Wireless Channel
From a technical point of view, the greatest distinction
between wireless and wireline communications lies in the physical properties of
wireless channels. These physical properties can be described in terms of
several distinct phenomena, including ambient noise, propagation losses,
multipath, interference, and properties arising from the use of multiple
antennas. Here we review these phenomena only briefly. Further discussion and
details can be found, for example, in [38, 46, 148, 216, 405, 450, 458, 465].
Like all practical communications channels, wireless channels
are corrupted by ambient noise. This noise comes from thermal motion of
electrons on the antenna and in the receiver electronics and from background
radiation sources. This noise is well modeled as having a very wide bandwidth
(much wider than the bandwidth of any useful signals in the channel) and no
particular deterministic structure (structured noise can be treated separately
as interference). A very common and useful model for such noise is additive
white Gaussian noise (AWGN), which as the name implies, means that it is
additive to the other signals in the receiver, has a flat power spectral
density, and induces a Gaussian probability distribution at the output of any
linear filter to which it is input. Impulsive noise also occurs in some wireless
channels. Such noise is similarly wideband but induces a non-Gaussian amplitude
distribution at the output of linear filters. Specific models for such impulsive
noise are discussed in Chapter
4.
Propagation losses are also an issue in wireless channels.
These are of two basic types: diffusive losses and shadow fading. Diffusive losses arise because of the open nature of
wireless channels. For example, the energy radiated by a simple point source in
free space will spread over an ever-expanding spherical surface as the energy
propagates away from the source. This means that an antenna with a given
aperture size will collect an amount of energy that decreases with the square of
the distance between the antenna and the source. In most terrestrial wireless
channels, the diffusion losses are actually greater than this, due to the
effects of ground-wave propagation, foliage, and so on. For example, in cellular
telephony, the diffusion loss is inverse square with distance within line of
sight of the cell tower, and it falls off with a higher power (typically, 3 or
4) at greater distances. As its name implies, shadow
fading results from the presence of objects (buildings, walls, etc.)
between the transmitter and receiver. Shadow fading is typically modeled by an
attenuation (i.e., a multiplicative factor) in signal amplitude that follows a
log-normal distribution. The variation in this fading is specified by the
standard deviation of the logarithm of this attenuation.
Multipath refers to the
phenomenon by which multiple copies of a transmitted signal are received at the
receiver, due to the presence of multiple radio paths between the transmitter
and receiver. These multiple paths arise due to reflections from objects in the
radio channel. Multipath is manifested in several ways in communications
receivers, depending on the degree of path difference relative to the wavelength
of propagation, the degree of path difference relative to the signaling rate,
and the relative motion between the transmitter and receiver. Multipath from
scatterers that are spaced very close together will cause a random change in the
amplitude of the received signal. Due to central-limit effects, the resulting
received amplitude is often modeled as being a complex Gaussian random variable.
This results in a random amplitude whose envelope has a Rayleigh distribution,
and this phenomenon is thus termed Rayleigh
fading. Other fading distributions also arise, depending on the physical
configuration (see, e.g., [396]). When the scatterers are spaced
so that the differences in their corresponding path lengths are significant
relative to a wavelength of the carrier, the signals arriving at the receiver
along different paths can add constructively or destructively. This gives rise
to fading that depends on the wavelength (or, equivalently, the frequency) of
radiation, which is thus called frequency-selective
fading. When there is relative motion between the transmitter and
receiver, this type of fading also depends on time, since the path length is a
function of the radio geometry. This results in time-selective fading. (Such motion also causes signal
distortion due to Doppler effects.) A related phenomenon arises when the
difference in path lengths is such that the time delay of arrival along
different paths is significant relative to a symbol interval. This results in
dispersion of the transmitted signal, and causes intersymbol interference (ISI); that is, contributions
from multiple symbols arrive at the receiver at the same time.
Many of the advanced signal transmission and processing methods
that have been developed for wireless systems are designed to contravene the
effects of multipath. For example, wideband signaling techniques such as spread
spectrum are often used as a countermeasure to frequency-selective fading. This
both minimizes the effects of deep frequency-localized fades and facilitates the
resolvability and subsequent coherent combining of multiple copies of the same
signal. Similarly, by dividing a high-rate signal into many parallel lower-rate
signals, OFDM mitigates the effects of channel dispersion on high-rate signals.
Alternatively, high-data-rate single-carrier systems make use of channel
equalization at the receiver to counteract this dispersion. Some of these issues
are discussed further in Section 1.3.
Interference, also a significant issue in many wireless
channels, is typically one of two types: multiple-access interference and
co-channel interference. Multiple-access
interference (MAI) refers to interference arising from other signals in
the same network as the signal of interest. For example, in cellular telephony
systems, MAI can arise at the base station when the signals from multiple mobile
transmitters are not orthogonal to one another. This happens by design in CDMA
systems, and it happens in FDMA or TDMA systems due to channel properties such
as multipath or to nonideal system characteristics such as imperfect
channelization filters. Co-channel interference
(CCI) refers to interference from signals from different networks, but operating
in the same frequency band as the signal of interest. An example is the
interference from adjacent cells in a cellular telephony system. This problem is
a chief limitation of using FDMA in cellular systems and was a major factor in
moving away from FDMA in second-generation systems. Another example is the
interference from other devices operating in the same part of the unregulated
spectrum as the signal of interest, such as interference from Bluetooth devices
operating in the same 2.4-GHz ISM band as IEEE 802.11 wireless LANs.
Interference mitigation is also a major factor in the design of transmission
techniques (e.g., the above-noted movement away from FDMA in cellular systems)
as well as in the design of advanced signal processing systems for wireless, as
we shall see in the sequel.
The phenomena we have discussed above can be incorporated into
a general analytical model for a wireless multiple-access channel. In
particular, the signal model in a wireless system is illustrated in Fig. 1.2. We can write the signal received
at a given receiver in the following form:
Equation 1.9
where gk(t,u) denotes the impulse response of a linear filter
representing the channel between the kth
transmitter and the receiver, i(·) represents
co-channel interference, and n(·) represents
ambient noise. The modeling of the wireless channel as a linear system seems to
agree well with the observed behavior of such channels. All of the quantities
gk(·, ·), i(·), and n(·) are, in
general, random processes. As noted above, the ambient noise is typically
represented as a white process with very little additional structure. However,
the co-channel interference and channel impulse responses are typically
structured processes that can be parameterized.
An important special case is that of a pure multipath channel,
in which the channel impulse responses can be represented in the form
Equation 1.10
where Lk is the
number of paths between user k and the receiver,
a
,k and t,k are
the gain and delay, respectively, associated with the th path of the
kth user, and d(·) denotes the Dirac delta function. Note that this
is the situation illustrated in Fig. 1.2,
in which we have written the time-invariant impulse response as gk(t) 
gk(t, 0). This model is an
idealization of the actual behavior of a multipath channel, which would not have
such a sharply defined impulse response. However, it serves as a useful model
for signal processor design and analysis. Note that this model gives rise to
frequency-selective fading, since the relative delays will cause constructive
and destructive interference at the receiver, depending on the wavelength of
propagation. Often, the delays {t, k}
are assumed to be known to the receiver or are spaced uniformly at the inverse
of the bulk bandwidth of the signaling waveforms. A typical model for the path
gains {a, k} is that they are independent
complex Gaussian random variables, giving rise to Rayleigh fading.
Note that, in general, the receiver will see the following
composite modulation waveform associated with the
symbol bk[i]:
Equation 1.11
If these waveforms are not orthogonal for different values of
i, ISI will result. Consider, for example, the
pure multipath channel of (1.10) with
signaling waveforms of the form
Equation 1.12
where sk(·) is a
normalized signaling waveform [
|sk(t)|2 dt = 1],
Ak is a complex amplitude, and T is the inverse of the single-user symbol rate. In
this case, the composite modulation waveforms are given by
Equation 1.13
with
Equation 1.14
If the delay spread (i.e., the maximum of the differences of
the delays {t, k} for different values of ) is significant
relative to T, ISI may be a factor. Note that for
a fixed channel, the delay spread is a function of the physical geometry of the
channel, whereas the symbol rate depends on the data rate of the transmitted
source. Thus, higher-rate transmissions are more likely to encounter ISI than
are lower-rate transmissions. Similarly, if the composite waveforms for
different values of k are not orthogonal, MAI
will result. This can happen, for example, in CDMA channels when the
pseudorandom code sequences used by different users are not orthogonal. It can
also happen in CDMA and TDMA channels, due to the effects of multipath or
asynchronous transmission. These issues are discussed further in the sequel as
the need arises.
This model can be further generalized to account for multiple
antennas at the receiver. In particular, we can modify (1.9) as follows:
Equation 1.15
where the boldface quantities denote (column) vectors with
dimensions equal to the number of antennas at the received array. For example,
the pth component of gk(t, u) is the
impulse response of the channel between user k
and the pth element of the receiving array. A
useful such model is to combine the pure multipath model of (1.10) with a model in which the spatial aspects of the
array can be separated from its temporal properties. This yields channel impulse
responses of the form
Equation 1.16
where the complex vector a, k describes the response of the
array to the th path of user k. The simplest
such situation is the case of a uniform linear
array (ULA), in which the array elements are uniformly spaced along a
line, receiving a single-carrier signal arriving along a planar wavefront and
satisfying the narrowband array assumption. The
essence of this assumption is that the signaling waveforms are sinusoidal
carriers carrying narrowband modulation and that all of the variation in the
received signal across the array at any given instant in time is due to the
carrier (i.e., the modulating waveform is changing slowly enough to be assumed
constant across the array). In this case, the array response depends only on the
angle f, k at which the corresponding
path's signal is incident on the array. In particular, the response of a P-element array is given in this case by
Equation 1.17
where j denotes the imaginary
unit and where 
, with l the carrier wavelength and d the interelement spacing (see [126, 266, 269, 404, 445, 450, 510] for further discussion of
systems involving multiple receiver antennas).
It is also of interest to model systems in which there are
multiple antennas at both the transmitter and receiver, called multiple-input/multiple-output (MIMO) systems. In this case the channel transfer functions
are matrices, with the number of rows equal to the number of receiving antennas
and the number of columns equal to the number of transmitting antennas at each
source. There are several ways of handling the signaling in such configurations,
depending on the desired effects and the channel conditions. For example,
transmitter beamforming can be implemented by transmitting the same symbol
simultaneously from multiple antenna elements on appropriately phased versions
of the same signaling waveform. Space-time coding can be implemented by
transmitting frames of related symbols over multiple antennas. Other
configurations are of interest as well. Issues concerning multiple-antenna
systems are discussed further in the sequel as they arise.
Sections
