2 The wireless channel
A good understanding of the wireless channel, its key physical parameters
and the modeling issues, lays the foundation for the rest of the book. This is
the goal of this chapter.
A defining characteristic of the mobile wireless channel is the variations
of the channel strength over time and over frequency. The variations can be
roughly divided into two types (Figure 2.1):
• Large-scale fading, due to path loss of signal as a function of distance
and shadowing by large objects such as buildings and hills. This occurs as
the mobile moves through a distance of the order of the cell size, and is
typically frequency independent.
• Small-scale fading, due to the constructive and destructive interference of the
multiple signal paths between the transmitter and receiver. This occurs at the
spatial scale of the order of the carrier wavelength, and is frequency dependent.
We will talk about both types of fading in this chapter, but with more
emphasis on the latter. Large-scale fading is more relevant to issues such as
cell-site planning. Small-scale multipath fading is more relevant to the design
of reliable and efficient communication systems – the focus of this book.
We start with the physical modeling of the wireless channel in terms of electromagnetic
waves. We then derive an input/output linear time-varying model
for the channel, and define some important physical parameters. Finally, we
introduce a few statistical models of the channel variation over time and over
frequency.
2.1 Physical modeling for wireless channels
Wireless channels operate through electromagnetic radiation from the transmitter
to the receiver. In principle, one could solve the electromagnetic
field equations, in conjunction with the transmitted signal, to find the
10
11 2.1 Physical modeling for wireless channels
Figure 2.1 Channel quality
varies over multiple
time-scales. At a slow scale,
channel varies due to
large-scale fading effects. At a
fast scale, channel varies due
to multipath effects.
Time
Channel quality
electromagnetic field impinging on the receiver antenna. This would have to
be done taking into account the obstructions caused by ground, buildings,
vehicles, etc. in the vicinity of this electromagnetic wave.1
Cellular communication in the USA is limited by the Federal Communication
Commission (FCC), and by similar authorities in other countries,
to one of three frequency bands, one around 0.9 GHz, one around 1.9 GHz,
and one around 5.8 GHz. The wavelength of electromagnetic radiation at
any given frequency f is given by = c/f , where c = 3×108m/s is the
speed of light. The wavelength in these cellular bands is thus a fraction of a
meter, so to calculate the electromagnetic field at a receiver, the locations of
the receiver and the obstructions would have to be known within sub-meter
accuracies. The electromagnetic field equations are therefore too complex to
solve, especially on the fly for mobile users. Thus, we have to ask what we
really need to know about these channels, and what approximations might be
reasonable.
One of the important questions is where to choose to place the base-stations,
and what range of power levels are then necessary on the downlink and uplink
channels. To some extent this question must be answered experimentally, but
it certainly helps to have a sense of what types of phenomena to expect.
Another major question is what types of modulation and detection techniques
look promising. Here again, we need a sense of what types of phenomena to
expect. To address this, we will construct stochastic models of the channel,
assuming that different channel behaviors appear with different probabilities,
and change over time (with specific stochastic properties). We will return to
the question of why such stochastic models are appropriate, but for now we
simply want to explore the gross characteristics of these channels. Let us start
by looking at several over-idealized examples.
1 By obstructions, we mean not only objects in the line-of-sight between transmitter and
receiver, but also objects in locations that cause non-negligible changes in the electromagnetic
field at the receiver; we shall see examples of such obstructions later.
12 The wireless channel
2.1.1 Free space, fixed transmit and receive antennas
First consider a fixed antenna radiating into free space. In the far field,2 the
electric field and magnetic field at any given location are perpendicular both
to each other and to the direction of propagation from the antenna. They
are also proportional to each other, so it is sufficient to know only one of
them ( just as in wired communication, where we view a signal as simply
a voltage waveform or a current waveform). In response to a transmitted
sinusoid cos 2 ft, we can express the electric far field at time t as
E f t r = s f cos 2 f t−r/c
r
(2.1)
Here, r represents the point u in space at which the electric field is
being measured, where r is the distance from the transmit antenna to u and
where represents the vertical and horizontal angles from the antenna
to u respectively. The constant c is the speed of light, and s f is the
radiation pattern of the sending antenna at frequency f in the direction ;
it also contains a scaling factor to account for antenna losses. Note that the
phase of the field varies with fr/c, corresponding to the delay caused by the
radiation traveling at the speed of light.
We are not concerned here with actually finding the radiation pattern for
any given antenna, but only with recognizing that antennas have radiation
patterns, and that the free space far field behaves as above.
It is important to observe that, as the distance r increases, the electric field
decreases as r−1 and thus the power per square meter in the free space wave
decreases as r−2. This is expected, since if we look at concentric spheres of
increasing radius r around the antenna, the total power radiated through the
sphere remains constant, but the surface area increases as r2. Thus, the power
per unit area must decrease as r−2. We will see shortly that this r−2 reduction
of power with distance is often not valid when there are obstructions to free
space propagation.
Next, suppose there is a fixed receive antenna at the location u = r .
The received waveform (in the absence of noise) in response to the above
transmitted sinusoid is then
Er f t u = f cos 2 f t−r/c
r
(2.2)
where f is the product of the antenna patterns of transmit and receive
antennas in the given direction. Our approach to (2.2) is a bit odd since we
started with the free space field at u in the absence of an antenna. Placing a
2 The far field is the field sufficiently far away from the antenna so that (2.1) is valid. For
cellular systems, it is a safe assumption that the receiver is in the far field.
13 2.1 Physical modeling for wireless channels
receive antenna there changes the electric field in the vicinity of u, but this
is taken into account by the antenna pattern of the receive antenna.
Now suppose, for the given u, that we define
H f
= f e −j2 fr/c
r
(2.3)
We then have Er f t u = H f e j2 ft . We have not mentioned it yet,
but (2.1) and (2.2) are both linear in the input. That is, the received field
(waveform) at u in response to a weighted sum of transmitted waveforms is
simply the weighted sum of responses to those individual waveforms. Thus,
H f is the system function for an LTI (linear time-invariant) channel, and its
inverse Fourier transform is the impulse response. The need for understanding
electromagnetism is to determine what this system function is. We will find in
what follows that linearity is a good assumption for all the wireless channels
we consider, but that the time invariance does not hold when either the
antennas or obstructions are in relative motion.
2.1.2 Free space, moving antenna
Next consider the fixed antenna and free space model above with a receive
antenna that is moving with speed v in the direction of increasing distance
from the transmit antenna. That is, we assume that the receive antenna is at
a moving location described as u t = r t with r t = r0
+vt. Using
(2.1) to describe the free space electric field at the moving point u t (for the
moment with no receive antenna), we have
E f t r0
+vt = s f cos 2 f t−r0/c−vt/c
r0
+vt
(2.4)
Note that we can rewrite f t −r0/c −vt/c as f 1−v/c t −fr0/c. Thus,
the sinusoid at frequency f has been converted to a sinusoid of frequency
f 1−v/c ; there has been a Doppler shift of −fv/c due to the motion of
the observation point.3 Intuitively, each successive crest in the transmitted
sinusoid has to travel a little further before it gets observed at the moving
observation point. If the antenna is now placed at u t , and the change of
field due to the antenna presence is again represented by the receive antenna
pattern, the received waveform, in analogy to (2.2), is
Er f t r0
+vt = f cos 2 f 1−v/c t−r0/c
r0
+vt
(2.5)
3 The reader should be familiar with the Doppler shift associated with moving cars. When an
ambulance is rapidly moving toward us we hear a higher frequency siren. When it passes us
we hear a rapid shift toward a lower frequency.
14 The wireless channel
This channel cannot be represented as an LTI channel. If we ignore the timevarying
attenuation in the denominator of (2.5), however, we can represent the
channel in terms of a system function followed by translating the frequency f
by the Doppler shift −fv/c. It is important to observe that the amount of shift
depends on the frequency f. We will come back to discussing the importance
of this Doppler shift and of the time-varying attenuation after considering the
next example.
The above analysis does not depend on whether it is the transmitter or
the receiver (or both) that are moving. So long as r t is interpreted as the
distance between the antennas (and the relative orientations of the antennas
are constant), (2.4) and (2.5) are valid.
2.1.3 Reflecting wall, fixed antenna
Consider Figure 2.2 in which there is a fixed antenna transmitting the sinusoid
cos 2 ft, a fixed receive antenna, and a single perfectly reflecting large fixed
wall. We assume that in the absence of the receive antenna, the electromagnetic
field at the point where the receive antenna will be placed is the sum of
the free space field coming from the transmit antenna plus a reflected wave
coming from the wall. As before, in the presence of the receive antenna, the
perturbation of the field due to the antenna is represented by the antenna pattern.
An additional assumption here is that the presence of the receive antenna does
not appreciably affect the plane wave impinging on the wall. In essence, what
we have done here is to approximate the solution of Maxwell’s equations by a
method called ray tracing. The assumption here is that the received waveform
can be approximated by the sum of the free space wave from the transmitter plus
the reflected free space waves from each of the reflecting obstacles.
In the present situation, if we assume that the wall is very large, the reflected
wave at a given point is the same (except for a sign change4) as the free space
wave that would exist on the opposite side of the wall if the wall were not present
(see Figure 2.3). Thismeansthat the reflectedwavefromthe wall has the intensity
of a free space wave at a distance equal to the distance to the wall and then
Figure 2.2 Illustration of a
direct path and a reflected
path.
Wall
Transmit
antenna
Receive antenna
r
d
4 By basic electromagnetics, this sign change is a consequence of the fact that the electric field is
parallel to the plane of the wall for this example.
15 2.1 Physical modeling for wireless channels
Figure 2.3 Relation of reflected
wave to wave without wall.
Transmit
antenna Wall
back to the receive antenna, i.e., 2d−r. Using (2.2) for both the direct and the
reflected wave, and assuming the same antenna gain for both waves, we get
Er f t = cos 2 f t−r/c
r
− cos 2 f t− 2d−r /c
2d−r
(2.6)
The received signal is a superposition of two waves, both of frequency f.
The phase difference between the two waves is
= 2 f 2d−r
c
+
− 2 fr
c
= 4 f
c
d−r + (2.7)
When the phase difference is an integer multiple of 2 , the two waves add
constructively, and the received signal is strong. When the phase difference
is an odd integer multiple of , the two waves add destructively, and the
received signal is weak. As a function of r, this translates into a spatial pattern
of constructive and destructive interference of the waves. The distance from
a peak to a valley is called the coherence distance:
xc
=
4
(2.8)
where
= c/f is the wavelength of the transmitted sinusoid. At distances
much smaller than
xc, the received signal at a particular time does not
change appreciably.
The constructive and destructive interference pattern also depends on the
frequency f: for a fixed r, if f changes by
1
2
2d−r
c
− r
c
−1
(2.9)
we move from a peak to a valley. The quantity
Td
= 2d−r
c
− r
c
(2.10)
is called the delay spread of the channel: it is the difference between the propagation
delays along the two signal paths. The constructive and destructive interference
pattern does not change appreciably if the frequency changes by an amount
much smaller than 1/Td. This parameter is called the coherence bandwidth.
16 The wireless channel
2.1.4 Reflecting wall, moving antenna
Suppose the receive antenna is now moving at a velocity v (Figure 2.4). As it
moves through the pattern of constructive and destructive interference created
by the two waves, the strength of the received signal increases and decreases.
This is the phenomenon of multipath fading. The time taken to travel from a
peak to a valley is c/ 4fv : this is the time-scale at which the fading occurs,
and it is called the coherence time of the channel.
An equivalent way of seeing this is in terms of the Doppler shifts of the
direct and the reflected waves. Suppose the receive antenna is at location r0
at time 0. Taking r = r0
+vt in (2.6), we get
Er f t = cos 2 f 1−v/c t−r0/c
r0
+vt
− cos 2 f 1+v/c t+ r0
−2d /c
2d−r0
−vt
(2.11)
The first term, the direct wave, is a sinusoid at frequency f 1−v/c , experiencing
a Doppler shift D1
=−fv/c. The second is a sinusoid at frequency
f 1+v/c , with a Doppler shift D2
=+fv/c. The parameter
Ds
= D2
−D1 (2.12)
is called the Doppler spread. For example, if the mobile is moving at 60 km/h
and f = 900 MHz, the Doppler spread is 100 Hz. The role of the Doppler
spread can be visualized most easily when the mobile is much closer to the
wall than to the transmit antenna. In this case the attenuations are roughly the
same for both paths, and we can approximate the denominator of the second
term by r = r0
+vt. Then, combining the two sinusoids, we get
Er f t ≈ 2 sin 2 f vt/c+ r0
−d /c sin 2 f t−d/c
r0
+vt
(2.13)
This is the product of two sinusoids, one at the input frequency f, which is typically
of the order of GHz, and the other one at fv/c=Ds/2, which might be of
the order of 50 Hz. Thus, the response to a sinusoid at f is another sinusoid at
f with a time-varying envelope, with peaks going to zeros around every 5 ms
(Figure 2.5). The envelope is at its widest when the mobile is at a peak of the
Figure 2.4 Illustration of a
direct path and a reflected
path.
Wall
Transmit
antenna
r (t)
d
υ
17 2.1 Physical modeling for wireless channels
Figure 2.5 The received
waveform oscillating at
frequency f with a slowly
varying envelope at frequency
Ds/2.
t
Er (t)
interference pattern and at its narrowest when the mobile is at a valley. Thus,
the Doppler spread determines the rate of traversal across the interference
pattern and is inversely proportional to the coherence time of the channel.
We now see why we have partially ignored the denominator terms in (2.11)
and (2.13). When the difference in the length between two paths changes by
a quarter wavelength, the phase difference between the responses on the two
paths changes by /2, which causes a very significant change in the overall
received amplitude. Since the carrier wavelength is very small relative to
the path lengths, the time over which this phase effect causes a significant
change is far smaller than the time over which the denominator terms cause
a significant change. The effect of the phase changes is of the order of
milliseconds, whereas the effect of changes in the denominator is of the order
of seconds or minutes. In terms of modulation and detection, the time-scales
of interest are in the range of milliseconds and less, and the denominators are
effectively constant over these periods.
The reader might notice that we are constantly making approximations in
trying to understand wireless communication, much more so than for wired
communication. This is partly because wired channels are typically timeinvariant
over a very long time-scale, while wireless channels are typically
time-varying, and appropriate models depend very much on the time-scales of
interest. For wireless systems, the most important issue is what approximations
to make. Thus, it is important to understand these modeling issues thoroughly.
2.1.5 Reflection from a ground plane
Consider a transmit and a receive antenna, both above a plane surface such
as a road (Figure 2.6). When the horizontal distance r between the antennas
becomes very large relative to their vertical displacements from the ground
18 The wireless channel
Figure 2.6 Illustration of a
direct path and a reflected
path off a ground plane.
Transmit antenna
Ground plane
Receive antenna
hr
hs
r2
r
r1
plane (i.e., height), a very surprising thing happens. In particular, the difference
between the direct path length and the reflected path length goes to zero
as r−1 with increasing r (Exercise 2.5). When r is large enough, this difference
between the path lengths becomes small relative to the wavelength c/f . Since
the sign of the electric field is reversed on the reflected path5, these two waves
start to cancel each other out. The electric wave at the receiver is then attenuated
as r−2, and the received power decreases as r−4. This situation is particularly
important in rural areas where base-stations tend to be placed on roads.
2.1.6 Power decay with distance and shadowing
The previous example with reflection from a ground plane suggests that the
received power can decrease with distance faster than r−2 in the presence of
disturbances to free space. In practice, there are several obstacles between
the transmitter and the receiver and, further, the obstacles might also absorb
some power while scattering the rest. Thus, one expects the power decay to
be considerably faster than r−2. Indeed, empirical evidence from experimental
field studies suggests that while power decay near the transmitter is like r−2,
at large distances the power can even decay exponentially with distance.
The ray tracing approach used so far provides a high degree of numerical
accuracy in determining the electric field at the receiver, but requires a precise
physical model including the location of the obstacles. But here, we are only
looking for the order of decay of power with distance and can consider an
alternative approach. So we look for a model of the physical environment with
the fewest parameters but one that still provides useful global information
about the field properties. A simple probabilistic model with two parameters
of the physical environment, the density of the obstacles and the fraction of
energy each object absorbs, is developed in Exercise 2.6. With each obstacle
5 This is clearly true if the electric field is parallel to the ground plane. It turns out that this is
also true for arbitrary orientations of the electric field, as long as the ground is not a perfect
conductor and the angle of incidence is small enough. The underlying electromagnetics is
analyzed in Chapter 2 of Jakes [62].
19 2.1 Physical modeling for wireless channels
absorbing the same fraction of the energy impinging on it, the model allows
us to show that the power decays exponentially in distance at a rate that is
proportional to the density of the obstacles.
With a limit on the transmit power (either at the base-station or at the
mobile), the largest distance between the base-station and a mobile at which
communication can reliably take place is called the coverage of the cell. For
reliable communication, a minimal received power level has to be met and
thus the fast decay of power with distance constrains cell coverage. On the
other hand, rapid signal attenuation with distance is also helpful; it reduces the
interference between adjacent cells. As cellular systems become more popular,
however, the major determinant of cell size is the number of mobiles in the
cell. In engineering jargon, the cell is said to be capacity limited instead of
coverage limited. The size of cells has been steadily decreasing, and one talks
of micro cells and pico cells as a response to this effect. With capacity limited
cells, the inter-cell interference may be intolerably high. To alleviate the
inter-cell interference, neighboring cells use different parts of the frequency
spectrum, and frequency is reused at cells that are far enough. Rapid signal
attenuation with distance allows frequencies to be reused at closer distances.
The density of obstacles between the transmit and receive antennas depends
very much on the physical environment. For example, outdoor plains have
very little by way of obstacles while indoor environments pose many obstacles.
This randomness in the environment is captured by modeling the density
of obstacles and their absorption behavior as random numbers; the overall
phenomenon is called shadowing.6 The effect of shadow fading differs from
multipath fading in an important way. The duration of a shadow fade lasts for
multiple seconds or minutes, and hence occurs at a much slower time-scale
compared to multipath fading.
2.1.7 Moving antenna, multiple reflectors
Dealing with multiple reflectors, using the technique of ray tracing, is in principle
simply a matter of modeling the received waveform as the sum of the responses
from the different paths rather than just two paths. We have seen enough examples,
however, to understand that finding the magnitudes and phases of these
responses is no simple task. Even for the very simple large wall example in
Figure 2.2, the reflected field calculated in (2.6) is valid only at distances from
the wall that are small relative to the dimensions of the wall. At very large distances,
the total power reflected from the wall is proportional to both d−2 and
to the area of the cross section of the wall. The power reaching the receiver is
proportional to d −r t −2. Thus, the power attenuation from transmitter to
receiver (for the large distance case) is proportional to d d−r t −2 rather
6 This is called shadowing because it is similar to the effect of clouds partly blocking sunlight.
20 The wireless channel
than to 2d − r t −2. This shows that ray tracing must be used with some
caution. Fortunately, however, linearity still holds in these more complex cases.
Another type of reflection is known as scattering and can occur in the
atmosphere or in reflections from very rough objects. Here there are a very
large number of individual paths, and the received waveform is better modeled
as an integral over paths with infinitesimally small differences in their lengths,
rather than as a sum.
Knowing how to find the amplitude of the reflected field from each type
of reflector is helpful in determining the coverage of a base-station (although
ultimately experimentation is necessary). This is an important topic if our
objective is trying to determine where to place base-stations. Studying this in
more depth, however, would take us afield and too far into electromagnetic
theory. In addition, we are primarily interested in questions of modulation,
detection, multiple access, and network protocols rather than location of
base-stations. Thus, we turn our attention to understanding the nature of the
aggregate received waveform, given a representation for each reflected wave.
This leads to modeling the input/output behavior of a channel rather than the
detailed response on each path.
2.2 Input/output model of the wireless channel
We derive an input/output model in this section. We first show that the multipath
effects can be modeled as a linear time-varying system. We then obtain
a baseband representation of this model. The continuous-time channel is then
sampled to obtain a discrete-time model. Finally we incorporate additive noise.
2.2.1 The wireless channel as a linear time-varying system
In the previous section we focused on the response to the sinusoidal input
t =cos 2 ft. Thereceived signal can be written as i ai f t t− i f t ,
where ai f t and i f t are respectively the overall attenuation and propagation
delay at time t from the transmitter to the receiver on path i. The
overall attenuation is simply the product of the attenuation factors due to the
antenna pattern of the transmitter and the receiver, the nature of the reflector,
as well as a factor that is a function of the distance from the transmitting
antenna to the reflector and from the reflector to the receive antenna. We have
described the channel effect at a particular frequency f. If we further assume
that the ai f t and the i f t do not depend on the frequency f, then we
can use the principle of superposition to generalize the above input/output
relation to an arbitrary input x t with non-zero bandwidth:
y t =
i
ai t x t− i t (2.14)
21 2.2 Input/output model of the wireless channel
In practice the attenuations and the propagation delays are usually slowly
varying functions of frequency. These variations follow from the time-varying
path lengths and also from frequency-dependent antenna gains. However, we
are primarily interested in transmitting over bands that are narrow relative
to the carrier frequency, and over such ranges we can omit this frequency
dependence. It should however be noted that although the individual attenuations
and delays are assumed to be independent of the frequency, the overall
channel response can still vary with frequency due to the fact that different
paths have different delays.
For the example of a perfectly reflecting wall in Figure 2.4, then,
a1 t =
r0
+vt
a2 t =
2d−r0
−vt
(2.15)
1 t = r0
+vt
c
− ∠ 1
2 f
2 t = 2d−r0
−vt
c
− ∠ 2
2 f
(2.16)
where the first expression is for the direct path and the second for the reflected
path. The term ∠ j here is to account for possible phase changes at the
transmitter, reflector, and receiver. For the example here, there is a phase
reversal at the reflector so we take 1
= 0 and 2
= .
Since the channel (2.14) is linear, it can be described by the response
h t at time t to an impulse transmitted at time t− . In terms of h t ,
the input/output relationship is given by
y t =
−
h t x t− d (2.17)
Comparing (2.17) and (2.14), we see that the impulse response for the fading
multipath channel is
h t =
i
ai t − i t (2.18)
This expression is really quite nice. It says that the effect of mobile users,
arbitrarily moving reflectors and absorbers, and all of the complexities of solving
Maxwell’s equations, finally reduce to an input/output relation between
transmit and receive antennas which is simply represented as the impulse
response of a linear time-varying channel filter.
The effect of the Doppler shift is not immediately evident in this representation.
From (2.16) for the single reflecting wall example,
i t = vi/c
where vi is the velocity with which the ith path length is increasing. Thus,
the Doppler shift on the ith path is −f
i t .
In the special case when the transmitter, receiver and the environment
are all stationary, the attenuations ai t and propagation delays i t do not
22 The wireless channel
depend on time t, and we have the usual linear time-invariant channel with
an impulse response
h =
i
ai − i (2.19)
For the time-varying impulse response h t , we can define a time-varying
frequency response
H f t
=
−
h t e−j2 f d =
i
ai t e−j2 f i t (2.20)
In the special case when the channel is time-invariant, this reduces to the
usual frequency response. One way of interpreting H f t is to think of the
system as a slowly varying function of t with a frequency response H f t
at each fixed time t. Corresponding, h t can be thought of as the impulse
response of the system at a fixed time t. This is a legitimate and useful
way of thinking about many multipath fading channels, as the time-scale
at which the channel varies is typically much longer than the delay spread
(i.e., the amount of memory) of the impulse response at a fixed time. In the
reflecting wall example in Section 2.1.4, the time taken for the channel to
change significantly is of the order of milliseconds while the delay spread is
of the order of microseconds. Fading channels which have this characteristic
are sometimes called underspread channels.
2.2.2 Baseband equivalent model
In typical wireless applications, communication occurs in a passband
fc
−W/2 fc
+W/2 of bandwidth W around a center frequency fc, the
spectrum having been specified by regulatory authorities. However, most
of the processing, such as coding/decoding, modulation/demodulation,
synchronization, etc., is actually done at the baseband. At the transmitter, the
last stage of the operation is to “up-convert” the signal to the carrier frequency
and transmit it via the antenna. Similarly, the first step at the receiver is to
“down-convert” the RF (radio-frequency) signal to the baseband before further
processing. Therefore from a communication system design point of view, it
is most useful to have a baseband equivalent representation of the system.
We first start with defining the baseband equivalent representation of signals.
Consider a real signal s t with Fourier transform S f , band-limited in
fc
−W/2 fc
+W/2 with W<2fc. Define its complex baseband equivalent
sb t as the signal having Fourier transform:
Sb f =
√
2S f +fc f+fc > 0
0 f +fc
≤ 0
(2.21)
23 2.2 Input/output model of the wireless channel
Figure 2.7 Illustration of the
relationship between a
passband spectrum S(f ) and
its baseband equivalent Sb(f ).
W
2
1
Sb ( f )
S( f )
f
f
–fc – W
2
fc – W
2
– fc
W
2
+ W
2
fc +
W
2
–
√2
Since s t is real, its Fourier transform satisfies S f =S∗ −f , which means
that sb t contains exactly the same information as s t . The factor of
√
2 is
quite arbitrary but chosen to normalize the energies of sb t and s t to be
the same. Note that sb t is band-limited in −W/2 W/2 . See Figure 2.7.
To reconstruct s t from sb t , we observe that
√
2S f = Sb f −fc +S∗
b −f −fc (2.22)
Taking inverse Fourier transforms, we get
s t = 1 √
2
sb t ej2 fct +s∗
b t e−j2 fct =
√
2 sb t ej2 fct (2.23)
In terms of real signals, the relationship between s t and sb t is
shown in Figure 2.8. The passband signal s t is obtained by modulating
sb t by
√
2 cos 2 fct and sb t by −
√
2 sin 2 fct and summing, to
get
√
2 sb t ej2 fct (up-conversion). The baseband signal sb t (respectively
sb t ) is obtained by modulating s t by
√
2 cos 2 fct (respectively
−
√
2 sin 2 fct) followed by ideal low-pass filtering at the baseband
−W/2 W/2 (down-conversion).
Let us now go back to the multipath fading channel (2.14) with impulse
response given by (2.18). Let xb t and yb t be the complex baseband
equivalents of the transmitted signal x t and the received signal y t ,
respectively. Figure 2.9 shows the system diagram from xb t to yb t . This
implementation of a passband communication system is known as quadrature
amplitude modulation (QAM). The signal xb t is sometimes called the
24 The wireless channel
Figure 2.8 Illustration of
upconversion from sb(t) to
s(t), followed by
downconversion from s(t)
back to sb(t). X
X
X
X
[sb(t)]
[sb(t)]
[sb(t)]
[sb(t)]
–√2 sin 2π fc t –√2 sin 2π fc t
√2 cos 2π fc √2 cos 2π fc t t
s(t)
–W
2
W2
–W
2
W2
1
1
+
Figure 2.9 System diagram
from the baseband transmitted
signal xb(t) to the baseband
received signal yb(t).
X
X
X
X
[xb(t)]
[xb(t)]
[yb(t)]
[yb(t)]
–W
2
W2
–W
2
W2
1
1
+
x(t) y(t)
h(τ, t)
–√2 sin 2π fc t –√2 sin 2π fc t
√2 cos 2π fc √2 cos 2π fc t t
in-phase component I and xb t the quadrature component Q (rotated
by /2). We now calculate the baseband equivalent channel. Substituting
x t =
√
2 xb t e j2 fct and y t =
√
2 yb t e j2 fct into (2.14) we get
yb t e j2 fct =
i
ai t xb t− i t e j2 fc t− i t
=
i
ai t xb t− i t e−j2 fc i t
e j2 fct
(2.24)
Similarly, one can obtain (Exercise 2.13)
yb t e j2 fct =
i
ai t xb t− i t e−j2 fc i t
e j2 fct
(2.25)
Hence, the baseband equivalent channel is
yb t =
i
ab
i t xb t− i t (2.26)
25 2.2 Input/output model of the wireless channel
where
ab
i t
= ai t e−j2 fc i t (2.27)
The input/output relationship in (2.26) is also that of a linear time-varying
system, and the baseband equivalent impulse response is
hb t =
i
ab
i t − i t (2.28)
This representation is easy to interpret in the time domain, where the effect
of the carrier frequency can be seen explicitly. The baseband output is the
sum, over each path, of the delayed replicas of the baseband input. The
magnitude of the ith such term is the magnitude of the response on the given
path; this changes slowly, with significant changes occurring on the order of
seconds or more. The phase is changed by /2 (i.e., is changed significantly)
when the delay on the path changes by 1/ 4fc , or equivalently, when the
path length changes by a quarter wavelength, i.e., by c/ 4fc . If the path
length is changing at velocity v, the time required for such a phase change
is c/ 4fcv . Recalling that the Doppler shift D at frequency f is fv/c, and
noting that f ≈ fc for narrowband communication, the time required for a
/2 phase change is 1/ 4D . For the single reflecting wall example, this is
about 5ms (assuming fc
= 900MHz and v = 60km/h). The phases of both
paths are rotating at this rate but in opposite directions.
Note that the Fourier transform Hb f t of hb t for a fixed t is simply
H f +fc t , i.e., the frequency response of the original system (at a fixed t)
shifted by the carrier frequency. This provides another way of thinking about
the baseband equivalent channel.
2.2.3 A discrete-time baseband model
The next step in creating a useful channel model is to convert the continuoustime
channel to a discrete-time channel. We take the usual approach of the
sampling theorem. Assume that the input waveform is band-limited to W.
The baseband equivalent is then limited to W/2 and can be represented as
xb t =
n
x n sinc Wt−n (2.29)
where x n is given by xb n/W and sinc t is defined as
sinc t
= sin t
t
(2.30)
This representation follows from the sampling theorem, which says that any
waveform band-limited to W/2 can be expanded in terms of the orthogonal
26 The wireless channel
basis sinc Wt−n n, with coefficients given by the samples (taken uniformly
at integer multiples of 1/W).
Using (2.26), the baseband output is given by
yb t =
n
x n
i
ab
i t sinc Wt−W i t −n (2.31)
The sampled outputs at multiples of 1/W, y m
= yb m/W , are then
given by
y m =
n
x n
i
ab
i m/W sinc m−n− i m/W W (2.32)
The sampled output y m can equivalently be thought of as the projection
of the waveform yb t onto the waveform Wsinc Wt−m . Let
= m−n.
Then
y m =
x m−
i
ab
i m/W sinc − i m/W W (2.33)
By defining
h m
=
i
ab
i m/W sinc − i m/W W (2.34)
(2.33) can be written in the simple form
y m =
h m x m− (2.35)
We denote h m as the th (complex) channel filter tap at time m. Its value
is a function of mainly the gains abi
t of the paths, whose delays i t are
close to /W (Figure 2.10). In the special case where the gains abi
t and the
delays i t of the paths are time-invariant, (2.34) simplifies to
h
=
i
ab
i sinc − iW (2.36)
and the channel is linear time-invariant. The th tap can be interpreted as
the sample /W th of the low-pass filtered baseband channel response hb
(cf. (2.19)) convolved with sinc(W ).
We can interpret the sampling operation as modulation and demodulation in
a communication system. At time n, we are modulating the complex symbol
x m (in-phase plus quadrature components) by the sinc pulse before the
up-conversion. At the receiver, the received signal is sampled at times m/W
27 2.2 Input/output model of the wireless channel
Figure 2.10 Due to the decay
of the sinc function, the ith
path contributes most
significantly to the th tap if
its delay falls in the window
/W −1/ 2W /W +
1/ 2W .
1
W
Main contribution l = 0
Main contribution l = 0
Main contribution l = 1
Main contribution l = 2
Main contribution l = 2
i = 0
i = 1
i = 2
i = 3
i = 4
0 1 2
l
at the output of the low-pass filter. Figure 2.11 shows the complete system.
In practice, other transmit pulses, such as the raised cosine pulse, are often
used in place of the sinc pulse, which has rather poor time-decay property
and tends to be more susceptible to timing errors. This necessitates sampling
at the Nyquist sampling rate, but does not alter the essential nature of the
model. Hence we will confine to Nyquist sampling.
Due to the Doppler spread, the bandwidth of the output yb t is generally
slightly larger than the bandwidth W/2 of the input xb t , and thus the output
samples y m do not fully represent the output waveform. This problem is
usually ignored in practice, since the Doppler spread is small (of the order
of tens to hundreds of Hz) compared to the bandwidth W. Also, it is very
convenient for the sampling rate of the input and output to be the same.
Alternatively, it would be possible to sample the output at twice the rate of
the input. This would recapture all the information in the received waveform.
28 The wireless channel
X X
X X
[x[m]]
sinc (Wt – n)
[x[m]]
sinc (Wt – n)
h(τ, t)
1
–W W
–W W
1
+
[xb(t)]
[y[m]]
[yb(t)] [y[m]]
[yb(t)]
x(t) y(t)
[xb(t)]
2 2
2 2
–√2 sin 2π fc t –√2 sin 2π fc t
√2 cos 2π fc √2 cos 2π fc t t
Figure 2.11 System diagram The number of taps would be almost doubled because of the reduced sample
from the baseband transmitted
symbol x[m] to the baseband
sampled received signal y[m].
interval, but it would typically be somewhat less than doubled since the
representation would not spread the path delays so much.
Discussion 2.1 Degrees of freedom
The symbol x m is the mth sample of the transmitted signal; there are
W samples per second. Each symbol is a complex number; we say that it
represents one (complex) dimension or degree of freedom. The continuoustime
signal x t of duration one second corresponds to W discrete symbols;
thus we could say that the band-limited, continuous-time signal has W
degrees of freedom, per second.
The mathematical justification for this interpretation comes from the
following important result in communication theory: the signal space of
complex continuous-time signals of duration T which have most of their
energy within the frequency band −W/2 W/2 has dimension approximately
WT. (A precise statement of this result is in standard communication
theory text/books; see Section 5.3 of [148] for example.)
This result reinforces our interpretation that a continuous-time signal
with bandwidth W can be represented by W complex dimensions per
second.
The received signal y t is also band-limited to approximately W (due
to the Doppler spread, the bandwidth is slightly larger than W) and has W
complex dimensions per second. From the point of view of communication
over the channel, the received signal space is what matters because it
dictates the number of different signals which can be reliably distinguished
at the receiver. Thus, we define the degrees of freedom of the channel
to be the dimension of the received signal space, and whenever we refer
to the signal space, we implicitly mean the received signal space unless
stated otherwise.
29 2.2 Input/output model of the wireless channel
2.2.4 Additive white noise
As a last step, we include additive noise in our input/output model. We make
the standard assumption that w t is zero-mean additive white Gaussian noise
(AWGN) with power spectral density N0/2 (i.e., E w 0 w t = N0/2 t .
The model (2.14) is now modified to be
y t =
i
ai t x t− i t +w t (2.37)
See Figure 2.12. The discrete-time baseband-equivalent model (2.35) now
becomes
y m =
h m x m− +w m (2.38)
where w m is the low-pass filtered noise at the sampling instant m/W.
Just like the signal, the white noise w t is down-converted, filtered at the
baseband and ideally sampled. Thus, it can be verified (Exercise 2.11) that
w m =
−
w t m 1 t dt (2.39)
w m =
−
w t m 2 t dt (2.40)
where
m 1 t
=
√
2W cos 2 fct sinc Wt−m
m 2 t
= −
√
2W sin 2 fct sinc Wt−m (2.41)
It can further be shown that m 1 t m 2 t m forms an orthonormal set of
waveforms, i.e., the waveforms are orthogonal to each other (Exercise 2.12).
In Appendix A we review the definition and basic properties of white Gaussian
random vectors (i.e., vectors whose components are independent and
identically distributed (i.i.d.) Gaussian random variables). A key property is
that the projections of a white Gaussian random vector onto any orthonormal
vectors are independent and identically distributed Gaussian random
variables. Heuristically, one can think of continuous-time Gaussian white
noise as an infinite-dimensional white random vector and the above property
carries through: the projections onto orthogonal waveforms are uncorrelated
and hence independent. Hence the discrete-time noise process w m
is white, i.e., independent over time; moreover, the real and imaginary
components are i.i.d. Gaussians with variances N0/2. A complex Gaussian
random variable X whose real and imaginary components are i.i.d. satisfies
a circular symmetry property: ej X has the same distribution as X for
any . We shall call such a random variable circular symmetric complex
30 The wireless channel
X
X X
X
[x[m]] [y[m]]
[x[m]] [y[m]]
[xb(t)] [yb(t)]
[xb(t)] [yb(t)]
sinc(Wt – n)
sinc(Wt – n)
w(t)
y(t)
x(t)
+ h(τ, t) +
W2
2
– W
2
W2
2
– W
2
–√2 sin 2π fc t –√2 sin 2π fc t
√2 cos 2π fc √2 cos 2π fc t t
Figure 2.12 A complete system Gaussian, denoted by 0 2 , where 2 = E X 2 . The concept of cirdiagram.
cular symmetry is discussed further in Section A.1.3 of Appendix A.
The assumption of AWGN essentially means that we are assuming that the
primary source of the noise is at the receiver or is radiation impinging on
the receiver that is independent of the paths over which the signal is being
received. This is normally a very good assumption for most communication
situations.
2.3 Time and frequency coherence
2.3.1 Doppler spread and coherence time
An important channel parameter is the time-scale of the variation of the
channel. How fast do the taps h m vary as a function of time m? Recall that
h m =
i
ab
i m/W sinc − i m/W W
=
i
ai m/W e−j2 fc i m/W sinc − i m/W W (2.42)
Let us look at this expression term by term. From Section 2.2.2 we gather that
significant changes in ai occur over periods of seconds or more. Significant
changes in the phase of the ith path occur at intervals of 1/ 4Di , where
Di
= fc
i t is the Doppler shift for that path. When the different paths
contributing to the th tap have different Doppler shifts, the magnitude of
h m changes significantly. This is happening at the time-scale inversely
proportional to the largest difference between the Doppler shifts, the Doppler
spread Ds:
Ds
= max
i j
fc
i t −
j t (2.43)
31 2.3 Time and frequency coherence
where the maximum is taken over all the paths that contribute significantly to
a tap.7 Typical intervals for such changes are on the order of 10 ms. Finally,
changes in the sinc term of (2.42) due to the time variation of each i t are
proportional to the bandwidth, whereas those in the phase are proportional
to the carrier frequency, which is typically much larger. Essentially, it takes
much longer for a path to move from one tap to the next than for its phase
to change significantly. Thus, the fastest changes in the filter taps occur
because of the phase changes, and these are significant over delay changes
of 1/ 4Ds .
The coherence time Tc of a wireless channel is defined (in an order of
magnitude sense) as the interval over which h m changes significantly as a
function of m. What we have found, then, is the important relation
Tc
= 1
4Ds
(2.44)
This is a somewhat imprecise relation, since the largest Doppler shifts may
belong to paths that are too weak to make a difference. We could also view a
phase change of /4 to be significant, and thus replace the factor of 4 above
by 8. Many people instead replace the factor of 4 by 1. The important thing
is to recognize that the major effect in determining time coherence is the
Doppler spread, and that the relationship is reciprocal; the larger the Doppler
spread, the smaller the time coherence.
In the wireless communication literature, channels are often categorized as
fast fading and slow fading, but there is little consensus on what these terms
mean. In this book, we will call a channel fast fading if the coherence time Tc
is much shorter than the delay requirement of the application, and slow fading
if Tc is longer. The operational significance of this definition is that, in a
fast fading channel, one can transmit the coded symbols over multiple fades
of the channel, while in a slow fading channel, one cannot. Thus, whether a
channel is fast or slow fading depends not only on the environment but also
on the application; voice, for example, typically has a short delay requirement
of less than 100 ms, while some types of data applications can have a laxer
delay requirement.
2.3.2 Delay spread and coherence bandwidth
Another important general parameter of a wireless system is the multipath
delay spread, Td, defined as the difference in propagation time between the
7 The Doppler spread can in principle be different for different taps. Exercise 2.10 explores
this possibility.
32 The wireless channel
longest and shortest path, counting only the paths with significant energy.
Thus,
Td
= max
i j
i t − j t (2.45)
This is defined as a function of t, but we regard it as an order of magnitude
quantity, like the time coherence and Doppler spread. If a cell or LAN has
a linear extent of a few kilometers or less, it is very unlikely to have path
lengths that differ by more than 300 to 600 meters. This corresponds to path
delays of one or two microseconds. As cells become smaller due to increased
cellular usage, Td also shrinks. As was already mentioned, typical wireless
channels are underspread, which means that the delay spread Td is much
smaller than the coherence time Tc.
The bandwidths of cellular systems range between several hundred kilohertz
and several megahertz, and thus, for the above multipath delay spread values,
all the path delays in (2.34) lie within the peaks of two or three sinc functions;
more often, they lie within a single peak. Adding a few extra taps to each
channel filter because of the slow decay of the sinc function, we see that
cellular channels can be represented with at most four or five channel filter
taps. On the other hand, there is a recent interest in ultra-wideband (UWB)
communication, operating from 3.1 to 10.6 GHz. These channels can have up
to a few hundred taps.
When we study modulation and detection for cellular systems, we shall see
that the receiver must estimate the values of these channel filter taps. The taps
are estimated via transmitted and received waveforms, and thus the receiver
makes no explicit use of (and usually does not have) any information about
individual path delays and path strengths. This is why we have not studied the
details of propagation over multiple paths with complicated types of reflection
mechanisms. All we really need is the aggregate values of gross physical
mechanisms such as Doppler spread, coherence time, and multipath spread.
The delay spread of the channel dictates its frequency coherence. Wireless
channels change both in time and frequency. The time coherence shows
us how quickly the channel changes in time, and similarly, the frequency
coherence shows how quickly it changes in frequency. We first understood
about channels changing in time, and correspondingly about the duration of
fades, by studying the simple example of a direct path and a single reflected
path. That same example also showed us how channels change with frequency.
We can see this in terms of the frequency response as well.
Recall that the frequency response at time t is
H f t =
i
ai t e−j2 f i t (2.46)
The contribution due to a particular path has a phase linear in f. For multiple
paths, there is a differential phase, 2 f i t − k t . This differential
33 2.3 Time and frequency coherence
10
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
–60
–50
–40
–30
–20
–10
0
0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76
0.45
0
–10
–20
–0.001
–0.0008
–0.0006
–0.0004
–0.0002
0
0.0002
0.0004
0.0006
0.0008
0.001
0 50 100 150 200 250 300 350 400 450 500 550
–30
–40
–50
–60
–0.006 –70
–0.005
–0.004
–0.003
–0.002
–0.001
0
0.001
0.002
0.003
0.004
0 50 100 150 200 250 300 350 400 450 500 550 0.5
(d)
Power spectrum
(dB)
Power specturm
(dB)
Amplitude
(linear scale)
Amplitude
(linear scale)
(b)
Time (ns)
Time (ns)
(a)
(c)
40 MHz
Frequency (GHz)
Frequency (GHz)
200 MHz
Figure 2.13 (a) A channel over phase causes selective fading in frequency. This says that Er f t changes
200MHz is frequency-selective,
and the impulse response has
many taps. (b) The spectral
content of the same channel.
(c) The same channel over
40MHz is flatter, and has for
fewer taps. (d) The spectral
contents of the same channel,
limited to 40MHz bandwidth.
At larger bandwidths, the same
physical paths are resolved into
a finer resolution.
significantly, not only when t changes by 1/ 4Ds , but also when f changes
by 1/ 2Td . This argument extends to an arbitrary number of paths, so the
coherence bandwidth, Wc, is given by
Wc
= 1
2Td
(2.47)
This relationship, like (2.44), is intended as an order of magnitude relation,
essentially pointing out that the coherence bandwidth is reciprocal to the
multipath spread. When the bandwidth of the input is considerably less than
Wc, the channel is usually referred to as flat fading. In this case, the delay
spread Td is much less than the symbol time 1/W, and a single channel
filter tap is sufficient to represent the channel. When the bandwidth is much
larger than Wc, the channel is said to be frequency-selective, and it has to
be represented by multiple taps. Note that flat or frequency-selective fading
is not a property of the channel alone, but of the relationship between the
bandwidth W and the coherence bandwidth Td (Figure 2.13).
The physical parameters and the time-scale of change of key parameters of
the discrete-time baseband channel model are summarized in Table 2.1. The
different types of channels are summarized in Table 2.2.
34 The wireless channel
Table 2.1 A summary of the physical parameters of the channel and the
time-scale of change of the key parameters in its discrete-time baseband
model.
Key channel parameters and time-scales Symbol Representative values
Carrier frequency fc 1 GHz
Communication bandwidth W 1MHz
Distance between transmitter and receiver d 1 km
Velocity of mobile v 64 km/h
Doppler shift for a path D = fcv/c 50 Hz
Doppler spread of paths corresponding to
a tap Ds 100 Hz
Time-scale for change of path amplitude d/v 1 minute
Time-scale for change of path phase 1/ 4D 5ms
Time-scale for a path to move over a tap c/ vW 20 s
Coherence time Tc
= 1/ 4Ds 2.5 ms
Delay spread Td 1 s
Coherence bandwidth Wc
= 1/ 2Td 500 kHz
Table 2.2 A summary of the types of wireless
channels and their defining characteristics.
Types of channel Defining characteristic
Fast fading Tc
delay requirement
Slow fading Tc
delay requirement
Flat fading W
Wc
Frequency-selective fading W Wc
Underspread Td
Tc
2.4 Statistical channel models
2.4.1 Modeling philosophy
We defined Doppler spread and multipath spread in the previous section as
quantities associated with a given receiver at a given location, velocity, and
time. However, we are interested in a characterization that is valid over some
range of conditions. That is, we recognize that the channel filter taps {h m }
must be measured, but we want a statistical characterization of how many
taps are necessary, how quickly they change and how much they vary.
Such a characterization requires a probabilistic model of the channel tap
values, perhaps gathered by statistical measurements of the channel. We are
familiar with describing additive noise by such a probabilistic model (as
a Gaussian random variable). We are also familiar with evaluating error
probability while communicating over a channel using such models. These
35 2.4 Statistical channel models
error probability evaluations, however, depend critically on the independence
and Gaussian distribution of the noise variables.
It should be clear from the description of the physical mechanisms generating
Doppler spread and multipath spread that probabilistic models for the
channel filter taps are going to be far less believable than the models for
additive noise. On the other hand, we need such models, even if they are
quite inaccurate. Without models, systems are designed using experience and
experimentation, and creativity becomes somewhat stifled. Even with highly
over-simplified models, we can compare different system approaches and get
a sense of what types of approaches are worth pursuing.
To a certain extent, all analytical work is done with simplified models. For
example, white Gaussian noise (WGN) is often assumed in communication
models, although we know the model is valid only over sufficiently small
frequency bands. With WGN, however, we expect the model to be quite good
when used properly. For wireless channel models, however, probabilistic
models are quite poor and only provide order-of-magnitude guides to system
design and performance. We will see that we can define Doppler spread, multipath
spread, etc. much more cleanly with probabilistic models, but the underlying
problem remains that these channels are very different from each other and
cannot really be characterized by probabilistic models. At the same time, there
is a large literature based on probabilistic models for wireless channels, and it
has been highly useful for providing insight into wireless systems. However,
it is important to understand the robustness of results based on these models.
There is another question in deciding what to model. Recall the continuoustime
multipath fading channel
y t =
i
ai t x t− i t +w t (2.48)
This contains an exact specification of the delay and magnitude of each path.
From this, we derived a discrete-time baseband model in terms of channel
filter taps as
y m =
h m x m− +w m (2.49)
where
h m =
i
ai m/W e−j2 fc i m/W sinc − i m/W W (2.50)
We used the sampling theorem expansion in which x m = xb m/W and
y m = yb m/W . Each channel tap h m contains an aggregate of paths,
with the delays smoothed out by the baseband signal bandwidth.
Fortunately, it is the filter taps that must be modeled for input/output
descriptions, and also fortunately, the filter taps often contain a sufficient path
aggregation so that a statistical model might have a chance of success.
36 The wireless channel
2.4.2 Rayleigh and Rician fading
The simplest probabilistic model for the channel filter taps is based on
the assumption that there are a large number of statistically independent
reflected and scattered paths with random amplitudes in the delay window corresponding
to a single tap. The phase of the ith path is 2 fc i modulo 2 . Now,
fc i
= di/ , where di is the distance travelled by the ith path and is the carrier
wavelength. Since the reflectors and scatterers are far away relative to the carrier
wavelength, i.e., di
, it is reasonable to assume that the phase for each
path is uniformly distributed between 0 and 2 and that the phases of different
paths are independent. The contribution of each path in the tap gain h m is
ai m/W e−j2 fc i m/W sinc − i m/W W (2.51)
and this can be modeled as a circular symmetric complex random variable.8
Each tap h m is the sum of a large number of such small independent
circular symmetric random variables. It follows that h m is the sum of
many small independent real random variables, and so by the Central Limit
Theorem, it can reasonably be modeled as a zero-mean Gaussian random
variable. Similarly, because of the uniform phase, h m ej is Gaussian
with the same variance for any fixed . This assures us that h m is in
fact circular symmetric 0 2
(see Section A.1.3 in Appendix A for an
elaboration). It is assumed here that the variance of h m is a function of the
tap , but independent of time m (there is little point in creating a probabilistic
model that depends on time). With this assumed Gaussian probability density,
we know that the magnitude h m of the th tap is a Rayleigh random
variable with density (cf. (A.20) in Appendix A and Exercise 2.14)
x
2
exp −x2
2 2
x≥ 0 (2.52)
and the squared magnitude h m 2 is exponentially distributed with density
1
2
exp −x
2
x≥ 0 (2.53)
This model, which is called Rayleigh fading, is quite reasonable for scattering
mechanisms where there are many small reflectors, but is adopted
primarily for its simplicity in typical cellular situations with a relatively small
number of reflectors. The word Rayleigh is almost universally used for this
8 See Section A.1.3 in Appendix A for a more in-depth discussion of circular symmetric
random variables and vectors.
37 2.4 Statistical channel models
model, but the assumption is that the tap gains are circularly symmetric
complex Gaussian random variables.
There is a frequently used alternative model in which the line-of-sight path
(often called a specular path) is large and has a known magnitude, and that
there are also a large number of independent paths. In this case, h m , at
least for one value of , can be modeled as
h m =
+1
ej + 1
+1
0 2
(2.54)
with the first term corresponding to the specular path arriving with uniform
phase and the second term corresponding to the aggregation of the large
number of reflected and scattered paths, independent of . The parameter
(so-called K-factor) is the ratio of the energy in the specular path to the
energy in the scattered paths; the larger is, the more deterministic is the
channel. The magnitude of such a random variable is said to have a Rician
distribution. Its density has quite a complicated form; it is often a better model
of fading than the Rayleigh model.
2.4.3 Tap gain auto-correlation function
Modeling each h m as a complex random variable provides part of the statistical
description that we need, but this is not the most important part. The more
important issue is how these quantities vary with time. As we will see in the rest
of the book, the rate of channel variation has significant impact on several aspects
of the communication problem. A statistical quantity that models this relationship
is known as the tap gain auto-correlation function, R n . It is defined as
R n
= h∗
m h m+n (2.55)
For each tap , this gives the auto-correlation function of the sequence of
random variables modeling that tap as it evolves in time. We are tacitly
assuming that this is not a function of time m. Since the sequence of random
variables h m for any given has both a mean and covariance function
that does not depend on m, this sequence is wide-sense stationary. We also
assume that, as a random variable, h m is independent of h m for all
= and all m m . This final assumption is intuitively plausible since paths
in different ranges of delay contribute to h m for different values of .9
The coefficient R 0 is proportional to the energy received in the th
tap. The multipath spread Td can be defined as the product of 1/W times
the range of which contains most of the total energy
=0 R 0 . This is
9 One could argue that a moving reflector would gradually travel from the range of one tap to
another, but as we have seen, this typically happens over a very large time-scale.
38 The wireless channel
somewhat preferable to our previous “definition” in that the statistical nature
of Td becomes explicit and the reliance on some sort of stationarity becomes
explicit. Now, we can also define the coherence time Tc more explicitly as
the smallest value of n > 0 for which R n is significantly different from
R 0 . With both of these definitions, we still have the ambiguity of what
“significant” means, but we are now facing the reality that these quantities
must be viewed as statistics rather than as instantaneous values.
The tap gain auto-correlation function is useful as a way of expressing the
statistics for how tap gains change given a particular bandwidth W, but gives
little insight into questions related to choice of a bandwidth for communication.
If we visualize increasing the bandwidth, we can see several things happening.
First, the ranges of delay that are separated into different taps become narrower
(1/W seconds), so there are fewer paths corresponding to each tap, and thus the
Rayleigh approximation becomes poorer. Second, the sinc functions of (2.50)
become narrower, andR 0 gives a finer grained picture of the amount of power
being received in the th delay window of width 1/W. In summary, as we try
to apply this model to larger W, we get more detailed information about delay
and correlation at that delay, but the information becomes more questionable.
Example 2.2 Clarke’s model
This is a popular statistical model for flat fading. The transmitter is fixed,
the mobile receiver is moving at speed v, and the transmitted signal is
scattered by stationary objects around the mobile. There are K paths, the
ith path arriving at an angle i
= 2 i/K, i = 0 K−1, with respect
to the direction of motion. K is assumed to be large. The scattered path
arriving at the mobile at the angle has a delay of t and a timeinvariant
gain a , and the input/output relationship is given by
y t =
K−1
i=0
a i
x t− i
t (2.56)
The most general version of the model allows the received power distribution
p and the antenna gain pattern to be arbitrary functions of
the angle , but the most common scenario assumes uniform power distribution
and isotropic antenna gain pattern, i.e., the amplitudes a
= a/
√
K
for all angles . This models the situation when the scatterers are located
in a ring around the mobile (Figure 2.14). We scale the amplitude of each
path by
√
K so that the total received energy along all paths is a2; for large
K, the received energy along each path is a small fraction of the total energy.
Suppose the communication bandwidth W is much smaller than the
reciprocal of the delay spread. The complex baseband channel can be
represented by a single tap at each time:
y m = h0 m x m +w m (2.57)
39 2.4 Statistical channel models
Rx
Figure 2.14 The one-ring model.
The phase of the signal arriving at time 0 from an angle is 2 fc 0
mod 2 , where fc is the carrier frequency. Making the assumption that
this phase is uniformly distributed in 0 2 and independently distributed
across all angles , the tap gain process h0 m is a sum of many small
independent contributions, one from each angle. By the Central Limit
Theorem, it is reasonable to model the process as Gaussian. Exercise 2.17
shows further that the process is in fact stationary with an autocorrelation
function R0 n given by:
R0 n = 2a2 J0 n Ds/W (2.58)
where J0 · is the zeroth-order Bessel function of the first kind:
J0 x
= 1
0
ejx cos d (2.59)
and Ds
= 2fcv/c is the Doppler spread. The power spectral density S f ,
defined on −1/2 +1/2 , is given by
S f =
4a2W
Ds
√
1− 2fW/Ds 2
−Ds/ 2W f +Ds/ 2W
0 else
(2.60)
This can be verified by computing the inverse Fourier transform of (2.60)
to be (2.58). Plots of the autocorrelation function and the spectrum for are
shown in Figure 2.15. If we define the coherence time Tc to be the value
of n/W such that R0 n = 0 05R0 0 , then
Tc
= J−1
0 0 05
Ds
(2.61)
i.e., the coherence time is inversely proportional to Ds .
40 The wireless channel
2000
2.5
3
3.5
1.5
1
0.5
0
–0.5
–1
–1.5
200 400 600 800 1000 1200 1400 1600 1800
2
R0 [n]
–1/2 1/2
S ( f )
–Ds / (2W ) 0 Ds / (2W )
Figure 2.15 Plots of the auto-correlation function and Doppler spectrum in Clarke’s model.
In Exercise 2.17, you will also verify that S f df has the physical
interpretation of the received power along paths that have Doppler shifts
in the range f f +df . Thus, S f is also called the Doppler spectrum.
Note that S f is zero beyond the maximum Doppler shift.
Chapter 2 The main plot
Large-scale fading
Variation of signal strength over distances of the order of cell sizes.
Received power decreases with distance r like:
1
r2 (free space)
1
r4 (reflection from ground plane)
Decay can be even faster due to shadowing and scattering effects.
41 2.4 Statistical channel models
Small-scale fading
Variation of signal strength over distances of the order of the carrier
wavelength, due to constructive and destructive interference of multipaths.
Key parameters:
Doppler spread Ds
←→coherence time Tc
∼ 1/Ds
Doppler spread is proportional to the velocity of the mobile and to the
angular spread of the arriving paths.
delay spread Td
←→coherence bandwidth Wc
∼ 1/Td
Delay spread is proportional to the difference between the lengths of the
shortest and the longest paths.
Input/output channel models
• Continuous-time passband (2.14):
y t =
i
ai t x t− i t
• Continuous-time complex baseband (2.26):
yb t =
i
ai t e−j2 fc i t xb t− i t
• Discrete-time complex baseband with AWGN (2.38):
y m =
h m x m− +w m
The th tap is the aggregation of the physical paths with delays in
/W −1/ 2W /W +1/ 2W .
Statistical channel models
• h m m is modeled as circular symmetric processes independent across
the taps.
• If for all taps,
h m ∼ 0 2
the model is called Rayleigh.
• If for one tap,
h m =
+1
ej + 1
+1
0 2
the model is called Rician with K-factor .
42 The wireless channel
• The tap gain auto-correlation function R n
= E h∗
0 h n models
the dependency over time.
• The delay spread is 1/W times the range of taps which contains most
of the total gain
=0 R 0 . The coherence time is 1/W times the range
of n for which R n is significantly different from R 0 .
2.5 Bibliographical notes
This chapter was modified from R. G. Gallager’s MIT 6.450 course notes on digital
communication. The focus is on small-scale multipath fading. Large-scale fading
models are discussed in many texts; see for example Rappaport [98]. Clarke’s model
was introduced in [22] and elaborated further in [62]. Our derivation here of the Clarke
power spectrum follows the approach of [111].
2.6 Exercises
Exercise 2.1 (Gallager) Consider the electric field in (2.4).
1. It has been derived under the assumption that the motion is in the direction of
the line-of-sight from sending antenna to receive antenna. Find the electric field
assuming that is the angle between the line-of-sight and the direction of motion
of the receiver. Assume that the range of time of interest is small enough so that
changes in can be ignored.
2. Explain why, and under what conditions, it is a reasonable approximation to ignore
the change in over small intervals of time.
Exercise 2.2 (Gallager) Equation (2.13) was derived under the assumption that
r t ≈ d. Derive an expression for the received waveform for general r t . Break the
first term in (2.11) into two terms, one with the same numerator but the denominator
2d−r0
−vt and the other with the remainder. Interpret your result.
Exercise 2.3 In the two-path example in Sections 2.1.3 and 2.1.4, the wall is on the
right side of the receiver so that the reflected wave and the direct wave travel in opposite
directions. Suppose now that the reflecting wall is on the left side of transmitter. Redo the
analysis. What is the nature of the multipath fading, both over time and over frequency?
Explain any similarity or difference with the case considered in Sections 2.1.3 and 2.1.4.
Exercise 2.4 A mobile receiver is moving at a speed v and is receiving signals arriving
along two reflected paths which make angles 1 and 2 with the direction of motion.
The transmitted signal is a sinusoid at frequency f.
1. Is the above information enough for estimating (i) the coherence time Tc; (ii) the
coherence bandwidth Wc? If so, express them in terms of the given parameters. If
not, specify what additional information would be needed.
2. Consider an environment in which there are reflectors and scatterers in all directions
from the receiver and an environment in which they are clustered within a small
43 2.6 Exercises
angular range. Using part (1), explain how the channel would differ in these two
environments.
Exercise 2.5 Consider the propagation model in Section 2.1.5 where there is a reflected
path from the ground plane.
1. Let r1 be the length of the direct path in Figure 2.6. Let r2 be the length of the
reflected path (summing the path length from the transmitter to the ground plane
and the path length from the ground plane to the receiver). Show that r2
−r1 is
asymptotically equal to b/r and find the value of the constant b. Hint: Recall that
for x small,
√
1+x ≈ 1+x/2 in the sense that
√
1+x−1 /x→1/2 as x→0.
2. Assume that the received waveform at the receive antenna is given by
Er f t = cos 2 ft−fr1/c
r1
− cos 2 ft−fr2/c
r2
(2.62)
Approximate the denominator r2 by r1 in (2.62) and show that Er
≈ /r2 for r−1
much smaller than c/f . Find the value of .
3. Explain why this asymptotic expression remains valid without first approximating
the denominator r2 in (2.62) by r1.
Exercise 2.6 Consider the following simple physical model in just a single dimension.
The source is at the origin and transmits an isotropic wave of angular frequency .
The physical environment is filled with uniformly randomly located obstacles. We
will model the inter-obstacle distance as an exponential random variable, i.e., it has
the density10
e− r r≥ 0 (2.63)
Here 1/ is the mean distance between obstacles and captures the density of the obstacles.
Viewing the source as a stream of photons, suppose each obstacle independently
(from one photon to the other and independent of the behavior of the other obstacles)
either absorbs the photon with probability or scatters it either to the left or to the
right (both with equal probability 1− /2).
Now consider the path of a photon transmitted either to the left or to the right with
equal probability from some fixed point on the line. The probability density function
of the distance (denoted by r) to the first obstacle (the distance can be on either side
of the starting point, so r takes values on the entire line) is equal to
q r
= e− r
2
r∈ (2.64)
So the probability density function of the distance at which the photon is absorbed
upon hitting the first obstacle is equal to
f1 r
= q r r ∈ (2.65)
10 This random arrangement of points on a line is called a Poisson point process.
44 The wireless channel
1. Show that the probability density function of the distance from the origin at which
the second obstacle is met is
f2 r
=
−
1− q x f1 r −x dx r ∈ (2.66)
2. Denote by fk r the probability density function of the distance from the origin
at which the photon is absorbed by exactly the kth obstacle it hits and show the
recursive relation
fk+1 r =
−
1− q x fk r −x dx r ∈ (2.67)
3. Conclude from the previous step that the probability density function of the distance
from the source at which the photon is absorbed (by some obstacle), denoted by
f r , satisfies the recursive relation
f r = q r + 1−
−
q x f r −x dx r ∈ (2.68)
Hint: Observe that f r =
k=1 fk r .
4. Show that
f r =
√
2
e−
√
r (2.69)
is a solution to the recursive relation in (2.68). Hint: Observe that the convolution
between the probability densities q · and f · in (2.68) is more easily represented
using Fourier transforms.
5. Now consider the photons that are absorbed at a distance of more than r from the
source. This is the radiated power density at a distance r and is found by integrating
f x over the range r if r > 0 and − r if r < 0. Calculate the radiated
power density to be
e−
√
r
2
(2.70)
and conclude that the power decreases exponentially with distance r. Also observe
that with very low absorption →0 or very few obstacles →0 , the power
density converges to 0.5; this is expected since the power splits equally on either
side of the line.
Exercise 2.7 In Exercise 2.6, we considered a single-dimensional physical model of a
scattering and absorption environment and concluded that power decays exponentially
with distance. A reading exercise is to study [42], which considers a natural extension
of this simple model to two- and three-dimensional spaces. Further, it extends the
analysis to two- and three-dimensional physical models. While the analysis is more
complicated, we arrive at the same conclusion: the radiated power decays exponentially
with distance.
45 2.6 Exercises
Exercise 2.8 (Gallager) Assume that a communication channel first filters the transmitted
passband signal before adding WGN. Suppose the channel is known and the
channel filter has an impulse response h t . Suppose that a QAM scheme with symbol
duration T is developed without knowledge of the channel filtering. A baseband filter
t is developed satisfying the Nyquist property that t−kT k is an orthonormal
set. The matched filter −t is used at the receiver before sampling and detection.
If one is aware of the channel filter h t , one may want to redesign either the
baseband filter at the transmitter or the baseband filter at the receiver so that there
is no intersymbol interference between receiver samples and so that the noise on the
samples is i.i.d.
1. Which filter should one redesign?
2. Give an expression for the impulse response of the redesigned filter (assume a
carrier frequency fc).
3. Draw a figure of the various filters at passband to show why your solution is
correct. (We suggest you do this before answering the first two parts.)
Exercise 2.9 Consider the two-path example in Section 2.1.4 with d = 2 km and the
receiver at 1.5 km from the transmitter moving at velocity 60 km/h away from the
transmitter. The carrier frequency is 900 MHz.
1. Plot in MATLAB the magnitudes of the taps of the discrete-time baseband channel
at a fixed time t. Give a few plots for several bandwidths W so as to exhibit both
flat and frequency-selective fading.
2. Plot the time variation of the phase and magnitude of a typical tap of the discretetime
baseband channel for a bandwidth where the channel is (approximately)
flat and for a bandwidth where the channel is frequency-selective. How do the
time-variations depend on the bandwidth? Explain.
Exercise 2.10 For each tap of the discrete-time channel response, the Doppler spread
is the range of Doppler shifts of the paths contributing to that tap. Give an example
of an environment (i.e. location of reflectors/scatterers with respect to the location of
the transmitter and the receiver) in which the Doppler spread is the same for different
taps and an environment in which they are different.
Exercise 2.11 Verify (2.39) and (2.40).
Exercise 2.12 In this problem we consider generating passband orthogonal waveforms
from baseband ones.
1. Show that if the waveforms t − nT n form an orthogonal set, then the
waveforms n 1 n 2 n also form an orthogonal set, provided that t is bandlimited
to −fc fc . Here,
n 1 t = t−nT cos 2 fct
n 2 t = t−nT sin 2 fct
How should we normalize the energy of t to make the t orthonormal?
2. For a given fc, find an example where the result in part (1) is false when the
condition that t is band-limited to −fc fc is violated.
Exercise 2.13 Verify (2.25). Does this equation contain any more information about
the communication system in Figure 2.9 beyond what is in (2.24)? Explain.
46 The wireless channel
Exercise 2.14 Compute the probability density function of the magnitude X of a
complex circular symmetric Gaussian random variable X with variance 2.
Exercise 2.15 In the text we have discussed the various reasons why the channel tap
gains, h m , vary in time (as a function of m) and how the various dynamics operate
at different time-scales. The analysis is based on the assumption that communication
takes place on a bandwidth W around a carrier frequency fc with fc
W. This
assumption is not valid for ultra-wideband (UWB) communication systems, where the
transmission bandwidth is from 3.1 GHz to 10.6 GHz, as regulated by the FCC. Redo
the analysis for this system. What is the main mechanism that causes the tap gains to
vary at the fastest time-scale, and what is this fastest time-scale determined by?
Exercise 2.16 In Section 2.4.2, we argue that the channel gain h m at a particular
time m can be assumed to be circular symmetric. Extend the argument to show that it
is also reasonable to assume that the complex random vector
h
=
h m
h m+1
h m+n
is circular symmetric for any n.
Exercise 2.17 In this question, we will analyze in detail Clarke’s one-ring model
discussed at the end of the chapter. Recall that the scatterers are assumed to be located
in a ring around the receiver moving at speed v. There are K paths coming in at angles
i
= 2 i/K with respect to the direction of motion of the mobile, i = 0 K−1
The path coming at angle has a delay of t and a time-invariant gain a/
√
K (not
dependent on the angle), and the input/output relationship is given by
y t = √a
K
K−1
i=0
x t− i
t (2.71)
1. Give an expression for the impulse response h t for this channel, and give an
expression for t in terms of 0 . (You can assume that the distance the mobile
travelled in 0 t is small compared to the radius of the ring.)
2. Suppose communication takes place at carrier frequency fc and over a narrowband
of bandwidth W such that the delay spread of the channel Td satisfies Td
1/W.
Argue that the discrete-time baseband model can be approximately represented by
a single tap
y m = h0 m x m +w m (2.72)
and give an approximate expression for that tap in terms of the a ’s and t ’s.
Hint: Your answer should contain no sinc functions.
3. Argue that it is reasonable to assume that the phase of the path from an angle at
time 0,
2 fc 0 mod 2
is uniformly distributed in 0 2 and that it is i.i.d. across .
47 2.6 Exercises
4. Based on the assumptions in part (3), for large K one can use the Central Limit
Theorem to approximate h0 m as a Gaussian process. Verify that the limiting
process is stationary and the autocorrelation function R0 n is given by (2.58).
5. Verify that the Doppler spectrum S f is given by (2.60). Hint: It is easier to show
that the inverse Fourier transform of (2.60) is (2.58).
6. Verify that S f df is indeed the received power from the paths that have Doppler
shifts in f f +df . Is this surprising?
Exercise 2.18 Consider a one-ring model where there are K scatterers located at
angles i
= 2 i/K, i = 0 K−1, on a circle of radius 1 km around the receiver
and the transmitter is 2 km away. (The angles are with respect to the line joining the
transmitter and the receiver.) The transmit power is P. The power attenuation along a
path from the transmitter to a scatterer to the receiver is
G
K
· 1
s2
· 1
r2 (2.73)
where G is a constant and r and s are the distance from the transmitter to the scatterer
and the distance from the scatterer to the receiver respectively. Communication takes
place at a carrier frequency fc
=1 9 GHz and the bandwidth is W Hz. You can assume
that, at any time, the phases of each arriving path in the baseband representation of
the channel are independent and uniformly distributed between 0 and 2 .
1. What are the key differences and the similarities between this model and the
Clarke’s model in the text?
2. Find approximate conditions on the bandwidth W for which one gets a flat fading
channel.
3. Suppose the bandwidth is such that the channel is frequency selective. For large
K, find approximately the amount of power in tap of the discrete-time baseband
impulse response of the channel (i.e., compute the power-delay profile.). Make any
simplifying assumptions but state them. (You can leave your answers in terms of
integrals if you cannot evaluate them.)
4. Compute and sketch the power-delay profile as the bandwidth becomes very large
(and K is large).
5. Suppose now the receiver is moving at speed v towards the (fixed) transmitter. What
is the Doppler spread of tap ? Argue heuristically from physical considerations
what the Doppler spectrum (i.e., power spectral density) of tap is, for large K.
6. We have made the assumptions that the scatterers are all on a circle of radius 1km
around the receiver and the paths arrive with independent and uniform distributed
phases at the receiver. Mathematically, are the two assumptions consistent? If not,
do you think it matters, in terms of the validity of your answers to the earlier parts
of this question?
Exercise 2.19 Often in modeling multiple input multiple output (MIMO) fading
channels the fading coefficients between different transmit and receive antennas are
assumed to be independent random variables. This problem explores whether this is
a reasonable assumption based on Clarke’s one-ring scattering model and the antenna
separation.
1. (Antenna separation at the mobile) Assume a mobile with velocity v moving away
from the base-station, with uniform scattering from the ring around it.
48 The wireless channel
(a) Compute the Doppler spread Ds for a carrier frequency fc, and the corresponding
coherence time Tc.
(b) Assuming that fading states separated by Tc are approximately uncorrelated, at
what distance should we place a second antenna at the mobile to get an independently
faded signal? Hint: How much distance does the mobile travel in Tc?
2. (Antenna separation at the base-station) Assume that the scattering ring has radius
R and that the distance between the base-station and the mobile is d. Further
assume for the time being that the base-station is moving away from the mobile
with velocity v . Repeat the previous part to find the minimum antenna spacing at
the base-station for uncorrelated fading. Hint: Is the scattering still uniform around
the base-station?
3. Typically, the scatterers are local around the mobile (near the ground) and far away
from the base-station (high on a tower). What is the implication of your result in
part (2) for this scenario?
