Diversity Reception in Spread Spectrum

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Diversity Reception in Spread Spectrum

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Title: Diversity Reception in Spread Spectrum

1
Chapter 4

Diversity Reception in Spread Spectrum

2
Diversity Reception in Spread Spectrum

Spread spectrum modulation techniques are mostly
employed in wireless communication systems.
In order to fully understand spread spectrum
communications, we need to have a basic idea on
the characteristics of wireless channels.
The behavior of a typical mobile wireless channel
is considerably more complex than that of an AWGN
channel.

3
4.1 Path loss

Besides the thermal noise at the receiver front
end (which is modeled by AWGN), there are several
other well studied channel impairments in a
typical wireless channel
Path Loss
Describes the loss in power as the radio signal
propagates in space.
Shadowing
Due to the presence of fixed obstacles in the
propagation path of the radio signal
Fading
Accounts for the combined effect of multiple
propagation paths, rapid movements of mobile
units (transmitters/receivers) and reflectors.

In any real channel, signals attenuate as they
propagate.
For a radio wave transmitted by a point source in
free space, the loss in power, known as path
loss, is given by
?is the wavelength of the signal.
d is the distance between the source and the
receiver.
The power of the signal decays as the square of
the distance.
In land mobile wireless communication
environments, similar situations are observed.

The mean power of a signal decays as the n-th
power of the distance
c is a constant
The exponent n typically ranges from 2 to 5 1.
The exact values of c and n depend on the
particular environment.
The loss in power is a factor that limits the
coverage of a transmitter.

7
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8
4.2 Shadowing

Shadowing is due to the presence of large-scale
obstacles in the propagation path of the radio
signal.
Due to the relatively large obstacles, movements
of the mobile units do not affect the short term
characteristics of the shadowing effect.
Instead, the nature of the terrain surrounding
the base station and the mobile units as well as
the antenna heights determine the shadowing
behavior.

Usually, shadowing is modeled as a slowly
time-varying multiplicative random process.
Neglecting all other channel impairments, the
received signal r(t) is given by
s(t) is the transmitted signal.
g(t) is the random process which models the
shadowing effect.
For a given observation interval, we assume g(t)
is a constant g, which is usually modeled 2 as
a lognormal random variable whose density
function is given by

We notice that ln g is a Gaussian random variable
with mean µ and variance s2.
This translates to the physical interpretation
that µ and s2 are the mean and variance of the
loss measured in decibels (up to a scaling
constant) due to shadowing.
For cellular and microcellular environments,s,
which is a function of the terrain and antenna
heights, can range 2 from 4 to 12 dB.

12
4.3 Fading

Fading
In a typical wireless communication environment,
multiple propagation paths often exist from a
transmitter to a receiver due to scattering by
different objects.
Signal copies following different paths can
undergo different attenuation, distortions,
delays and phase shifts.
Constructive and destructive interference can
occur at the receiver.
When destructive interference occurs, the signal
power can be significantly diminished.
The performance of a system (in terms of
probability of error) can be severely degraded by
fading.

Very often, especially in mobile communications,
not only do multiple propagation paths exist, but
they are also time-varying.
The result is a time-varying fading channel.
Communication through these channels can be
difficult.
Special techniques may be required to achieve
satisfactory performance.

14
4.3.1 Parameters of fading channels

The general time varying fading channel model is
too complex for the understanding and performance
analysis of wireless channels.
Fortunately, many practical wireless channels can
be adequately approximated by the wide-sense
stationary uncorrelated scattering (WSSUS) model
2, 3.
The time-varying fading process is assumed to be
a wide-sense stationary random process.
The signal copies from the scatterings by
different objects are assumed to be independent.

The following parameters are often used to
characterize a WSSUS fading channel.
Multipath spread
Coherence bandwidth
Doppler spread
Coherence time

Multipath spread Tm
Suppose that we send a very narrow pulse in a
fading channel.
We can measure the received power as a function
of time delay as shown in Figure 4.1.

The average received power P(t) as a function of
the excess time delaytis called the multipath
intensity profile or the delay power spectrum.
Excess time delay time delay- time delay of
first path
The range of values oftover which P(t) is
essentially non-zero is called the multipath
spread of the channel, and is often denoted by
Tm.
Tm essentially tells us the maximum delay between
paths of significant power in the channel.
For urban environments, Tm can 1 range from
to .

Coherence bandwidth
In a fading channel, signals with different
frequency contents can undergo different degrees
of fading.
The coherence bandwidth, denoted by ,
gives an idea of how far apart in frequency for
signals to undergo different degrees of fading.
Roughly speaking, if two sinusoids are separated
in frequency by more than , then they
would undergo different degrees of (often assumed
to be independent) fading.
It can be shown that is related to Tm by

Doppler spread Bd
Due the time-varying nature of the channel, a
signal propagating in the channel may undergo
Doppler shifts (frequency shifts).
When a sinusoid of frequency is transmitted
through the channel, the received power spectrum
can be plotted against the Doppler shift as in
Figure 4.2.

The result is called the Doppler power spectrum.
The Doppler spread, denoted by Bd, is the range
of values that the Doppler power spectrum is
essentially non-zero.
It essentially gives the maximum range of Doppler
shifts.

Coherence time
In a time-varying channel, the channel impulse
response varies with time.
The coherence time, denoted by , gives a
measure of the time duration over which the
channel impulse response is essentially invariant
(or highly correlated) .
Therefore, if a symbol duration is smaller than
, then the channel can be considered as
time invariant during the reception of a symbol.
Of course, due to the time-varying nature of the
channel, different time-invariant channel models
may still be needed in different symbol
intervals.

and Bd are related by

23
4.3.2 Classification of fading channels

Based on the parameters of the channels and the
characteristics of the signals to be transmitted,
time-varying fading channels can be classified
as
Frequency non-selective versus frequency
selective
Frequency non-selective (also called flat fading)
Channel
If the bandwidth of the transmitted signal is
small compared with , then all frequency
components of the signal would roughly undergo
the same degree of fading.

We notice that because of the reciprocal
relationship between and Tm and the one
between bandwidth and symbol duration, in a
frequency non-selective channel, the symbol
duration is large compared with Tm.
In this case, delays between different paths are
relatively small with respect to the symbol
duration.
We can assume that we would receive only one copy
of the signal, whose gain and phase are actually
determined by the superposition of all those
copies that come within Tm.

Frequency selective channel
On the other hand, if the bandwidth of the
transmitted signal is large compared with
, then different frequency components of the
signal (that differ by more than would
undergo different degrees of fading.
Due to the reciprocal relationships, the symbol
duration is small compared with Tm.
Delays between different paths can be relatively
large with respect to the symbol duration.
We then assume that we would receive multiple
copies of the signal.

Slow fading versus fast fading
Slow fading channel
If the symbol duration is small compared with
, then the channel is classified as slow
fading.
Slow fading channels are very often modeled as
time-invariant channels over a number of symbol
intervals.
Moreover, the channel parameters, which is slow
varying, may be estimated with different
estimation techniques.

fast fading (also known as time selective
fading).
On the other hand, if is close to or
smaller than the symbol duration, the channel is
considered to be fast fading.
In general, it is difficult to estimate the
channel parameters in a fast fading channel.

We notice that the above classification of a
fading channel depends transmitted signal.
The two ways of classification give rise to four
different types
Frequency non-selective slow fading
Frequency selective slow fading
Frequency non-selective fast fading
Frequency selective fast fading
If a channel is frequency non-selective slow
fading (also known as non-dispersive), then the
following relationships must be satisfied, where
T is the symbol duration.
and

or
The product TmBd is called the spread factor of
the physical channel.
If TmBd lt 1, the physical channel is underspread.
If TmBd gt 1, the physical channel is overspread.
Therefore, if a channel is classified as
frequency non-selective slow fading, the physical
channel must be underspread.

30
4.3.3 Common fading channel models

Based on the classification in Section 4.3.2, we
can develop mathematical models for different
kind of fading channels to facilitate the
performance analysis of communication systems in
fading environments.

31
Frequency non-selective fading channel

First, let us consider frequency non-selective
fading channels.
Suppose that the signal s(t) is sent.
Frequency non-selectiveness implies that we can
assume only one copy of the signal is received

In (4.6), the complex gain imposed by the fading
channel is represented by
where a(t) and ?(t) are the overall (real) gain
and the overall phase shift resulting, actually,
from the superposition of many copies with
different gains and phase shifts.
In general,a(t) and ?(t) are modeled as WSS
random processes.

For a slow fading channel,a(t) and ?(t) can be
assumed to be invariant over an observation
period less that .
Therefore, they can be simply replaced by random
variables.
Denoting the corresponding random variables
byaand ?, we have
Since the gainsacos(?) andasin(?) on the in-phase
and the quadrature channels result from the
superposition of large number of contributions,
they can be modeled as Gaussian random variables.
Very often, they are modeled as iid zero mean
Gaussian random variables.

Thus the complex gain is a zero-mean symmetric
complex Gaussian random variable.
This also implies that a is Rayleigh distributed
and ? is uniformly distributed on 0 2p).
The resulting model is called a frequency
non-selective slow Rayleigh fading channel.
This model is accurate when there is no
direct-line-of-sight path between the transmitter
and the receiver.
In some cases, especially when there is a
dominant propagation path from the transmitter to
the receiver, is better modeled by a Rician
random variable.
The result is a frequency non-selective slow
Rician fading channel.

For a fast fading channel, the characterizations
of the random processesa(t) and?(t) depend on the
Doppler power spectrum which, in turn, depends on
the physical channel environment, such as
The heights of the transmitter and receiver
antennae
The polarization of the radio wave
The speed of the mobile
The speed and geometry of the scatters.
Considering the received signal at a mobile unit
for special case where a vertical monopole
antenna is employed at the mobile unit with a
ring of scatters, the WSS processß(t) is modeled
4 as a zero-mean complex Gaussian process with
autocorrelation function

The Doppler spread Bd is given by
v is the speed of the mobile in the direction
toward the base station
fc is the carrier frequency
c is the speed of light
The Doppler spectrum is the Fourier transform of
the autocorrelation function and is given by

37
Frequency selective fading channel

In a frequency selective fading channel, many
distinct copies of the transmitted signal are
received at the receiver.
For the slow fading case, the received signal can
be expressed as
are the complex gains for the received
paths.
the number of distinct paths L, the gain of each
distinct path al, the phase shift of each
distinct path ?l, and the relative delay of each
distinct path tl are all random variables.
In the fast fading channel case, all these random
variables become random processes.

38
4.4 Diversity reception

We can see from (4.6) that the received signal
power reduces greatly when the channel is in deep
fades.
This causes a significant increase in the symbol
error probability.
To overcome the detrimental effect of fading, we
often make use of diversity.
The idea of diversity is to make use of multiple
copies of the transmitted signal, which undergo
independent fading, to reduce the degradation
effect of fading.
As a motivation to study diversity techniques, we
start by quantifying how much degradation on the
symbol error performance fading can cause for a
non-dispersive channel.

39
4.4.1 Performance under non-dispersive fading

Let us consider a BPSK system.
The transmitted signal is given by
is the data symbol.
The transmitted energy per symbol is
Under a frequency non-selective slow
(non-dispersive) Rayleigh fading channel, the
received signal is
n(t) represents AWGN with power spectral density
N0.
a is Rayleigh distributed.
?is uniformly distributed on 0 2p).

The received energy per symbol is
We define the received SNR?by
It can be shown 3 that?is chi-square
distributed with density function
the average received SNR
Suppose that we can accurately estimate?so that
optimal coherent detection can be performed.
Then the conditional symbol error probability
given is (see Section 1.4.1)

By averaging over?, we can show 3 that the
unconditional symbol error probability is
For Ps can be approximated by
.
An important observation is that Ps decreases
only inversely with the average received SNR .
On the other hand, when there is no fading, Ps
decreases exponentially with the received SNR
(which is a constant).
Therefore, a much larger amount of energy is
required to lower the probability of error in a
fading channel.
The same situation occurs with other types of
modulation under a frequency non-selective slow
Rayleigh fading channel.

42
4.4.2 Diversity Techniques

Diversity techniques can be used to improve
system performance in fading channels.
Instead of transmitting and receiving the desired
signal through one channel, we obtain L copies of
the desired signal through L different channels.
The idea is that while some copies may undergo
deep fades, others may not.
We might still be able to obtain enough energy to
make the correct decision on the transmitted
symbol.
There are several different kinds of diversity
which are commonly employed in wireless
communication systems

43
Frequency diversity

One approach to achieve diversity is to modulate
the information signal through L different
carriers.
Each carrier should be separated from the others
by at least the coherence bandwidth so
that different copies of the signal undergo
independent fading.
At the receiver, the L independently faded copies
are optimally combined to give a statistic for
decision.
The optimal combiner is the maximum ratio
combiner, which will be introduced later.
Frequency diversity can be used to combat
frequency selective fading.

44
Temporal diversity

Another approach to achieve diversity is to
transmit the desired signal in L different
periods of time, i.e., each symbol is transmitted
L times.
The intervals between transmission of the same
symbol should be at least the coherence time
so that different copies of the same symbol
undergo independent fading.
Optimal combining can also be obtained with the
maximum ratio combiner.
We notice that sending the same symbol L times is
applying the (L, 1) repetition code.
Actually, non-trivial coding can also be used.
Error control coding, together with interleaving,
can be an effective way to combat time selective
(fast) fading.

45
Spatial diversity

Another approach to achieve diversity is to use L
antennae to receive L copies of the transmitted
signal.
The antennae should be spaced far enough apart so
that different received copies of the signal
undergo independent fading.
Different from frequency diversity and temporal
diversity, no additional work is required on the
transmission end, and no additional bandwidth or
transmission time is required.
However, physical constraints may limit its
applications.
Sometimes, several transmission antennae are also
employed to send out several copies of the
transmitted signal.
Spatial diversity can be employed to combat both
frequency selective fading and time selective
fading.

46
Multipath diversity

As discussed before, the received signal consists
of multiple copies of the transmitted signal when
the channel is under frequency selective fading.
If the fading on different paths are independent,
we can combine the contributions from different
paths to enhance the total received signal power.
A receiver structure that performs this operation
is known as the Rake receiver.
W need to increase the signal bandwidth in order
to obtain the resolution required to separate
different transmission paths.
Therefore, spread spectrum techniques are usually
employed together with the Rake receiver.
Sometimes, different artificial transmission
paths are created in order to achieve multipath
diversity in the absence of frequency selective
fading.

47
4.5 Diversity combining methods

As discussed in Section 4.4.2, the idea of
diversity is to combine several copies of the
transmitted signal, which undergo independent
fading, to increase the overall received power.
Different types of diversity call for different
combining methods.
Here, we review several common diversity
combining methods.
In particular, we discuss maximal ratio combining
and Rake receiver in detail.

48
4.5.1 Maximal ratio combining

For simplicity, let us restrict our discussion to
non-dispersive fading channels and BPSK signals.
Figure 4.3 Block diagram representation of
combining approach used in maximal
ratio combining .

Assume that the transmitter equivalent lowpass
signal is given by u(t)
The signal on each of the L branches at the input
to the receiver is given by
(4.16)
where is the complex channel gain of
the kth path and is the additive noise in
the kth receiver branch .
The most general linear combining rule is
(4.17)

For the maximal ratio combining case , it is
desirable to maximize the instantaneous signal to
noise (SNR) ratio and therefore, hopefully,
minimize the probability of error at the output
of the combiner .
This may be done by proper selection of the
combiner coefficients, as will now be
demonstrated .
The signal s(t) and noise n(t) components at the
output of the combiner are, respectively,
(4.18)

Assuming that the nk(t) are independent , the
instantaneous SNR is then given by
(4.19)
where Nk denotes the variance of nk(t).
To maximize the instantaneous SNR ?, the Schwarz
inequality for complex quantities may be employed
, which is of the form
(4.20)

For complex quantities ak and bk. Equality, and
hence maximization, is achieved, if bk Kak for
any arbitrary complex constant K.
Letting
(4.21)
then the Schwarz inequality is maximized if and
only if
(4.22)

The maximal ratio combining rule is just a
weighted summation of each of the L branches,
whereby each branch is multiplied by the complex
conjugate of the channel gain and divided by
the noise variance of that channel .
Hence the receiver must be capable of determining
the channel gain and phase as well as the noise
variance for each branch .
The decision statistic at the output of the
matched filter detector following the maximal
ratio combiner is given by
where E is the energy per transmitted bit .

54
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55

The resulting conditional symbol error
probability is

In order to apply maximal ratio combining, we
need to have knowledge of the fading coefficients
ßk and the noise power spectral densities Nk of
the L channels.
We note that these channel parameters are usually
obtained by estimation and the errors in this
estimation process may sometimes affect the
effectiveness of the maximal ratio combining
scheme.
Extension of the maximal ratio combining scheme
to spread spectrum modulations is trivial if the
spread bandwidth is smaller than the coherence
bandwidth of the channel, i.e, the flat fading
assumption is still valid.
When the spread bandwidth is larger than the
coherence bandwidth of the channel, the spread
spectrum signal will experience frequency
selective fading.

In this case, one can still employ the form of
maximal ratio combining depicted in Figure 4.3 by
choosing, for example, the strongest path in each
channel. However, this may not be the best
strategy.
Next, we look at the performance gain obtained by
maximal ratio combining.
Let us consider the simple case where the noise
power spectral densities are equal, i.e, N1 N2
NL N0, and the L channels undergo
identical, independent Rayleigh fading.

From (4.23), the conditional symbol error
probability,
is Rayleigh distributed.
It can be shown 3 that ? is chi-squared
distributed with 2L degrees of freedom and its
density function is

Thus the unconditional symbol error probability
is
Compared to the case of no diversity (L 1), we
see that the symbol error probability decreases
with the L-th power of instead of .
This significantly reduces the loss in
performance due to fading when L is large.

60
4.5.2 Rake receiver

In a fading environment, the principal means for
a direct-sequence system to obtain the benefits
of diversity combining is by using a rake
receiver.
A rake receiver provides path diversity by
coherently combining resolvable multipath
components that are often present during
frequency-selective fading.
This receiver is the standard type for
direct-sequence systems used in mobile
communication networks.

Consider a multipath channel with
frequency-selective fading slow enough that its
time variations are negligible over a signaling
interval.
The receiver selects among the M baseband signals
or complex envelopes
(4.29)
T the duration of the transmitted signal.
Td the multipath delay spread.
L the number of multipath components.
the delay of component i
ci the channel parameter which is a complex
number representing the attenuation and
phase shift of component.

An idealized sketch of the output of a baseband
matched filter that receives three multipath
components of the signal to which it is matched
is shown in Figure 4.4.
Figure 4.4 Response of matched filter to input
with three resolvable multipath components.
If a signal has bandwidth W, then the duration of
the matched-filter response to this signal is on
the order of 1/W.
A necessary condition for at least two resolvable
multipath components is that duration 1/W is less
than the delay spread Td.

is required, which implies that
frequency-selective fading and resolvable
multipath components are associated with wideband
signals.
There are at most resolvable
components.
The receiver uses a separate baseband matched
filter or correlator for each possible desired
signal including its multipath components.
Thus, if is the symbol waveform,
then the
matched filter is matched to the signal in (4.29)
with the symbol
duration.
Each matched-filter output sampled at provides a
decision variable.

The decision variable is
(4.30)
where is the received signal, including
the noise, after down conversion to baseband.
An alternative form that requires only a single
transversal filter and M matched filters is
derived by changing variables in (4.30) and using
the fact that is zero outside the
interval
The result is
(4.31)

For frequency-selective fading and resolvable
multipath components, a simplifying assumption is
that each delay is an integer multiple of 1/W.
Accordingly, L is increased to equal the maximum
number of resolvable components, and we set
where is the maximum delay.
As a result, some of the may be equal to
zero.
The decision variables become
(4.32)

A receiver based on these decision variables,
which is called a rake receiver, is diagrammed in
Figure 4.5.
Figure 4.5 Rake receiver for M orthogonal
pulses. MF denotes a matched filter.

An alternative configuration to that of Figure
4.5 uses a separate transversal filter for each
decision variable and has the corresponding
matched filter in the front, as shown in Figure
4.6(a).
The matched-filter or correlator output is
applied to parallel fingers, the outputs of which
are recombined and sampled to produce the
decision variable.
The number of fingers Ls where is
equal to the number the resolvable components
that have significant power.
The matched filter produces a number of output
pulses in response to the multipath components,
as illustrated in Figure 4.4.
Each finger delays and weights one of these
pulses by the appropriate amount so that all the
finger output pulses are aligned in time and can
be constructively combined after weighting, as
shown in Figure 4.6(b).

Figure 4.6 Rake receiver (a) basic
configuration for generating a decision variable
and (b) a single finger.

69
4.5.3 Other diversity combining methods

There are several possible diversity combining
methods other than maximal ratio combining and
Rake receiver.
Suppose L independent non-dispersive fading
channels are available.
Instead of weighting the received signal from the
k-th channel by we can weight its
contribution by
This method is known as equal-gain combining and
it gives a conditional error probability of
which is suboptimal compared to the conditional
error probability given by the maximal ratio
combining in (4.23).

The advantage of equal-gain combining is that we
need only to estimate the phases of the L
channels.
The fading amplitudes and the noise power
spectral densities are not needed.
If we employ a non-coherent modulation scheme, we
can perform non-coherent equal-gain combining for
which the phases are also unnecessary.
Of course, further trade-off in the symbol error
probability performance is incurred in this case.

When the noises in the L channels are correlated,
maximal ratio combining is no longer optimal.
Multiple access interference (interference from
other users signals) across the L channels in
CDMA systems give a common example of correlated
noises.
In this case, a noise-whitening approach can be
employed to combine the contributions from the L
channels (see 7 for example).
Finally, if a code is applied across the L
channels, diversity combining should be applied
in conjunction with the decoding process of the
error-control code.
A common example of this is the use of
error-control coding and interleaving to combat
fast fading 3.

72
4.6 References

1 W. C. Y. Lee, Mobile celluar
Telecommunication System, McGraw-Hill, Inc.,
1989.
2 R. L. Peterson, R. E. Ziemer, and D. E.
Borth, Introduction to Spread Spectrum
Communications, Prentice Hall, Inc., 1995.
3 J. G. Proakis, Digital Communications, 3rd
Ed., McGraw-Hill, Inc., 1995.
4 W. C. Jakes, Microwave Mobile
Communications,Wiley, New York, 1974.
5 G. L. Turin, Introduction to spread-spectrum
antimultipath techniques and their application to
urban digital radio, Proc. IEEE, vol. 68, pp.
328353, Mar. 1980.
6 J. S. Lehnert and M. B. Pursley, Multipath
diversity reception of spread-spectrum
multipleaccess communications, IEEE Trans.
Commun., vol. 35, no. 11, pp. 11891198, Nov.
1987.
7 T. F. Wong, T. M. Lok, J. S. Lehnert, and M.
D. Zoltowski, A Linear Receiver for
Direct-Sequence Spread-Spectrum Multiple-Access
Systems with Antenna Arrays and Blind
Adaptation, IEEE Trans. Inform. Theory, vol. 44,
no. 2, pp. 659676, Mar. 1998.