Simulation of Multipath Fading Effects in Mobile Radio Systems

Microwave Journal, Online Paper, October 2010
Simulation of Multipath Fading Effects in Mobile Radio Systems
Firas Mohammed Ali Al-Raie
Department of Electronic Engineering, The Polytechnic Higher Institute, Yefren, Libya
E-mail: firas@ieee.org
Abstract
RF signals transmitted via wireless mobile channels suffer from several effects like small-
scale fading and signal dispersion and distortion. This paper reviews these effects and
simulates Rayleigh and Rician multipath fading channels with a comparison between them in
terms of the effect of RF signal random fluctuations, average received signal level, outage
probability, and effect of Doppler shift. In addition to that, signal dispersion occurring to
pulses transmitted through these types of channels has also been discussed and simulated.
1. Introduction
In mobile communication systems, the RF signal propagates from the transmitter to the
receiver via multiple different paths due to the obstacles and reflectors existing in the wireless
channel. These multipaths are caused by mechanisms of reflection, diffraction, and scattering
from buildings, structures, and other obstacles existing in the propagation environment [1].
Multipath propagation is usually described by line of sight (LOS) path and non line of sight
(NLOS) paths as shown in Figure 1.
Figure 1: Multipath Propagation
1

When the mobile unit is considerably far from the base station, the LOS signal path does not
exist and reception occurs mainly from the indirect signal paths. These multiple paths have
different propagation lengths, and thus will cause amplitude and phase fluctuations and time
delay in the received signal. Therefore, the main effect of multipath propagation can be
described in terms of fading and delay spread [2].
When the waves of multipath signals are out of phase, reduction of the signal strength at the
receiver can occur. This causes significant fluctuations in the received signal amplitude with
time leading to a phenomenon known as multipath fading or small scale fading. A
representation of multipath fading is shown in Figure 2.
Figure 2: Representation of Multipath Fading
Small-scale fading is also called Rayleigh fading because if the multiple reflective paths are
large in number and there is no line-of-sight signal component, the envelope of the received
signal is statistically described by Rayleigh distribution. When there is a dominant nonfading
signal component present, such as a line-of-sight propagation path, the small scale fading
envelope is described by Rician distribution and, thus, is referred to as Rician fading [3].
When the mobile unit is moving, its velocity causes a shift in the frequency of the signal
transmitted along each signal path. This phenomenon is known as the Doppler shift. Signals
traveling along different paths can have different Doppler shifts, corresponding to different
rates of change in phase. The difference in Doppler shifts between different signal
components contributing to a single fading signal component is known as the Doppler spread.
Channels with a large Doppler spread have signal components that are each changing
independently in phase over time [4]. If the Doppler spread is significant relative to the
bandwidth of the transmitted signal, the received signal will undergo fast fading. On the other
2

hand, if the Doppler spread of the channel is much less than the bandwidth of the baseband
signal, the signal undergoes slow fading [5]. So, the terms slow and fast fading refer to the
rate at which the magnitude and phase change imposed by the channel on the signal changes.
Because multiple reflections of the transmitted signal may arrive at the receiver at different
times, this can result in inter-symbol interference (ISI) due to the crashing of bits into one
another. This time dispersion of the channel is called multipath delay spread and is an
important parameter to asses the performance capabilities of wireless communication systems
[6].
2. Modeling of Rayleigh Fading
As stated previously, Rayleigh fading results from the multiple NLOS paths of the signal
propagating from transmitter to receiver. If the transmitted signal s(t) is assumed to be
unmodulated carrier, then it may take the form:
)2cos()( tfts cπ= (1)
where fc is carrier frequency of the radio signal.
After propagation over N reflected and scattered paths, the received signal may be considered
as the sum of these N components with random amplitude and phase for each component.
Thus, when the receiving station is stationary, the received signal r(t) can be written as [5]:
∑=
+=
N
i
ici tfatr
1
)2cos()( ϕπ (2)
where ai is a random variable corresponding to the amplitude of the ith
signal component, and
φi is another uniformly distributed random variable corresponding to the phase angle of the ith
signal component.
Using the trigonometric identity:
βαβαβα sin.sincos.cos)cos( −=+ (3)
equation (2) can be re-written in the form:
∑ ∑= =
−=
N
i
N
i
iiciic atfatftr
1 1
sin).2sin(cos).2cos()( ϕπϕπ (4)
Equation (4) can be expressed as [5]:
)2sin(.)2cos(.)( tfYtfXtr cc ππ −= (5)
where:
3

∑=
=
N
i
iiaX
1
cosϕ (6)
(7)
X and Y can be considered as two identical, independent Gaussian random variables when N
tends to a large value [7].
∑=
=
N
i
iiaY
1
sinϕ
Equation (5) represents the received RF signal when the receiver is stationary. If the mobile
unit is moving with a speed of v meters per second relative to the base station, the received
signal will acquire a Doppler shift in frequency. The maximum Doppler shift is given by [3]:
cd f
c
v
f = (8)
The instantaneous Doppler shift in frequency is dependent on the angle of arrival of the
incoming signal path component as shown in Figure 3 [3].
Figure 3: A Mobile Unit Moving at Speed v
The instantaneous value of the Doppler shift fdi can be expressed as [5]:
iddi ff αcos= (9)
where αi is the angle of arrival for the ith
path signal component.
On the other hand, the instantaneous frequency of the received RF signal becomes [5]:
idci fff αcos+= (10)
Accordingly, the received signal can thereby expressed in the form:
∑=
++=
N
i
idici tffatr
1
))(2cos()( ϕπ (11)
Equation (11) can alternatively be written in other form using the trigonometric identity (3):
∑ ∑= =
+−+=
N
i
N
i
idiicidiic tfatftfatftr
1 1
)2sin().2sin()2cos().2cos()( ϕππϕππ (12)
The received signal can also be formulated as [5]:
4

)().2sin()().2cos()( tYtftXtftr cc ππ −= (13)
where:
∑=
+=
N
i
idii tfatX
1
)2cos()( ϕπ (14)
(15)
and
∑=
+=
N
i
idii tfatY
1
)2sin()( ϕπ
iddi ff αcos= .
X(t) and Y(t) are also known as the in-phase and quadrature components of the received
signal respectively. It is seen from equation (13) that the received signal is like a quadrature
modulated carrier. The envelope of the received signal is given by:
[ 2/122
)()()( tYtXtA += ] (16)
It can be shown that the probability density function (pdf) of the envelope A(t) of the received
signal is Rayleigh distributed [7].
The instantaneous power of the received signal is given by:
22
)()()( tYtXtP += (17)
On the other hand, the average value of the received power Pav is the statistical mean of P(t):
[ ] 〉〈== )()( tPtPmeanPav (18)
At the receiver side, the in-phase and quadrature components X(t) and Y(t) can be obtained by
demodulating the received signal r(t).
4. Modeling of Rician Fading
When the received signal is made up of multiple reflective paths plus a significant line of
sight (non-faded) component, the received signal is said to be Rician faded signal because the
pdf of the RF signal's envelope follows Rician distribution [3]. The received RF signal in this
case can be written as:
∑=
++++⋅=
N
i
idicidcLOS tffatffKtr
1
))(2cos())(2cos()( ϕππ (19)
where KLOS is the amplitude of the direct (LOS) component, fd is the Doppler shift in
frequency along the LOS path, and fdi is the Doppler shift in frequency along the ith
NLOS
path signal component.
The received signal can be written in terms of in-phase and quadrature components as:
5

)()2sin()()2cos())(2cos()( tYtftXtftffKtr ccdcLOS ⋅+⋅++⋅= πππ (20)
where X(t) and Y(t) are given by equations (14) and (15) respectively.
5. Outage Probability
In a fading radio channel, the transmitted signal suffers deep fades that can lead to a complete
loss of the signal or outage of the signal. The outage probability is a measure of the quality of
the transmission in a mobile radio system. Outage occurs when the received signal power
drops below a certain threshold level [5]. If f(p) is the probability density function of the
received power , then the outage probability is calculated theoretically from [5]:
∫=
thP
outage dppfP
0
)( (21)
6. Signal Dispersion
In addition to fading, multipath radio channels affect also the shape of the signals sent from
transmitter to receiver. If there are N indirect (NLOS) paths of the radio channel, and a pulse
s(t) is being transmitted through this channel, then the received pulse will be:
∑=
−⋅=
N
i
dii ttsatr
1
)()( (22)
where ai is the amplitude of the ith
path component, and tdi is the delay time associated with
the ith
path.
As shown from equation (22), the received pulse consists of N delayed versions of the
transmitted pulse with different amplitudes. This will distort the received pulse by broadening
and spreading it [5]. The dispersive characteristics of the multipath radio channel to a
transmitted signal can be illustrated by evaluating the impulse response of the channel. When
a very narrow pulse, like the delta function δ(t), is sent through the channel, then the received
signal will approach the impulse response of the radio channel according to the convolution
integral.
)()()( tsthtr c ⊗=
∫
∞
∞−
⋅−= τττ dsthc )()( (23)
where hc(t) is the impulse response of the channel.
When the transmitted pulse s(t) approaches the Dirac delta function δ(t), then the received
signal r(t) will equal the impulse response of the channel hc(t).
6

)()()()( ttsc thtr δ== (24)
7. Results of Simulation
To illustrate the fading effects of a typical mobile radio channel, some MATLAB script files
were written and used for this purpose. The Rayleigh faded signal of equation (13) has been
generated with a carrier frequency of 900MHz (typical GSM system), and a number of NLOS
paths N=10. The amplitudes of the signal components ai were generated as random numbers
ranging from 0 to 1. The phase angles of the signal paths φi were generated as uniform
distributed random numbers lying between 0 and 2π. The received signal vector was
calculated with 2000 samples using a sampling time interval of 0.2 ns. The envelope of the
received signal was generated by first extracting the in-phase component vector X, and the
quadrature component vector Y by demodulating the received signal vector r using the QAM
method. The envelope vector A has been calculated using equation (16).
In a similar manner, the Rician faded signal was generated by means of equation (20) and
using different values of the direct path signal component amplitude KLOS.
Figure 4 presents the simulated Rayleigh and Rician faded signals and envelopes with a
mobile speed v = 0 (stationary mobile unit). As shown from this simulation, the RF signals
fluctuate randomly with time with different rates of change. The fluctuations in the Rayleigh
faded envelope are higher when compared to the Rician faded envelope. In this case, it was
assumed that the direct component gain of the Rician channel KLOS=1. It is obvious that the
lower rate of change in the Rician envelope is due to the presence of the direct path
component in the received signal.
7

Figure 4: The Simulated Rayleigh and Rician Faded RF Signals and Their Envelopes
It can also be seen from Figure 4 that the average value of the Rician received signal
envelope is relatively higher than that of the Rayleigh envelope. This surely depends on the
value of the direct path gain KLOS.
In Figure 5, the Rayleigh faded signal envelope was simulated with different mobile speeds.
It can be noted how Doppler shift affects the received envelope by the increase in the rate of
fluctuations when the mobile speed is increased.
Figure 6 depicts the effect of increasing the amplitude of the direct path component on the
shape of the Rician faded received signal envelope. The higher the value of KLOS, the higher
the average value of the received signal power. Besides, the rate of random fluctuations in the
received signal envelope is reduced by the increase of KLOS.
In Figure 7, a comparison between the received power of the Rayleigh faded signal and the
Rician faded signal is presented assuming stationary mobile receiver. The received power
8

vector P was evaluated using equation (17), while the average power was calculated from
equation (18). A similar threshold power level of -5 dBW was assumed in this simulation.
Figure 5: Simulated Rayleigh Faded Envelope with Different Mobile Speeds
Figure 6: Simulation of the Rician Signal Envelope with Different Values of KLOS
9

Figure 7: Simulated Received Power for the Rayleigh and Rician Fading Channels
It can be shown also from Figure 7 that the average received power is higher in the case of
Rician fading channel, while the fade durations in the Rayleigh received signal are wider
leading to larger outage probability.
The outage probability of the received signal power can be estimated by counting the sample
values of the received signal power vector which are less than the threshold power level. The
outage probability is then obtained by dividing the counted number of samples by the total
number of samples of the simulated power vector [8]. Figure 8 shows the variation of the
estimated outage probability with mobile speed for both the simulated Rayleigh and Rician
faded signals. The outage probability is slightly increased with the increase of the mobile
speed due to Doppler shift. The outage probability is generally higher in the case of Rayleigh
fading.
10

Figure 8: Variation of Outage Probability with Mobile Speed for Two Different
Threshold Power Levels
To demonstrate the effect of the fading channel on the shape of the transmitted pulses through
it, a normalized Gaussian pulse has been sent through a multipath channel. The normalized
Gaussian pulse takes the form [9]:
22
/
)( σπ t
ets ⋅−
= (25)
where σ is a parameter representing the pulse width.
Figure 9 shows a typical Gaussian pulse with σ = 1ms sketched using MATLAB. This pulse
has been transmitted through a multipath dispersive channel with N=10, and a maximum
delay time of 2ms. The received signal was estimated using equation (22). Figure 10 presents
11

four consequent simulations for the received pulse compared with the transmitted pulse in
each case. It was assumed that the paths' amplitude coefficients of the channel take random
values from 0 to 0.25, and paths' time delays take values from 0 to 2ms.
Figure 9: The Gaussian Pulse
Figure 10: Comparison between Transmitted and Received Pulses via a Multipath
Channel for Different Simulations
As shown from these simulations, the multipath channel has the effect of spreading or
broadening the transmitted pulse, distorting it, and time shifting it. The random change in the
shape of the received signal in these consequent simulations confirms the time varying
characteristics of the mobile radio channel.
In order to compare the impulse responses of two types of mobile radio channels, a very
narrow pulse (with σ = 10 µs) was sent through two mediums. The first medium represents a
rural area, while the second one represents an urban region. In the rural area, it was assumed
12

that the maximum delay time is 0.5ms, while the maximum delay time in the urban area was
assumed to be 4ms. Figure 11 presents the received pulses in the two kinds of the channels
which represent in this case the impulse responses of these channels as stated in equation (24).
As shown from Figure 11, the received pulses reach in a shorter time in the case of the rural
area. The urban area, on the other hand, causes spreading for the received pulses.
Figure 11: Comparison between the Channel Impulse Responses for the Rural and
Urban Regions
8. Summary
This paper has reviewed the fading effects associated with mobile radio channels. MATLAB
was used as a tool for simulating the received signal patterns for both Rayleigh and Rician
fading channels. It was demonstrated that the degrading effects of the Rician channel on the
received signals are relatively smaller than those in the Rayleigh channel. The examined
parameters included average received power level, rate of received signal fluctuations, outage
probability, and Doppler shift. The dispersive characteristics of multipath fading channels
have also been simulated by showing the effects of these types of channels on the pulses
transmitted throughout them.
References
[1] J. B. Andersen, T. S. Rappaport, and S. Yoshida," Propagation Measurements and Models
for Wireless Communications Channels", IEEE Communications Magazine, January
1995, pp. 42-49.
[2] B. Sklar, "Rayleigh Fading Channels in Mobile Digital Communication Systems - Part I:
Characterization", IEEE Communications Magazine, July 1997, pp. 90-100.
13

[3] T. S. Rappaport, Wireless Communications: Theory and Practice, Prentice-Hall, 2nd
Edition, 2002.
[4] D. Tse, and P. Viswanath, Fundamentals of Wireless Communication, Cambridge
University Press, 2005.
[5] P. M. Shankar, Introduction to Wireless Systems, John Wiley & Sons, 2002.
[6] J. Lempiäinen, and M. Manninen, Radio Interface System Planning for GSM / GPRS /
UMTS, Kluwer Academic Publishers, 2002.
[7] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd Edition,
McGraw Hill, New York, 1991.
[8] G. S. Prabhu, and P. M. Shankar, "Simulation of Flat Fading Using MATLAB for
Classroom Instruction", IEEE Transactions on Education, Vo. 45, Issue 1, February 2002,
pp. 19-25.
[9] S. Haykin, Communication Systems, 3rd
Edition, John Wiley & Sons, New York, 1994.
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