CHAPTER 4

Mobile Radio Propagation:

Small-Scale Fading and
Multipath

mall-scale fading, or simply fading, is used

to describe the rapid fluctuation of the amplitude of a radio signal over a short
period of time or travel distance, so that large-scale path loss effects may be
ignored. Fading is caused by interference between two or more versions of the
transmitted signal which arrive at the receiver at slightly different times. These
waves, called rnultipath waves, combine at the receiver antenna to give a result-
ant signal which can vary widely in amplitude and phase, depending on the dis-
tribution of the intensity and relative propagation time of the waves and the
bandwidth of the transmitted signal.

4.1 Small-Scale Multipath Propagation

Multipath in the radio channel creates small-scale fading effects. The three
most important effects are:
Rapid changes in signal strength over a small travel distance or time inter-
val
Random frequency modulation due to varying Doppler shifts on different
multipath signals
Time dispersion (echoes) caused by multipath propagation delays.

In built-up urban areas, fading occurs because the height of the mobile
antennas are well below the height of surrounding structures, so there is no sin-
gle line-of-sight path to the base station. Even when a line-of-sight exists, multi-
path still occurs due to reflections from the ground and surrounding structures.
The incoming radio waves arrive from different directions with different propa-
gation delays. The signal received by the mobile at any point in space may con-
sist of a large number of plane waves having randomly distributed amplitudes,

139

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 140 Ct,. 4 Mobile Rado Propagation: Smafi-Scale Fading and Multpath

phases, and angles of arrival. These multipath components combine vectorially

at the receiver antenna, and can cause the signal received by the mobile to dis-
tort or fade. Even when a mobile receiver is stationary, the received signal may
fade due to movement of surrounding objects in the radio channel.
If objects in the radio channel are static, and motion is considered to be only
due to that of the mobile, then fading is purely a spatial phenomenon. The spa-
tial variations of the resulting signal are seen as temporal variations by the
receiver as it moves through the multipath field. Due to the constructive and
destructive effects of multipath waves summing at various points in space, a
receiver moving at high speed can pass through several fades in a small period of
time. In a more serious case, a receiver may stop at a particular location at which
the received signal is in a deep fade. Maintaining good communications can then
becom? very thificult, although passing vehicles or people walking in the vicinity
of the mobile can often disturb the field pattern, thereby diminishing the likeli-
hood of the received signal remaining in a deep null for a long period of time.
Antenna space diversity can prevent deep fading nulls, as shown in Chapter 6.
Figure 3.1 shows typical rapid variations' in the received signal level due to
small-scale fading as a receiver is moved over a distance of a few meters.
Due to the relative motion between the mobile and the base station, each
multipath wave experiences an apparent shift in frequency. The shift in received
signal frequency due to motion is called the Doppler shift, and is directly propor-
tional to the velocity and direction of motion of the mobile with respect to the
threction of arrival of the received multipath wave.

4.1.1 Factors Influencing Small-Scale Fading

Many phy'sical factors in the radio propagation channel influence small-
scale fading. These include the following:
Multipath propagation The presence of reflecting objects and scatterers
in the channel creates a constantly changing environment that dissipates
the signal energy in amplitude, phase, and time. These effects result in mul-
tiple versions of the transmitted signal that arrive at the receiving antenna,
displaced with respect to one another in time and spatial orientation. The
random phase and amplitudes of the different multipath components cause
fluctuations in signal strength, thereby inducing small-scale fading, signal
distortion, or both. Multipath propagation often lengthens the time required
for the baseband portion of the signal to reach the receiver which can cause
signal smearing due to intersymbol interference.
Speed of the mobile The relative mtion between the base station and
the mobile results in random frequency modulation due to different Doppler
shifts on each of the multipath components. Doppler shift will be positive or
negative depending on whether the mobile receiver is moving toward or
away from the base station.

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 Small-Scale Multipath Propagation 141

Speed of surrounding objects If objects in the radio channel are in

motion, they induce a time varying Doppler shift on inultipath components.
If the surrounding objects move at a greater rate than the mobile, then this
effect dominates the small-scale fading. Otherwise, motion of surrounthng
objects may be ignored, and only the speed of the mobile need b? considered.
The transmission bandwidth of the signal If the tran iniitted radio
signal bandwidth is greater than the "bandwidth" of the multipath channel,
the received signal will be distorted, but the received signal strength will not
fade much over a local area (i.e., the small-scale signal fading will not be sig-
nificant). As will be shown, the bandwidth of the channel can be quantified
by the coherence bandwidth which is related to the specific multipath struc-
ture of the channel. The coherence bandwidth is a measure of the maximum
frequency difference for which signals are still strongly correlated in ampli-
tude. If the transmitted signal has a narrow bandwidth as compared to the
channel, the amplitude of the signal will change rapidly, but the signal will
not be distorted in time. Thus, the statistics of small-scale signal strength
and the likelihood of signal smearing appearing over small-scale distances
are very much related to the specific amplitudes and delays of the multipath
channel, as well as the bandwidth of the transmitted signal.

4.1.2 Doppler Shift

Consider a mobile moving at a constant velocity v, along a path segment
ha'ving length d between points X and Y, while it receives signals from a remote
source S, as illustrated in Figure 4.1. The difference in path lengths traveled by
the wave from source S to the mobile at points X andY is = dcosO = vAtcosO.
where at is the time required for the mobile to travel from X to Y, and $ is
assumed to be the same at points X and Y since the source is assumed to be very
far away. The phase change in the received signal due to the difference in path
lengths is therefore

a = 21L\l
= (4.1)

and hence the apparent change in frequency, or Doppler shift, is given by fd'
where

(42)
Equation (4.2) relates the Doppler shift to the mobile velocity and the spa-
tial angle between the direction of motion of the mobile and the direction of
arrival of the wave. It can be seen from equation (4.2) that if the mobile is mov-
ing toward the direction of arrival of the wave, the Doppler shift is positive (i.e.,
the apparent received frequency is increased), and if the mobile is moving away
from the direction of arrival of the wave, the Doppler shift is negative (i.e. the

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 142 Ch. 4 Mobile Ratio Propagation: Small-Scale Fading and Muftipath

apparent received frequency is decreased). As shown in section 4.7.1, multipath

components from a CW signal which arrive from different directions contribute
to Doppler spreading of the received signal, thus increasing the signal band-
width.
S
.4

/ / I
/
/
/ I
/

Figure 4.1
Illustration of Doppler effect.

Example 4.1
Consider a transmitter which radiates a sinusoidal carrier frequency of 1850
MHz. For a vehicle moving 60 mph, compute the received carrier frequency if
the mobile is moving (a) directly towards the transmitter, (b) directly away
from the transmitter, (c) in a direction which is perpendicular to the direction
of arrival of the transmitted signal.

Solution to Example 4.1

Given:
Carrier frequency = 1850MHz

x
Therefore, wavelength k = c/fe = = 0.i62m
1850x 10
Vehicle speed v = 60 mph = 26.82 rn/s

(a) The vehicle is moving directly towards the transmitter.

The Doppler shift in this case is positive and the received frequency is given
by equation (4.2)

tIJfd = = 1850.00016MHz

(b) The vehicle is moving directly away from the transmitter.

The Doppler shift in this case is negative and hence the received frequency
is given by

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 Impulse Response Model of a Multipath Channel 143

= = 1849.999834MHz

(c) The vehicle is moving perpendicular to the angle of arrival of the transmit-
ted signal.
In this case, S = 90, cosS = 0, and there is no Doppler shift.
The received signal frequency is the same as the transmitted frequency of
1850 MHz.

4.2 Impulse Response Model of a Multipath Channel

The small-scale variations of a mobile radio signal can be directly related to
the impulse response of the mobile radio channel. The impulse response is a
wideband channel characterization and contains all information necessary to
simulate or analyze any type of radio transmission through the channel. This
stems from the fact that a mobile radio channel may be modeled as a linear filter
with a time varying impulse response, where the time variation is due to
receiver motion in space. The filtering nature of the channel is caused by the
summation of amplitudes and delays of the multiple arriving waves at any
instant of time. The impulse response is a useful characterization of the channel,
since it may be used to predict and compare the performance of many different
mobile communication systems and transmission bandwidths for a particular
mobile channel condition.
lb show that a mobile radio channel may be modeled as a linear filter with
a time varying impulse response, consider the case where time variation is due
strictly to receiver motion in space. This is shown in Figure 4.2.

if
spatial position
Figure 4.2
The mobile radio channel as a function of time and space.

In Figure 4.2, the receiver moves along the ground at some constant veloc-
ity v. For a fixed position d, the channel between the transmitter and the receiver
can be modeled as a linear time invariant system. However, due to the different
multipath waves which have propagation delays which vary over different spa-
tial locations of the receiver, the impulse response of the linear time invariant
channel should be a function of the position of the receiver. That is, the channel
impulse response can be expressed as h(d,t). Let x(t) represent the transmitted
signal, then the received signal y(d,t) at position d can be expressed as a convo-
lution of x (t) with h(d,t).

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 144 Cli. 4 Mobile Rado Propagation: Small-Scale Fading and Multipath

y(d, t) = x(t) h(d, t) = t - (4.3)

For a causal system, h(d, t) = 0 for t.cO, thus equation (4.3) reduces to

y(d, t) x(t)h(d, t t)d'r (4.4)

= J
Since the receiver moves along the ground at a constant velocity v, the posi-
tion of the receiver can by expressed as
d = ut (4.5)
Substituting (4.5) in (4.4), we obtain

y(vt,t) = Jx(t)h(vt,t_t)dt (4.6)

Since v is a constant, y(ut, t) is just a function oft. Therefore, equation (4.6)

can be expressed as

y(t) = t t)d'r = x(t) 0 h(ut, t) = x(t) 0 h(d, t) (4.7)

From equation (4.7) it is clear that the mobile radio channel can be modeled as a
linear time varying channel, where the channel changes with time and distance.
Since u may be assumed constant over a short time (or distance) interval,
we may let x(t) represent the transmitted bandpass waveform, y(t) the
received waveform, and h(t, v) the impulse response of the time varying multi-
path radio channel. The impulse response h (t, t) completely characterizes the
channel and is a function of both t and t. The variable t represents the time
variations due to motion, whereas t represents the channel multipath delay for
a fixed value of t - One may think of t as being a vernier adjustment of time. The
received signal y(t) can be expressed as a convolution of the transmitted signal
x(t) with the channel impulse response (see Figure 4.3a).

y(t) x(t)h(t, t)dx = x(t) 0 h(t, t) (4.8)

=
If the multipath channel is assumed to be a bandlixnited bandpass channel,
which is reasonable, then h(t,t) may be equivalently described by a complex
baseband impulse response hb(t, t) with the input and output being the com-
plex envelope representations of the transmitted and received signals, respec-
tively (see Figure 43b). That is,

r(t) = c(t) 0 t) (4.9)

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 Impulse Response Model of a Multipath Channel 145

hut) = Re{hh(t. t)e ,y(t)

}

y(t) =
(a)
y(t) = x(t)h(t)

ca) t) , r(t)
=

(b)

Figure 4.3
(a)Bandpass channel impulse response model.
(b) Baseband equivalent channel impulse response model.

where c(t) and .r(t) are the complex envelopes of x(t) and y(t), defined as
x(t) = (4.10)

y(t) = (4.11)
The factor of 1/2 in equation (4.9) is due to the properties of the complex
envelope, in order to represent the passband radio system at baseband. The low-
pass characterization removes the high frequency variations caused by the car-
rier, making the signal analytically easier to handle. It is shown by Couch
[Cou93] that the average power of a bandpass signal x2(t) is equal to
where the overbar denotes ensemble average for a stochastic signal, or time
average for a deterministic or ergodic stochastic signal.
It is useful to discretize the multipath delay axis 'r of the impulse response
into equal time delay segments called excess delay bins, where each bin has a
time delay width equat tot) + I where t0 is equal to 0, and represents the
first arriving signal at the receiver. Letting i = 0, it is seen that r to is equal to

the time delay bin width given by At. For convention, t0 = o, t1 = At, and
= iAt, for i = 0 to N , where N represents the total number of possible
I

equally-spaced multipath components, including the first arriving component.

Any number of multipath signals received within the i th bin are represented by
a single resolvable multipath component having delay ;. This technique of
quantizing the delay bins determines the time delay resolution of the channel
model, and the useful frequency span of the model can be shown to be l/(2At)
That is, the model may be used to analyze transmitted signals having band-
widths which are less than I /(2At) Note that t0 = 0 is the excess time delay
.

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 146 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and Multipath

of the first arriving multipath component, and neglects the propagation delay
between the transmitter and receiver Excess delay is the relative delay of the
i th multipath component as compared to the first arriving component and is
given by t1. The maximum excess delay of the channel is given by NAt.
Since the received signal in a multipath channel consists of a series of
attenuated, time-delayed, phase shifted replicas of the transmitted signal, the
baseband impulse response of a multipath channel can be expressed as
N-I
h6(t, 'c) = a1(t, + $1(t, r))j8(t (4.12)

where a1(t, t) and are the real amplitudes and excess delays, respectively,
of i th multipath component at time t [Tur721. The phase term
+ t)in (4.12) represents the phase shift due to free space propaga-
tion of the i th multipath component, plus any additional phase shifts which are
encountered in the channel. In general, the phase term is simply represented by
a single variable t) which lumps together all the mechanisms for phase
shifts of a single multipath component within the ith excess delay bin. Note that
some excess delay bins may have no multipath at some time t and delay t1,
since t) may be zero. In equation (4.12), N is the total possible number of
multipath components (bins), and S(.) is the unit impulse function which deter-
mines the specific multipath bins that have components at time t and excess
delays t1. Figure 4.4 illustrates an example of different snapshots of
h5(t, t), where t varies into the page, and the time delay bins are quantized to
widths of At.
4

1(13)

1(h)

to
to t1 t2 t3 t4 TNI

Figure 4.4
An example of the time varying discrete-time impulse response model for a multipath radio channel.

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 Impulse Response Model ol a Multipath 147

If the channel impulse response is assumed to be time invariant, or is at

least wide sense stationary over a small-scale time or distance interval, then the
channel impulse response may be simplified as

hb(r) = (4.13)

When measuring or predicting hb(r,), a probing pulse p (t) which approxi-

mates a delta function is used at the transmitter. That is
p(t) (4.14)
is used to sound the channel to determine (t).
For small-scale channel modeling, the power delay profile of the channel is
found by taking the spatial average of jh,, (t;t) 2 over a local area. By makthg
several local area measurements of h5 (t;r) 2 in different locations, it is possible
to build an ensemble of power delay profiles, each one representing a possible
small-scale multipath channel state [Rap9la}.
Based on work by Cox [Cox72], [Cox75], if p (t) has a time duration much
smaller than the impulse response of the multipath channel, p (t) does not need
to be deconvolved from the received signal r (t) in order to determine relative
multipath signal strengths. The received power delay profile in a local area is
given by

F (t;t) (t;t) ;2
kjhb (4.15)
and many snapshots of hb (t;t) are typically averaged over a local (small-
2

scale) area to provide a single time-invariant multipath power delay profile

P (t) The gain k in equation (4.15) relates the transmitted power in the prob-
.

ing pulse p (t) to the total power received in a multipath delay profile.

4.2.1 Relationship Between Bandwidth and Received Power

In actual wireless communication systems, the impulse response of a multi-
path channel is measured in the field using channel sounding techniques. We
now consider two extreme channel sounding cases as a means of demonstrating
how the small-scale fading behaves quite differently for two signals with differ-
ent bandwidths in the identical multipath channel.
Consider a pulsed, transmitted RF signal of the krm
x (t) = Re {p (t) exp
where p (t) is a repetitive baseband pulse train with very narrow pulse width
Tb,, and repetition period TREP which is much greater than the maximum mea-
sured excess delay Tmax in the channel. Now let
p(t) = for

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 148 Oh. 4 Mobile Radio Propagation: Small-Scale Fading and Mulbpath

and let p (t) be zero elsewhere for all excess delays of interest. The low pass
channel output r (t) closely approximates the impulse response hb (t) and is
given by

r(t) (exp .
p (t
=
(4.16)

=
-

Th determine the received power at some time to, the powerjr (t0) 2 is mea-
sured. The quantity V (t0) 12 is called the instantaneous rnultipath power delay
profile of the channel, and is equal to the energy received over the time duration
of the multipath delay divided by tmax That is, using equation (4.16)

tmax

jrCt0)j2 = r(t) xr*(t)dt (4.17)

tm ax J
0
tmax NtNI 1
= _!_
max ,f
o j=Ot=0
Note that if all the multipath components are resolved by the probe p(t), then
>T66 and
Tmax (N I

j2
jr (t0) = (t0)p2 (t dt (4.18)
max j'
o k=O
N I
2
2 T
j
1

= {J. dt
max k=0 66
o

a wideband probing signal p(t), T66 is smaller than the delays between
multipath components in the channel, and equation (4.18) shows that the total
received power is simply related to the sum of the powers in the individual mul-
tipath components, mad is scaled by the ratio of the probing pulse's width and
amplitude, and the maximum observed excess delay of the channel. Assuming
that the received power from the multipath components forms a random process
where each component has a random amplitude and phase at any time t, the
average small-scale received power for the wideband probe is found from equa-
tion (4.17) as

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 Impulse Response Model of a Multipath Channel 149

= EQ
t: (4.19)

In equation (4.19), E09 [.1 denotes the ensemble average over all possible val-
ues of a1 and in a local area, and the overbar denotes sample average over a
local measurement area which is generally measured using multipath measure-
ment equipment. The striking result of equations (4.18) and (4.19) is that if a
transmitted signal is able to resolve the multipaths, then the small-scale
received power is simply the sum of the powers received in each multipath compo-
nent. In practice, the amplitudes of individual multipath components do not fluc-
tuate widely in a local area. Thus, the received power of a wideband signal such
as p (t) does not fluctuate significantly when a receiver is moved about a local
area [Rap89].
Now, instead of a pulse, consider a CW signal which is transmitted into the
exact same channel, and let the complex envelope be given by c (t) = 2. Then,
the instantaneous complex envelope of the received signal is given by the phasor
sum

r(t) = (4.20)

and the instantaneous power is given by

2
jr(t) U01 (t, t) (4.21)
= :aiexP
As the receiver is moved over a local area, the channel changes, and the
received signal strength will vary at a rate governed by the fluctuations of a
and As mentioned earlier, a1 varies little over local areas, but e. will vary
-

greatly due to changes in propagation distance over space, resulting in large fluc-
tuations of r (t) as the receiver is moved over small distances (on the order of a
wavelength). That is, since r (t) is the phasor sum of the individual multipath
components, the instantaneous phases of the multipath components cause the
large fluctuations which typifies small-scale fading for CW signals. The average
received power over a local area is then given by

Ea& = Ea a1exp (4.22)

a +a + + aN 1eJON
-
(4 23)
( JO0
-J9N
x +a1g + +aNIe

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 150 Ch. 4. MoSe Radio Propagation: Small-ScaLe Fading and Multipath

NI N
E08[Pcw] (4.24)
i=O i=O &j*i
where is the path amplitude correlation coefficient defined to be
= (4.25)
and the overbar denotes time average for CW measurements made by a mobile
receiver over the local measurement area [RapS9]. Note that when
cos and/or
= 0 = 0, then the average power for a CW signal is
equivalent to the average received power for a wideband signal in a small-scale
region. This is seen by comparing equation (4.19) and equation (4.24). This can
occur when either the multipath phases are identically and independently clis-
tributed (i.i.d uniform) over [0, 2nJ or when the path amplitudes are uncorre-
lated. The !.i.d uniform distribution of 9 is a valid assumption since niultipath
components traverse differential path lengths that measure hundreds of wave-
lengths and are likely to arrive with random phases. If for some reason it is
believed that the phases are not independent, the average wideband power and
average CW power will still be equal if the paths have uncorrelated amplitudes.
However, if the phases of the paths are dependent upon each other, then the
amplitudes are likely to be correlated, since the same mechanism which affects
the path phases is likely to also affect the amplitudes. This situation is highly
unlikely at transmission frequencies used in wireless mobile systems.
Thus it is seen that the received local ensemble average power of wideband
and narrowband signals are equivalent. When the transmitted signal has a
bandwidth much greater than the bandwidth of the channel, then the multipath
structure is completely resolved by the received signal at any time, and the
received power varies very Little since the individual multipath amplitudes do
not change rapidly over a local area. However, if the transmitted signal has a
very narrow bandwidth (e.g., the baseband signal has a duration greater than
the excess delay of the channel), then multipath is not resolved by the received
signal, and large signal fluctuations (fading) occur at the receiver due to the
phase shifts of the many unresolved multipath components.
Figure 4.5 illustrates actual indoor radio channel measurements made
simultaneously with a wideband probing pulse having Tbb = IOns, and a CW
transmitter. The carrier frequency was 4 GHz. It can be seen that the CW signal
undergoes rapid fades, whereas the wideband measurements change little over
the 5?. measurement track. However, the local average received powers of both
signals were measured to be virtually identical EHaw9l].
Example 4.2
Assume a discrete channel impulse response is used to model urban radio
channels with excess delays as large as 100 gs and microcellular channels
with excess delays no larger than 4 j.&s. If the number of multipath bins is fixed

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 Impulse Respqnse Model of a Multipath Channel 151

Figure 4.5
Measured wideband and narrowband received signals over aSk (0.375 in) measurement track inside
a building. Carrier frequency is 4 GHz. Wideband power is computed using equation (4.19), which
can be thought of as the area under the power delay profile.

at 64, find (a) at, and (b) the maximum bandwidth which the two models can
accurately represent. Repeat the exercise for an indoor channel model with
excess delays as large as 500 ns. As described in section 4.7.6, SIRCIM and
SMRCIM are statistical channel models based on equation (4.12) that use
parameters in this example.

Solution to Example 4.2

The maximum excess delay of the channel model is given by TN = NAt. There-
fore, for TN = lOOMS, and N = 64 we obtain At = TN/N = 1.5625 is. 'The
maximum bandwidth that the SMRCIM model can accurately represent is
equal to
l/(2At) = =0.32MHz.
l/(2(l.56251.zs))
For the SMRCJM urban microcell model, t& = 4iiS, At = TN/N = 62.5 ns.
The maximum bandwidth that can be represented is
l/(2At) = l/(2(62.Sns)) =8MHz.

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 152 Ch. 4 Mobile Radio Propagation: Small- Scale Fading and Multipath

9
= 500x10
Similarly, for indoor channels, Ar = tN/N = 7.8 125 ns.

The maximum bandwidth for the indoor channel model is

l/(2Ar) = ns)) 64MHz.

Example 4.3
Assume a mobile traveling at a velocity of 10 mIs receives two multipath com-
ponents at a carrier frequency of 1000 MHz. The first component is assumed to
arrive at t = 0 with an initial phase of 00 and a power of -70 dBm, and the
second component which is 3 dE weaker than the first component is assumed
to arrive at r = pss, also with an initial phase of 00. If the mobile moves
1

directly towards the direction of arrivgl of the first component and directly
away from the direction of arrival of the second component, compute the nar-
rowband instantaneous power at time intervals of 0.1 s from 0 5 to 0.5 S. Com-
pute the average narrowband power received over this observation interval.
Compare average narrowband and wideband received powers over the inter-
val.

Solution to Enmple 4.3

Given u = 10 m/s, time intervals of 0.18 correspond to spatial intervals of 1 m.
The carrier frequency is given to be 1000 MHz, hence the wavelength of the
signal is
3x108
Oim
1 iooox:o6
The narrowband instantaneous power can be computed using equation (4.21).
Note -70 dBm = 100 pW. At time t =0, the phases of both multipath compo-
nents are 00, hence the narrowband instantaneous power is equal to

jr(t)12

= fllo0pWxexp(0) +./5OpWx = 29! pW

Now, as the mobile moves, the phase of the two multipath components changes
in opposite directions.
At t = 0.1 the phase of the first component is
2nd2irut2nxlQ(mJs)xO.Is
A 0.3m
= 20.94 rad = 2.09 rad = 120
Since the mobile moves towards the direction of arrival of the first component,
and away from the direction of arrival of the second component, is positive,
and 02 is negative.
Therefore, at t = 0.ls, = 120, and 2 = 120, and the instantaneous
power is equal to

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 Small-Scale Multipath Measurements i

r(t)12 =

= = 78.2pW

Similarly, at t = 0.2 s, = 240, and 02 = 240, and the instantaneous

power is equal to

r(t)12 =

= L/lOOpw exp(j240) +]5OpWx exp(j240)12 = 81.5 pW

Similarly at t = 0.3 s, = 360 = 0, and 0, = 360 = 0 and the instan-

taneous power is equal to
N-] 2

lr (t) 2 = U01 (t. t)

= = 29) pW

It follows that at t = 0.4 s, Ir(t)I2 = 78.2 2W, and at t = 0.5 s. rU)1 = 81.5
pW.

The average narrowband received power is equal to

(2) (291) + (2) (78.2) + (2) (81.5)
pW = 150.233 pW

Using equation (4.19), the widehand power is given by

E UOt)12j

Eao[PWBJ = 100 pW + 50 pW = ISO pW

As can be seen, the narrowband and wideband received power are virtually
identical when averaged over 0.5 s (or S m). While the CW signal fades over the
observation interval, the wideband signal power remains constant.

4.3 Small-Scale Multipath Measurements

Because of the importance of the multipath structure in determining the
small-scale fading effects, a number of wideband channel sounding techniques
have been developed. These techniques may be classified as direct pulse mea-
surements, spread spectrum sliding correlator measurements, and swept fre-
quency measurements.

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 154 Ch. Mobile Radio Propagation: Small-Scale Fading and Muftipatli

4.3.1 Direct RF Pulse System

A simple channel sounding approach is the direct RE pulse system (see Fig-
ure 4.6). This technique allows engineers to determine rapidly the power delay
profile of any channel, as demonstrated by Rappaport and Seidel ER.ap89],
[Rap9O]. Essentially a wide band pulsed bistatic radar, this system transmits a
repetitive pulse of width tbb s, and uses a receiver with a wide bandpass filter
(BW = 2/tbbHz). The signal is then amplified, detected with an envelope detec-
tor, and displayed and stored on a high speed oscilloscope. This gives an immedi-
ate measurement of the square of the channel impulse response convolved with
the probing pulse (see equation (4.17)). If the oscilloscope is set on averaging
mode, then this system can provide a local average power delay profile. Another
attractive aspect of this system is the lack of complexity, since off-the-shelf
equipment may be used.
Tx 4 r REP *
- - . - -.

tbb
' Pulse Width =

Rr
a-- Resolution = Pulse Width

Figure 4.6
Direct RF channel impulse response measurement system.

The minimum resolvable delay between multipath components is equal to

the prdbing pulse width tb,, The main problem with this system is that it is sub-
-

to interference and noise, due to the wide passband filter required for multi-
path time resolution. Also, the pulse system relies on the ability to trigger the
oscilloscope on the first arriving signal. If the first arriving signal is blocked or
fades, severe fading occurs, and it is possible the system may not trigger prop-
erly. Another disadvantage is that the phases of the individual multipath
nents are not received, due to the use of an envelope detector. However, use of a
coherent detector permits measurement of the multipath phase using this tech-
nique.

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 Small-Scale Measurements 155

4.3.2 Spread Spectrum Sliding Correlator Channel Sounding

The basic block diagram of a spread spectrum channel sounding system is
shown in Figure 4.7. The advantage of a spread spectrum system is that, while
the probing signal may be wideband, it is possible to detect the transmitted sig-
nal using a narrowband receiver preceded by a wideband mixer, thus improving
the dynamic range of the system as compared to the direct RF pulse system.
In a spread spectrum channel sounder, a carrier signal is "spread" over a
large bandwidth by mixing it with a binary pseudo-noise (PN) sequence having a
chip duration and a chip rate equal to I Hz. The power spectrum
envelope of the transmitted spread spectrum signal is given by iDix84j as
2
S(f)= (4.26)
[
and the null-to-null bandwidth is
(4.27)

Tx

Resolution (rms pulse width)

BW=

Wideband Filter Narrowband Filter @

Figure 4.7
Spread spectrum channel impulse response measurement system.

The spread spectrum signal is then received, filtered, and despread using a
PN sequence generator identical to that used at the transmitter. Although the
two PN sequences are identical, the transmitter chip clock is run at a slightly
faster rate than the receiver chip clock. Mixing the chip sequences in this fashion
implements a sliding correlator [Dix84]. When the PN code of the faster chip

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 156 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and Multipath

clock catches up with the PN code of the slower chip clock, the twa chip
sequences will be virtually identicallyaligned, giving maximal correlation. When
the two sequences are Qot maximally correlated, mixing the incoming spread
spectrum signal with the unsynchronized receiver chip sequence will spread this
signal into a bandwidth at least as large as the receiver's reference PN sequence.
In this way, the narrowband filter that follows the correlator can reject almost all
of the incoming signal power. This is how processing gain is realized in a spread
spectrum receiver and how it can reject passband interference, unlike the direct
RF pulse sounding system.
Processing gain (PG) is given as
= =
PG = (4.28)
Rbb
where tbb = 1/Rbb, is the period of the baseband information. For the ease of a
sliding correlator channel sounder, the baseband information rate is equal to the
frequency offset of the PN sequence clocks at the transmitter and receiver.
When the incoming signal is correlated with the receiver sequence, the sig-
nal is collapsed back to the original bandwidth (i.e., "despread"), envelope
detected, and displayed on an oscilloscope. Since different incoming multipaths
will have different time delays, they will maximally correlate with the receiver
PN sequence at different times. The energy of these individual paths will pass
through the correlator depending on the time delay. Therefore, after envelope
detection, the channel impulse response convolved with the pulse shape of a sin-
gle chip is displayed on the oscilloscope. Cox [Cox72] first used this method to
measure channel impulse responses in outdoor suburban environments at 910
MHz. Devasirvatham [Dev86], [Dev9Oa] successfully used a direct sequence
spread spectrum channel sounder to measure time delay spread of multipath
components and signal level measurements in office and residential buildings at
850 MHz. Bultitude [Bul891 used this technique for indoor and microcellular
channel sounding work, as did Landron [Lan92].
The time resolution (Ar) of multipath components using a spread spec-
trum system with sliding correlation is

At = = (4.29)

In other words, the system can resolve two multipath components as long
as they are equal to dr greater than seconds apart. In actuality, multipath
components with interarrival times smaller than 27', can be resolved since the
nns pulse width is smaller than the absolute width of the triangular correlation
pulse, and is on the order of
The sliding correlation process gives equivalent time measurements that
are updated every time the two sequences are maximally correlated. The time
between maximal correlations (7') can be calculated from equation (4.30)

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 Small-Scale Multipath Measurements 157

= Tjl = (4.30)

where =chip period (s)

= chip rate (Hz)
y = slide factor (dimensionless)
1 = sequence length (chips)
The slide fator is defined as the ratio between the transmitter chip clock
rate and the difference between the transmitter and receiver chip clock rates
[Dev86]. Mathematically, this is expressed as

-r (4.31)
=
where a = transmitter chip clock rate (Hz)
= receiver
chip clock rate (Hz)
For a maximal length PN sequence, the sequence length is
= (4.32)
where n is the number of shift registers in the sequence generator [Dix841.
Since the incoming spread spectrum signal is mixed with a receiver PN
sequence that is slower than the transmitter sequence, the signal is essentially
down-converted ("collapsed") to a low-frequency narrowband signal. In other
words, the relative rate of the two codes slipping past each other is the rate of
information transferred to the oscilloscope. This narrowband signal allows nar-
rowband processing, eliminating much of the passband noise and interference.
The processing gain of equation (4.28) is then realized using a narrowband filter
(BW = 2(af3) ).
The equivalent time measurements refer to the relative times of mu.ltipath
components as they are displayed on the oscilloscope. The observed time scale on
the oscilloscope using a sliding correlator is related to the actual propagation
time scale by

Actual Propagation Time = Observed Time (433)

This effect is due to the relative rate of information transfer in the sliding
correlator. For example, of equation (4.30) is an observed time measured on
an oscilloscope and not actual propagation time. This effect, known as time dila-
tion, occurs in the sliding correlator system because the propagation delays are
actually expanded in time by the sliding correlator.
Caution must be taken to ensure that the sequence length has a period
which is greater than the longest multipath propagation delay. The PN sequence
period is
tPNseq = (4.34)

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 158 Ch. 4. Mobile Radio Propagation: Small-Scale Fading and Multipath

The sequence period gives an estimate of the maximum unambiguous

range of incoming multipath signal components. This range is found by multiply-
ing the speed of light with in equation (4.34).
There are several advantages to the spread spectrum channel sounding sys-
tem. One of the key spread spectrum modulation characteristics is the ability to
reject passband noise, thus improving the coverage range for a given transmitter
power. Transmitter and receiver PN sequence synchronization is eliminated by
the sliding correlator. Sensitivity is adjustable by changing the sliding factor and
the post-correlator filter bandwidth. Also, required transmitter powers can be
considerably lower than comparable direct pulse systems due to the inherent
"processing gain" of spread spectrum systems.
A disadvantage of the spread spectrum system, as compared to the direct
pulse system, is that measurements are not made in real time, but they are com-
piled as the PN codes slide past one another. Depending on system parameters
and measurement objectives, the time required to make power delay profile mea-
surements may be excessive. Another disadvantage of the system described here
that a noncoherent detector is used, so that phases of individual multipath
components can not be measured. Even if coherent detection is used, the sweep
time of a spread spectrum signal induces delay such that the phases of individ-
ual multipath components with different time delays would be measured at sub-
stantially different times, during which the channel might change.

4.33 Frequency Domain Channel Sounding

Because of the dual relationship between time domain and frequency
domain techniques, it is possible to measure the channel impulse response in the
frequency domain. Figure 4.8 shows a frequency domain channel sounder used
for measuring channel impulse responses. A vector network analyzer controls a
synthesized frequency sweeper, and an S-parameter test set is used to monitor
the frequency response of the channel. The sweeper scans a particular frequency
band (centered on the carrier) by stepping through discrete frequencies. The
number and spacings of these frequency steps impact the time resolution of the
impulse response measurement. For each frequency step, the S-parameter test
set transmits a known signal level at port 1 and monitors the received signal
level at port 2. These signal levels allow the analyzer to determine the complex
response (i.e., transmissivity S21(w)) of the channel over the measured frequency
range. The transmissivity response is a frequency domain representation of the
channel impulse response. This response is then converted to the time domain
using inverse discrete Fourier transform (IDFT) processing, giving a band-lim-
ited version of the impulse response. In this technique works well and
indirectly provides amplitude and phase information in the time domain. How-
ever, the system requires careful calibration and hardwired synchronization
between the transmitter and receiver, making it useful only for very close mea-

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 _________ __________

Parameters of Mobile Multipath Channels 159

surements (e.g., indoor channel sounthng). Another limitation with this

is the non-real-time nature of the measurement. For time varying channels, the
channel frequency response can change rapidly, giving an erroneous impulse
response measurement. To mitigate this effect, fast sweep times are necessary to
keep the total swept frequency response measurement interval as short as possi-
ble. A faster sweep time can be accomplished by reducing the number of fre-
quency steps, but this sacrifices time resolution and excess delay range in the
time domain. The swept frequency system has been used successfully for indoor
propagation studies by Pahlavan [Pah95] and Zaghloul, et.al. [Zag9laJ,
[Zag9lb].

Vector Net-work Analyzer

with
Swept Frequency Oscillator
X(w) Y(w)
S-parameter test set
Port! Port2

S21(w)zH(o))

Inverse
DFT Processor

I
h(t) FT'[W(u)}
Figure 4.8
Frequency domain channel impulse response measurement system.

4.4 Parameters of Mobile Multipath Channels

Many multipath channel parameters are derived from the power delay pro-
file, given by equation (4.18). Power delay profiles are measured using the tech-
niques discussed in Section 4.4 and are generally represented as plots of relative
received power as a function of excess delay with respect to a fixed time delay
reference. Power delay profiles are found by averaging instantaneous power
delay profile measurements over a local area in order to determine an average
small-scale power delay profile. Depending on the time resolution of the probing
pulse and the type of multipath channe}s studied, researchers often choose to
sample at spatial separations of a quarter of a wavelength and over receiver
movements no greater than 6 m in outdoor channels and no greater than 2 in in
indoor channels in the 450 MHz - 6 GHz range. This small-scale sampling avoids

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 ____

160 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and MuFtipath

large-scale averaging bias in the resulting small-scale statistics. Figure 4.9

shows typical power delay profile plots from outdoor and indoor channels, deter-
mined from a large number of closely sampled instantaneous profiles.

4.4.1 Time Dispersion Parameters

In order to compare different multipath channels and to develop some gen-
eral design guidelines for wireless systems, parameters which grossly quantify
the multipath channel are used. The mean excess delay, rms delay spread, and
excess delay spread (X dB) are multipath channel parameters that can be deter-
mined from a power delay profile. The time dispersive properties of wide band
multipath channels are most commonly quantified by their mean excess delay
(0 and rms delay spread (ar). The mean excess delay is the first moment of the
power dela)? profile and is defined to be

=
(4.35)

The rms delay spread is the square root of the second central moment of the
power delay profile and is defined to be

= (t)2 (4.36)
where

k k
(4.37)

These delays are measured relative to the first detectable signal arriving at the
receiver at to = 0. Equations (4.35) - (4.37) do not rely on the absolute power
level of P (t), but only the relative amplitudes of the multipath components
within P (t). Typical values of rms delay spread are on the order of microsec-
onds in outdoor mobile radio channels and on the order of nanoseconds in indoor
radio channels. Table 4.1 shows the typical measured values of rms delay spread.
It is important to note that the rms delay spread and mean excess delay are
defined from a single power delay profile which is the temporal or spatial aver-
age of consecutive impulse response measurements collected and averaged over
a local area. I5rpically, many measurements are made at many local areas in
order to determine a statistical range of multipath channel parameters for a
mobile communication system over a large-scale area [Rap9O].
The maximum excess delay (X dB) of the power delay profile is defined to be
the time delay during which multipath energy falls to X dB below the mat-

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 Parameters of Mobile Channels 161

85

m
-go
C
0
C-
w
a -95
V

S. -ioo
S.
-J
-4
C

U,
-jos
V
5,
.4
U,
U
HO
aSI

"5

Excess Duisy Tim.. (microacondsl

(a)
H

3&Qmpath
18 dE Atta2
2 mV/div
100 na/dIv
51.7 RicE
ns

50050 150 350 450

Delay (mis)

(b)
Fgure 4.9
Measured mukipath power delay profUes
a) From a 900 MHz cellular system in San Francisco [From [Rap9OJ IEEE].
b) Inside a grocery store at 4 GHz [From [Haw9lJ IEEE].

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 162 Ch. 4 Mobile Radio Propagatloil: Small-Scale Fading and Muftipath

Table 4.1 TypIcal Measured Values of RMS Delay Spread

Environment Frequency RMS Delay Reference

Notes
(MHz) Spread

Urban 910 1300 ns avg. New York City [0ox751

600 ns st. dev.
3500 ns max.
Urban 892 10-25 Ps Worst case San Francisco [Rap9O]
Suburban 910 200-310 ns Averaged typical case ECox72I
Suburban 910 1960-2110 ns Averaged extreme case [Cox72]
Indoor 1500 10-50 ns Office building [Sa187]
25 ns median
Indoor 850 270 ns max. Office building [Dev9Oa]
Indoor 1900 70-94 ns avg. Three San Francisco [Sei92aJ
1470 ns max. buildings

mum. In other words, the maximum excess delay is defined as t0 , where t0

is the first arriving signal and is the maximum delay at which a multipath
component is within X dB of the strongest arriving multipath signal (which does
not necessarily arrive at t0). Figure 4.10 illustrates the computation of the max-
imum excess delay for multipath components within 10 dB of the maximum. The
maximum excess delay (X dB) defines the temporal extent of the multipath that
is above a particular threshold. The value of is sometimes called the excess
delay spread of a power delay profile, but in all cases must be specified with a
threshold that relates the multipath noise floor to the maximum received multi-
path component.
In practice, values for P, and depend on the choice of noise threshold
used to process P (t). The noise threshold is used to differentiate between
received multipath components and thermal noise. If the noise threshold is set
too low, then noise will be processed as multipath, thus giving rise to values of
t, P. and that are artificially high.
It should be noted that the power delay proffle and the magnitude fre-
quency response (the spectral response) of a mobile radio channel are related
through the Fourier transform. It is therefore possible to obtain an equivalent
description of the channel in the frequency domain using its frequency response
characteristics. Analogous to the delay spread parameters in the time domain,
coherence bandwidth is used to characterize the channel in the frequency
domain. The rms delay spread and coherence bandwidth are inversely propor-
tional to one another, although their exact relationship is a function of the exact
multipath structure.

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 Parameters of Mobile Multipath Channels 163

I
RMSDeIaySprsad-46.40rs

MSmumExcsuD.IayclOdBsB4ns
-10

Threshold Level -20dB

-20

Mean Excess D.tsy = 45.05 ns

30
I

50 0 50 100 150 200 250 300 350 400 450

Excess Deby (ris)

Figure 4.10
Example of an indoor power delay profile; nns delay spread, mean excess delay, maximum excess
delay (10 dB), and threshold level are shown.

4.4.2 Coherence Bandwidth

While the delay spread is a natural phenomenon caused by reflected and
scattered propagation paths in the radio channel, the coherence bandwidth,
is a defined relation derived from the rms delay spread. Coherence bandwidth is
a statistical measure of the range of frequencies over which the channel can be
considered "flat" (i.e., a channel which passes all spectral components with
approximately equal gain and linear phase); In other words, coherence band-
width is the range of frequencies over which two frequency components have a
strong potential for amplitude correlation. Two sinusoids with frequency separa-
tion greater than are affected quite differently by the channel. If the coher-
ence bandwidth is defined as the bandwidth over which the frequency correlation
function is above 0.9, then the coherence bandwidth is approximately [Lee89b}

(4.38)
t
If the definition is relaxed so that the frequency correlation function is above 0.5.
then the coherence bandwidth is approximately

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 164 Cl,. 4 Mobile Radio Propagation: Sniall-Scale Fading and Muhipath

(4.39)
5

It is important to note that an exact relationship between coherence band-

width and nns delay spread does not exist, and equations (4.38) and (4.39) are
"ball park estimates". In general, spectral analysis techniques and simulation
are required to determine the exact impact that time varying multipath has on a
particular transmitted signal [Chu87], [Fun93], [Ste941. For this reason, accu-
rate multipath channel models must be used in the design of specific modems for
wireless applications ERap9la], [Woe941.

Example 4.4
Calculate the mean excess delay, rms delay spread, and the maximum excess
delay (10 dE) for the multipath profile given in the figure below. Estimate the
50% coherence bandwidth of the channel. Would this channel be suitable for
AMPS or GSM service without the use of an equalizer?
P/r)
0 dB-
-10 dB

-20dB

-30 dR

0 12 J
(las)
Figure E4.4

Solution to Example 4.4

The rms delay spread for the given multipath profile can be obtained using
equations (4.35) (4.37). The delays of each profile are measured relative to
the first detectable signal. The mean excess delay for the given profile
(l)(5)+(O.l)(I)+(0.I)(2)+(0.01)(0) 438
[0.01+0.1+0.1+1]
-

The second moment for the given power delay profile can be calculated
as
(5)2k
= (1) (0.1) (1)2+ (0.1) (2)2+ (0.01) (0) 2
- 2107 .

1.21

Therefore the rins delay spread, = ./21.07 (4.38)2

=
The coherence bandwidth is found from equation (439) to be

= 146kHz
=

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 Parameters of Mobile Multipath Channels 165

Since is greater than 30 kHz. AMPS will work without an equalizer. How-
ever, GSM requires 200 kHz bandwidth which exceeds thus an equalizer
would be needed for this channel.

4.4.3 Doppler Spread and Coherence Time

Delay spread and coherence bandwidth are parameters which describe the
time dispersive nature of the channel in a local area. However, they do not offer
information about the time varying nature of the channel caused by either rela-
tive motion between the mobile and base station, or by movement of objects in
the channel. Doppler spread and coherence time are parameters which describe
the time varying nature of the channel in a small-scale region.
Doppler spread BD is a measure of the spectral broadening caused by the
time rate of change of the mobile radio channel and is defined as the range of fre-
quencies over which the received Doppler spectrum is essentially non-zero.
When a pure sinusoidal tone of frequency is transmitted, the received signal
spectrum, called the Doppler spectrum, will have components in the range

to + where fd is the Doppler shift. The amount of spectral broadening

depends on which is a function of the relative velocity of the mobile, and the
angle 9 between the direction of motion of the mobile and direction of arrival of
the scattered waves. If the baseband signal bandwidth is much greater than BD,
the effects of Doppler spread are negligible at the receiver. This is a slow fading
channel.
Coherence time T, is the time domain dual of Doppler spread and is used
to characterize the time varying nature of the frequency dispersiveness of the
channel in the time domain. The Doppler spread and coherence time are
inversely proportional to one another. That is,

(4.40.a)
= /.
Coherence tine is actually a statistical measure of the time duration over
which the channel impulse response is essentially invariant, and quantifies the
similarity of the channel response at different times. In other words, coherence
time is the time duration over which two received signals have a strong potential
for amplitude correlation. If the reciprocal bandwidth of the baseband signal is
greater than the coherence time of the channel, then the channel will change
during the transmission of the baseband message, thus causing distortion at the
receiver. If the coherence time is defined as the time over which the time correla-
tion function is above 0.5, then the coherence time is approximately iSte94l

Tc- ifm (4.40.b)

where Tm is the maximum Doppler shift given by Tm = v/k. In practice, (4.40.a)

suggests a time duration during which a Rayleigh fading signal may fluctuate

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 166 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and

wildly, and (4.40.b) is often too restrictive. A popular rule of thumb for modem
digital communications is to define the coherence time as the geometric mean of
equations (4.40.a) and (4.40.b). That is,
= 0.423
= f (4.40.c)
I'm

The definition of coherence time implies that two signals arriving with a
time separation greater than are affected differently by the channel. For
example, for a vehicle traveling 60 mph using a 900 MHz carrier, a conservative
value of can be shown to be 2.22 ms from (4.40.b). If a digital transmission
system is used, then as long as the symbol rate is greater than I = 454 bps,
the channel will not cause distortion due to motion (however, distortion could
result from multipath time delay spread, depending on the channel impulse
response). Using the practical formula of (4.40.c), = 6.77 ms and the symbol
rate must exceed 150 bits/s in order to avoid distortion due to frequency disper-
sion.

Ernniple 4.5
Determine the proper spatial sampling interval required to make small-scale
propagation measurements which assume that consecutive samples are highly
correlated in time. How many samples will be required over 10 m travel dis-
tance if 4 = 1900 MHz and v = 50 in/s. How long would it take to make these
measurements, assuming they could be made in real time from a moving vehi-
cle? What is the Doppler spread B0 for the channel?

Solution to Example 4.5

For correlation, ensure that the time between samples is equal to Tc/2 , and
use the smallest value of for conservative design.
Using equation (4.40.b)
9A 9c =
lfltfm l6iw
16 x 3.14 x 50 x 1900 x iob

Tc =
Taking time samples at less than half
corresponds to a spatial sampling interval of
= = SOXS6SMS
= 0.014l25m = 1.41cm

Therefore, the number of samples required over a 10 m travel distance is

N2 = 708 samples
= = 0.014125
The time taken to make this measurement is equal to lOm = 0.2 s
50 m's
The Doppler spread is

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 Types of Small-Scale Fading 167

i9oo x io6
BD=tm___

3xl08
=316.66Hz

4.5 Types of Small-Scale Fading

Section4.3 demonstrated that the type of fading experienced by a signal
propagating through a mobile radio channel depends on the nature of the trans-
mitted signal with respect to the characteristics of the channel. Depending on
the relation between the signal parameters (such as bandwidth, symbol period,
etc.) and the channel parameters (such as rms delay spread and Doppler spread),
different transmitted signals will undergo different types of fading. The time dis-
persion and frequency dispersion mechanisms in a mobile radio channel lead to
four possible distinct effects, which are manifested depending on the nature of
the transmitted signal, the channel, and the velocity. While multipath delay
spread leads to time dispersion and frequency selective fading, Doppler spread
leads to frequency dispersion and time selective fading. The two propagation
mechanisms are independent of one another. Figure 4.11 shows a tree of the four
different types of fading.

Small-Scale Fading
(Based on multipath time delay spread)

Flat Fading Frequency Selective Fading

1. 13W of signal <BW of channel 1. SW of signal > SW of channel
2. Delay spread <Symbol period 2. Delay spread> Symbol period

Small-Scale Fading
(Based on Doppler spread)

Fast Fading Slow Fading

1. High Doppler spread 1. Low Doppler spread
2. Coherence time c Symbol period 2. Coherence time> Symbol period
3. Channel variations faster than base- 3. Channel variations slower than
band signal variations baseband signal variations

Figure 4.11
Types of small-scale fading.

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 168 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and Mullipath

4.5.1 Fading Effects Due to Multipath lime Delay Spread

Time dispersion due to multipath causes the transmitted signal to undergo
either flat or frequency selective fading.
4.5.1.1 Flat fading
If the mobile radio channel has a constant gain and linear phase response
over a bandwidth which is greater than the bandwidth of the transmitted signal,
then the received signal will undergo flat fading. This type of fading is histori-
cally the most common type of fading described in the technical literature. In flat
fading, the multipath structure of the channel is such that the spectral charac-
teristics of the transmitted signal are preserved at the receiver. However the
strength of the received signal changes with time, due to fluctuations in the gain
of the channel caused by multipath. The characteristics of a flat fading channel
are illustrated in Figure 4.12.

sO)
h('z.t,)

s(t) hO, t) r(t)

0 i ot 0

Figure 4.12
Flat fading channel characteristics.

It can be seen from Figure 4.12 that if the channel gain changes over time,
a change of amplitude occurs in the received signal. Over time, the received sig-
nal r (t) varies in gain, but the spectrum of the transmission is preserved. In a
flat fading channel, the reciprocal bandwidth of the transmitted signal is much
larger than the multipath time delay spread of the channel, and h,, (t, r) can be
approximated as having no excess delay (i.e., a single delta function with r = 0).
Flat fading channels are also known as amplitude uarying channels and are
sometimes referred to as narrowband channels, since the bandwidth of the
applied signal is narrow as compared to the channel flat fading bandwidth. Typi-
cal flat fading channels cause deep fades, and thus may require 20 or 30 dB more
transmitter power to achieve low bit error rates during times of deep fades as

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 Types ol Smafi-Scale Fading 169

compared to systems operating over non-fading channels. The distribution of the

instantaneous gain of flat fading channels is important for designing radio links,
and the most common amplitude distribution is the Rayleigh distribution. The
Rayleigh flat fading channel model assumes that the channel induces an ampli-
tude which varies in time according to the Rayleigh distribution.
To summarize, a signal undergoes flat fading if
(4.41)
and
Ts>> (4.42)
where is the reciprocal bandwidth (e.g., symbol period) and B5 is the band-
width, respectively, of the transmitted modulation, and and are the ntis
delay spread and coherence bandwidth, respectively, of the channel.
4.5.1.2 Frequency Selective Fading
If the channel possesses a constant-gain and linear phase response over a
bandwidth that is smaller than the bandwidth of transmitted signal, then the
channel creates frequency selective fading on the received signal. Under such
conditions the channel impulse response has a multipath delay spread which is
greafer than the reciprocal bandwidth of the transmitted message waveform.
When this occurs, the received signal includes multiple versions of the transmit-
ted waveform which are attenuated (faded) and delayed in time, and hence the
received signal is distorted. Frequency selective fading is due to tine dispersion
of the transmitted symbols within the channel. Thus the channel induces
intersymbol interference (IS!). Viewed in the frequency domain, certain fre-
quency components in the received signal spectrum have greater gains than oth-
ers.
Frequency selective fading channels are much more difficult to model than
flat fading channels since each multipath signal must be modeled and the chan-
nel must be considered to be a linear filter. It is for this reason that wideband
multipath measurements are made, and models are developed from these mea-
surements. When analyzing mobile communication systems, statistical impulse
response models such as the 2-ray Rayleigh fading model (which considers the
impulse response to be made up of two delta functions which independently fade
and have sufficient time delay between them to induce frequency selective fading
upon the applied signal), or computer generated or measured impulse responses,
are generally used for analyzing frequency selective small-scale fading. Figure
4.13 illustrates the characteristics of a frequency selective fading channel.
For frequency selective fading, the spectrum S (I) of the transmitted signal
has a bandwidth which is greater than the coherence bandwidth Bc of the chan-
nel. Viewed in the frequency domain, the channel becomes frequency selective,
where the gain is different for different frequency components. Frequency selec-

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 170 Ch. 4 Mobile Radio Propagation: Small-Scale Fadng and Muttipath

r(r)

s(t)

01;
fl,0
ho, t)

t
r(t)

or,

Figure 4.13
Frequency selective fading channel characteristics.

tive fading is caused by multipath delay&which approach or exceed the symbol

period of the transmitted symbol. Frequency selective fading channels are p1so
known as wideband channels since the bandwidth of the signal s(t) is wider
than the bandwidth of the channel impulse response. As time varies, the channel
varies in gain and phase across the spectrum of s(t), resulting in time varying
distortion in the received signal r(t). To summarize, a signal undergoes frequency
selective fading if
B5>
and
(4.44)
A common rule of thumb is that a channel is frequency selective if
although this is dependent on the specific type of modulation used. Chapter 5
presents simulation results which illustrate the impact of time delay spread on
bit error rate (BEll).

4.5.2 Fading Effects Due to Doppler Spread

4.5.2.1 Fast Fading
Depending on how rapidly the transmitted baseband signal changes as
compared to the rate of change of the channel, a channel may be classified either
as a fast fading or slow fading channel. In a fast fading channel, the channel
impulse response changes rapidly within the symbol duration. That is, the coher-
ence time of the channel is smaller than the symbol period of the transmitted
signal. This causes frequency dispersion (also called time selective fading) due to
Doppler spreading, which leads to signal distortion. Viewed in the frequency
domain, signal distortion due to fast fading increases with increasing Doppler

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 Types of Small-Scale Fading 171

spread relative to the bandwidth of the transmitted signal. Therefore, a signal

undergoes fast fading if
(4.45)
and
(4.46)
It should be noted that when a channel is specified as a fast or slow fading
channel, it does not specify whether the channel is flat fading or frequency selec-
tive in nature. Fast fading only deals with the rate of change of the channel due
to motion. En the case of the flat fading channel, we can approximate the impulse
response to be simply a delta function (no time delay). Hence, a flat fading, fast
fading channel is a channel in which the amplitude of the delta function varies
faster than the rate of change of the transmitted baseband signal. In the case of
a frequency selective, fast fading channel, the amplitudes, phases, and time
delays of any one of the multipath components vary faster than the rate of
change of the transmitted signal. In practice, fast fading only occurs for very low
data rates.
4.5.2.2 Slow Fading
In a slow fading channel, the channel impulse response changes at a rate
much slower than the transmitted baseband signal s(t). In this case, the channel
may be assumed to be static over one or several reciprocal bandwidth intervals.
In the frequency domain, this implies that the Doppler spread of the channel is
much less than the bandwidth of the baseband signal. Therefore, a signal under-
goes slow fading if
i's a (4.47)
and
B5 >> BD (4.48)
It should be clear that the velocity of the mobile (or velocity of objects in the
channel) and the baseband signaling determines whether a signal undergoes
fast fading or slow fading.
The relation between the various multipath parameters and the type of fad-
ing experienced by the signal are summarized in Figure 4.14. Over the years,
some authors have confused the tenns fast and slow fading with the terms large-
scale and small-scale fading. It should be emphasized that fast and slow fading
deal with the relationship between the time rate of change in the channel and
the transmitted signal, and not with propagation path loss models.

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 172 Ch, 4 Mobile Radio Propagation: Small-Scale Fading and Multipath

'E T
FlatSiow ' Flat Fast
0UD
Fading I
Fading

tl . I
Frequency Selective1 Frequency Selective
Slow Fading Fast Fading

Transmitted Symbol Period

(a)

B8
C I

I Frequency Selective I
Fast Fading
Frequency Selective
Slow Fading --

P Flat Fast Flat Slow

J Fading Fading

Bd

Transmitted Baseband Signal Bandwidth

(b)
Figure 4.14
Matrix illustrating type of fading experienced by a signal as a function of
(a) symbol period
(b) baseband signal bandwidth.

4.6 RayleIgh and Ricean Distributions

4.6.1 Rayleigh Fading Distribution

In mobile radio channels, the Rayleigh distribution is commonly used to
describe the statistical time varying nature of the received envelope of a flat fad-
ing signal, or the envelope of an individual multipath component. It is well
known that the envelope of the sum of two quadrature Gaussian noise signals
obeys a Rayleigh distribution. Figure 4.15 shows a Rayleigh distributed signal
envelope as a function of time. The Rayleigh distribution has a probability den-
sity function (pdf) given by

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 Rayleigh and Ricean Distribulions 173

Typical simulated Rayleigh fading at the carrier

Receiver speed = 120 km/hr
10
5

0
-5
a
-a
0
'U
15
0.)
> 20
0)
-J
-25
C
a,

35

-40
0 50 I DO 1 50 200 250
Elapsed Time (rns)

Figure 4.15
A typical Rayleigh fading envelope at 900 MHz (From 1jun93) IEEE].

(
fT r
jexp
p(r) = 12 (4.49)
0 (rcO)
where a is the rms value of the received voltage signal before envelope detection,
and is the time-average power of the received signal before envelope detec-
tion. The probability that the envelope of the received signal does not exceed a
specified value R is given by the corresponding cumulative distribution function
(CDF)
R
1 2"
P(R) = Pr(rR) = fp(r)dr = I (4.50)
2a)
The mean value rmean of the Rayleigh distribution is given by

= E[r} = frp (r)dr = = I .2533a (4.51)

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 174 Ch. 4-Mobile Radio Propagation: Small-Scale Fading and Multipath

and the variance of the Rayleigh distribution is given by which represents

the ac power in the signal envelope

= E[r2) E2(r] = (4.52)

= = 0.4292a2

The rms value of the envelope is the square root oTthe mean square, or lie.
The median value of r is found by solving

= I
2 .v0
p(r)dr (4.53)

and is
= l177a
rmedian (4.54)
Thus the mean and the median differ by only 0.55 dB in a Rayleigh fading
signal. Note that the median is often used in practice, since fading data are usu-
ally measured in the field and a particular distribution cannot be assumed. By
using median values instead of mean values it is easy to compare different fad
ing distributions which may have widely varying means. Figure 4.16 illustrates
the Rayleigh pdf. The corresponding Rayleigh cumulative distribution function
(CDF) is shown in Figure 4.17.

0.6065
a
/
p(r)

/ I

Received signal envelope voltage r (volts)

Figure 4.16
Rayleigh probability density function (pdfi.

4.6.2 Ricean Fading Distribution

When there is a dominant stationary (nonfading) signal component
present, such as a line-of-sight propagation path, the small-scale fading envelope

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 Rayleigh and Ricean Distributions 175

10
U OBS light clutter, Site C
50
S LOS heavy clutter, Site E
* LOS light cluter, Sue D

20
U

10

% Probability
5 .
Signal Level . *1
<Abscissa . *:
2 .
. t.
1 U

0.5

0.2 Log-normal dB
Rayleigh
Rician K=6 dB
0.1
-30 -20 -10 0 10

Signal Lent (dli sborat media)

Figure 4.17
Cumulative distribution for three small-scale fading measurements and their fit to Rayleigh, Ricean,
and log-normal distributions [From [RapS9] IEEEI.

distribution is Ricean. In such a situation, random multipath components arriv-

ing at different angles are superimposed on a stationary dominant signal. At the
output of an envelope detector, this has the effect of adding a dc component to the
random multipath.
Just as for the case of detection of,a sine wave in thermal noise [Ric48], the
effect of a dominant signal arriving with many weaker multipath signals gives
rise to the Ricean distribution. As the dominant signal becomes weaker, the com-
posite signal resembles a noise signal which has an envelope that is Rayleigh.
Thus, the Ricean distribution degenerates to a Rayleigh distribution when the
dominant component fades away.

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 176 Ch. 4. Mobite Radio Propagation: Small-Scale Fading and Multipath

The Ricean distribution is given by

2
(r A )2

2a
p(r) = jL..e
(455)
0 for(rcO)
The parameter A denotes the peak amplitude of the dominant signal and
(.) is the modified Bessel function of the first kind and zero-order. The Ricean
distribution is often described in terms of a parameter K which is defined as the
ratio between the deterministic signal power and the variance of the multipath.
ItisgivenbyK =A2 /(2a) or,intermsofdB
2

A2
K(dB) = dB (456)
2a
The parameter K is known as the Ricean factor and completely specifies
the Ricean distribution. As A *0, K * dB, and as the dominant path
decreases in amplitude, the Ricean distribution degenerates to a Rayleigh distri-
bution. Figure 4.18 shows the Ricean pdf The Ricean CDF is compared with the
Rayleigh CDF in Figure 4.17.

K=-xdB

dB
p(r):
/

Received signal envelope voltage r (volts)

Figure 4.18
Probability density function of Ricean distributions: K = dB (Rayleigh) and K = 6 dB. For
K>>1, the Ricean pdf is approximately Gaussian about the mean.

4.7 Statistical Models for Multipath Fading Channels

Several multipath models have been suggested to explain the observed sta-
tistical nature of a mobile channel. The first model presented by Ossana [0ss64]
was based on interference of waves incident and reflected from the fiat sides of
randomly located buildings. Although Ossana's model [0ss64] predicts flat fad-
ing power spectra that were in agreement with measurements in suburban

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 Models for Muftipath Fading Channels 177

areas, it assumes the existence of a direct path between the transmitter and
receiver, and is limited to a restricted range of reflection angles. Ossana's model
is therefore rather inflexible and inappropriate for urban areas where the direct
path is almost always blocked by buildings or other obstacles. Clarke's model
[C1a681 is based on scattering and is widely used.

4.7.1 Clarke's Model for Flat Fading

Clarke [C1a681 developed a model where the statistical characteristics of
the electromagnetic fields of the received signal at the mobile are deduced from
scattering. The model assumes a fixed transmitter with a vertically polarized
antenna. The field incident on the mobile antenna is assumed to be comprised of
N azimuthal plane waves with arbitrary carrier phases, arbitrary azimuthal
angles of arrival, and each wave having equal average amplitude. It should be
noted that the equal average amplitude assumption is based on the fact that in
the absence of a direct line-of-sight path, the scattered components arriving at a
receiver will experience similar attenuation over small-scale distances.
Figure 4.19 shows a diagram of plane waves incident on a mobile traveling
at a velocity v, in the x-direction. The angle of arrival is measured in the x-y
plane with respect to the direction of motion. Every wave that is incident on the
mobile undergoes a Doppler shift due to the motion of the receiver and arrives at
the receiver at the same time. That is, no excess delay due to multipath is
assumed for any of the waves (flat fading assumption). For the n th wave arriv-
ing at an angle to the x-axis, the Doppler shift in Hertz is given by

f,, = (4.57)

where X is the wavelength of the incident wave.

y
in x-y plane

Figure 4.19
Illustrating plane waves arrivingat random angles.

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 178 Ch. 4 Mobile Radio Propagation: Small-Scale Fadin9 and Multipath

The vertically polarized plane waves arriving at the mobile have E and H
field components given by

= (4.58)

= - + (4.59)

=- (4.60)

where E0 is the real amplitude of local average E-field (assumed constant),

is a real random variable representing the amplitude of individual waves, fl is
the intrinsic impedance of free space (37711), and is the carrier frequency. The
random phase of the n th arriving component is given by
= (4.61)
The amplitudes of the E-and H-field are normalized such that the ensemble
average of the 's is given by

= I (4.62)

Since the Doppler shift is very small when compared to the carrier fre-
quency, the three field components may be modeled as narrow band random pro-
cesses. The three components and H,, can be approximated as Gaussian
random variables if N is sufficiently large. The phase angles are assumed to
have a uniform probability density funccion (pdf) on the interval (0,2it]. Based on
the analysis by Rice [Ric481 the E-field can be expressed in an in-phase and
quadrature form
= (4.63)
where

= E0Z C1 + (4.M)

and

= (4.65)

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 Statistical Models for Multipath Fading Channels 179

Both (t) and (t) are Gaussian random processes which are denoted
as and T9, respectively, at any time t. and are uncorrelated zero-mean
Gaussian random variables with an equal variance given by

(4.66)
where the overbar denotes the ensemble average.
The envelope of the received E-field, E/t), is given by

EZ(t)I = = r(t) (4.67)

Since and are Gaussian random variables, it can be shown through a
Jacobean transformation [Pap9l] that the random received signal envelope r
has a Rayleigh distribution given by

=
p(r) (4.68)

where a2 = E0/2 2

4.7.1.1 Spectral Shape Due to Doppler Spread in Clarke's Model

Gans [Gan72] developed a spectrum analysis for Clarke's model. Let
p (a) da denote the fraction of the total incoming power within da of the angle
a, and let A denote the average received power with respect to an isotropic
antenna. As N * p (a) da approaches a continuous, rather than a discrete,
distribution. If G (a) is the azimuthal gain pattern of the mobile antenna as a
function of the angle of arrival, the total received power can be expressed as

= JAG(a)p(a)da (4.69)

where AG (a)p (a)da is the differential variation of received power with angle.
If the scattered signal is a CW signal of frequency then the instantaneous fre-
quency of the received signal component arriving at an angle a is obtained using
equation (4.57)

[(a) = [ +f = (4.70)
where is the maximum Doppler shift. It should be noted that [(a) is an even
function of a, (i.e., /(a) = fT-a)).
If S (I) is the power spectrum of the received signal, the differential varia-
tion of received power with frequency is given by
S(f)Id)9 (4.71)

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 180 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and Multipath

Equating the differential variation of received power with frequency to the

differential variation in received power with angle, we have
S(f)Id/9 = A[p(a)G(a) +p(a)G(--a)]IdaI (4.72)
Differentiating equation (4.70), and rearranging the terms, we have
Idfl = IdaIjsinalfm (4.73)
Using equation (4.70), a can be expressed as a function off as

a= cos-1
[t'fc] (4.74)

This implies that

sina = (4.75)

Substituting equation (4.73) and (4.75) into both sides of (4.72), the power spec-
tral density S (/) can be expressed as

s n = A[p(a)G(cz) +p(a)G(--a)] (4.76)

/
I,n I

where
S(/) = 0,
Vft! >fm (4.77)
The spectrum is centered on the carrier frequency and is zero outside the
limits of fm Each of the arriving waves has its own carrier frequency (due to
its direction of arrival) which is slightly offset from the center frequency. For the
case of a vertical A/4 antenna (G(a) = 1.5), and a uniform distribution
p(a) = 1/2n over 0 to 2E ,the output spectrum is given by (4.76) as
= 1.5
5E (f) (4.78)

f U

In equation (4.78) the power spectral density at f = fm is infinite, i.e.,

Doppler components arriving at exactly 0 and 1800 have an infinite power
spectral density. This is not a problem since a is continuously distributed and
the probability of components arriving at exactly these angles is zero.
Figure 4.20 shows the power spectral density of the resulting RF signal due
to Doppler fading. Smith [Smi751 demonstrated an easy way to simulate Clarke's
model using a computer simulation as described Section 4.7.2.
After envelope detection of the Doppler-shifted signal, the resulting base-
band spectrum has a maximum frequency of 2fm. It can be shown {Jak74] that
the electric field produces a baseband power spectral density given by

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 Statistical Models for Multipath Fading Channels 181

I
I&fm fcilm

Figure 4.20
Doppler power spectrum for an unmodulated CW carrier [From [Gan721] IEEE].

(1)2]
= 8fmK[1I
where K [.} is the complete elliptical integral of the first kind. Equation (4.79) is
not intuitive and is a result of the temporal correlation of the received signal
when passed through a nonlinear envelope detector. Figure 4,21 illustrates the
baseband spectrum of the received signal after envelope detection.
The spectral shape of the Doppler spread determines the time domain fad-
ing waveform and dictates the temporal correlation and fade slope behaviors.
Rayleigh fading simulators must use a fading spectrum such as equation (4.78)
in order to produce realistic fading waveforms that have proper time correlation.

4.7.2 Simulation of Clarke and Gans Fading Model

It is often useful to simulate multipath fading channels in hardware or soft-
ware. A popular simulation method uses the concept of in-phase and quadrature
modulation paths to produce a simulated signal with spectral and temporal char-
acteristics very close to measured data.
As shown in Figure 4.22, two independent Gaussian low pass noise sources
are used to produce in-phase and quadrature fading branches. Each Gaussian
source may be formed by summing two independent Gaussian random variables
which are orthogonal (i.e., g = a +jb, where a and b are real Gaussian random
variables and g is complex Gaussian). By using the spectral filter defined by
equation (4.78) to shape the random signals in the frequency domain, accurate
time domain waveforms of Doppler fading can be produced by using an inverse
fast Fourier transform (IFF'l') at the last stage of the simulator.

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 182 Cli. 4 Mobile Radio Propagation: Small-Scale Fading and Multipath

0dB

-1dB

-2dB

-3dB

-5dB

2 -6dB

-7dB

-8dB
0.001 0.01 0.1 1.0 2.0 10
1/fm

Figure 421
Baseband power spectral density of a CW Doppler signal after envelope detection.

Smith [3mi751 demonstrated a simple computer program that implements

Figure 4.22(b). His method uses a complex Gaussian random number generator
(noise source) to produce a baseband line spectrum with complex weights in the
positive frequency band. The maximum frequency component of the line spec-
trum is /'m Using the property of real signals, the negative frequency compo-
nents are constructed by simply conjugating the complex Gaussian values
obtained for the positive frequencies. Note that the IFfl of this signal is a
purely real Gaussian random process in the time domain which is used in one of
the quadrature arms shown in Figure 4.22. The random valued line spectrum is
then multiplied with a discrete frequency representation of (15 having the
same number of points as the noise source. Th handle the case where equation
(4.78) approaches infinity at the passband edge, Smith truncated the value of
SE (fin) by computing the slope of the function at the sample frequency just
pri'or to the passband edge and extended the slope to the passband edge. Simula-
tions using the architecture in Figure 4.22 are usually implemented in the fre-
quency domain using complex Gaussian line spectra to tt'ke advantage of easy
implementation of equation (4.78). This, in turn, implies that the low pass Gaus-

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 Statistical Models for Multipath Fading Channels 183

C
a
C

sin 2irQ

(a)

sin 2nf0t

(b)
Figure 4.22
Simulator using quadrature amplitude modulation with (a) RY Doppler filter and
(b) baseband Doppler filter.
sian noise components are actually a series of frequency components (line spec-
trum from -4, to In)' which are equally spaced and each have a complex
Gaussian weight. Smith's simulation methodology is shown in Figure 4.23.
'lb implement the simulator shown in Figure 4.23, the following steps are
used:
(1) Speci5r the number of frequency domain points (N) used to represent
JSE (I') and the maximum Doppler frequency shift The value used for
N is usually a power of 2.
(2) Compute the frequency spacing between adjacent spectral lines as
11= 24,/(N_ I). This defines the time duration of a fading waveform,
T = l/Af.
(3) Generate complex Gaussian random variables for each of the N/2 positive
frequency components of the noise source.
(4) Construct the negative frequency components of the noise source by conju-

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 184 Ch. 4 Mobile Radio Propagation: Small-Scale Facing and Multipath

gating positive frequency values and assigning these at negative frequency

values.
(5) Multiply the in-phase and quadrature noise sources by the fading spectrum
JSE(f).
(6) Perform an IFFT on the resulting frequency domain signals from the in-
phase and quadrature arms to get two N-length time series, and add. the
squares of each signal point in time to create an N-point time series like
under the radical of equation (4.67).
(7) Take the square root of the sum obtained in step 6 to obtain an N point time
series of a simulated Rayleigh fading signal with the proper Doppler spread
and time correlation.

Figure 4.23
Frequency domain implementation of a Rayleigh fading simulator at baseband

Several Rayleigh fading simulators may be used in conjunction with vari-

able gains and time delays to produce frequency selective fading effects. This is
shown in Figure 4.24.
By making a single frequency component dominant in amplitude within
,jSE(f). the fading is changed from Rayleigh to Ricean. This can be used to
alter the probability distributions of the individual multipath components in the
simulator of Figure 4.24.
To determine the impact of flat fading on an applied signal s(t), one
merely needs to multiply the applied signal by r(t), the output of the fading
simulator. To determine the impact of more than one multipath component, a
convolution must be performed as shown in Figure 4.24.

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 Statistical Models for Multipath Fading Channels 185

Signal s(t)
under test

nt)
Figure 4.24
A signal may be applied to a Rayleigh fading simulator to determine performance in a wide range of
channel ccnditions. Both flat and frequency selective fading conditions may be simulated, depending
on gain and time delay settings.

4.7.3 Level Crossing and Fading Statistics

Rice computed joint statistics for a mathematical problem which is similar
to Clarke's fading model [C1a68], and thereby provided simple expressions for
computing the average number of level crossing and the duration of fades. The
level crossing rate (LCR) and average fade duration of a Rayleigh fading signal
are two important statistics which are useful for designing error control codes
and diversity schemes to be used in mobile communication systems, since it
becomes possible to relate the time rate of change of the received signal to the
signal level and velocity of the mobile.
The level crossing rate (LCR) is defined as the expected rate at which the
Rayleigh fading envelope, normalized to the local rms signal level, crosses a
specified level in a positive-going direction. The number of level crossings per
second is given by

NH = = (4.80)

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 186 Ch. 4 Mobile Radio Propagation: Snail-Scale Fading and Muftipath

where r is the time derivative of r (t) (i.e., the slope), p (R, r') is the joint den-
sity function of r and f at r = R, the maximum Doppler frequency and
f,,, is
p = R/Rrms is the value of the specified level R, normalized to the local rms
amplitude of the lading envelope EJakl4]. Equation (4.80) gives the value of NR,
the average number of level crossings per second at specified R. The level cross-
ing rate is a function of the mobile speed as is apparent from the presence of fm
in equation (4.80). There are few crossings at both high and low levels, with the
maximum rate occurring at p = I / ,fi, (i.e., at a level 3 dB below the rms level).
The signal envelope experiences very deep fades only occasionally, but shallow
fades are frequent.
Example 4.6
For a Rayleigh fading signal, compute the positive-going level crossing rate for
p = 1, when the maximum Doppler frequency ) is 20 Hz. What is the max-
imum velocity of the mobile for this Doppler frequency if the carrier frequency
is 900 MHz?

Solution to Example 4.6

Using equation (4.80), the number of zero level crossings is
NR = ,11(20) (I)e = 18.44 crossings per second
The maximum velocity of the mobile can be obtained using the Doppler rela-
tionfdmax = v/X.
Therefore velocity of the mobile at = 20 Hz is
v= = 2QHz(1/3 m) = 6.66m/s = 24km/hr

The average fade duration is defined as the average period of time for which
the received signal is below? a specified level R. For a Rayleigh fading signal, this
is given by

t = jrPr[rR] (4.81)

where Pr [r R] is the probability that the received signal r is less than R and
is given by

Fr[rR] = (4.82)

where is the duration of the fade and T is the observation interval )f the fad-
ing signal. The probability that the received signal r is less than the threshold
R is found from the Rayleigh distribution as

Fr[rRI = fp(r)dr = lexp(p2) (4.83)

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 Statistical Models for Multipath Fading Channels 187

where p (r) is the pdf of a Rayleigh distribution. Thus, using equations (4.80),
(4.81), and (4.83), the average Vade duration as a function of p and fm can be
expressed as

= e
(4.84)
PIm
The average duration of a signal fade helps detennine the most likely num-
ber of signaling bits that may be lost during a fade. Average fade duration pnma-
rily depends upon the speed of the mobile, and decreases as the maximum
Doppler frequency fm becomes large. If there is a particular fade margin built
into the mobile communication system, it is appropriate to evaluate the receiver
performance by determining the rate at which the input signal falls below a
given level B, and how long it remains below the level, on average. This is use-
ful for relating SNR during a fade to the instantaneous BElt which results.
Example 4.7
Find the average fade duration for threshold levels p = 0.01, p = 0.1, and
p = I, when the Doppler frequency is 200 Hz.

Solution to Example 4.7

Average fade duration can be found by substituting the given values in equa-
tion (4.75)
For p = 001
0.012
e L
=19.9ps
(0.01)
For p = 0.1

= l
=200ps
0. 1) 200
For p = I

3.43ms

Example 4.8
Find the average fade duration for a threshold level of p = 0.707 when the
Doppler frequency is 20 Hz. For a binary digital modulation with bit duration
of 50 bps, is the Rayleigh fading slow or fast? What is the average number of
bit errors per second for the given data rate. Assume that a bit error occurs
whenever any portion of a bit encounters a fade for which p c 0.1

Solution to Example 4.8

The average fade duration can be obtained using equation (4.84).

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 188 CIt 4 Mobile Radio Propagation: Small-Scale Fading and Multipath

0 707
e I
183

For a data rate of 50 bps, the bit period is 20 ms. Since the bit period is greater
than the average fade duration, for the given data rate the signal undergoes
fast Rayleigh fading. Using equation (4.84), the average fade duration for
p 0.1 is equal to 0.002 s. This is less than the duration of one bit. Therefore,
only one bit on average will be lost during a fade. Using equation (4.80), the
number of level crossings for p = 0.1 is N,. = 4.96 crossings per seconds.
Since a bit error is assumed to occur whenever a portion of a bit encounters a
fade, and since average fade duration spans only a fraction of a bit duration,
the total number of bits in error is 5 per second, resulting in a BER = (5/50) =
0.1.

4.7.4 Two-ray Rayleigh Fading Model

Clarke's model and the statistics for Rayleigh fading are for flat fading con-
ditions and do not consider multipath time delay. In modern mobile communica-
tion systems with high data rates, it has become necessary to model the effects of
multipath delay spread as well as fading. A commonly used multipath model is
an independent Rayleigh fading 2-ray model (which is a specific implementation
of the generic fading simulator shown in Figure 4.24). Figure 4.25 shows a block
diagram of the 2-ray independent Rayleigh fading channel model. The impulse
response of the model is represented as
Fib (t) =
exp (j41)6(t) + a2exp
a1 (4.85)
where a1 and a2 are independent and Rayleigh distributed, and 42 are inde-
Rendent and uniformly distributed over [0, 2a] , and r is the time delay
the two rays. By setting a, equal to zero, the special case of a flat Ray-
leigh fading channel is obtained as
hb(t) = (4.86)
By varying r, it is possible to create a wide range of frequency selective fad-
ing effects. The proper time correlation properties of the Rayleigh random vari-
ables px and a2 are guaranteed by generating two independent waveforms,
each produced from the inverse Fourier transform of the spectrum described in
Section 4.7.2.

4.1.5 Saleh and Valenzuela Indoor Statistical Model

Saleh
and Valenzuela [Sal87] reported the results of indoor propagation
measurements between two vertically polarized omni-directional antennas
located on the same floor of a medium sized office building. Measurements were
made using 10 ns, 1.5 GHz, radar-like pulses. The method involved averaging
the square law detected pulse response while sweeping the frequency of the

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 Statistical Models for Multipath Fading Channels 189

a2exp(j42)

Figure 4.25
Two-ray Rayleigh fading model.

transmitted pulse. Using this method, multipath components within 5 ns were

resolvable.
The results obtained by Saleh and Valenzula show that: (a) the indoor chan-
nel is quasi-static or very slowly time varying, and (b) the statistics of the chan-
nel impulse response are independent of transmitting and receiving antenna
polarization, if there is no line-of-sight path between them. They reported a max-
imum multipath delay spread of 100 ns to 200 ns within the rooms of a building,
and 300 ns in hallways. The measured mis delay spread within rooms had a
median of 25 ns and a maximum of 50 ns. The large-scale path loss with no line-
of-sight path was found to vary over a 60 dB range and obey a log-distance power
law (see equation (3.68)) with an exponent between 3 and 4.
Saleh and Valenzuela developed a simple multipath model for indoor chan-
nels based on measurement results. The model assumes that the multipath com-
ponents arrive in clusters. The amplitudes of the received components are
independent Rayleigh random variables with variances that decay exponentially
with cluster delay as well as excess delay within a cluster. The corresponding
phase angles are independent uniform random variables over [0,2it}. The clus-
ters and multipath components within a cluster form Poisson arrival processes
with different rates. The clusters and multipath components within a cluster
have exponentially distributed interarrival times. The formation of the clusters
is related to the building structure, while the components within the cluster are
formed by multiple reflections from objects in the vicinity of the transmitter and
the receiver.

4.7.6 SIRCIM and SMRCIM Indoor and Outdoor Statistical Models

Rappaport and Seidel [Rap9la] reported measurements at 1300 MHz in
five factory buildings and carried out subsequent measurements in other types of

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 190 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and Multipatti

buildings. The authors developed an elaborate, empirically derived statistical

model based on the discrete impulse response channel model and wrote a coin-
puter program called SIRCIM (Simulation of Indoor Radio Channel Impulse-
response Models). SIRCIM generates realistic samples of small-scale indoor
channel impulse response measurements [Rap9laJ. Subsequent work by Huang
produced SMRCIM (Simulation of Mobile Radio Channel Impulse-response Mod-
els), a similar program that generates small-scale urban cellular and microcellu-
lar channel impulse responses [Rap93aJ. These programs are currently in use at
over 100 institutions throughout the world.
By recording power delay profile impulse responses at A/4 intervals on a 1
m track at many indoor measurement locations, the authors were able to charac-
terize local small-scale fading of individual multipath components, and the
small-scale variation in the number and arrival times of multipath components
within a local area. Thus, the resulting statistical models are functione of multi-
path time delay bin the small-scale receiver spacing, X1, within a 1 m local
area, the topography Sm which is either line-of-sight (LOS) or obstructed, the
large-scale T-R separation distance and the particular measurement loca-
tion Therefore, each individual baseband power delay profile is expressed in
a manner similar to equation (4.13), except the random amplitudes and time
delays are random variables which depend on the surrounding environment.
Phases are synthesized using a pseudo-deterministic model which provides real-
istic results, so that a complete time varying complex baseband channel impulse
response hb (t;r1) may be obtained over a local area through simulation.

hb (t, X1,

S Sm,

In equation (4.87), is the average multipath receiver power within a discrete

excess delay interval of 7.8125 ns.
The measured multipath delays inside open-plan buildings ranged from 40
ns to 800 ns. Mean multipath delay and rms delay spread values ranged from 30
ns to 300 ns, with median values of 96 ns in LOS paths and 105 ns in obstructed
paths. Delay spreads were found to be uncorrelated with T-R separation but
were affected by factory inventory; building construction materials, building age,
wall locations, and ceiling heights. Measurements in a food processing factory
that manufactures dry-goods and has considerably less metal inventory than
other factories had an rms delay spread that was half of those observed in facto-
ries producing metal products. Newer factories which incorporate steel beams
and steel reinforced concrete in the building structure have stronger inultipath
signals and less attenuation than older factories which used wood and brick for
perimeter walls. The data suggested that radio propagation in buildings may be
described by a hybrid geometric/statistical model that accounts for both reflec-

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 Statistical Models for Multipath Fading Channels 191

tions from walls and ceilings and random scattering from inventory and equip-
ment.
By analyzing the measurements from fifty local areas in many buildings, it
was found that the number of multipath components, arriving at a certain
location is a function ofX, and and almost always has a Gaussian distri-
bution. The average number of multipath components ranges between 9 and 36,
and is generated based on an empirical fit to measurements. The probability that
a multipath component will arrive at a receiver at a particular excess delay T4 in
a particular environment Sm is denoted as (Ti, 5m) This was found from
measurements by counting the total number of detected multipath components
at a particular discrete excess delay time, and dividing by the total number of
possible multipath components for each excess delay interval. The probabilities
for multipath arriving at a particular excess delay values may be modeled as
piecewise functions of excess delay, and are given by
T
1fl67
(T.II0)
= <0.65
4

360
(110 nscT.<200 ns) (4.88)
for LOS
(T.200)
[0.22 (200 ns cT <500 ns)

= 0.55 < 100 ns)

667
(4.89)
forOBS . .

008+O67exp (IOOnscT,<SOOns)
where corresponds to the LOS topography, and corresponds to obstructed
topography. SIRCIM uses the probability of arrival distributions described by
equation (4.88) or (4.89) along with the probability distribution of the number of
multipath components, (X, 5m' Pa), to simulate power delay profiles over
small-scale distances. A recursive algorithm repeatedly compares equation (4.88)
or (4.89) with a uniformly distributed random variable until the proper is
generated for each profile [Hua9l], [Rap9la].
Figure 4.26 shows an example of measured power delay profiles at 19 dis-
crete receiver locations along a 1 m track, and illustrates accompanying narrow-
band information which SIRCIM computes based on synthesized phases for each
multipath component [Rap9laJ. Measurements reported in the literature pro-
vide excellent agreement with impulse responses predicted by SIRCIM.
Using similar statistical modeling techniques, urban cellular and microcel-
lular multipath measurement data from [Rap9O}, [Sei9l], [Sei92a] were used to
develop SMRCIM. Both large cell and microcell models were developed. Figure

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 192 Ch. 4' Mobile Radio Propagation: Small-Scale Fading and Multipath

4.27 shows an example of SMRCIM output for an outdoor microcell environment

[Rap93aII.

Figure 4.26
Indoor wide band impulse responses simulated by SIRCIM at 1.3 GHz. Also shown are the distribu-
tions of the ntis delay spread and the narrowband signal power distribution. The channel is simu-
lated as being obstructed in an open-plan building, T-R separation is 25 m. The rms delay spread is
137.7 ns. All multipath components and parameters are stored on disk [From [Rap93a] IEEE).

4.8 Problems
4.1 Draw a block diagram of a binary spread spectrum sliding correlator multipath
measurement system. Explain in words how it is used to measure power delay
profiles.
(a) If the transmitter chip period is 10 ns, the PN sequence has length 1023,
and a 6 0Hz carrier is used at the transmitter, find the time between
maximal correlation and the slide factor if the receiver uses a PN
sequence clock that is 30 kI-Iz slower than the transmitter.
(b) If an oscilloscope is used to display one complete cycle of the PN sequence
(that is, if two successive maximal correlation peaks are to be displayed
on the oscilloscope), and if 10 divisions are provided on the oscilloscope
time axis, what is the most appropriate sweep setting (in secorids/divi-

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 Problems i

Figure 4.27
Urban wideband impulse responses simulated by SMRCIM at 1.3 GHz. Also shown are the distribu-
tions of the rms delay spread and the narrowband fading. T-R separation is 2.68 km. The ruts delay
spread is 3.8 All multipath components and parameters are saved on disk. [From [Rap93a]
IEEE
sion) to be used?
(c) What is the required IF passband bandwidth for this system? How is this
much better than a direct pulse system with similar time resolution?
4.2 If a particular modulation provides suitable BELt performance whenever
0.!, determine the smallest symbol period (and thus the greatest
symbol rate) that may be sent through RF channels shown in Figure P4.2,
without using an equalizer.
4.3 For the power delay profiles in Figure P4.2, estimate the 90% correlation and
50% correlation coherence bandwidths.
4.4 Approximately how large can the rms delay spread be in order for a binary
modulated signal with a bit rate of 25 kbps to operate without an equalizer?
What about an 8-PSK system with a bit rate of 75 kbps?
4.5 Given that the probability density function of a Rayleigh distributed envelope

is given by p(r) = ;exp[4] , where a2 is the variance, show that the

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 194 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and Multipath

P/t) PJr)
0dB: indoor
0dB
-10dB -

2OdB -20dB-

-30dB- -30dB-

0 50 75 100 excess 0 5 10
(a) delay (ns) (b)
Figure P4.2: Two channel responses for Problem 4.2

cumulative distnbution function is gwen as p (r <R) = I expj Find

I
the percentage of tine that a signal is 10 dB or more below the rms value for a
Rayleigh fading signal.
4.6 The fading characteristics of a CW carrier in an urban area are to be mea-
sured. The following assumptions are made:
(1) The mobile receiver uses a simple vertical monopole
(2) Large-scale fading due to path loss is ignored.
(3) The mobile has no line-of-sight path to the base station.
(4) The pdf of the received signal follows a Rayleigh distribution.
(a) Derive the ratio of the desired signal level to the rms signal level that
maximizes the level crossing rate. Express your answer in dB.
(b) Assuming the maximum velocity of the mobile is 50 km/br, and the car-
rier frequency is 900 MHz, determine the maximum number of times the
signal envelope will fade below the level found in (a) during a 1 minute
test.
(c) How long, on average, will each fade in (b) last?
4.7 A vehicle receives a 900 MHz transmission while traveling at a constant veloc-
ity for 10 s. The average fade duration for a signal level 10 dB below the rms
level is 1 ms. How far does the vehicle travel during the 10 s interval? How
many fades does the signal undergo at the rms threshold level during a 10 s
interval? Assume that the local mean remains constant during travel.
4.8 An automobile moves with velocity u('t) shown in Figure P4.8. The received
mobile signal experiences inultipath Rayleigh fading on a 900 MHz CW carrier.
What is the average crossing rate and fade duration over the 100 s interval?
Assume p = 0.1 and ignore large-scale fading effects.

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 Ch. 4. Mobile Radio Propagation: Small-Scale Fading and Multipath 195

ta)
(in mis)

30/.

0 10 90 100
time (s)
Figure P4.8 Graph of velocity of mobile.

4.9 For a mobile receiver operating at frequency of 860 MHz and moving at 100
kmIhr
(a) sketch the Doppler spectrum if a CW signal is transmitted and indicate
the maximum and minium frequencies
(b) calculate the level crossing rate and average fade duration if p = 20 dB-

4.10 For the following digital wireless systems, estimate the maximum rms delay
spread for which no equalizer is required at the receiver (neglect channel cod-
ing, antenna diversity, or use of extremely low power levels).
System RF Data Rate Modulation
USDC 48.6 kbps ,ti4DQPSK
GSM 270.833 kbps GMSK
DECT 1152 kbps GMSK

4.11 Derive the RF Doppler spectrum for a 5i8k vertical monopole receiving a CW
signal using the models by Clarke and Gang. Plot the RF Doppler spectrum
and the corresponding baseband spectrum out of an envelope detector. Assume
isotropic scattering and unit average received power.
4.12 Show that the magnitude (envelope) of sum of two independent identically
distributed complex (quadrature) Gaussian sources is Rayleigh distributed.
Assume that the Gaussian sources are zero mean and have unit variance.
4.13 Using the method described in Chapter 4, generate a time sequence of 8192
sample values of a Rayleigh fading signal for
(a) Td = 20 Hzand(b) Id = 200 Hz.
4.14 Generate 100 sample functions of fading data described in Problem 4.13, and
compare the simulated and theoretical values of RRMS, NR, and t for p = 1,
0.1, and 0.01. Do your simulations agree with theory?
4.15 Recreate the plots of the DFs shown in Figure 4.17, starting with the pdfs fo;
Rayleigh, Ricean, and log-normal distributions.
4i6 Plot the probability density function and the CDF for a Ricean distribution
having (a) K = 10 dB and (b) K = 3 dB. The abscissa of the CDF plot should be
labeled in dE relative to the median signal level for both plots. Note that the
median value for a Ricean distribution changes as K changes.

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 196 Ch. 4 Mobile Radio Propagation: Small-Scale Fading and Mulhpath

4.17 Based on your answer in Problem 4.16, if the median SNR is -70 dBm, what is
the likelihood that a signal greater than -80 dBm will be received in a Ricean
fading channel having (a) K = 10 dB, and (b) K = 3 dE?
4.18 A local spatial average of a power delay profile is shown in Figure P4.18.

Prtr)
0 dB

-10dB
-20dB

-30 dB

0
(ps)
Figure P4.18. Power delay profile.

(a) Determine the rms delay spread and mean excess delay for the channel.
(b) Determine the maximum excess delay (20 dB).
(c) If the channel is to be used with a modulation that requires an equalizer
whenever the symbol duration T is less than determine the maxim urn
RF symbol rate that can be supported without requiring an equalizer.
(d) If a mobile traveling at 30 km/lu receives a signal through the channel,
determine the time over which the channel appears stationary (or at least
highly correlated).
4.19 A flat Rayleigh fading signal at 6 GHz is received by a mobile traveling at 80
kmthr.
(a) Determine the number of positive-going zero crossings about the ms value
that occur over a 5s interval.
(b) Determine the average duration of a fade below the rms level.
(c) Determine the average duration of a fade at a level of 20 dB below the ms
value.
4.20 Using computer simulation, create a Rayleigh fading simulator that has 3
independent Rayleigh fading multipath components, each having variable
multipath time delay and average power. Then convolve a random binary bit
stream through your simulator and observe the time waveforms of the output
stream. You may wish to use several samples for each bit (7 is a good number).
Observe the effects of multipath spread as you vary the bit period and time
delay of the ehannel.
4.21 Based on concepts taught in this chapter, propose methods that could be used
by a base station to determine the vehicular speed of a mobile user. Such meth-
ods are useful for handoff algorithms.

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