THE Nakagami- fading channel model [1] is one of the

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1 24 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 1, JANUARY 2005 On the Crossing Statistics of Phase Processes and Random FM Noise in Nakagami-q Mobile Fading Channels Neji Youssef, Member, IEEE, Wissem Elbahri, Matthias Pätzold, Senior Member, IEEE, and Sadok Elasmi, Member, IEEE Abstract The crossing statistics of phase processes and random frequency modulation (FM) noise are studied for Nakagamifading channels. Closed-form expressions are first derived for the probability density function (PDF) and the cumulative distribution function (CDF) of random FM noise. The crossing rate of the phase process is then obtained for any crossing level of the phase. Moreover, the conditional PDF of random FM noise and envelope processes conditioned on the crossings of an arbitrary level of the phase are investigated. Since the Rayleigh fading channel is a special case of the Nakagami- fading channel, the derived expressions are verified by comparison with results known for Rayleigh fading channels. In addition, it is shown that the derived analytical results are in excellent agreement with those obtained by computer simulations. The presented results are useful, for example, for studying the statistics of noise spikes occurring in limiter-discriminator FM receivers and for investigating the cycle slipping phenomenon in phase-locked-loop schemes when considering the transmission over Nakagami- mobile fading channels. Index Terms Level crossing theory, Nakagami- fading channels, phase processes, random frequency modulation (FM) noise. I. INTRODUCTION THE Nakagami- fading channel model [1] is one of the often used channel models for the description of the statistics of envelope fading and phase fluctuations in narrowband mobile communication systems. The model has been proposed originally as a suitable stochastical model to describe the distribution of the signal amplitude recorded on satellite links subject to ionospheric scintillation [2]. Recently, it has been used more and more frequently in performance analysis of mobile radio communications [3], [4]. Furthermore, it is shown in [5] and [6] that this model is applicable for describing the statistics of the fading envelope of real-world mobile radio channels. Besides the statistics of the fading envelope, the statistical properties of the phase process and its time derivative, known as random frequency modulation (FM) noise, are also of interest in some applications. For example, the phase behavior characterization is useful in the design of optimal carrier recovery schemes needed in the synchronization subsystem of coherent receivers [7]. Another notable example involves the performance of FM receivers using a limiter-discriminator for detection, where random FM spikes generated by phase jumps deteriorate the error-rate performance [8]. In all cases, the level Manuscript received July 11, 2003; revised December 3, 2003; accepted January 9, The editor coordinating the review of this paper and approving it for publication is I. B. Collings. N. Youssef, W. Elbahri, and S. Elasmi are with the Ecole Supérieure des Communications de Tunis, Ariana, Tunisia ( neji.youssef@supcom.rnu.tn). M. Pätzold is with the Agder University College, Faculty of Engineering and Science, N-4876 Grimstad, Norway ( matthias.paetzold@hia.no). Digital Object Identifier /TWC crossing theory plays a central role in the determination of the statistical properties of the channel phase and random FM noise. Important quantities are, e.g., the average number of noise spikes and the cycle slipping events [9], [7], as well as the shape of noise spikes [10]. Further references to the statistical characterization of phase processes and random FM noise over Rayleigh and Rice channels are [11] [14]. As far as the authors are aware, there are no studies reported on the statistics of phase processes and random FM noise in the presence of Nakagamichannels. The intention of this paper is to fill this gap. The objective of this paper is to contribute to the above topic by analyzing the statistics of random FM noise and investigating the crossing statistics of phase processes encountered in Nakagami- channels. We derive closed-form expressions for the PDF and the cumulative distribution function (CDF) of random FM noise. Closed-form expressions are also derived for the crossing rate of the phase process, as well as for various conditional PDFs of the signal envelope and random FM noise conditioned on the crossings of the phase through a fixed phase level. The verification of the derived expressions is carried out by comparing the theoretical results with the corresponding computer simulations. The remainder of this paper is organized as follows. Section II is devoted to the derivation of the PDF and the CDF of random FM noise. The crossing statistics of the phase process is presented in Section III. Section IV deals with the derivation of the conditional PDFs of the signal envelope and random FM noise. Numerical examples are given in Section V, where the derived analytical results are illustrated and compared with the corresponding simulation results. Finally, in Section VI, we come to the conclusion of the paper. II. PDF AND CDF OF RANDOM FM NOISE The received signal in the equivalent complex baseband when transmitting an unmodulated carrier over a narrowband Nakagami- fading channel is described by the zero-mean complex Gaussian random process according to where and are uncorrelated zero-mean lowpass Gaussian processes with variances and, respectively. The autocorrelation function of will be denoted by. In the Nakagami- model, the autocorrelation functions and can have different shapes. Hence, since, it follows that can be different from. Also in (1), is the Nakagami- fading amplitude, while (1) /$ IEEE

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 1, JANUARY Fig. 1. Phasor diagram of the complex process X(t) =X (t) +jx (t) = R(t)exp(j#(t)). Angle represents an arbitrary level of the phase. where, and. It should be mentioned that the quantity can be expressed as, i.e., represents the negative curvature of the autocorrelation function at. For the classical Jakes Doppler PSD [11], the quantity may be written as, where denotes the maximum Doppler frequency of the Gaussian process. Although the assumption that the Gaussian processes and may have different maximum Doppler frequencies lacks of a clear physical basis, it allows us to increase the flexibility of the model and enables a good fit to measurement data [6]. The PDF of the random FM noise can be obtained by substituting (2) in is the corresponding random phase process. A summary of the first-order statistics of the random processes and can be found in [1]. In Fig. 1, the relationships between, and are indicated with the help of a phasor diagram, where a trajectory of the tip of the resultant phasor, denoted as point P, is also shown. The time derivative of the channel phase, denoted as, 1 is known as random FM noise [11]. For the purpose of deriving the PDF of random FM noise and the analysis of the crossing statistics, the joint PDF of the processes, and is required. This joint PDF can be obtained from the joint PDF of the Gaussian processes, and, by transforming the Cartesian coordinates to polar coordinates. This results in Performing the integration with respect to and using [15, ( )] yields It should be mentioned that in the case of Rayleigh fading, where and hold, the above equation reduces to the result given in [11, (1.4-1)]. It follows from (4) that the CDF of may be written as (3) (4), where is the Jacobian of the transformation. For a symmetrical Doppler power spectral density (PSD), where and are in pairs uncorrelated [11], it can be shown that, where is the variance of the process. The application of that transformation then results in the following expression for the desired joint PDF (5) Again, for the case where and, (5) simplifies to the result given in [11, (1.4-4)], as expected. The fact that is an even function in leads evidently to. Further, it can be verified that the second moment of the process equals (6) 1 Throughout the paper, the overdot refers to time derivative. (2) Thus, the variance of, defined as, isinfinite. Finally, we add for completeness that Rice [16] investigated the quantity in his analysis of random FM noise encountered in Rice fading channels, in order to obtain a measure of the spread of the PDF, because its standard deviation was shown to be infinite as well.

3 26 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 1, JANUARY 2005 By proceeding similarly to Rice in [16], we obtain in case of the Nakagami- fading model the following quantity: As will be shown in the next section, the above expression corresponds to the crossing rate of the phase process averaged over all possible values of the phase-crossing level. III. CROSSING RATE OF THE PHASE PROCESS The up-crossing rate of the phase process, denoted by, is the expected number of times the phase process goes through a specified phase level with a positive slope. A general expression for can be obtained from [11] and [16] as where is the joint PDF of and at the same time and at the phase level. This joint PDF can be obtained from (2) as where and. Now, after substituting (9) in (8) and performing the integration over, we finally obtain the following closed-form expression for the up-crossing rate of the phase process (7) (8) (9) (10) The derivation of the down-crossing rate of the phase process is performed similarly as above. The obtained result shows that is identical to, i.e.,, as expected. This property is a consequence of the independence of the Gaussian processes and. Note also that in case of and, we obtain from (10) the known average up- and down-crossing rate of the phase of Rayleigh processes [16], [17], i.e.,, where is the so-called radius of gyration of the Doppler PSD of and. In this case, the phase follows a uniform distribution; therefore, is independent of the phase-crossing level. Also, for the particular phase-crossing levels and, we obtain from (10) the crossing rates, where is the radius of gyration of the Doppler PSD of. Note that the quantity is a quarter of the expected number of zeros per second of the Gaussian process [16]. The equivalence of these results may be interpreted by considering the phasor diagram shown in Fig. 1. Referring to this figure, we can see that the occurrence of a -crossing is related to the condition that, while has to cross the axis. The fact that the condition holds with 50% probability leads to the confirmation of the above result. It may be worth mentioning that is the average number of positive (or negative) phase shifts in, which corresponds to the rate of occurrence of positive (or negative) noise spikes that would be observed at the output of a frequency detector. Thus, the mean time interval,, separating two consecutive noise spikes is given by. Similarly, for the phase-crossing levels, we obtain, where is the radius of gyration of the Doppler PSD of. It should be noted that is a quarter of the expected number of zeros per second of the Gaussian process. The underlying physical interpretation of the equivalence of these results can be obtained, again, from Fig. 1. If the Gaussian processes and have the same maximum Doppler frequency, then is independent of the variance and depends only on the maximum Doppler frequency of, so that the up- and down-crossing rates reduce to. In other words, if and are different only through and, then the crossing statistics of the phase process of Nakagami- fading channels is identical to the crossing statistics of the phase process of Rayleigh fading channels. The assumption that the Gaussian processes and have different maximum Doppler frequencies increases, therefore, the flexibility of the model and enables a better fitting to measurement data [6]. IV. CONDITIONAL PDFS OF AND In this section, we derive the conditional PDFs of the envelope and the random FM noise conditioned on the crossing events occurring at an arbitrary phase-crossing level. These conditional PDFs allow us to describe the statistics of and at the time instants when, as illustrated in Fig. 1. These quantities have been derived in [10] and [18] for the purpose of studying the statistics of FM click noise occurring in FM receivers [9]. In the following, we denote the joint conditional CDF of and given the phase-crossing level by, and designates the corresponding joint conditional PDF. The subscripts and in these symbols refer to the up- and down-crossings, respectively. Next, we derive the conditional PDF for the up-crossings. Our starting point is the joint conditional CDF as defined in [19] and [10] crosses crosses in in (11) where denotes probability. Replacing in the above equation by (10) and by the quantity deduced

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 1, JANUARY from (2), and then performing the integration with respect to, allows us to obtain the following: (12) where and. Now, the desired conditional PDF is obtained from (12) as Fig. 2. Average up-crossing rate N ( ) of the phase process #(t) versus the phase-crossing level. similarly as above. They are found to be given by, and. These symmetry relations are a consequence of the assumption that the Doppler PSD has a symmetrical shape. (13) where. This result can be identified with a one-sided Gaussian density with variance. Analogously, the conditional PDF can be derived by using (12) as follows: (14) As one would expect, for the particular phase-crossing levels, and, the conditional PDF reduces to the one-sided Gaussian density given by, where the subscript refers to the phase-crossing levels and, and refers to. For, we obtain the expression, which is seen to be independent of the parameter. In other words, the maximum Doppler frequency has no influence on the conditional PDF. This follows from Section II, where it was pointed out that itself depends on. Analogously, we obtain the conditional PDF, which is independent of the parameter and therefore independent of the maximum Doppler frequency. Finally, the PDFs conditioned on a down-crossing through an arbitrary phase-crossing level are obtained by proceeding V. NUMERICAL RESULTS AND DISCUSSIONS The PDF of the random FM noise, the phase-crossing rate, as well as the conditional PDFs of the envelope process and the random FM noise have been evaluated for the parameters given in [6]: s, and s. These values have been obtained by fitting the first- and second-order statistics of the envelope of the Nakagami- model to measurement data of an equivalent mobile satellite channel for a heavy shadowing environment. To confirm the correctness of the presented theoretical results, computer simulations have been performed using the deterministic simulation model described in [20]. The parameters of the channel simulator have been determined by applying the method of exact Doppler spread [20]. From the simulation, we obtained the measured average value of as s, which is in good agreement with the corresponding theoretical quantity, s, computed according to (7). It should be mentioned, however, that in the determination of the measured value, the FM spikes (corresponding to phase shifts) have not been considered. When taking the FM noise spikes into account, the simulation value of the mean of is found to be s. Therefore, (7) is valid only for the continuous part of the process. In Fig. 2, the theoretical up-crossing rate versus the phase-crossing level are shown together with the simulation results. The results for the range can be obtained from the symmetry of, since (10) shows that holds. As can be seen from Fig. 2, the simulation results are in good agreement with the theoretical results. The influence of the phase-crossing level on the conditional PDF is shown in Fig. 3, where the PDF of the Nakagami- process [1] is also plotted for the purpose of comparison. As can be seen, small values of the envelope occur with higher probability when the phase-crossing level increases from 0 to. We observe, however, the reverse on the tail of. Again, good agreement is found between

5 28 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 1, JANUARY 2005 Fig. 3. Conditional PDF p (z ) for the phase-crossing levels = 0;=4, and =2 in comparison with the Nakagami-q density p (z ) according to [1, (3.4)]. Fig. 4. Conditional PDF p ( _ ) for the phase-crossing levels =0;=4, and =2 in comparison with p ( _ ) according to (4). the derived analytical expression and the simulation results. The conditional PDF of the random FM noise is shown in Fig. 4 for three values of the phase-crossing level (, and ). This figure also shows the PDF according to (4) for only, as (4) shows that. It can be observed from this example that when is smaller than approximately 200, then the probability of occurrence of decreases, as the phase-crossing level increases from 0 to. The agreement between theory and simulation is well verified and serves to validate both the theoretical expressions derived here and the accuracy of the deterministic simulation method [20]. Finally, for illustration purposes, Fig. 5 displays waveform examples of the computer simulated processes, and, where a negative and a positive phase jump are shown causing a negative and a positive spike, respectively, in the simulated random FM noise. VI. CONCLUSION In this paper, we studied the crossing statistics of phase processes and random FM noise encountered in Nakagami- fading channels. The PDF of the random FM noise, the phase-crossing rate, as well as the conditional PDFs of the envelope process and the random FM noise have been derived. The derived ana- Fig. 5. Waveform examples of the simulated processes: (a) fading amplitude R(t), (b) phase process #(t), and (c) random FM noise #(t). _ lytical results are found to be in excellent agreement with those obtained by computer simulations. The presented theoretical results are useful for analyzing the statistics of FM spikes and cycle slipping phenomena in the case of data transmission over Nakagami- fading channels. REFERENCES [1] R. S. Hoyt, Probability functions for the modulus and angle of the normal complex variate, Bell Syst. Tech. J., vol. 26, pp , Apr

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