The Aerospace Corporation, P.O. Box 92957, Los Angeles, CA 90009-2957, USA
In this paper, we lay down the basic mathematical models that can be used to determinethe statistical characteristics of the scintillation process. We focus here on the f−4-Dopplerspectrum model, but the methodology is general. We compare the real-time fading modelwith the two-state Gilbert-Elliot Markov model or hidden Markov chain, and with theMarkov process model, with the purpose of finding the range of parameters where thesimplified models approach the actual real-time process. We also explore the deviationfrom Markovian-ness of the process as function of the coherence time.
Nomenclature
Symbolsτ0 Coherence time
CX(τ) Autocovariance function (AVF) of random process X(t)F Fading marginRX(τ) Autocorrelation function (ACF) of random process X(t)SX(f) Power spectral density of random process X(t)
I. Introduction
Satellite communication systems are being designed to provide a reliable quality of service (QoS) incommunications over satellite links, even during adverse conditions. Scintillation due to nuclear explosions
in areas surrounding the satellites can cause fluctuations in the instantaneous signal-to-noise ratio of thereceived signal that in turn result in a degradation in communication performance. Particularly insidiousare fades of long duration, which can affect a large number of consecutive bits or even packets. Traditionalmitigation techniques, such as channel coding with interleaving, require large delays and buffering in orderto achieve the error performance needed for the specific application. In some cases, such as voice or video,the maximum allowable delay is often upper-bounded, thus precluding the use of large interleavers. Storagelimitations may similarly impose restrictions on the buffer sizes. Different applications, such as file transfer,may impose a different set of requirements on the QoS of the communication.
As an alternative to coding and interleaving, packet retransmission techniques can be employed whenthe instantaneous signal-to-noise ratio at the receiver falls below the decoder’s failure threshold. In thissituation, the receiver requests a retransmission of the failed packet. These techniques, commonly referredto as automatic repeat requests (ARQs), also add complexity to the system, in terms of both computationand storage, and engender transmission delays. Clearly, ARQ techniques work best when retransmissionsare uncorrelated with each other and when there is a reasonable chance that the re-sent packet encountersa more benign channel. Combined approaches in which ARQ is used in combination with error correctioncoding are also possible.
The trade-offs between transmitted power, delay, complexity, storage, etc., need to be carefully analyzedand understood in order to select the best mitigation approach for the scenarios where the communicationsystem is designed to operate. The trade space is exceedingly large and the exhaustive exploration isa daunting task. Simulations require an extremely large amount of computational power and may run∗Engineering Specialist, Comm. Systems Eng. Dept., The Aerospace Corporation
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26th International Communications Satellite Systems Conference (ICSSC)10 - 12 June 2008, San Diego, CA
uninterrupted for many hours or even days. Analytical models have the advantage of providing a syntheticview of the problem and quantifiable performance measures as function of all the parameters of interest.They can also help drastically reduce the computation time. Another benefit is the ability to cross-check thesimulation results or direct the simulations to explore the effects of certain parameters in their most criticalranges.
In this paper, we try to lay down the basic mathematical models that can be used to determine the sta-tistical characteristics of the scintillation process. Among the different approaches proposed in the literature,we focus on the so-called f−4-Doppler spectrum model, which is the most appropriate to describe the fadinginduced by nuclear scintillation.
This document is organized as follows. The continuous-time model is introduced first, including manyof its statistical characteristics. The performance of communication channels in fading is often analyzed byalso discretizing the amplitude of the received signal. The correlation among subsequent samples is used todefine an appropriate Markov chain, as explained in §III. Hidden Markov models and semi-Markov processesare also considered in the following sections, in order to increase the flexibility and improve the accuracy ofthe model. The report concludes with some comparisons among the different approaches.
II. Continuous-time Model
The scintillation or fading process, X(t), can be modeled as a filtered, or colored, Gaussian process.The process X(t) is the output of a real linear filter with impulse response h(t), when the input, U(t), is astationary zero-mean complex white Gaussian process with power spectral density SU (f) = σ2
U , and can bewritten as
X(t) = h(t) ∗ U(t),
where ∗ denotes a convolution operation.The filter h(t) that is used to model an f−4-Doppler spectrum scintillation process1 (the reason for this
denomination will become clear shortly) is a double-pole filter with system function
H(s) =H0
(s+ pF )2, such that H(f) = H(s)|s=i2πf ,
equivalent to a cascade of two identical RC filters with time constant RC = τF . The pole is pF = 1/τF andthe constant H0 is adjusted so as to have output power equal to σ2
U when the input is white. This yields
SX(f) = σ2U |H(f)|2 =
4σ2UτF
1 + 2(2πfτF )2 + (2πfτF )4, (1)
which decays as f−4 (as opposed to, for example, the exponential decay of the Gaussian Doppler spec-trum SX(f) = σ2
U
√πτ0 exp{−(πτ0f)2}). The autocorrelation function (ACF) of X(t) is equal to RX(τ) =
σ2URh(τ), where
Rh(τ) = e−|τ |/τF
(1 +|τ |τF
)= e−α|τ |/τ0
(1 + α
|τ |τ0
). (2)
Note that the value of τ = τ0 such that Rh(τ0) = e−1Rh(0) is τ0 = ατF , α ≈ 2.146193. The quantity τ0 isreferred to as coherence time.
The autocovariance function (AVF) of R2(t), CR2(τ) is thus
CR2(τ) = σ4UR2
h(τ). (3)
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0.5 1.0 1.5 2.0 2.5 3.0
Τ
Τ0
0.2
0.4
0.6
0.8
1.0
CHΤL
Figure 1. Normalized covariance functions of X(t)(blue), R(t) = |X(t)| (yellow) and R2(t) = |X(t)|2 (red).
As for the envelope, R(t) = |X(t)| = (X2R(t) +
X2I (t))1/2, it is known that when the real and imag-
inary components of the stationary complex Gaus-sian process X(t) are mutually independent andzero-mean and have identical ACF, 1
2RX(τ), thenthe envelope is a stationary Rayleigh process, withfirst-order Rayleigh distribution and ACF2,3
RR(τ) =σ2Uπ
4 2F1
(− 1
2 ,−12 ; 1; (Rh(τ))2
), (4)
where 2F1(·, ·; ·; ·) is the hypergeometric function,defined by
2F1(a, b; c; z) =∞∑n=0
a(n)b(n)
c(n)
zn
n!,
and x(n) = x(x+ 1) · · · (x+ n− 1) is the rising fac-torial of x. Because E{R(t)} = σU
√π/2, then the
AVF of R(t) isCR(τ) = σ2
U
π
4− π[2F1
(− 1
2 ,−12 ; 1; (Rh(τ))2
)− 1].
Figure 1 shows the normalized AVFs of X(t), R(t) = |X(t)| and R2(t) = |X(t)|2. Note that at τ = τ0, CR(τ)is 12.5% of its maximum, and it reaches its 1/e point at about 0.61τ0.
The PDF of R(t), the envelope or amplitude of the complex Gaussian process, is known to be Rayleigh.Given a Rayleigh fading process, the mean duration of a fade can be computed as5
Tfade =τ0∆
√πF
2
(e1/F − 1
), where ∆ =
√−τ
20
2d2
dτ2Rh(τ)
∣∣∣∣τ=0
(5)
and F is the fade margin. For an f−4-spectrum fading process, ∆ = α√2≈ 1.5176. The mean duration of a
flare and the mean separation of fades are respectively given by
Tflare =τ0∆
√πF
2, TS =
τ0∆
√πF
2e1/F .
Table 1 shows the values of Tfade, Tflare, and TS , normalized against τ0, for different values of the fadingmargin.
Table 1. Mean durations of fade and flare and mean separation of fades vs. fading margin.
The Nakagami-m distribution spans a wide range of fading distributions via the parameter m. It includesthe one-sided Gaussian distribution (m = 1/2) and the Rayleigh distribution (m = 1) as special cases. For
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m →∞, the Nakagami-m channel converges to a non-fading AWGN channel. When m > 1 the Nakagami-m distribution closely approximates a Rice distribution where the Rician K factor and the Nakagami mparameter are related by
K =√m2 −m+m− 1, m =
(1 +K)2
1 + 2K,
and can thus be used to model propagation paths consisting of one strong line-of-sight (LOS) componentand many weaker random components.
The mean durations of a fade and a flare and the mean separation between fades can be computed asfollows:12
Tfade =τ0∆
√πF 2m−1
2m−mem/FP (m,m/F ),
Tflare =τ0∆
√πF 2m−1
2m−mem/F [1− P (m,m/F )],
TS =τ0∆
√πF 2m−1
2m−mem/F ,
where ∆ is given in Eq. (5) and P (·, ·) is the regularized lower-incomplete gamma function.
III. Markov Models
A flat-fading fading channel can be modeled with a first-order amplitude-based finite-state Markov chain(AFSMC).2 Higher-order Markov chains are highly complex and difficult to analyze. Many different versionsof AFSMCs have been proposed to model wireless channels, with varying degrees of success. The chainis identified by the state probabilities and the transition probabilities. The amplitude range of interest isdivided into M intervals, and each is assigned to a state. The thresholds, τi, can be selected in manydifferent ways, e.g., so that the initial state probabilities be approximately equal (they are computed byintegrating the Rayleigh PDF over the selected intervals). Alternatively, they can be chosen according to agiven set of preselected SNR levels. The corresponding amplitude levels, ri, are typically the midpoints ofthe selected intervals. Each state is in turn associated with a binary symmetric channel (BSC). The crossoverprobabilities of each BSC determine the symbol error probability at the corresponding SNR level.
The Markov chain’s transition probabilities can be computed from the bivariate Rayleigh PDF of jointrandom variables |X(t)| and |X(t − Ts)|, where we take Ts to be the sampling period, commonly set equalto the block transmission time:2
Pij =
∫ τj
τj−1
∫ τi
τi−1fR1,R2(u, v) dudv∫ τi
τi−1fR(u) du
,
normalized so that Pij = Pij/∑Mk=1 Pik.
2. The Gilbert-Elliot Model
The simplest possible AFSMC is a two-state model, generally known as the Gilbert-Elliot model, where thetwo states are labeled as good (high SNR) or bad (deep fade). Two-state models have been widely used toanalyze the block error process and the behavior of upper-layer protocols in wireless channels.9,10 In thetwo-state model, the transition probability is given by
P =
[p 1− pq 1− q
],
where p (q) is the probability that the block at time t has been transmitted successfully, given that theprevious block was transmitted successfully (unsuccessfully). The steady-state distribution π = [πG, πB]T,obtained by solving the eigenvalue problem πT = πTP, is given by
πG =q
1− p+ q, πB =
1− p1− p+ q
,
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which yields a probability of block error equal to
Pe = 1− q
1− p+ q= πB.
Also note that the average length of an error burst is `b = q + 2(1 − q)q + 3(1 − q)2q + · · · = 1/q. Thequantities Pe and `b have physical significance and can be used to determine the transition probabilities ofthe Markov chain. It is assumed that a block error is made when the fading amplitude R is below a certainthreshold σU/
√F (where F is sometimes referred to as fading margin) and correct reception occurs when
the amplitude is above σU/√F . The fading margin can be related to the SNR, γ, by considering that in a
slow flat-fading situation the fading envelope remains constant R = r for the duration of the block and theinstantaneous SNR is given by γ = r2Ts/N0, Es = σ2
UTs being the energy of the signal transmitted over theblock (packet, symbol, or frame).
With these definitions, the quantities Pe and q can be computed as follows:
Pe = P{R < σU/√F} = FR(σU/
√F ) = 1− e−1/F ,
where FR(u) is the distribution of the envelope, and
The latter can be computed by using for FR1,R2(u, v) the joint bivariate Rayleigh or Nakagami-m cumulativedistribution.4
Figure 2 shows the behavior of the transition probabilities p and q versus the Nakagami m parameterfor a sampling period of τ0 and a fading margin of 10 dB: p decreases slightly with increasing m, but q (theprobability of moving from the bad to the good state) markedly increases.
0.5 1 1.50.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
m
p, q vs. m
Figure 2. Behavior of the transition probabilities p(blue) and q (green) in the Gilbert-Elliot model as theparameter m of the Nakagami-m distribution changesfrom 0.5 to 1.5.
The ACF of the envelope of the discrete-timescintillation process can be computed using the fol-lowing:
RR[m] = [r(i)]TPmr,
CR[m] = [r(i)]TPmr− ([π(i)]Tr)(πTr), (6)
where r(i) = [rGπ(i)G , rBπ
(i)B ]T and π
(i)G = 1 − π(i)
B isthe initial distribution. Figure 3 on the next pageshows the behavior of the normalized AVF CR[m] fordifferent values of fading margin and different sam-pling times. The mismatch between the autocorre-lations cannot be reduced to zero by increasing thenumber of states in the Markov chain, because theMarkov model implies a geometric time distribution.Improvements in the model have been suggested.11
Figure 4 on the following page displays the be-havior of Pe, `b, πG = 1−Pe and p versus the fadingmargin, F . The different curves in the same subplotshow the effect of choosing different sampling times,0.1τ0, 0.5τ0, τ0, and 1.5τ0 (which correspond to cor-relation factors of 0.98, 0.7089, 0.3679, and 0.1687,respectively). Note that Pe is solely a function ofthe fading margin.
3. Hidden Markov Models
An alternative to higher-order AFSMCs are hidden Markov models (HMMs),13,14 stochastic processes withobservable outputs, which can be statistical functions of the underlying Markov chains. HMMs are definedby a set of states, the state distribution, and the set of transition probabilities, as well as the output symbol
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Figure 3. Comparison of normalized autocovariance functions: simulated by Gilbert-Elliot model (solid) andtrue (dotted). The different curves correspond to sampling times of 0.1τ0 (blue), 0.5τ0 (green), τ0 (red), and1.5τ0 (cyan).
Figure 4. Probability of packet error, average length of error burst, probability of correct reception, andprobability of remaining in a good state vs. fading margin. The different curves correspond to sampling timesof 0.1τ0 (blue), 0.5τ0 (green), τ0 (red), and 1.5τ0 (cyan).
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probability distribution. The Baum-Welch algorithm is used to match the HMM parameters to the observedoutputs. HMMs offer the ability to model different kinds of fading. One promising feature of HMMs is thatthe PSDs they generate are rational functions of z = e−i2πf , as explained below.
An HMM is defined by {S,X,π,P,B(x)}, where S is the set of the M Markov chain states, X is the setof observable variables X[n], π is the initial state distribution, P = (Pij) is the transition matrix, with Pij =P{Sn = j|Sn−1 = i} (Sn ∈ S is the random variable denoting the state at time n), and B(x) = diag(b(x)),where b(x) = [b1(x), . . . , bM (x)]T, is the matrix of conditional probabilities bj(x) = P{X[n] = x|Sn = j}.Let P(x) = PB(x)and define (for a discrete-valued X)
E(x) =∑x∈X
xP(x), E(x2) =∑x∈X
x2P(x), E0(x) =∑x∈X
xB(x), E0(x2) =∑x∈X
x2B(x).
With these definitions,13
RX [0] = πTE0(x2)1,RX [∞] = (πTE0(x)1)2,
RX [m] = πTE0(x)Qm−1E(x)1 + RX [∞], m ≥ 1,
where Q = P− 1πT. The PSD of the HMM process is equal to
SX(f) = RX [∞]δ(f) +(πTE0(x2)1− RX [∞]
)+ 2πTE0(x)Re
{[Iei2πf −Q]−1
}E(x)1.
As an example, let us consider the two-state HMM, M = 2, where the two states will be labeled goodand bad as in the Gilbert-Elliot model. The set X represents the observables, for instance X = {0, 1}, wherethe observable 0 represents an error-free transmission and 1 represents a frame error. Upon carrying out thecomputations, the ACF turns out to be equal to
RX [m] =
x2, m = 0
(p− q)|m|(πTRvR)(vTLR1) + x2, m 6= 0,
(7)
where x is the average observed output, and x2 its second moment. Even in a two-state HMM, the additionalparameters, ε, can help fine-tune the ACF (or AVF, PSD, etc.) to the desired quantities. The set X can bemade to represent the error event or the observed envelope or power of the process, etc.
4. Markov and Semi-Markov Models
An extension of AFSMCs that shows the promise of providing better results in the medium-speed fadingprocesses is based on semi-Markov processes (SMPs).15 SMPs are an extension of both finite-state Markovchains and renewal processes, where the intertransition intervals are drawn from a set of distributions. Thedistribution of these intertransition intervals unfortunately is very hard to compute. This problem is closelyrelated to the computation of the distribution of the time between level crossings.16 This problem is knownto be extremely difficult, and for non-arithmetic distributions there is no closed-form expression even fornormal processes.
An alternative is to assume that the process is Markov, i.e., a semi-Markov process with exponentiallydistributed intertransition intervals, and to compute the parameters of the SMP accordingly, as follows.Given a fading process, the mean duration of a fade can be computed as shown in §II.
Let p = [pG, pB]T, with
pG =Tflare
Tflare + Tfade= e−1/F , pB = 1− e−1/F .
The transition probabilities in time, P(t) can be computed from the Kolmogorov differential equations, andare equal to
P(t) = P∞ + Ptre−(νG+νB)t,
where P∞ = 1pT and
Ptr =
[pB −pB
−pG pG
].
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The ACF of the resulting process can be computed with the same methodology that was used to deriveEq. (7), and is equal to
RX(τ) = pTE0(x)P(τ)E0(x)1.
The AVF is equal toCX(τ) = pTE0(x)PtrE0(x)1e−(νG+νB)τ .
Note that the AVF decays exponentially and reaches 1/e of its maximum value at
τ ′0 =1
νG + νB=τ0∆
√πF
2
(1− e−1/F
).
Figure 5. Behavior of τ ′0 as function of the noise margin.
This value, normalized to τ0, depends only onthe fade margin, as shown in figure 5. The dashedcurve in the same figure shows the behavior of the1/e point of (p − q)τ , which is the continuous-timefunction that approximates the behavior of the AVFobtained via the Markov chain or hidden Markovmodel approach, see Eq. (7), for a sampling time ofτ0 (green) and 0.5τ0 (red).
IV. Comparisons
The three models that we consider here are thetwo-state AFSMC (or the two-state HMM), thetwo-state Markov process model and a continuous-time two-state model based on the f−4-spectrumcontinuous-time model (these models are here la-beled DT, MP, and CT). In the DT model, the tran-sition matrix is
P =
[p 1− pq 1− q
],
which implies that time is discrete and proceeds in multiples of Ts. This model assumes that the n-steptransition matrix is given by
PDT[n] = Pn.
The MP model assumes that the intertransition time is exponentially distributed and the transitionmatrix is equal to
PMP(t) = P∞ + Ptre−(νG+νB)t,
with the parameters defined in §4. The CT model represents what is believed to be the actual behavior,based on the f−4 scintillation model. In this case, the transition probabilities are computed as in the DTmodel, but the transition matrix PCT(t) changes continuously with time: the change is due to the fact that pand q are functions of the correlation coefficient ρ, which itself is a function of time (see §2). Figure 6 on thenext page shows how p and q, i.e., the (1,1) and (2,1) elements of the three transition matrices, change withtime in the three models, with fading margins of 0 and 10 dB. As seen from these figures, the three modelsare in good agreement for this choice of parameters, up to the second-moment statistics. Note that thecalculation of p and q requires knowledge of the process’s bivariate distribution (see §2 and §1). Nonetheless,if the DT or MP models are used to determine the performance of a communication system that prescribesmultiple successive transmissions (as in an ARQ-based system), then we need to quantify the deviation ofthe real process from a memoryless (Markov) model. In order to do that, we can compute the conditionalprobability of encountering a fade, given that the previous transmissions were also in a fade. Let f(n) bethe probability of a fade on the nth transmission, given a fade on transmissions 1, . . . , n − 1. Obviously,f(2) = 1 − q (f(1) is the unconditional probability of fade, f(1) = 1 − e−1/F ). In a Markov system (such
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0 1 2 3 4 50
0.5
1p vs. time, F=10dB
t/τ0
0 1 2 3 4 50
0.5
1q vs. time, F=10dB
t/τ0
0 1 2 3 4 50
0.5
1p vs. time, F=0dB
t/τ0
0 1 2 3 4 50
0.5
1q vs. time, F=0dB
t/τ0
Figure 6. Temporal behavior of p and q in the three models, DT (blue), MP (green), and CT (red).
as models DT and MP above), f(n) = f(2) for all n ≥ 2. The calculation of f(n) requires the n-variateRayleigh distribution, Fn(u1, . . . , un), because at fading margin F
f(n) = P{fade|n− 1 previous fades} =Fn(σU/
√F , . . . , σU/
√F )
Fn−1(σU/√F , . . . , σU/
√F ).
In the calculation of f(4) instead of using an approximate expression17,18 where it is assumed that (R−1X )1,4 =
0, we will numerically compute the integral of the multivariate Gaussian PDF. Figure 7 shows the behaviorof f(n), n = 2, 3, 4 versus fade margin. It is clear that the actual process is very close to memoryless for theparameters used, and the assumption f(n) = f(2), n ≥ 3 is very good. Figure 8 on the next page showsthe ratios of the probabilities of going from good/bad to good (good/bad to bad) after one, two, or threetransitions (each of duration τ0) to the steady-state probability of good (bad). Again, the curves are veryclose to each other, suggesting that a model of memory one is a good approximation of the real process whenthe observations are Ts seconds apart. The missing points have been removed from the plots because of the
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lower reliability of the integral computation at low vaues of SNR. Note that for points that are separated intime by fractions of the coherence time, the assumption of memory one (which underlies the Markov models)is not accurate.
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Figure 8. Ratio of the probabilities of going from bad to bad (solid), good to good (dashed), good to bad(dotted), and bad to good (dot-dashed) after one (blue), two (green), and three (red) transmissions to thesteady-state probabilities of good/bad. Transmissions are τ0 seconds apart, and fading margins range from0 dB to 25 dB.
References
1Bogusch, R. L., “Digital Communications in Fading Channels: Modulation and Coding,” Tech. Rep. MRC-R-1043, AirForce Weapons Laboratory, March 1987.
2Tan, C. C. and Beaulieu, N. C., “On First-Order Markov Modeling for the Rayleigh Fading Channel,” IEEE Tr. Comm.,Vol. 48, No. 12, 2000, pp. 2032–2040.
3Davenport, W. B. J. and Root, W. L., An Introduction to the Theory of Random Signals and Noise, IEEE Press, NewYork, 1958.
4Simon, M. K. and Alouini, M.-S., Digital Communication over Fading Channels, J. Wiley & Sons, 2nd ed., 2005.5Dana, R. A., “Statistics of Sampled Rayleigh Fading,” Tech. Rep. DNA-TR-89-5, Defense Nuclear Agency, April 1988.6Baddour, K. E. and Beaulieu, N. C., “Autoregressive Modeling for Fading Channel Simulation,” IEEE Tr. Comm., Vol. 4,
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