Diversity Reception in Spread Spectrum

Tan F. Wong: Spread Spectrum & CDMA

4. Diversity in Spread Spectrum

Chapter 4 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. Besides the thermal noise at the receiver front end (which is modeled by AWGN), there are several other well studied channel impairments [1] in a typical wireless channel: Path Loss which describes the loss in power as the radio signal propagates in space Shadowing which is due to the presence of fixed obstacles in the propagation path of the radio signal Fading which accounts for the combined effect of multiple propagation paths, rapid movements of mobile units (transmitters/receivers) and reflectors. We will give a brief introduction on these three impairments. Our focus is on fading and how spread spectrum techniques can help to combat fading in wireless channels.

4.1 Path loss 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

L=(

4d )2;

4.1

(4.1)


where


is the wavelength of the signal, and 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:

L = cdn ; where

(4.2)

c is a constant and 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.

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:

r(t) = g (t)s(t),

(t) is the transmitted signal and 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 where s

lognormal random variable whose density function is given by 8 >
:

We notice that

p21g exp

0

(ln g )2 22

g0 g < 0:

(4.3)

ln g is a Gaussian random variable with mean and variance 2. This translates to the

physical interpretation that and

2 are the mean and variance of the loss measured in decibels (up to a scaling constant) due to shadowing. For cellular and microcellular environments, , which is a

4 12 dB.

function of the terrain and antenna heights, can range [2] from to

4.2



4.3 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. This phenomenon is called fading. 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.

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]. In the WSSUS model, the time-varying fading process is assumed to be a wide-sense stationary random process and 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 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

( ) as a function of the

excess time delay1 is called the multipath intensity profile or the delay power spectrum. The range of values of over which P

( ) is essentially non-zero is called the multipath spread of the channel, and

is often denoted by Tm . It essentially tells us the maximum delay between paths of significant power in the channel. For urban environments, Tm can [1] range from 1

Excess time delay = time delay

time delay of first path

4.3

0:5s to 5s.



average received power

excess delay τ

t Tm

transmitted impulse

Figure 4.1: Multipath delay profile 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

(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 by more than

(f )c 1=Tm: Coherence time

(4.4)

(t)c

In a time-varying channel, the channel impulse response varies with time. The coherence time, denoted

(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

by

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. 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 4.4



average received power

single tone

f

Doppler shift Bd Figure 4.2: Doppler power spectrum

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. It can shown that

(t)c and Bd are related by (t)c 1=Bd:

(4.5)

4.3.2 Classification of fading channels Based on the parameters of the channels and the characteristics of the signals to be transmitted, timevarying fading channels can be classified as: Frequency non-selective versus frequency selective If the bandwidth of the transmitted signal is small compared with

(f )c, then all frequency compo-

nents of the signal would roughly undergo the same degree of fading. The channel is then classified as frequency non-selective (also called flat fading). 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 . 4.5



On the other hand, if the bandwidth of the transmitted signal is large compared with different frequency components of the signal (that differ by more than

(f )c, then

(f )c) would undergo differ-

ent degrees of fading. The channel is then classified as frequency selective. 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 If the symbol duration is small compared with

(t)c, 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. On the other hand, if

(t)c is close to or smaller than the symbol duration, the channel is considered

to be fast fading (also known as time selective 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 on the properties of the transmitted signal. The two ways of classification give rise to four different types of channel:

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 relationships

Tm < T and

T < (t)c 4.6



must be satisfied, where T is the symbol duration. Therefore,

Tm < (t)c or

Tm Bd < 1: The product Tm Bd is called the spread factor of the physical channel. If Tm Bd channel is underspread. If Tm Bd

< 1, the physical

> 1, the physical channel is overspread. Therefore, if a channel is

classified as frequency non-selective slow fading, the physical channel must be underspread.

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. 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:

r(t) = (t)s(t): In (4.6), the complex gain imposed by the fading channel is represented by

(4.6)

(t)

= (t) exp(j(t)),

(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, (t) and (t) are where

modeled as WSS random processes. For a slow fading channel,

(t) and (t) can be assumed to be invariant over an observation period

(t)c. Therefore, they can be simply replaced by random variables. Denoting the corresponding random variables by and , we have (t) = = exp(j ). Since the gains cos( ) and sin( ) less that

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

4.7



mean Gaussian random variables. Thus the complex gain is a zero-mean symmetric complex Gaussian random variable. This also implies that is Rayleigh distributed and is uniformly distributed on

[0; 2). 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 processes

(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, and the speed and geometry of the scatterers. 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

(t) is modeled [4] as a zero-mean complex Gaussian process with autocorrelation function R ( ) = J0 (2Bd ). The Doppler spread Bd is given by of scatterers, the WSS process

Bd = vfc =c;

(4.7)

where v is the speed of the mobile in the direction toward the base station, fc is the carrier frequency, and

c is the speed of light. The Doppler spectrum is the Fourier transform of the autocorrelation

function and is given by

(f ) =

8 > > < > > :

1

Bd

0

p1

1 (f=Bd )2

for jf j < Bd

(4.8)

elsewhere.

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

r(t) = where l

L X l=1

l s(t l );

(4.9)

= l exp(jl ), for l = 1; 2; : : : ; L, are the complex gains for the received paths. In (4.9), the

number of distinct paths L, the gain of each distinct path l , the phase shift of each distinct path l , 4.8



and the relative delay of each distinct path l are all random variables. In the fast fading channel case, all these random variables become random processes.

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 degradating 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.

4.4.1 Performance under non-dispersive fading Let us consider a BPSK system. The transmitted signal is given by

s(t) = b0 pT (t); where

b0

2 f1g is the data symbol.

(4.10)

The transmitted energy per symbol is

E = T=2.

Under a

frequency non-selective slow (non-dispersive) Rayleigh fading channel, the received signal is

r(t) = ej b0 pT (t) + n(t);

(4.11)

n(t) represents AWGN with power spectral density N0 , is Rayleigh distributed and is uniformly distributed on [0; 2 ). The received energy per symbol is 2 E . We define the received SNR

by

= 2 E =N0: (4.12) where

It can be shown [3] that is chi-square distributed with density function 8 >
:

1

0

exp(

)

for

0

otherwise,

4.9

(4.13)


where the average received SNR


= E [ ] = E [2]E =N0. 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) q Pr(symbol errorj ) = Q( 2 ): (4.14) By averaging over , we can show [3] that the unconditional symbol error probability is

1 Ps = 1 2

s

1 +

!

:

(4.15)

>> 1, Ps can be approximated by 41 . 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 For

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.

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: 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.

4.10



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

(t)c 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. 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. 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. As mentioned in Section 2.6.3, we 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. Other approaches, including polarization diversity and different combinations of frequency, tem-

4.11



poral and spatial diversity, may also be used.

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.

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

s(t) = b0 pT (t), 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 rk (t) = k s(t) + nk (t); where 1 ; 2 ; : : : ; L are the complex fading gains on the L independent channels and n1

(4.16)

(t), n2(t), : : :,

nL (t) are L independent AWGN processes with power spectral densities N1 ; N2 ; : : : ; NL . The block diagram representing this diversity reception model is given in Figure 4.3. We note that we can employ this model for the cases of frequency, temporal, and spatial diversity. Suppose we want to linearly combine the received signals from the L channels to form a decision statistic:

r(t)

= =

L X k=1

b0

ck rk (t) L X k=1

where

n(t) =

!

ck k pT (t) + n(t);

L X k=1

ck nk (t)

4.12

(4.17)

(4.18)



is an AWGN process with power spectral density

PL

k=1

jck j2Nk .

From Section 1.1, the maximum

^ = sgn(Re(z)), where

likelihood receiver of this combined scheme is the one that decides b0

z

L X

=

k=1 L X

=

k=1

Z

!

ck k

T

r(t) dt

0

X L

!

ck k

l=1

cl

Z

T

0

rl (t) dt:

(4.19)

Let us define the signal to noise ratio by

Ej

=

PL

PL

k=1 ck k

k=1

j2 ;

jck j2Nk

(4.20)

E = T=2 is the transmitted symbol energy. The symbol error probability given all the fading p coefficients is Q( 2 ). Since the Q-function is monotone decreasing, we know that the conditional where

symbol error probability is minimized when we choose the combining coefficients ck ’s to maximize . From the Schwartz inequality, 2 we know that, L X

k=1

Equality in (4.21) holds when ck choosing ck

2 k

ck

=

j k j2

L X

!

Nk

k=1

k=1

0

2

@

=

!

jck j Nk : 2

(4.21)

C k =Nk for some constant C . Thus we can maximize by

= k=Nk and the resulting decision rule is L 2 j j k b^ = sgn Re 0

L X

4

sgn Re

!

X

k=1 Nk " L X k Z T l=1

Nk

0

31

k Z T rl (t) dt5A l=1 Nk 0

L X

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 k =Nk and then add up all the contributions from the

L channels to form the decision

statistic. The resulting conditional symbol error probability is

Q When

Nk

=

N0 for all k

0v u u @t

2E

L X k=1

1

j k j2 : N A

k

(4.23)

= 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. In order to apply PL 2 k=1

2

L ja j2 PL jb j2 . Equality holds if and only if a k k=1 k k=1 k p p obtain (4.21), we let ak = ck Nk and bk = k = Nk .

ak bk

P

4.13

= Cbk for some constant C .

To



T

s(t) 0

β1

dt β*1 /N 1

n1(t) T

s(t) 0

β2

dt β*2 /N 2

n2(t)

Σ

z

T

s(t) 0

βL

dt β*L /N L

nL(t)

Figure 4.3: Maximal ratio combiner for BPSK modulation

4.14

sgn(Re( ))

^ b0



maximal ratio combining, we need to have knowledge of the fading coefficients k 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, q

Pr(symbol errorj ) = Q( 2 ); where

E

=

and k

PL

(4.24)

k=1 k 2

(4.25)

N0

= j k j is Rayleigh distributed. It can be shown [3] that is chi-squared distributed with 2L

degrees of freedom and its density function is 8 >
:

1 (L 1)!

L

0

L

1

exp(

= ) 0

< 0;

(4.26)

= NE E [k2 ] for k = 1; : : : ; L. Thus the unconditional symbol error probability is

where

0

Ps

=

1

Z

2 )p ( ) d LL 1 1

= 2 1 1 + k=0 0

"

1,

When

q

Q(

s

!#

X

1 + k 1 1 + k 2 1 + !"

L

2L 1 1 Ps 4 L !

4.15

s

!#k

:

(4.27)

!L

(4.28)



Compared to the case of no diversity (L the L-th power of

= 1), we see that the symbol error probability decreases with

1= instead of 1= . This significantly reduces the loss in performance due to fading

when L is large.

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 r(t) =

L X l=1

l s(t l ) + n(t);

(4.29)

s(t) is the transmitted signal and n(t) is AWGN with power spectral density N0 . The incremental differences between excess delays 1 ; 2 ; : : : ; L 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 where

diversity. First, let us assume that a single BPSK symbol is sent, i.e,

s(t)

= b0 pT (t).

Given all the fading

coefficients, the maximum likelihood receiver is the one that gives the following decision statistic:

z

assuming L

> L

1

>

> 1.

Z 1 l r(t)pT (t 1 l=1

=

L X

=

L X l=1

l

Z

L +T L

r(t

l ) dt

(L

l )) dt

(4.30)

The corresponding receiver, shown in Figure 4.4, is the so-called

Rake receiver. It can be shown easily that the conditional symbol error probability given and 1 ; : : : ; L is

Q

0v u u @t

2E

L X

N0 k=1

1 ; : : : ; L

1

j k j2 ; A

(4.31)

T , for k = 2; : : : ; L, which is our frequency selective fading assumption. For independent Rayleigh fading with a uniform multipath intensity profile, i.e, j k j are iid Rayleigh random variables with NE E [j k j2 ] = for k = 1; : : : ; l, the average symbol error probability is given provided k

k

1

0

by (4.27). This means that the performance gain of multipath diversity using the Rake receiver is the

4.16


τ L−τL-1

s(t)

β*L


τ L-1−τL-2

* βL-1

β*2

Σ τ

delay by

τ 2 −τ1

τ L+T τL

τ

dt

z

Figure 4.4: Rake receiver for BPSK modulation

4.17

β*1

sgn(Re( ))

^ b0



same as that of the maximal ratio combining with L independent channels. For other types of multipath intensity profiles, the performance gains will be different. 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. 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-of-phase autocorrelation magnitude [5, 6]. The Rake receiver structure in Figure 4.4 can be easily modified (see Homework 4) to accommodate DS-SS signaling. 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 k

k

1

T 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. Let us consider a simple case where the delay spread Tm is much smaller than T 3 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 and we can simply assume that one symbol is sent, i.e,

s(t)

= b0a(t)pT (t), where a(t) is the BPSK spreading signal given by (2.12).

The Rake receiver is

still optimal in this case and the decision statistic z is

z

= =

3

1 r(t)a(t l )pT (t l ) dt 1 l=1 Z 1 L X L X b0 l k a(t k )a(t l )pT (t 1 l=1 k=1 L X

l

Z

l )pT (t k ) dt

In this case, we cannot employ multipath diversity and Rake receiver without spreading since the channel is frequency

non-selective without spreading.

4.18


+

L X l=1

l

Z

(t

R We note that the integral 11 a


1 n(t)a(t 1

k )a(t

l )pT (t l ) dt: l )pT (t

l )pT (t

(4.32)

k ) dt 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 6= k, its value should be much smaller than T if the

sequence is properly chosen. The second term in (4.32) is a zero-mean Gaussian random variable, zn , with variance equal to

N0

L X L X l=1 k=1

Z

l k

which is approximately equal to N0 T

1 a(t 1 PL

l=1

k )a(t l )pT (t l )pT (t k ) dt

(4.33)

j lj2 by the same argument above. Hence,

z b0 T

L X l=1

j l j2 + zn;

(4.34)

when the sequence is chosen properly. With this approximation, the conditional error probability is again given by (4.31). 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: DS-SS alleviates the detrimental effect of intersymbol interference on the Rake receiver and enhances the multipath resolution. Therefore, multipath diversity and Rake receiver usually come along with DS-SS signaling.

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 k =Nk , we can weight its contribution by

exp( jk ).

This method is known as equal-gain combining and it gives a conditional error probability of (see Homework 4)

Q

0v u u Bu Bt @

2E

PL

k=1

PL

j k j

k=1 Nk

2

1 C C A

(4.35)

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. 4.19



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].

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 DirectSequence Spread-Spectrum Multiple-Access Systems with Antenna Arrays and Blind Adaptation,” IEEE Trans. Inform. Theory, vol. 44, no. 2, pp. 659–676, Mar. 1998.

4.20

Diversity Reception in Spread Spectrum

Diversity Reception in Spread Spectrum

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