1688 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 11, NOVEMBER 1999
Finite-State Markov Model for
Rayleigh Fading Channels
Qinqing Zhang, Member, IEEE, and Saleem A. Kassam, Fellow, IEEE
Abstract—In this paper we form a finite-state Markov channel
model to represent Rayleigh fading channels. We develop and
analyze a methodology to partition the received sisgnal-to-noise
ratio (SNR) into a finite number of states according to the
time duration of each state. Each state corresponds to different
channel quality indicated by bit-error rate (BER). The number of
states and SNR partitions are determined by the fading speed of
the channel. Computer simulations are performed to verify the
accuracy of the model.
Index Terms—Bit-error rate, Markov model, packet transmis-
sion, Rayleigh fading, state duration.
I. INTRODUCTION
I
N PREVIOUS work communication performance has been
evaluated for the nonstationary fading channel with per-
fect interleaving, and characterized in terms of long-term
parameters such as the average bit-error rate (BER). How-
ever, interleaving introduces complexity and delay. Perfect
interleaving is not possible in any practical system, especially
for real-time multimedia services. The focus of this paper is
the design of a useful finite-state Markov model to represent
the Rayleigh fading channel for the purpose of performance
evaluation.
There is a large literature dealing with the representation and
analysis of burst-error channels using simple Markov models.
The classical two-state Gilbert–Elliott model [1], [2] for burst
noise channels has been widely used and analyzed. In [3], a
multistate quasi-stationary Markov channel model was used
to characterize the wireless nonstationary channel. The model
was formed based on experimental measurements of some real
channels. In [4], a finite-state Markov channel (FSMC) was
built by partitioning the received instantaneous signal-to-noise
ratio (SNR) into
intervals. However, the choice of number
of states and SNR partitions was somewhat arbitrary.
In this paper we establish the relationship between a phys-
ical channel and its finite-state Markov model for a packet
transmission system. We adopt the view of [4] and develop a
methodology in forming a finite-state Markov channel model
Paper approved by B. Vucetic, the Editor for Modulation of the IEEE
Communications Society. Manuscript received September 17, 1998; revised
April 30, 1998, May 19, 1998, and September 9, 1998. This work was
supported by the NSF under Grant NCR96-28240. This paper was presented in
part at the IEEE Conference on Information Sciences and Systems (CISS’98),
Princeton, NJ, March 1998.
Q. Zhang is with Bell Laboratories, Lucent Technologies, Holmdel, NJ
07733 USA.
S. A. Kassam is with the Moore School of Electrical Engineering, Univer-
sity of Pennsylvania, Philadelphia, PA 19104 USA.
Publisher Item Identifier S 0090-6778(99)08952-7.
to reflect the Rayleigh fading channel. The received SNR
values are partitioned into a finite number of states according
to a criterion based on the average duration of each state.
This model is very useful and enables one to avoid slow
bit-level simulations and focus on overall system design. We
have applied this model to analyze the performance of, and
help design adaptive error control schemes for, wireless video
transmission systems [5].
II. F
INITE-STATE MARKOV MODEL FOR THE
WIRELESS FADING CHANNEL
Let
denote the state space of a stationary
Markov chain with
states [6]. The state space is that of
different channel states with corresponding bit-error rate
Let be the state at time
Let be the state transition probability and
be the steady-state probability.
If we assume that the transitions only happen between
adjacent states, we get
if (1)
In a typical multipath propagation environment, the re-
ceived signal envelope has the Rayleigh distribution. With
additive Gaussian noise, the received instantaneous SNR
is distributed exponentially with probability density function
(2)
Here
is the average SNR.
The fading characteristics of the signal envelope are deter-
mined by the Doppler frequency due to the motion of a mobile
terminal. Let
be the maximum Doppler frequency caused
by motion at a certain speed. Now consider the level crossing
rate of the instantaneous SNR process
It is the average
number of times per unit interval that a fading signal crosses
a given signal level
For a random distribution of direction
of motion providing a maximum Doppler frequency
,we
can show, as in [8], that the level crossing rate of level
for the SNR process, in the positive direction only (or in the
negative direction only), is
(3)
A finite-state Markov channel model can be built to repre-
sent the time-varying behavior of the Rayleigh fading channel.
Let
be received SNR thresholds in
0090–6778/99$10.00 1999 IEEE