that do not converge.

Spread-time code division multiple access (ST-CDMA) was proposed as an alternative CDMA scheme to spread spectrum (SS) (see Crespo, P.M. et al., Proc. IEEE Globecom, p.836-40, 1991; US Patents No. 5177768, 5175744, 5175743, 5173923; IEEE Trans. Commun., vol.43, no.6, p.2139-48, 1995). ST-CDMA can be considered as the time-frequency dual of SS-CDMA. In ST-CDMA, pseudorandom (PN) sequences are assigned to each user, and the Fourier transform of the transmitted pulse for a given user is determined by modulating the amplitude of the desired spectrum by the user's PN sequence. The main advantage of ST-CDMA is the flexibility with which power-limited pulses can be designed. We examine the effect of fading channels on the performance of spread-time. We compute the average signal-to-interference-plus-noise ratio in two different fading channels both for ST-CDMA and SS-CDMA. We show that ST-CDMA has 2.1 dB better performance than SS-CDMA with a rectangular pulse chip.

In this paper we consider two multistage multiuser DS/CDMA detectors in multipath fading channels. The first detector is a modified version of detector proposed in Hui and Letaief (1998), and the second detector is a combination of the first one used as a PIC (parallel interference cancellation) detector and a detector called SIC (serial interference cancellation). These detectors take the advantage of multipath diversity to improve the signal to noise ratio (SNR). We also use an adaptive method to compute the threshold levels. We obtain an analytical equation for BER (bit error rate). The results show that the mentioned methods achieve much better performance than taht of Hui and Letaief.

The conventional single user detector in DS/CDMA (direct sequence code division multiple access) systems involves multiple access interference and near-far effect which cause the limitation in capacity. The complexity of optimum multiuser detectors also grows up exponentially with the number of users. There has been a lot of interest in suboptimal multiuser detectors with less complexity and reasonable performance. In this paper, we apply decision based neural network (DBNN), fuzzy decision neural network (FDNN), and discriminative learning and backpropagation neural networks employing multilayer perceptron for detection of signals of users in DS/CDMA systems in additive white Gaussian noise (AWGN) channel. We also show that FDNN and discriminative learning neural net have almost the same performance.

The method used in Aghamohammadi and Meyr (1990) for finding the error probability of linearly modulated signals on Rayleigh frequency-flat fading channels has been applied to the more general case of Ricean fading. A signal received on a fading channel is subject to a multiplicative distortion (MD) and to the usual additive noise. Following a compensation of the MD, the signal provided to the detector may be thought to include only a single additive distortion term ("final noise"), which comprises the effects of the original additive noise, the MD, and the error in MD compensation. An exact expression for the probability density function of the final noise is derived. This allows calculation of error probability for arbitrary types of linear modulations. Results for many cases of interest are presented. Furthermore, as special cases of Ricean fading, error probability for Rayleigh fading and non-fading channels are obtained which either match the results or complete the approximate derivations formerly known from the literature.>