تجزیه و تحلیل عملکرد از سیستم های DS / CDMA با ​​سایه و محو شدن مسطح

A new method is developed for evaluating the error probability (Pe) for direct sequence, code division multiple access (DS/CDMA) wireless systems that includes the effects of shadowing and fading. The method is based on saddle point integration (SPI) of the test statistic's moment generating function (MGF) in the complex plane. The SPI method is applicable to both ideal and wireless channels. For wireless channels, a Padé approximation (PA) of the MGF, which is derived from the moments of the channel's shadowing and fading distributions, allows efficient evaluation of the Pe. The SPI method can be used to model independent channels using separate shadowing and fading moments for each individual channel. The relative error between the probability density function (PDF) of the composite variate representing log-normal shadowing and Rayleigh fading and the PDF found from the inverse Laplace transform of the PA is negligible. Results show that log-normal shadowing increases the Pe by 100–1000% compared to channels exhibiting fading only.

Much research effort has been directed towards the performance evaluation of direct sequence, code division multiple access (DS/CDMA) communication systems during the past two decades. Methods for computing the error probability, Pe, for DS/CDMA systems have been proposed for both ideal channels [13], [14], [16] and [18], and fading channels [3] and [4]. Recently, DS/CDMA models have been improved to include the effects of channel coding and multiuser receivers [10], [11] and [12]. It is well known that wireless channels are affected by shadowing, which is a long-term variation in the mean envelope averaged over several wavelengths, in addition to fading [17]. Due to the mathematical complexity, previous methods for computing the Pe for wireless channels have not included the effects of both shadowing and fading. In this paper, we develop a new method for evaluating error probabilities for DS/CDMA wireless systems including the effects of shadowing as well as flat fading. In the process, we also present the method for ideal channels. Saddle point integration (SPI) is used to efficiently evaluate chip-asynchronous, DS/CDMA error probabilities. SPI is based on numerical contour integration of the test statistic's moment generating function (MGF) in the complex plane. The SPI method has been used to compute the error probability resulting from intersymbol and cochannel interference [6], radar detection probabilities [8], and K-distributions [5]. SPI is superior to the characteristic function method presented in [3] because it is less susceptible to roundoff error due to the integrand oscillations, and this property is particularly valuable for automated system modeling tools. We use Padé approximations (PAs) [1] and [2] to model the effects of shadowing and flat fading channels. The unconditioned MGF, averaged over the shadowing and fading distributions, is approximated by its PA. This PA is determined from the moments of the shadowing and fading distributions. In this work, we consider mean envelope, log-normal shadowing as well as Rayleigh and Ricean fading. However, the model is applicable to any channel given the exact moments of the shadowing and fading distributions. The SPI method is completely general and does not place any restrictions on the channel statistics. Independent channels can be modeled using separate shadowing and fading moments for each individual channel. As a result, one can model different types of channels simultaneously such as Rayleigh fading with shadowing and Ricean fading without shadowing. Previous results in the literature have considered the average Pe based on the statistics of the source, channel, and additive white Gaussian noise (AWGN). However, the systems engineer must also determine the worst case Pe when analyzing a wireless communication system for a particular environment. For the reverse link from the mobile station (MS) to the base station (BS), we compare the average and worst case Pe given rectangular chips where the average Pe is determined by modeling the multiple access interference (MAI) sources with asynchronous chips while the worst case Pe is found assuming synchronous chips for the MAI sources. This analysis does not apply to the forward link from the BS to the MS in cellular DS/CDMA, which is also synchronous, because the separate information sequences are spread at the BS with orthogonal, Walsh–Hadamard sequences prior to transmission. This paper is organized as follows. In Section 2, we present the chip-asynchronous DS/CDMA system model. The SPI method is derived for random signature sequences for ideal channels in Section 3 and wireless channels with shadowing and fading in Section 4. Finally in Section 5, we present numerical results which evaluate the error probabilities for all channel models

Evaluating error probabilities for DS/CDMA systems over wireless channels, which include the effects of shadowing and fading, is an extremely challenging mathematical task if approached directly using PDFs. However, using PAs based on the moment matching approach, we are able to develop a simple and computationally efficient solution by evaluating the contour integral of system's MGF. The fading and shadowing moments in the infinite series given in (16) grow exponentially. As a result, simply truncating the infinite series leads to large errors when evaluating the Pe. Using PAs eliminates this error by providing a rational approximation of the infinite sum. Including the effects of shadowing in the Pe simply involves multiplying individual fading moments by the shadowing moments before computing the PA. Results show that failure to account for the shadowing term typically results in an error in the Pe of 100–1000% but can be more than two orders of magnitude for channels with Ricean fading depending on the shadowing standard deviation and the Rice factor. The SPI method for evaluating the Pe for DS/CDMA systems over both ideal and wireless channels proves to be efficient under both assumptions that all of the MAI sources are either chip-asynchronous or chip-synchronous with respect to the arrival of the reference signal at the receiver. Modeling chip-asynchronous, MAI sources occurs before evaluating the PA for the system which only increases the processing time by a small percentage. From the results, the systems engineer can expect the Pe to increase by a factor ranging between 2 and 10 when comparing the average Pe to the worst-case Pe under the assumption of rectangular chips. The near–far problem occurs in the reverse link where the higher received powers from the MSs which are closer to the BS cause higher probabilities of error for the MSs which are located further away in the cell. In cellular DS/CDMA, fast transmission power control (TPC) combats the near–far problem by controlling the transmit power for each of the MSs so that the received powers at the BS for each MS are approximately equal. Given that neither open-loop or closed-loop TPC can produce equal received powers for the MSs at the BS, TPC minimizes the effects of distance, shadowing, and fading. For shadowing and fading, TPC reduces the variances of these effects so that the true Pe will be somewhere between the results for ideal channels and wireless channels presented in this paper for both cases of fading only and fading with shadowing. Thus, the SPI method can be used to analyze the contribution of the TPC errors on the Pe in the case of shadowing by using results similar to those found in Fig. 11. For the particular scenario presented in the figure, we can evaluate the effects of the TPC error on the Pe for a given shadowing standard deviation resulting from misestimating the shadowing effect. Finally in some wireless systems, TPC may not be practical or even desired. For example in a battle field communications system, it may not be desired for a base station to continuously transmit for the sole purpose of power control thereby giving away its location. Also, TPC may not be an option in industrial, scientific, and medical (ISM) bands where multiple, independent DS/CDMA networks could be operating simultaneously. TPC may be used for one sub-network, but the other sub-networks will be producing uncontrolled interference. Moreover, TPC does not work in peer-to-peer networks such as two- or three-way, walkie-talkie systems in a crowded event where, again, multiple sub-networks are operating independently. Therefore, the system designer should take into account the effects of shadowing when analyzing the system Pe, and the SPI method can be used to model shadowing in a wireless network.