Discrete Rayleigh Fading Channel Modeling

Transcription

1 Discrete Rayleigh Fading Channel Modeling Julio Aráuz March 22 Department of Information Sciences and Telecommunications University of Pittsburgh 35 N. Bellefield Ave. Pittsburgh PA, 526 Abstract Diverse methods that have been proposed to model Rayleigh fading channels will be explored in this tutorial. Preliminary concepts related to wireless communications channel modeling are identified before introducing the models. The first model presented is the finite state Markov channel model which is based on the side information given by the received signal to noise ratio. The Markovian validity of the model is described along with the adequate conditions under which such validity holds. The relationships between the model and modulation schemes, error control protocols and channel coding are also described. Additionally hidden Markov models proposed to model Rayleigh fading channels are also studied along with their validity assumptions.. Introduction The field of mobile communications is changing the way people interact in their daily lives. The wireless industry has developed and deployed an infrastructure that aims at providing diverse services for the market. New wireless communications services offer people the possibility of being connected almost anywhere they go. However the design, production and deployment of such technological infrastructure come along with a high cost. This high cost may hinder manufacturers from building actual systems to test their initial designs. Therefore manufacturers look at different alternatives to avoid high costs; one of these alternatives is simulating a real wireless system. The advantage of simulations is that they could allow less expensive testing of designs, although they could require previous investments on computing resources. A simulation of a wireless system could depend on various components. For example an end to end simulation of a discrete system could include blocks to study voice encoding, channel coding, interleaving and modulation issues. A key component of a wireless system simulation is the wireless channel model. The modeling of the wireless channel in a simulation will dictate, for example, how bits or packets are lost. These factors will definitely influence the overall performance of the system.

2 Different approaches could be used to simulate diverse types of channels and their conditions. For example some models can be used to study and represent physical layer conditions like SNR or BER while others may be used to study higher layer issues like packet or segment error rates as a function of time. Wireless channel models are commonly used to study, via simulations, the performance of transport or link layer protocols. For example, Chaskar et al [9] study the performance of a transport protocol (TCP) over wireless links; Chiani et al [] perform an analytical study of TCP/IP over wireless links. Labiod [5] studies the performance of error correcting codes over wireless links. In all of these cases a model for the losses of frames at the link layer level has been implemented. Furthermore, the performance of other communication protocols, such as ATM [2] [24], over wireless links has also been studied via simulations. The validity of these performance studies is influenced by that of the underlying channel model. The common use of channel models gives rise to questions about their validity. This originates this tutorial s motivation, to study these models and the conditions under which they can be used. This document presents a tutorial on current methods being used to model flat Rayleigh fading channels. Since the models that will be described include a common basis of concepts, the tutorial starts by identifying and describing them. After the basic concepts have been introduced a common model will be presented, this is the Finite State Markov Channel (FSMC) model [36]. Based on this model, current and past literature elaborate on the analysis of the model s capacity, application areas and correctness; therefore the main results of these analyses will be included. Along with this assessment, several points will be highlighted by the author to illustrate where there could be a possibility for future work on the topic. Following the details on the general Markov modeling of Rayleigh fading wireless channels a different approach will be introduced. This approach makes use of hidden Markov models (HMM) to approximate the characteristic of the channel. Two key advantages of using HMM is that current literature presents powerful methods for fitting the models to experimental data [3] and that it is possible to obtain closed solutions for several parameters of the model. Both of these advantages could result in better approximation of the model to specific conditions. 2. Preliminary concepts It is relevant to remember that the main goal of this tutorial is to present methods to model a Rayleigh fading channel. Therefore it is important to briefly review the concepts behind fading. In a wireless communication system the signals may travel through multiple paths between a transmitter and a 2

3 receiver [26]. This effect is called multipath propagation. Due to the multiple paths, the receiver of the signal will observe variations of amplitude, phase and angle of arrival of the transmitted signal. These variations originate the phenomenon referred as multipath fading. The variations are characterized by two main manifestations [26], large-scale and small-scale fading. Furthermore these manifestations give rise to specific types of degradations of the signal. Based on [26] figure presents the fading manifestations and its associated degradations. Fading Manifestations Large Scale Fading Small Scale Fading Signal dispersion Time variance of the channel Degradations Frequency selective fading Flat fading Fast Fading Slow Fading Figure. Channel fading manifestations and degradations The first fading manifestation, large-scale fading, refers to path loss caused by the effects of the signal traveling over large areas. Large-scale fading characterizes the losses due to considerably big physical objects in the signal s path like hills or forests. The path loss is characterized [26] by a mean loss (due to the distance between the transmitter and the receiver and the propagation environment characteristics) and a variation around the mean loss. On the other hand small-scale fading characterizes the effects of small changes in the separation between a transmitter and a receiver. These changes can be caused by mobility of the transmitter, receiver or the intermediate objects in the path of the signal. Small scale changes result in considerable variations of signal amplitude and phase. Small-scale fading is also known as Rayleigh fading since the The mathematical expression for the Rayleigh probability distribution is: r r 2b p( r) e 2 = for r and (r) b p is zero for r<. 3

4 fluctuation of the signal envelope is Rayleigh distributed when there is no predominant line of sight between the transmitter and receiver. When there is a predominant line of sight between the transmitter and receiver the fluctuations are statistically described by a Rician p.d.f [23]. Figure shows two manifestations of small scale fading. The first one, signal dispersion, refers to the time spreading of the signal. Dispersion causes the underlying digital pulses transmitted in the signal to spread in time. The second manifestation reflects the time variant behavior of the channel that is due to relative mobility between a transmitter and a receiver or the objects in the path of the signal. Both of these manifestations can be characterized in the time and frequency domain by fading degradation types. As shown in figure, the degradation types of the dispersion manifestation are frequency selective fading and flat fading. From the time domain point of view, frequency selective fading occurs when the maximum spread in time of a symbol is greater than the duration of the symbol. Consequently, another name for this fading degradation is channel induced intersymbol interference. From the frequency domain point of view, frequency selective fading occurs when the spectral components of a signal are affected in different ways by the channel. In particular, frequency selective fading occurs when the channel s coherence bandwidth (the channel s bandwidth in which all components experience approximately the same fading characteristics) is smaller than the signal s bandwidth. When the conditions described above, for frequency selective fading are not met, the degradation is referred as flat fading. In this case the channel characteristics are approximately flat for all frequencies. Figure also shows the degradation types of the channel s time variance manifestation. These are fast and slow fading. From the time domain point of view, fast fading refers to the condition in which the channel s coherence time (an expected time duration during which the channel s response is invariant) is smaller than the symbol duration. Before describing fast fading in the frequency domain it is necessary to introduce the Doppler frequency concept. The Doppler frequency (f m ) characterizes the maximum Doppler frequency shift of the signals in a mobile environment. This is computed as f m = v / λ. Where v is the relative velocity between the transmitter and receiver and λ is the wavelength of the transmitted signal. With this basis one can indicate that from the frequency domain point of view, fast fading occurs when the signal bandwidth is less than the maximum frequency Doppler shift. Both types of small scale fading can be present in a wireless system. In this tutorial we will look at flat fading with the associated time variance manifestation. Large scale fading is reflected only on the strength of the received signal and will not be considered here. Having presented some relevant preliminary information we can proceed to the central topic of this document, modeling of flat Rayleigh fading channels. 4

5 3. Wireless channel models Among the common models proposed to characterize a flat Rayleigh fading channel one can find the Gilbert-Elliot channel, the FSMC [36] and hidden Markov models (HMM) [33]. The Gilbert-Elliot channel is a special case of a FSMC; on the other hand HMMs use a different approach to model the channel. The models in this section will be divided in FSMC models and HMMs. 3. Finite State Markov Channels Wang and Moayeri [36] propose the modeling of a Rayleigh fading channel by the use of a Markov process 2 with a finite number of states referred to as the Finite State Markov Channel (FSMC) model. The FSMC channel originated as an extension of a simpler model proposed earlier, the Gilbert- Elliot channel model. This tutorial will first present the FSMC and then it will proceed to describe the Gilbert-Elliot model. Following the approach of [36] this tutorial will start by establishing the general basis for Markov modeling of wireless channels and then proceed to link this basis with specific physical characteristics of the real channel. In order to avoid mathematical particularities, the actual mathematical expressions of the model will not be developed but only described. The theory behind the FSMC model is that of constant Markov processes. Constant Markov processes have the property that the state transition probabilities are independent of the time at which they occur. These processes can be defined by a finite number of possible states that are usually represented by a set S= {s o, s,, s n- } and a sequence of states {S k }, k=,,2.. Each of the elements of the set S will later be associated with a characteristic of the channel; for now let s just affirm that each state represents a condition in which we have some probability of having an error in a symbol being transmitted over the channel. For the moment, we are not linking this condition to a physical concept. There are some basic particularities that should be established to fully describe the mathematical model. Nevertheless, remember we are temporarily trying to detach the mathematical details from the channel s physical reality. The next table summarizes, as described in [36], what is necessary to mathematically describe the FSMC, the reader should notice the terminology and notation used is the same used for Markov processes. The table shows the elements of a n state FSMC model. 2 A Markov process with a discrete state space is referred to as a Markov chain [5, pg. 2] 5

6 Component Notation Description for a n state FSMC model Transition probability matrix P A n n matrix representing the probability of transition between states or into the same current state. Steady state probability vector π A n vector representing the steady state probability of being in any of the n states (remember that πp=p) Crossover probability vector e A n vector representing the different probabilities of have a symbol in error in any of the n states Table. Elements necessary to describe the Finite State Markov Channel (FSMC) It is important to understand that with the elements defined Table, it is possible (under certain assumptions that will be described later) to mathematically obtain relationships between each state, their duration and its physical meaning [38]. The elements shown in Table have some constraints. Any probability of the matrix P should be between and ; the rows of P should add up to one and the elements of e should be between and.5. Having mathematically described the FSMC, let s draw what the model proposes. p p2 p p State State State2 State n- BSCs { -p p p -p p p2 -p2 p2 -pn- pn- Figure 2. The Finite State Markov Channel model representation In figure 2 one can observe a FSMC with n states. The figure shows the in-state transition probabilities (pii), adjacent state transition probabilities (pij) and in its lower part the probability of a symbol error in each state (-p i ). At this point we can make the first connection between this model and the reality being modeled. This connection lies on the elements of the vector e and each state. The elements of the vector e, called crossover probabilities, represent the crossover probabilities of n binary symmetric channels (BSC). The n BSCs are drawn in the lower part of figure and illustrate how a symbol being transmitted, for example a zero or a one, can be received in error in any of the n states. As 6

7 shown in the figure a BSC is associated with each of the n states and its goal is to relate the varying nature of the channel, and its error processes characteristics with the FSMC. Of course there are additional restrictions over the concepts already introduced. Not any set of P, π and e that satisfy the mathematical expressions for the FSMC [36] will adequately approximate the model to a real physical channel. Therefore it is necessary to establish relationships between these abstract elements and reality. Additionally, it is important to mention that the FSMC characterization will be restricted to a discrete channel structure case, this is a digital modulation scheme will be assumed in order to compute all the necessary values for the parameters. The first relationship that must be drawn is that between the states and the actual channel. At this point it is necessary to link each of the states to a physical concept, received signal to noise ratio (SNR). The SNR is an important side parameter that is used to represent a channel s quality [37]. For example given a specific modulation scheme it is possible to compute the probability of having a symbol in error as a function of the received SNR [8]. Since the received signal envelope has a known probability distribution (i.e. Rayleigh distributed) and the received SNR is proportional to the square of the signal envelope one can compute the p.d.f of the received SNR. Reference [2] shows that the p.d.f of the SNR in a Rayleigh fading environment with Gaussian noise is exponentially distributed. Based on this fact, the FSMC model partitions the received SNR into a finite number of intervals (n) and represents each interval as a state of the Markov process. The next figure shows a graph of a received signal to noise ratio as a function of time at a receiver and how the partitioning could be performed. Received SNR Interval assigned to state n Interval assigned to state Interval assigned to state time Figure 3. Partitioning the received SNR and assigning each interval to a state of the FSMC The partitioning in figure 3 has created n SNR intervals, the first one starting at level of zero SNR and the last one including all received SNR values greater than a certain threshold. Wang and Moayeri [36] show that with the partition intervals defined (although no information on how the partitioning should be done is given) and assuming BPSK modulation, it is possible to obtain closed form expressions for the elements of the steady state probability vector π and the elements of the crossover probability vector e. These relationships are important because they later on help in verifying 7

8 how closely they match the simulation results. In summary, up to this point, with the partitioning information and the modulation scheme one can compute, as shown in detail in [36] and described later, the values for the elements π and e. For the model to be complete it is necessary to compute the values of the transition probabilities between states (elements of the matrix P). As shown in [36], assuming that the fading is slow enough helps in determining these probabilities, that way the received SNR remains constant during a channel symbol. Under slow fading conditions the level crossing rate at any particular SNR partitioning level is very small compared to the total time spent in that state. Reference [] shows that given the maximum Doppler frequency (f m ) it is possible to compute the number of times (N a ) that the received SNR passes downward across a certain level. Additionally the symbols per second transmitted in any state (and thus the total amount of time spent in a state) can be approximated with the elements of the vector π and the transmission rate. Following the procedure described above the symbols per second transmitted during the time spent in state k, noted as R, k is equal to (transmission_rate π k ), Therefore the transition probabilities can be approximated by the relation between the crossing rate and the rate symbol in each state. For example the probability of going from state k to state k- is approximately equal to N k /R k. Reference [36] verifies through simulations the appropriateness of these approximations. As mentioned elsewhere the FSMC is a generalization of the Gilbert-Elliot channel model. This model includes only two states. These two states are commonly referred as good and bad states. The good state is called noiseless since the crossover probability of the associated binary symmetric channel is zero. This means that during the good state no bits transmitted through the channel arrive with errors. The bad state is called totally noisy since the crossover probability of the associated binary symmetric channel is.5. The following figure illustrates a Gilbert-Elliot channel model and its associated BSCs. µ State State2 -p p=.5 λ -p2 p2= Figure 4 Gilbert Elliot Channel as usually implemented in simulations 8

9 Figure 4 shows a common Gilbert-Elliot channel representation in which there are no in state transitions. This model is usually used for simulations [9,, 2, 5, 8] because of the simplicity associated with its implementation. It is important to mention that, for an actual implementation in simulations the transition probabilities are not directly used, but the time spent in each state. In relation to this fact, the arrows in the figure indicate the transition rates µ and λ between states. The mean time in each state is computed based on the duration of a fade of the signal [] in relation to an acceptable level. Received signals that are below an acceptable level are modeled by the bad state. The good state represents strong enough signals that are above a certain level. Up to this point this document has presented the methods for modeling FSMC, but little has been said about its validity, accuracy, appropriate partitioning intervals or relationship with modulation or coding schemes. Current literature reviews these areas but also leaves some open questions about the appropriate conditions under which the model is valid. 3.. Validity and Accuracy of the FSMC The author believes that the most important way of validating any model should consist in comparing its results with experimental data. For example a comparison between the distributions of the time spent in each state could be performed between the model and experimental data from a sample function of the underlying random process. Previous literature mainly presents validations like this, but by comparing simulated Rayleigh fading channels with the FSMC results [36]. Later results [5] also show that experimental channel data appears to be suitable for modeling it with a Markov process. In regards to validation, ther are two common approaches. One common type of validation that has been thoroughly developed is based on an information theory analysis. The second one compares the correlations of the processes under analysis. For both cases, as will be presented here, simulations have been used in the literature to study the approaches. Information Theory Validation Analysis Before presenting the analysis for validating the FSMC, it is relevant to emphasize that the proposed model conforms to the Markov property. In general, the Markov property can be expressed as [3]: p [ S( tn ) = sn S( tn ) = sn, S( tn 2 ) = sn 2,..., S( to ) = s ] = p[ S( tn ) = sn S( tn ) = sn ] This property indicates that the probability of transition at a time n to a new state only depends on the state at time n- (also referred to as first-order assumption). For the FSMC we are therefore assuming that the history of the previous channel states, besides the previous one, does not carry significant 9

10 information about the next state. Without any further analysis it is difficult to visualize if the Rayleigh fading channel can be modeled following this assumption (later on we will describe why this assumption is valid and under what conditions). Furthermore it could appear to be more desirable to have a model that includes higher order assumptions and therefore maybe increase its accuracy [2, 3]. However, the problem with higher order models is that the complexity of its analysis and implementation increases considerably. Wang and Chang [37] proposed a mutual information metric to verify the accuracy of the first order Markovian assumption for a Rayleigh FSMC model. The goal of the metric is to confirm that given the information about the previous symbol, the uncertainty of the current one should be negligible. This uncertainty is measured in terms of average mutual information 3 of the received amplitudes. Let s proceed to describe the common information theory analysis of FSMC. Let A i (where i is the time index) be the received SNR received amplitude of the i th symbol. The information contained in A i given by the two consecutive (and previous) amplitudes A i- A i-2 is quantified by the average mutual information I(A i ;A i- A i-2 ). As proposed in [7], this can be expressed in terms of the average conditional mutual information I(A i ;A i-2 A i- ) which can be expanded as follows. I(A i ;A i- A i-2 ) = I(A i ;A i- ) + I(A i ;A i-2 A i- ). Wang and Chang s goal is to compute the value for the ratio I(A i ;A i-2 A i- ) / I(A i ;A i- A i-2 ), which as they show is a function of the joint p.d.f of A i, A i- and A i-2. Additionally they show how this p.d.f depends on the symbol transmission rate and other channel characteristics. If the ratio I(A i ;A i-2 A i- ) / I(A i ;A i- A i-2 ) is much smaller than one, then the average mutual information I(A i ;A i- A i-2 ) mainly depends on the first term, I(A i ;A i- ). This would mean that the information of A i would mainly depend on the previous symbol A i-. If this happens, the first order assumption for the FSMC would be verified. Since the joint p.d.f of A i, A i- and A i-2 depends on physical characteristics it is important to describe what these are and their ranges in order to maintain the FSMC validity. The results presented in [37] show that for the range of the product f m τ (where / τ is the symbol transmission rate) going from 2-4 to 4-3 the value of the ratio I(A i ;A i-2 A i- ) / I(A i ;A i- A i-2 ) is less than %. This value is even smaller for small values of f m τ, since as fading gets slower the information of A i is basically a function of A i- only. Therefore using higher order models will not improve the accuracy of the FSMC. On the other hand, for cases in which fast fading is observed the value of the ratio indicates that this is not negligible and the first order assumption is not longer valid. We will explore the actual limits for slow and fast fading validity of the FSMC later in this tutorial. 3 The average mutual information I(X;YZ) is a metric that indicates the amount of information about X that can be provided by YZ.

11 Stochastic Validation Analysis With the results presented in [37] the accuracy of the first order model is verified but nevertheless as pointed out by Tan and Beaulieu [3] the fact that one has small mutual information is not a sufficient condition to indicate a process is Markovian. Reference [3] indicates that I(A i ;A i-2 A i- ) can approach zero in two cases. The first case is when the samples at i, i- and i-2 are independent or when they are highly correlated. Under very slow fading conditions, such as those explored in [37], this is the case. Tan and Beaulieu [3] propose that an appropriate way of verifying the accuracy of the first order Markovian assumption is to analyze I(A i ;A i-,a i-2, A i-3,, A - ). The original validation in which Wang and Chang [37] analyze the value of I(A i ;A i-2 A i- ) only indicates that a second order Markovian model is marginally better than a first order one, but does not indicate that even higher order models are not better than the first order one. The tractability of the joint p.d.f motivates the usage of a different type of analysis. Reference [3] proposes a second type of analysis, a stochastic analysis, of the FSMC. In the stochastic analysis the autocorrelation functions of the FSMC 4 model and a ISORA 5 (isotropic scattering, omnidirectional receiving antenna) model are compared. The comparison of the autocorrelation functions of these two models provides an insight on how well the FSMC matches a generic real model. The results presented in [3 Fig. 2,3] indicate that the autocorrelation of a FSMC in general significantly differs from the ISORA model. These differences between the autocorrelation functions are more noticeable as the fading speed increases. For example, at a normalized value of f m τ of.2 the two models appear to be more consistent with one another. While at a value of f m τ of.2 the differences are quite noticeable. Additionally, from [3] it can be inferred that for slow fading conditions the two autocorrelation functions tend to match each other as the number of states (n) of the FSMC increases. Reference [3] uses a FSMC with a number of states that is in the range of 5 to states. The autocorrelation functions shown by Tan and Beaulieu also suggests that first order Markov chains, like the first order FSMC, are appropriate for very slowly fading channels but only for very slowly fading applications. Very slowly fading applications are those that require analysis over a short duration of time. An example of a very slowly fading application could be the analysis of error correction code block-error rates, which according to [3] requires analysis over a moderate number of consecutive samples. By analyzing how the autocorrelation functions diverge over an increasing separation between sampling points, [3] arrives to the conclusion of how the FSMC is valid for very 4 Reference 6 refers to the FSMC as AFSMC (Amplitude Based FSMC), in this tutorial FSMC will be used to maintain consistency. 5 The ISORA model refers to a complex Gaussian process with an envelope described by a stationary process with a Rayleigh first-order distribution.

12 slowly fading applications. Tan and Beaulieu do not elaborate on the fact that these very slowly fading applications if analyzed with the FSMC should also be analyzed under slow fading conditions in order for the model to be valid. For very slowly fading applications, the functions of both the ISORA and the FSMC are very similar for distinct values of sample separations [3 Fig. 5]. Having described the accuracy of a FSMC model under very slowly fading conditions we can proceed to portray what happens under fast fading conditions. As mentioned before under fast fading the autocorrelation function of the FSMC and the ISORA model differ but both tend to approach the conditions of an uncorrelated model over any fixed sample separation. This leads to conclude that an uncorrelated model is suitable under fast fading conditions. The details of how this model should be formulated are not given in [3] although it is pointed out that it is implementation and analysis is much simpler than that of the FSMC Partitioning, modulation and coding concerns in the FSMC model Partitioning Up to this point we have outlined the mathematical definitions for a FSMC model and related them to real physical characteristics of a Rayleigh fading channel. We then proceeded to elaborate on the validity and accuracy of a first order model and detailed under what conditions the model can be applied. In order to construct the actual model one of the parameters that must be specified is how the partitioning of the received SNR could be done. Wang and Moayeri [36] completely describe the channel model but do not get involved with the details of the partitioning of the SNR. In the literature a common approach for the partitioning is to select the thresholds in such a way that the steady state probabilities of being in any state are equal. In terms of the terminology used elsewhere this means that π = π π n- = / n (for a n state FSMC). Right away one could appreciate that the simplicity of this partitioning does not take into account the non linearity between SNR and the individual symbol probabilities of the BSCs of the model. Therefore, some kind of optimization can be performed. Current literature does not elaborate extensively on how to select a partitioning scheme. An open area to explore could be that of determining the effects on the model of placing a higher number of states in the regions where the average SNR value falls more often. This of course will depend on the specific average value of the SNR. Furthermore, the number of upward crossings through a certain level and fade duration vary as a function of the SNR level [, pg 35,37] therefore the effect of different partitioning schemes that take this into consideration can also be investigated. 2

13 An optimization of how the partitioning can be done is proposed in [35]. In addition to the simple partitioning method mentioned above, [35] suggests two other partitioning schemes for the SNR. A first approach suggests to make π i = 2 π i- and π = /(2 n -). This way the probability of being in a higher level doubles at each higher level. A third scheme that has been studied proposes to make π i = i π and π = 2/(n 2 +n). The way how these specific partitioning schemes are obtained or how they are helpful in approximating real channel conditions is not clearly specified in [35]. Reference [35] studies the impact of these different partitioning schemes by analyzing the capacity of the FSMC. The capacity they compute is the average capacity, in bits per channel use, over k all n states, this is = n π ( h( k = k e k )). Where h is the binary entropy function 6. Under average error probabilities ranging from.5 to. it was found that while keeping the number of states fixed, the capacity differences between the two later schemes are minimal (less than %). Greater differences in capacity are observed between the first and two other partitioning schemes mentioned in the previous paragraph. The literature however does not mention any other comparisons between the real channel behavior and the partitioning schemes. It is not specified in the available literature how the partitioning schemes approximate the conditions of the real physical channel. A different approach for computing the partitioning thresholds and the number of states is given in [38]. In general one could select the SNR ranges to be large enough so that a received symbol (or packet) is completely received during the associated state. On the other hand, it is also desirable that within a packet s duration, similar SNR will be experienced so that the packet will experience similar BER conditions, which are similar to the BER of the state. It is straightforward to understand that under these guidelines the average duration of each state is a critical design parameter in order to make a one step transition in the model after one packet time period. Zhang and Kassam [38] formulate a system of linear equations that compute the states duration as a multiple of the duration of a packet (in particular each state is assigned equal average duration). By solving a system of equations developed in [38] one obtains the values for the SNR partitioning thresholds for a given number of states. Reference [38] does not elaborate on what is an appropriate number of states. There are other references that have studied the behavior of a FSMC with different number of states. Babich and Lombardi [5] studied a two threshold (i.e. three states) FSMC model in a quantized Rayleigh fading environment. They show that a first-order FSMC with three states gives a good approximation of the fading process under sufficiently slow fading (f m τ <.2). Under fast fading 6 The binary entropy h(.) as a function of the average error probability is defined as: n h( e) = e log + ( e) log, where e = p k e k. e e k = 3

14 conditions (f m τ >.4), an uncorrelated model (zero-order model) proved to adequately approximate the fading process. For intermediate values of fading the authors in [6] suggest a higher order model. Nevertheless as mentioned before, current literature does not elaborate extensively on the selection process of the number of states. Up to this point we have detailed those results based on first order FSMC models. As pointed out by [6], for intermediate fading conditions a higher order model could be more suitable to describe the fading process. In reference [7], Babich et al propose a method for fitting Markov models of unspecified order to narrow-band fading channels with additive Gaussian noise. The proposed method uses a context tree pruning 7 [3] algorithm to fit experimental or simulated fading sequences to a Markov model. Modulation and coding The parameterization of the FSMC model requires the specification of the symbol error probabilities for the associated BSCs to each state. Since given a digital modulation scheme, the average error probability is a function of the received SNR it is possible to compute crossover probabilities of each associated BSC. In general fairly simply modulation schemes have been used in the study of FSMC models. For example [36] studies the behavior of the model using BPSK, while other use π/4 DQPSK [38]. No further relationships between the model and the modulation schemes are taken into consideration in the parameterization. How modulation influences a function of the fading process (statistics of block errors) is also relevant, especially in simulations. Zorzi and Rao [39, 4] study the statistics of block errors when transmitting data over fading channels. In [39] it was found that a simple two state first order Markov model that describes the success/failure of transmitted blocks gives results that agree with those from a detailed simulation (under slow fading conditions). Reference [4] studies the appropriateness of a Markov approximation for the block error process. This modeling of the error process is necessary when one studies the performance of upper layer protocols. In [4] the block error process is studied with binary Markov models both at the block (threshold model) and symbol level. At the block level, a block is considered in error when the value of the fading envelope is below a certain threshold as done in [39]. At the symbol level, a symbol is considered in error with certain a probability that depends both on the modulation scheme and the average SNR. Additionally assuming in [4] the effect of using a (N,k) block code is studied. In the model, a block is assumed to be correctly received when it contains fewer errors than those that the code is able correct. By tracking the fading envelope at the symbol level the authors tried to include the effects of a 7 The Context Tree Pruning CTP algorithm estimates the best-fitting Markov model to a discrete sequence. According to [36] is optimal in the sense that it provides the smallest parameterization. 4

15 varying envelope during the transmission of a block. The authors do not elaborate on the effect of interleaving. Interleaving will have an effect on the error process since it spreads the errors and could allow a more effective protection against burst of errors. Using two modulation schemes, BPSK and FSK the authors in [4] investigate the sensitivity of the block error process to coding and modulation schemes. They show that a Markov approximation for the block error process is a very good model for a broad range of parameters. For example, for block sizes (ranging from to 2 symbols per block), several error correcting capabilities and distinct levels of modulations the authors show [4 Fig. 5, 6] that the threshold model greatly approximates the results obtained by the symbol by symbol tracking process. These results indicate that for tracking error processes at the block level under very slow fading conditions it is sufficient to use the two state first order Markov model, this is the threshold model. The threshold model proves to be only sensitive to the normalized Doppler frequency. The following table summarizes the main approaches and proposals of different authors in regards to the FSMC models. Authors [Reference] Wang and Moayeri [36] Wang and Chang [37] Tan and Bealieu [3] Wang and Moayeri [35] Zang and Kassam [38] Zorzi and Rao [4] Type of study Relevant results, comments States, Partitioning used FSMC mathematical characterization and simulation Validation using information theory analysis Validation using information theory analysis and stochastic analysis Capacity modeling of the FSMC with different number of states and partitions Methodology for partitioning the SNR Block error process: Symbol by symbol tracking vs. Threshold model Verified the accuracy of the mathematical expressions for matrix P with simulated results Show how a st order FSMC is marginally better than a 2 nd order one. Indicate that results from [37] are not enough to prove the process is Markovian. Determine the ranges and applications of a first order FSMC Show the impact of the number of states on the capacity and the effect of different partitioning schemes on the capacity Does not elaborate on the fact that SNR and e have a non linear relationship Threshold model approximates the symbol by symbol model over a wide range of parameters Table 2. Summary of FSMC proposals 8 state model, Equal probabilities π =π π n- = / n Not applicable 5 through states, Equal probabilities 2 through 6 states,. Equal probabilities 2. π i = 2 π i- and π = /(2 n -). 3. π i = i π and π = 2/(n 2 +n). 2 through 6 states Equal probability method Quantization method 2 states 3.2 Hidden Markov Models As previously mentioned the application of first order FSMC is adequate under very slowly fading applications. Whenever there is a need to include the effect of very long channel memory the FSMC model is not longer appropriate. This is the case of a study of fade duration distributions in fading 5

16 channels [34]. In this case there is a need of Markov chains with larger memory, but since the number of states grows exponentially with the process memory [34] the approach is no longer practical. In such cases other methods such as those that use Hidden Markov Models can be used. Hidden Markov Models (HMM) [22] are probabilistic functions of Markov chains (also known as Markov sources). These models can be used to study the fading process of a Rayleigh fading channel. We will first start by defining the general characteristics and concepts related to HMMs. Then the tutorial will proceed to describe how they are used to model fading. A common discrete Markov process, like the one used in FSMC, is a stochastic process in which the outputs are observable. The outputs in this case are the set of states at each instant of time. Additionally each state corresponds to some physical and observable event. An observable Markov model could be too restrictive for some applications [22]. The observable models can be extended to include the case where the observation is a probabilistic function of the state. This results in a doubly embedded stochastic process called Hidden Markov Model. One of the stochastic process is not observable, hence the name Hidden Markov Model Characterization of HMMs A HMM is characterized with the following elements.. The number of states n in the model. It is important to note that even though the states are hidden in practical applications these are associated with some physical event. The set of the Markov chain states can therefore be represented by a set S = {s, s 2,, s n } 2. The discrete alphabet size m. The alphabet corresponds to the set of outputs of the model in any given state. The set X of the outputs can therefore be represented by as set X = {x, x 2,, x m } 3. The state transition probability distribution matrix A = {a ij }, where a ij = p[s j s i ]. 4. The observed symbol probability distribution matrix B. B is a diagonal matrix whose elements b j represent the probability p{x sj}, x є X. (if X is discrete). 5. The initial state probability vector π. This tutorial does not have as a goal to fully describe the characterizations of HMMs, but to relate these models to fading processes. As it will be explained later this relationship is similar to that of FSMC models; here it will also be necessary to partition the received SNR and assign states to the partitions. With the model already established, reference [34] shows how to compute the autocorrelation functions and other statistics of HMM. We will elaborate more on this later. In this tutorial we are interested in describing the methods that can be used to fit a HMM to a specific fading process. The first fitting method than can be used is the method of moments [34]. In this 6

17 method the parameters of the model are found by equating the moments of the two models (i.e. HMM and ISORA for example). This method has the problem that its system of equations is ill posed. This means that the moments are the same for very different models. Additionally the selection of moments, according to [34], is in general arbitrary. For example finding a HMM with an autocorrelation function that resembles that of the fading process does not guarantees that the multidimensional probabilities associated with these processes are close. The method of the moments is generally used to obtain a first approximation that will be refined later with more advanced statistical methods. A second fitting method consists in approximating multidimensional probability densities [34]. This method tries to answer the question of how to adjust the model parameters in order to maximize the probability of having a certain observable sequence. In more specific terms, if the observation sequence O is given, O= O, O 2,, O T how do we best describe it? (Based on the model s parameters). This means we are trying to maximize the probability p(o λ), where the model is λ = (A, B, π). The observation sequence used to compute the model parameters is called training sequence. There is no absolute optimal manner of estimating the model parameters to solve for the second model fitting method. However there are methods to locally maximize p(o λ) using an iterative procedure. One of these iterative procedures is the Baum-Welch method (also called EM, expectation maximization method) [22]. Reference [34] details several procedures that can be applied to optimize the computational efficiency of the method. Additionally, an advantage of HMM modeling of fading process is that it provides means to compute closed-form expressions for fade duration and level crossing number distributions [34]. These expressions could be useful in the implementation of simulations. Up to this point the actual HMM parameters have not been related to any real physical characteristic of the fading channel. References [28] and [34] illustrate how this is done. The physical reality is again connected to the HMM via the set of states S. As in the FSMC the fading amplitude needs to be quantized and an element of the set S is assigned to each quantization level. In the references there are no guidelines on how to select the threshold levels. Via simulation of the fading envelope it is possible to compute the transition probability matrix A. In a similar way the probability of the outputs of the model, this is matrix B, can computed. After this, [34] proceeds to compute the state duration distribution of a Rayleigh fading channel using the Baum-Welch algorithm. As shown in [34 Fig. ] the approximation of the state distribution closely resembles that obtained from simulation. In a similar manner, and as mentioned before, the fade duration distribution and level crossing number distributions are also computed in [34] and their validity is compared against simulated data. 7

18 The advantage of using HMMs is that they provide enough flexibility to model different types of fading [32] [34]. Additionally if fading is modeled with a HMM then bit errors and block errors occurring over fading channels can also be modeled with HMMs [3] [32]. 4. Discussion Throughout this tutorial we have analyzed two main approaches for the modeling of discrete Rayleigh fading channels. These approaches are the FSMC channel and the HMMs. Several open issues have been highlighted, among these, how current literature does not detail how to optimize the model in regards to the number of states. Another interesting issue is that of partitioning the received SNR. In this tutorial several partitioning approaches were introduced. The criteria to determine which partitioning approach is better and under what circumstances have not been completely developed. The two open topics mentioned before are relevant when a practical implementation of the model is necessary. The reader should remember that for performance studies of higher level protocols the simulation of the Rayleigh fading process is not the main issue, but the generation of a frame error process associated with the underlying fading one. In current literature real physical channels are not thoroughly compared with those obtained from simulation. Therefore the tuning criteria are not well understood. One exception is the study presented in [5] where the order of the model and the number of quantization levels are already given, but no detailed relationships on how the number of partitions affects the performance of the model are given. The concepts introduced in this tutorial have carried along an assumption that has not been extensively mentioned. This is that of flat fading. As described in the introduction, throughout the tutorial we have assumed that the fading is frequency non selective. This is not necessarily the case, especially if the bandwidth used by the signal is big enough like in the case of spread spectrum signals. For example it is not well understood how the models presented in the tutorial would describe the frame error process of the wireless LAN signals like those from the IEEE 82.b standard. Reference [4] deals with the characterization of frequency selective Rayleigh fading channels, nevertheless comparisons with real physical channels and tuning guidelines are not thoroughly developed. REFERENCES [] A. Abdi, Correspondence from the IEEE Transactions on Vehicular Technology, Vol. 48, No. 5, September 999, p [2] F. Babich, G. Lombardi, A Markov Model for the Mobile Propagation Channel, IEEE Transactions on Vehicular Technology, Vol. 49, No., January 2, pp [3] F. Babich, G. Lombardi, On Verifying a First-Order Markovian Model for the Multi-Threshold Success/Failure Process for Rayleigh Channel, PIMRC 97, Vol., 997, pp

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