Transcript 投影片 1

Chapter 4
Diversity Reception in
Spread Spectrum
1
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.
2
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.
3
4
• 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
– λis 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.
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• 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.
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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.
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• 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
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• We notice that ln g is a Gaussian random variable with mean μ
and variance σ2.
• This translates to the physical interpretation that μ and σ2 are the
mean and variance of the loss measured in decibels (up to a
scaling constant) due to shadowing.
• For cellular and microcellular environments,σ, which is a
function of the terrain and antenna heights, can range [2] from 4
to 12 dB.
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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.
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• 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.
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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 widesense stationary random process.
– The signal copies from the scatterings by different objects are
assumed to be independent.
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• The following parameters are often used to characterize a
WSSUS fading channel.
– Multipath spread
• Coherence bandwidth
– Doppler spread
• Coherence time
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• 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.
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– The average received power P(τ) as a function of the excess
time delayτis called the multipath intensity profile or the delay
power spectrum.
• Excess time delay = time delay- time delay of first path
– The range of values ofτover which P(τ) 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 0.5 s to 5 s .
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• Coherence bandwidth (f )c
– In a fading channel, signals with different frequency contents
can undergo different degrees of fading.
– The coherence bandwidth, denoted by (f )c , 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 (f )c , then they would undergo
different degrees of (often assumed to be independent)
fading.
– It can be shown that (f )c is related to Tm by
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• 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.
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• 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.
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• Coherence time (t )c
– In a time-varying channel, the channel impulse response
varies with time.
– The coherence time, denoted by (t )c , 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 (t )c , 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.
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• (t )c and Bd are related by
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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 (f )c, then all frequency components of
the signal would roughly undergo the same degree of
fading.
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• We notice that because of the reciprocal relationship
between (f )c 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.
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– Frequency selective channel
• On the other hand, if the bandwidth of the transmitted
signal is large compared with (f )c , then different
frequency components of the signal (that differ by more
than (f )c 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.
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• 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 timeinvariant channels over a number of symbol intervals.
• Moreover, the channel parameters, which is slow varying,
may be estimated with different estimation techniques.
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– 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.
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• 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
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or
• The product TmBd is called the spread factor of the physical
channel.
• If TmBd < 1, the physical channel is underspread.
• If TmBd > 1, the physical channel is overspread.
• Therefore, if a channel is classified as frequency non-selective
slow fading, the physical channel must be underspread.
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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.
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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:
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• In (4.6), the complex gain imposed by the fading channel is
represented by
– where α(t) and θ(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,α(t) and θ(t) are modeled as WSS random processes.
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• For a slow fading channel,α(t) and θ(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 byαand θ, we have
• Since the gainsαcos(θ) andαsin(θ) 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.
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• Thus the complex gain is a zero-mean symmetric complex
Gaussian random variable.
• This also implies that α is Rayleigh distributed and θ is uniformly
distributed on [0; 2π).
• 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.
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• For a fast fading channel, the characterizations of the random
processesα(t) andθ(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 processβ(t) is modeled [4] as a
zero-mean complex Gaussian process with autocorrelation
function
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• 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
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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 αl,
the phase shift of each distinct path θl, and the relative delay
of each distinct path τl are all random variables.
• In the fast fading channel case, all these random variables
become random processes.
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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.
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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.
– α is Rayleigh distributed.
– θis uniformly distributed on [0; 2π).
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• The received energy per symbol is
• We define the received SNRγby
• It can be shown [3] thatγis chi-square distributed with density
function
– the average received SNR
• Suppose that we can accurately estimateθso that optimal coherent
detection can be performed.
• Then the conditional symbol error probability given is (see
Section 1.4.1)
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• By averaging overγ, we can show [3] that the unconditional
symbol error probability is
1
4
• 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.
41
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:
42
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 (f )c 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.
43
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.
44
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.
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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.
46
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.
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4.5.1 Maximal ratio combining
• For simplicity, let us restrict our discussion to non-dispersive
fading channels and BPSK signals.
• Generalizations to other types of fading and modulation are
possible.
• Suppose the transmitted signal is
where b0 is the
binary data symbol.
• At the receiver, L copies of the transmitted signal are received
and the received signal from the k-th channel, for k = 1, 2, …, L,
is
–
are the complex fading gains on the L
independent channels.
–
are L independent AWGN processes with
power spectral densities N1, N2,…, NL.
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• We note that we can employ this model for the cases of frequency,
temporal, and spatial diversity.
49
• Suppose we want to linearly combine the received signals from
the L channels to form a decision statistic:
• Where
– is an AWGN process with power spectral density
50
• From Section 1.1, the maximum likelihood receiver of this
combined scheme is the one that decides
• Let us define the signal to noise ratio γ by
• whereε = T/2 is the transmitted symbol energy.
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• The symbol error probability given all the fading coefficients is
• Since the Q-function is monotone decreasing, we know that the
conditional symbol error probability is minimized when we
choose the combining coefficients ck’s to maximize γ.
• From the Schwartz inequality, we know that,
• Equality in (4.21) holds when
for some constant C.
52
• Thus we can maximize γ by choosing
resulting decision rule is
2 *
  L
 L *

k
bˆ0  sgn  Re  
  k
  k 1 N k  l 1 Nl

 
  L  k*
 sgn  Re  
  l 1 Nl

T
0

T
0

rl (t )dt   .

and the

rl (t )dt  
 

(4.22)
• This decision rule is shown in Figure 4.3.
• In short, we weight the matched filter output of the k-th channel
by
and then add up all the contributions from the L
channels to form the decision statistic.
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• The resulting conditional symbol error probability is
• When Nk = N0 for all k = 1, 2, …, L, we see from (4.23) that the
received powers from the L channels are added up.
• This combining method is called maximal ratio combining.
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• In order to apply maximal ratio combining, we need to have
knowledge of the fading coefficients βk 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.
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• 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.
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• From (4.23), the conditional symbol error probability,
–
is Rayleigh distributed.
• It can be shown [3] that γ is chi-squared distributed with 2L
degrees of freedom and its density function is
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• 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.
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4.5.2 Rake receiver
• When the transmission bandwidth, W, exceeds the coherence
bandwidth of the channel, the signal experiences frequency
selective fading and multiple transmission paths exist.
• For a slow fading channel, the received signal r(t) is given by
– where s(t) is the transmitted signal and n(t) is AWGN with
power spectral density N0.
• The incremental differences between excess delays
should have magnitudes at least of the order of 1/W because of
the frequency selective fading assumption.
• In this case, we can utilize multipath diversity.
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• First, let us assume that a single BPSK symbol is sent,
i.e,
• Given all the fading coefficients, the maximum likelihood
receiver is the one that gives the following decision statistic:
• The corresponding receiver, shown in Figure 4.4, is the so-called
Rake receiver.
60
61
• It can be shown easily that the conditional symbol error
probability given
and
is
provided
which is our
frequency selective fading assumption.
• For independent Rayleigh fading with a uniform multipath
intensity profile, i.e,
are iid Rayleigh random variables with
• The average symbol error probability is given by (4.27).
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• This means that the performance gain of multipath diversity using
the Rake receiver is the same as that of the maximal ratio
combining with L independent channels.
• In practice, we send a train of symbols instead of a single one.
• Unless consecutive symbols are separated by a guard interval
which is larger than the delay spread of the channel, they will
interfere with each other.
• However, the insertion of guard intervals greatly reduces the
spectral efficiency (number of symbols transmitted per second
per unit frequency).
• To avoid unnecessary waste of bandwidth, we usually pack data
symbols tightly together in practice.
• This means that the Rake receiver in Figure 4.4 cannot be applied
directly when a train of symbols is sent.
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• This problem can be solved by employing DS-SS modulation.
• The transmitted bandwidth of the DS-SS system is determined by
the chip duration, which is usually much smaller than the symbol
duration.
• Thus we can still resolve multipaths and employ multipath
diversity using the Rake receiver.
• Intersymbol interference is not a severe problem in DS-SS as
long as the delay spread is smaller than the period of the
spreading sequence, which is designed to have a small out-ofphase autocorrelation magnitude [5, 6].
• The Rake receiver structure in Figure 4.4 can be easily modified
(see Homework 4) to accommodate DS-SS signaling.
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• With the use of DS-SS, the transmission bandwidth increases to
the order of 1/Tc, where Tc is the chip duration.
• Therefore we are able to resolve multipaths separated by
incremental delays larger than Tc.
• Since the symbol duration T = NTc and N is usually large, the
conditions
assumed in (4.31) do not hold
anymore.
• Thus one may not be able to use (4.31) to obtain the conditional
symbol error probability.
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• Let us consider a simple case where the delay spread Tm is much
smaller than T and the period of the spreading sequence is N (i.e,
a short sequence).
• If the sequence is properly chosen, intersymbol interference is not
a big concern.
• we can simply assume that one symbol is sent, i.e,
– where a(t) is the BPSK spreading signal given by (2.12).
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• The Rake receiver is still optimal in this case and the decision
statistic z is
• We note that the integral
can be expressed in terms of the aperiodic autocorrelation
function of the sequence (see (3.15)) in a way similar to (3.14).
• In particular, for l = k, it takes on the value T.
For l ≠ k, its value should be much smaller than T if the sequence
is properly chosen.
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• The second term in (4.32) is a zero-mean Gaussian random
variable, zn, with variance equal to
– which is approximately equal to
argument above.
• Hence, when the sequence is chosen properly.
by the same
• With this approximation, the conditional error probability is again
given by (4.31).
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• In summary, we can employ DS-SS to enhance the multipath
resolution and combine the powers from different paths in an
optimal manner.
• The advantage of DS-SS is two-fold:
– Alleviates the detrimental effect of intersymbol interference
on the Rake receiver
– Eenhances the multipath resolution.
• Therefore, multipath diversity and Rake receiver usually come
along with DS-SS signaling.
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 (see Homework 4)
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• 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.
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• 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].
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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. 328–353, Mar.
1980.
[6] J. S. Lehnert and M. B. Pursley, “Multipath diversity reception of spreadspectrum multipleaccess communications,” IEEE Trans. Commun., vol. 35, no.
11, pp. 1189–1198, 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.
659–676, Mar. 1998.
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