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Chapter 4

Diversity Reception in Spread Spectrum

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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.

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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.

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• 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 wide-sense 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 to .

5 s0.5 s

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• 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

( )cf

( )cf

( )cf

( )cf

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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– 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.

( )ct

( )ct

( )ct

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• and Bd are related by( )ct

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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 , then all frequency components of the signal would roughly undergo the same degree of fading.

( )cf

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• 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.

( )cf

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( )cf

– 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.

( )cf

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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 time-invariant 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

• 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.

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4

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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:

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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.

( )cf

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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.

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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.

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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.

• Figure 4.3: Block diagram representation of combining approach used in maximal ratio combining .

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• 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)

( )k t

( )kx t

( )kn t

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• 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)

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• 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 γ, the Schwarz inequality for complex quantities may be employed , which is of the form

(4.20)

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• 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)

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• 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 .

k

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• The resulting conditional symbol error probability is

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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

• 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.

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• 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 compone

nt.

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• 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.

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• 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.

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• 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)

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• 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)

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• 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.

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• 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).

sL L

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

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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).

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• 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 spread-spectrum 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.