NTIA Technical Memorandum TM-05-423 

Analysis of the Markov Character 

of a General Rayleigh Fading Channel 

Roger Dalke 

George Hufford 

U.S. DEPARTMENT OF COMMERCE 

Carlos M. Gutierrez, Secretary

Michael D. Gallagher, Assistant Secretary 

for Communications and Information 

April 2005

CONTENTS

Page

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Mutual Information for Complex Normal Multivariates . . . . . . . . 3

2.2 Conditional Mutual Information . . . . . . . . . . . . . . . . . . . . . 4

2.3 The Nearly Markov Condition . . . . . . . . . . . . . . . . . . . . . . 5

2.4 An Example—The Mobile Channel . . . . . . . . . . . . . . . . . . . 6

3. DISCUSSION AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . 7

4. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

iii

ANALYSIS OF THE MARKOV CHARACTER

OF A GENERAL RAYLEIGH FADING CHANNEL

Roger Dalke and George Hufford∗

It has been proposed that first-order Markov channel models can be used to adequately predict the behavior of a mobile “Rayleigh” fading channel and hence

improve the reliability of bidirectional mobile communications systems. Previous

authors have addressed this question by applying information theory to the amplitude statistics of a stationary mobile communications channel. The previous work

required numerical analysis to show that for a particular covariance function and

range of relevant parameters (i.e., Doppler frequency, symbol period), the channel is

approximately first-order Markov. In our analysis, both amplitude and phase information are used to obtain analytic expressions which can easily be used to determine

if a non-stationary arbitrary Rayleigh channel is necessarily first-order Markov. The

analytic results are given in terms of arbitrary covariance functions that can readily

be applied to measurements. In particular, our results show that the previously

studied mobile channel is not first-order Markov in character.

Key words: Gaussian process; information theory; Markov process; mobile communications;

Rayleigh fading; reliability

1. INTRODUCTION

The performance of networks using wireless channels can be significantly affected by channel

impairments such as multipath fading. Various techniques have been proposed to mitigate

the fading problem. An important example is the mobile channel which is characterized by

slow Rayleigh fading. Improved reliability in bidirectional wireless communications networks

utilizing mobile channels can be achieved by employing forward-error-correction (FEC) combined with automatic-repeat-request (ARQ) hybrid protocols [1]. Such techniques require

channel models that can be used to predict fading characteristics of the channel. In practice,

this is accomplished by assuming a finite-state first-order Markov channel model based on

the assumption that current information about the state of the channel (e.g., signal-to-noise

ratio) can be used to reliably predict the state of the channel in the future (i.e., when the

next symbol is received) [2].

Information theoretic concepts of entropy and mutual information have been used to demonstrate the validity of the first-order Markov assumption for a Rayleigh fading channel [2]. In

∗The authors are with the Institute for Telecommunication Sciences, National Telecommunications and

Information Administration, U. S. Department of Commerce, Boulder, Colorado 80305.
that work, the theoretical mobile channel described by Jakes [3] was used to calculate the

mutual information relating the channel amplitude for a particular symbol to the channel

amplitude at the time of the previous two symbols. The resulting mathematical expressions are quite complicated, requiring numerical analysis and simulations to demonstrate

the authors’ assertion that the first-order Markov model is valid. The results presented are

not general, in that this mobile channel is stationary and has a particular autocorrelation

function.

In what follows, the same information theoretic concepts are used to analyze the Markov

character of an arbitrary nonstationary Rayleigh process (i.e., the autocorrelation function

is not specified). The advantage of the approach presented in this paper is that it yields

straightforward analytic expressions that can be applied to arbitrary Rayleigh channels and

can be used in conjunction with channel measurements. As an example, the results of this

analysis are applied to the mobile channel described in [3]. It is found that contrary to the

results given in [2], this particular channel is not approximately Markov.

2
2. ANALYSIS

For the purposes of this analysis, it is assumed that the communication system determines

the Rayleigh channel’s amplitude and phase at a sequence of discrete times and uses this

information to predict the channel’s behavior one period into the future. This can be accomplished, for example, by the use of pilot tones as described in [4]. At baseband, the

channel amplitude and phase at the time of the kth period are represented as a zero mean

normally distributed complex stochastic process Xk. The mutual information relating the

kth channel value to the previous two values is given by the relative entropy between the

joint distribution and the product of the marginal distributions [5]:

I(Xk; Xk−1, Xk−2) = E

(

log2

f(Xk−2, Xk−1, Xk)

f(·, ·, Xk)f(Xk−2, Xk−1, ·)

)

(1)

where E denotes expectation, f is the joint density function and the dots in place of arguments indicate marginal distributions in the remaining variables. Following [2], the first-order

Markov approximation is considered valid if the mutual information relating Xk and Xk−1 is

approximately the same as the mutual information relating Xk and the combination of both

Xk−2 and Xk−1. More explicitly, using the chain rule for mutual information [5], we wish to

determine conditions for which

I(Xk; Xk−1, Xk−2) − I(Xk; Xk−1) = I(Xk; Xk−2|Xk−1) ≈ 0 (2)

where

I(Xk; Xk−2|Xk−1) = E

(

log2

f(Xk−2, Xk|Xk−1)

f(Xk−2, ·|Xk−1)f(·, Xk|Xk−1)

)

(3)

and the conditional densities have the usual meaning, e.g.,

f(xk−2, xk|xk−1) = f(xk−2, xk−1, xk)

f(·, xk−1, ·)

. (4)

2.1 Mutual Information for Complex Normal Multivariates

To proceed further, it is useful to obtain general expressions for the mutual information

of complex Gaussian multivariate processes. Let U and V be complex multivariate random

vectors of dimension n and m with normally distributed zero mean components. The mutual

information I(U; V ) is obtained by extending the results presented in [6] to complex normal

variates as outlined below.

Let Z = (U, V ) and W be the (m+n)×(m+n) covariance matrix with components E{ZiZ

∗

j }.

The asterisk ∗ denotes complex conjugate (and Hermitian transpose in the case of matrices).

W can be written as a partitioned matrix

W =

 

Wu S

∗

S Wv

!

(5)

3
where Wu has components E{UiU

∗

j } and Wv has components E{ViV

∗

j }. The density functions

required to calculate the mutual information are

f(u, v) = 1

π

n+m det W

e

−z

∗W−1z

(6)

f(u, ·) = 1

π

n det Wu

e

−u

∗W−1

u u

(7)

f(·, v) = 1

πm det Wv

e

−v

∗W−1

v v

(8)

hence,

I(U; V ) = E

(

log2

f(U, V )

f(U, ·)f(·, V )

)

= log2

 

det Wu det Wv

det W

!

− E{Z

∗W−1Z} + E{U∗W−1

u U} + E{V

∗W−1

v V }.

(9)

For any n-dimensional complex multivariate vector Z and any n × n matrix A,

E{Z

∗AZ} =

X

i,j

AijE{Z

∗

i Zj} =

X

i,j

AijWji

= Trace{AW}

(10)

and consequently

E{Z

∗W−1Z} − E{U∗W−1

u U} − E{V

∗W−1

v V } = 0 (11)

whence,

I(U; V ) = log2

 

det Wu det Wv

det W

!

. (12)

2.2 Conditional Mutual Information

In this section, we use the results of the previous section to obtain an expression for the

conditional mutual information given in Equation 3. Without loss of generality, we simplify

the notation by setting the index k = 3 with the caveat that in general the parameters of

the process are a function of time. Using the results from the previous section, let U = X3

and V = (X2, X1), then

I(X3; X2, X1) = log2

det Wu det Wv

det W

(13)

and setting Vˆ = X2, Zˆ = (U, Vˆ ), and Wˆ

ij = E{ZˆiZˆ∗

j } we obtain

I(X3; X2) = log2

det Wu det Wvˆ

det Wˆ

. (14)

4
The desired conditional mutual information is

I(X3; X1|X2) = log2

det Wv det Wˆ

det Wvˆ det W

. (15)

At this point it is useful to adopt the following notation: σ

2

k = E{|Xk|

2} and σiσjρij =

E{XiX∗

j } where, of course, the σi are real and ρji = ρ

∗

ij . Then

det W = σ

2

1σ

2

2σ

2

3

(1 + 2<{ρ12ρ23ρ31} − |ρ12|

2 − |ρ23|2 − |ρ13|2

)

= σ

2

1σ

2

2σ

2

3

[(1 − |ρ12|

2

)(1 − |ρ23|

2

) − |ρ13 − ρ12ρ23|

2

]

(16)

det Wˆ = σ

2

2σ

2

3

(1 − |ρ23|

2

) (17)

det Wv = σ

2

1σ

2

2

(1 − |ρ12|

2

) (18)

det Wvˆ = σ

2

2

. (19)

The conditional mutual information can be written as

I(X3; X1|X2) = log2

(1 − |ρ12|

2

)(1 − |ρ23|

2

)

(1 − |ρ12|

2

)(1 − |ρ23|

2

) − |ρ13 − ρ12ρ23|

2

. (20)

A further simplification is achieved by defining

γ

2 =

|ρ13 − ρ12ρ23|

2

(1 − |ρ12|

2

)(1 − |ρ23|

2

)

(21)

yielding

I(X3; X1|X2) = − log2(1 − γ

2

). (22)

Clearly, the quantity γ

2 must satisfy 0 ≤ γ2 < 1, and note what happens if the process is

Markov. In that case the Chapman-Kolmogorov equation holds and may be written as

f(x3|x1) = ZZ∞

−∞

f(x3|x2)f(x2|x1) dxr2dxi2 (23)

(where xr2 and xi2 are the real and imaginary parts). From this there follows, for example,

that E{X3|X1} = E{E{X3|X2}|X1}. But the conditional density functions, such as f(x3|x1),

are again Gaussian with, however, new means and variances. The mean turns out to be, for

example, (σ1/σ3)ρ31X1, and we quickly find ρ13 = ρ12ρ23 and thus that γ

2 = 0. One expects

this, for then also I(X3; X1|X2) = 0—if one knows X2 then the value of X1 contributes no

information towards estimating X3.

2.3 The Nearly Markov Condition

Following [2, 7], we would say that a process is “nearly Markov” if the ratio

M =

I(X3; X1|X2)

I(X3; X2, X1)

 1. (24)

5
Using the results from the previous section,

I(X3; X2, X1) = − log2[(1 − |ρ23|

2

)(1 − γ

2

)] (25)

whence

M =

− log2(1 − γ

2

)

− log2[(1 − |ρ23|

2

)(1 − γ

2

)]. (26)

We note that 0 ≤ M ≤ 1, and that when M is nearly 0 then γ must be much smaller than

|ρ23|. In some sense the ratio M is a normalized way to say ρ13 ≈ ρ12ρ23.

Conversely, if the process is nearly Markov, then γ

2

is nearly 0 and, provided ρ23 is not also

small, there would follow

M ≈

γ

2

− log2(1 − |ρ23|

2

)

(27)

which ought also to be small.

When the process is stationary, the correlations become one dimensional so that we could

write ρjk = ρk−j. Our results are simplified a little bit and we find ρ12 = ρ23 = ρ1 and

ρ13 = ρ2 and

γ

2 =

|ρ2 − ρ

2

1

|

2

(1 − |ρ1|

2

)

2 M =

− log2(1 − γ

2

)

− log2[(1 − |ρ1|

2

)(1 − γ

2

)]. (28)

2.4 An Example—The Mobile Channel

As our example, consider the mobile channel described by [3] and analyzed in [2]. The

channel is stationary and the correlations are given by ρk = J0(kωmτ ) where ωm = 2πv/λ

is the maximum Doppler frequency, and where v is the speed of the mobile vehicle, λ is the

wavelength of the radio transmission, τ is the sample period, and J0 is the zero-order Bessel

function. The idea suggested by [2] is to sample at every symbol. This would mean that τ

is small. Indeed, the authors of [2] would say that .0002 < ωmτ < .004; and if this is true

then the Bessel functions can be reasonably approximated as

J0(z) ≈ 1 −

z

2

4

+

z

4

64

.

Then from Equation 28 we find

γ

2 ≈

1

4

(ωmτ )

4

(1 −

5

8

(ωmτ )

2

)

1

4

(ωmτ )

4

(1 −

3

8

(ωmτ )

2

)

≈ 1 −

1

4

(ωmτ )

2

M ≈

− log2[(ωmτ )

2/4]

− log2[(ωmτ )

4/8] ≈

1

2

− log2 ωmτ + 1

− log2 ωmτ + 3/4

≈

1

2

(29)

This result indicates that the channel is not approximately Markov. In fact, only about half

the mutual information I(X3; X1, X2) is obtained from the previous sample X2 alone. This

conclusion is not consistent with the result given in [2].

6
3. DISCUSSION AND CONCLUSIONS

The importance of predicting the future state of a Rayleigh fading channel based on present

channel conditions has been described in the literature (e.g., [2, 3, 7, 8]). In essence, such

knowledge can be used in conjunction with error-control coding and feedback systems to

improve the reliability of mobile communications systems. Owing to their simplicity, it is

desirable to obtain and use finite-state first-order Markov models to predict the state of

the channel one symbol period into the future. To this end, technical papers have been

published [2, 7] that “verify” the validity of the first-order Markov assumption for a mobile

communications channel. From our point of view, this analysis seemed to be quite limited

in that the approach used required numerical analysis techniques to calculate the mutual

information. In addition, only channel amplitudes were used to calculate relative entropy.

As a consequence, the results are limited to the particular channel that was analyzed and

cannot be applied to more general nonstationary Rayleigh channels with arbitrary covariance

functions.

In studying this problem, we found that by considering both the amplitude and the phase

of a general Rayleigh fading channel it was in fact possible to obtain analytic expressions

describing the efficacy of the first-order Markov assumption. The results of this analysis

are given in Section 2.2. The advantage of this approach is that the results can readily be

applied to nonstationary channels with arbitrary covariance functions. The only assumption

used in our analysis is that the fading is assumed to be approximately frequency flat with

Rayleigh amplitude distributions such as may be found in a typical mobile communications

environment. The primary value of our analysis is that it can easily be used to characterize

a particular mobile communications environment.

As an example, the results of our analysis were applied to the mobile channel used in [2]. It

is worthwhile to note that our results contradict the previous conclusion [2] that the mobile

channel is approximately first-order Markov in character. Specifically, we found that only

about half of the available information was obtained using just one channel state. This result

may be understood heuristically by considering a slowly varying Rayleigh channel with a

particular trend (i.e., the channel is fading, coming out of a fade, or relatively flat) at the

time of interest. If one considers the channel state at a particular symbol reception time and

uses it to predict the channel state one symbol period into the future without any additional

information, the best guess is to perhaps assume that the future state will be about the

same as the current state. Knowing only one state, it is not possible to predict which trend

is most likely. It seems obvious that given an additional past state one can better predict

the trend and hence obtain a better estimate of the future state. This is consistent with

what was found in the case of the mobile channel where, considering the total information

available about a future state based on the current and a previous state, only about half the

information is available from knowing the current state alone.

7
4. REFERENCES

[1] B. S. Wicker, “Reed-Soloman error control coding for Rayleigh fading channels with

feedback,” IEEE Trans. Vehicular Tech., vol. VT-41, no. 2, pp. 124–133, May 1992.

[2] H. S. Wang and P. Chang, “On verifying the first-order Markovian assumption for a

Rayleigh fading channel model,” IEEE Trans. Vehicular Tech., vol VT-45, no.2, pp.

353–357, May 1996.

[3] W. C. Jakes, ed., Microwave Mobile Communications, ch.1, New York: Wiley, 1974.

[4] F. Davarian, “Mobile digital communications via tone calibration,” IEEE Trans. Vehicular Tech., vol VT-36, pp. 55–62, May 1987.

[5] T. M. Cover and J. A. Thomas, Elements of Information Theory, ch. 2, New York: Wiley,

1991.

[6] S. Kullback, Information Theory and Statistics, ch. 9, New York: Wiley, 1959.

[7] H. S. Wang and N. Moayeri, “Finite-state Markov channel—A useful model for radio

communication channels,” IEEE Trans. Vehicular Tech., vol. VT-44, pp. 163–171, Feb.

1995.

[8] M. Zorzi, R. R. Rao, and L. B. Milstein, “Error statistics in data transmission over

fading channels,” IEEE Trans. Communications, vol. COM-46, no. 11, pp. 1468–1477,

Nov. 1998.

8