Chap13 Fading Channels I

7/30/2019 Chap13 Fading Channels I

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Digital CommunicationsChapter 13 Fading Channels I: Characterization and Signaling

Po-Ning Chen

Institute of Communication Engineering

National Chiao-Tung University, Taiwan

Digital Communications: Chapter 13 Ver 2010.09.06 Po-Ning Chen 1 / 110


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13.1 Characterization of fading multipath

channels



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The multipath fading channels with additive noise



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Time spread phenomenon of multipath channels(Unpredictable) Time-variant factors

DelayNumber of spreads

Size of the receive pulses



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Transmitted signal

s(t) = Res(t)e2fct

Received signal in absence of additive noiser

(t

)=

c

(; t

)s

(t

)d

=

c(; t)Res(t )e2fc(t)d= Re

c(; t)e 2fcs(t )d e2fct= Res(t) c(; t)e 2fc

c(;t) e2fct



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Note that now it is not appropriate to write s(t) c(t)because t and are now specifically for time argument andconvolution argument!

We should perhaps write s(t) c() and s(t) c(; t),which respectively denote:

s(t) c() =

c()s(t )dand

s

(t

) c

(; t

)=

c

(; t

)s

(t

)d.

From the previous slide, we know

c

(; t

)= c

(; t

)e 2fc and c

(; t

)=

c

(; t

).



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Rayleigh and Rician

The measurement suggests that c

(; t

)is a 2-D Gaussain

random process in t (not in ), which can be supported by thecentral limit theorem (CLT) because it is the the sum effectof many paths.

If zero mean,

c

(; t

)is Rayleigh distributed. The

channel is said to be a Rayleigh fading channel

If nonzero mean, c(; t) is Rician distributed. Thechannel is said to be a Rician fading channel

When diversity technique is used, c(; t) can be well modeledby Nakagami m-distribution.

Detail of these distributions can be found in Section 2.3.



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13.1-1 Channel correlation functions and

power spectra



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Assumption (WSS)

Assume c

(; t

)is WSS in t.

Rc ( , ; t) = E{c(; t+t)c (; t)}is only a function of time difference t.

Assumption (Uncorrelated scattering or US of a WSS channel)

For , assume c(; t1) and c(; t2) are uncorrelated forany t1, t2.

is the convolution argument and actually represents the

delay for a certain path.

Assumption (Math definition of US)

Rc ( , ; t

)= Rc(

; t

)

(

)Digital Communications: Chapter 13 Ver 2010.09.06 Po-Ning Chen 9 / 110


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Discussions

It may appear unnatural to define the autocorrelationfunction of a channel impulse response using the Dirac

delta function.

However, we have already learned that is theconvolution argument, and

(

)is the impulse response

of the identity channel. This hints somehow theconnection between channel impulse response andDirac delta function.

Recall that a WSS white (noise) process z() is definedbased onE

[z

(

)z

(+

)]=

N0

2

(

).



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Discussions

We can extensionally view that the autocorrelation functionof a 2-dimensional WSS white noise z

(; t

)is defined as

E[z(; t1)z(+ ; t2)] = N0(t1, t2)2

().US indicates that the accumulated power correlation from all

other paths is essentially zero!

Some researchers interpret US as zero-correlationscattering. So, from this, they dont interpret it as

E[z(; t1)z

(+ ; t2)] = E[z(; t1)]E[z

(+ ; t2)] = 0,which requires zero-mean assumption but simply sayE

[z

(; t1

)z

(+ ; t2

)]= 0 when 0.


M l i h i i fil f US WSS h l


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Multipath intensity profile of an US-WSS channel

The multipath intensity profile or delay powerspectrum for a US-WSS multipath fading channel is

given by: Rc () = Rcell (; t= 0) .It can be interpreted as the average signal powerremained after delay :


M l i h d f US WSS h l


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Multipath spread of an US-WSS channel

The multipath spread or delay spread of a US-WSSmultipath fading channel

The range of over which Rc() is essentially nonzero;it is usually denoted by Tm.



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Each corresponds to one path.No Tx power will remain at Rx after for path with delay > Tm.


T f f ti f lti th f di h l


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Transfer function of a multipath fading channel

The transfer function of a channel impulse response c(; t

)is

the Fourier transform with respect to the convolutionalargument :

C

(f; t

)=

c

(; t

)e 2f d

Property: If c

(; t

)is WSS; then so is C

(f; t

).

The autocorrelation function of WSS C(f; t) is equal to:RC(f, f; t) = EC(f; t+t)C (f; t)



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With an additional US assumption,

RC

(f, f; t

)= EC(f; t+t)C

(f; t)= E

c(; t+t)e 2f d

c (; t)e2f d=

Rc

(; t

)

(

)e2(ff) dd

=

Rc (; t) e 2ff1 d1= RC

(f; t

)For a US-WSS multipath fading channel,RC(f; t) = EC(f+f; t+t)C (f; t)This is often called spaced-frequency, spaced-timecorrelation function of a US-WSS channel.


C h t b d idth


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Coherent bandwidth

RC (f; t) = Rc (; t) e 2(f) dFor the case of t= 0, we have

RC (f)spaced-frequencycorrelation function

=

Rc () e 2(f) dRecall that Rc() = 0 outside [0,Tm).In freq domain, (f)c = 1Tm is correspondingly calledcoherent bandwidth.



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Example


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Example.

Give Rc

(

)= 107

(107

)for 0 < 100 ns. Then,

RC(f) = 107

42(f)2 e 2107f 1 12f .

RC(f)RC(0) Rc

(

)Digital Communications: Chapter 13 Ver 2010.09.06 Po-Ning Chen 19 / 110 Coherent bandwidth


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Coherent bandwidth

Since the channel output due to input s(t

)is equal to:

r(f) = s(f)C(f; t),where we abuse the notations to denote the Fourier transformsof r

(t) and s(t) as r(f) and s(f), we would sayr(f) = s(f)C(f; t)will have weak (power) affection on

r(f+f) = s(f+f)C(f+f; t)when f>

(f

)c.



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If signal transmitted bandwidth

B> (f)c, the channel is calledfrequency selective.

If signal transmitted bandwidthB< (f)c, the channel is calledfrequency non-selective.



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For frequency selective channels, the signal shape is moreseverely distorted than that of frequency non-selectivechannels.

Criterion for frequency selectivity:

B> (f)c 1T

>1

Tm T < Tm.


Time varying characterization: Doppler


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Time varying characterization: Doppler

Doppler effect appears via the argument t.


Doppler power spectrum of a US-WSS channel


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Doppler power spectrum of a US WSS channel

The Doppler power spectrum is

SC() =

RC(f= 0; t)e 2(t)d(t),where is referred to as the Doppler frequency.

Bd = Doppler spread is the range such that SC() isessentially zero.(t)c = 1Bd is called the coherent time.If symbol period T >

(t

)c

, the channel is classified as

Fast Fading.I.e., channel statistics changes within one symbol!

If symbol period T < (t)c

, the channel is classified asSlow Fading.



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Scattering function


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Scattering function

Summary:

Rc(; t) Channel autocorrelation function1-D FT:

RC

(f; t

)= F

{Rc

(; t

)}Spaced-freqspaced-timecorrelation func

S(;) = Ft{Rc(; t)} Scattering function2D FT: ??? = F,t{Rc(; t)}RC

(f; t

)1-D FT: ??? = Ff{RC(f; t)}SC(f;) = Ft{RC(f; t)} Doppler powerspectrum (f= 0)2D FT: S

(;

)= Ff,t

{RC

(f; t

)}Digital Communications: Chapter 13 Ver 2010.09.06 Po-Ning Chen 26 / 110 Scattering function


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Scattering function

The scattering function can be used to identify delayspread and Doppler spread at the same time.


Example. Medium-range tropospheric scatter


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Example. Medium range tropospheric scatter

channel


Example study of delay spread


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p y y p

The median delay spread is the 50% value, meaning that50% of all channels has a delay spread that is lower than themedian value. Clearly, the median value is not so interestingfor designing a wireless link, because you want to guaranteethat the link works for at least 90% or 99% of all channels.

Therefore the second column gives the measured maximumdelay spread values. The reason to use maximum delayspread instead of a 90% or 99% value is that many papersonly mention the maximum value. From the papers that do

present cumulative distribution functions of their measureddelay spreads, it can be deduced that the 99% value is only afew percent smaller than the maximum delay spread.



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Measured delay spreads in frequency range of 800M to 1.5GHz (surveyed by Richard van Nee)



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Measured delay spreads in frequency range of 1.8 GHz to 2.4GHz (surveyed by Richard van Nee)



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Measured delay spreads in frequency range of 4 GHz to 6 GHz(surveyed by Richard van Nee)

Conclusion by Richard van Nee: Measurements done atdifferent frequencies show the multipath channelcharacteristics are almost the same from 1 to 5 GHz.


Jakes model: Example 13.1-3


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p

Jakes modelA widely used model for Doppler power spectrum is the

so-called Jakes model (Jakes, 1974)RC(t) = J0(2fm(t))

and

SC() = 1

fm

1

1

(fm)

2,

fm

0, otherwise

where

fm = vfc

c is the maximum Doppler shift

v is the vehicle speed (m/s)

c is the light speed (3 108 m/s)fc is the carrier frequency

J0

(

)is the zero-order Bessel function

of the first kind.


Jakes model: Example 13.1-3


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Difference in path length

L = (L sin())2 + (L cos() + v t)

2 L

= L2 + v2(t)2 + 2L v t cos() LPhase change = 2 L(cfc)



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Estimated Doppler shift

m = limt0

12

t

=1

c

fc

limt0

L2 + v2(t)2 + 2L v t cos() L

t

= vfcc

cos() = fm cos()Example. v = 108 km/hour, fc = 5 GHz and c= 1.08 10

9

km/hour.

m = 500cos() Hz.This is ok because 500 Hz5GHz = 0.1 ppm.


Jakes model


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Here, a rough (and no so rigorous) derivation is provided for

Jakes model.

Just to give you a rough idea of how this model is obtained.

Suppose

(t

)is the delay

of some path.

(t) = limt0 (t+t)(t)t= limt0

L+Lc

L

c

t

= limt0L

ct

= vc

cos() (t) v

ccos()t +


A ( ) ( )( ) d i i d d f ( )


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Assume c(; t) c(t)() and is independent of c(t).c

(; t

) c

(t

)

(

)e 2fc(t)

= c(t)()e 2fc vc cos()t+= c(t)()e 2fce 2fm cos()t

Rc(; t) =

E [c(; t + t)c

(; t)] d= E c(t + t)()e 2fce 2fm cos()(t+t)c

(t

)

(

)e2fce2fm cos()t

= E [c(t)c(t + t)]E e 2fm cos()t ()Recall Rc( , ; t) = Rc(; t)( ).



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RC(f= 0; t)=

Rc(; t)d=

E

[c

(t

)c

(t+ t

)]E

e 2fm cos()t

(

)d

= E [c(t)c(t + t)]E e 2fm cos()t= E [c(t)c(t + t)] J0(2fm(t)),

where the last step is valid if is uniformly distributed over

[, ).


can be treated as uniformly distributed over [, ) and


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y [ , )independent of attenuation and delay path .


Channel model from IEEE 802.11 Handbook


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A consistent channel model is required to allowcomparison among different WLAN systems.

The IEEE 802.11 Working Group adopted the followingchannel model as the baseline for predicting multipath for

modulations used in IEEE 802.11a and IEEE 802.11b,which is ideal for software simulations.

The phase is uniformly distributed.

The magnitude is Rayleigh distributed with averagepower decaying exponentially.


imax1


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c(; t) = i=0

ie i( iTs)

where

Ts sampling periodie i N(0, 2i 2) + N(0, 2i 2)2

i= 20 e

iTsrms20 = 1 e

Tsrms


R ( ) E [ ( ) ( )]d


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Rc() =

E [c(; t)c (; t)]d=

imax1

i=0

E 2i ( iTs)( )d

=imax1

i=0

E 2i ( iTs)=

imax1

i=0 2

0e

iTs

rms

( iTs)By this example, I want to introduce the rms delay. By definition,the effective rms delay is

T2rms =

2Rc()d Rc()d

Rc()d Rc()d

2

=imax1i=0 (iTs)220eiTsrmsimax1i=0 20eiTsrms

imax1i=0 (iTs)20eiTsrmsimax1i=0 20eiTsrms

2


We wish to choose imax such that Trms rms.


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Let rms =rmsTs

, Trms =Trms

Tsand imax =

imaxrms

.

We obtain

Trms =

e1rms

(1 e1rms

)2

i2maxe

imax

(1 eimax

)2

1

2rms

= 2rms 112 + 1240 12rms + i2maxeimax(1 eimax)2 12rmsTaking Trms =

2rms 1

12 andi2maxe

imax

(1eimax

)2= 1240 yield

imax = 10.1072 . . . (or equivalently, imax = 10rms

Ts).



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Typical Multipath Delay Spread for Indoor environment (Table 8-1of IEEE 802.11 Handbook)



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13.1-2 Statistical models for fading

channels



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In addition to zero-mean Gaussian (Rayleigh), non-zero-meanGaussian (Rice) and Nakagami-m distributions, there are other

models for c(; t) have been proposed in literature.Example.

Channels with a direct path and a single multipath

component, such as airplane-to-ground communications

c(; t) = () + (t)( 0(t))where controls the power in the direct path and is

named specular component, and (t) is modeled aszero-mean Gaussian.Digital Communications: Chapter 13 Ver 2010.09.06 Po-Ning Chen 47 / 110


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Example.

Microwave LOS radio channels used for long-distance

voice and video transmission by telephone companies inthe 6 GHz band (Rummler 1979)

c

(

)=

(

) e2f0

( 0

)where

overall attenuation parameter

shape parameter due to multipath components

0 time delay

f0 frequency of the fade minimum, i.e., f0 = arg minfR C(f)Digital Communications: Chapter 13 Ver 2010.09.06 Po-Ning Chen 48 / 110

Rummler found that

1


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1

2 f

(

)

(1

)2.3

3

log() Gaussian distributed (i.e., lognormaldistributed)4 0 6.3 ns

Deep fading phenomenon: At f= f0, the so-called deep fading occurs.



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13.2 The effect of signal characteristics on

the choice of a channel model


Usually, we prefer slowly fading and frequency


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y g y

non-selectivity.

So we wish to choose symbol time T and transmissionbandwidth B such that

T W2

In such case, we shall add a lowpass filter at the Rx.

where L(f) = 1, f W20 otherwise

.

Digital Communications: Chapter 13 Ver 2010 09 06 Po Ning Chen 92 / 110


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r t =

s f C f e2ftdf+ zW t

Digital Communications: Chapter 13 Ver 2010 09 06 Po Ning Chen 93 / 110

For a bandlimited C f , sampling theorem gives:


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( )

c(t) = n=

c nW sincWt nWC

(f

)=

c

(t

)e 2ftdt

=

1W

n=

c nW e 2fnW, f W20, otherwise

Digital Communications: Chapter 13 Ver 2010.09

.06 Po Ning Chen 94 / 110

r(t) =

s(f)C(f)e2ftdf+ zW(t)=

1 c n W2

s f e 2f(tnW)df + zW t


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W

n=

c

W

W

2

s

(f

)e df+ zW

(t

)= 1W

n=

c nW

s t nW

+ zW(t)=

n=

cn s

t

n

W

+ zW

(t

), where cn =

1

Wc

n

W

For a time-varying channel, we replace c() and C(f) by c(; t)and C

(f; t

)and obtain

r(t) = n= cn(t) s t nW + zW(t)where cn =

1W

cn

W; t .



Statistically, c() = 0 for > Tm and < 0.So, c() is assumed band-limited and is also statistically time-limited!Hence, cn = 0 for n < 0 and n > TmW (since = nW > Tm).

tTm W t t n t


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r

(t

)=

n=0

cn

(t

) s

t

W+ zW

(t

)

For convenience, the text re-index the system as

r t =L

k=1

ck t s t k

W+ zW t .




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13.5-2 The RAKE demodulator

Di it l C i ti s Ch t 13 V 2010.09

.06 P Ni Ch 97 / 110

Assumption (Gaussian and US (uncorrelated scattering)){ck(t)}Lk=1 complex i.i.d. Gaussian and can be perfectly estimatedby Rx.


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So the Rx can regard the transmitted signal as one of

v1,(t) =Lk=1 ck(t) s1, t kW

vM,

(t

)=Lk=1 ck

(t

) sM,

t k

W

So slide 4-162 said:Coherent MAP detectionm = arg max

1mMRe

r

vm,

= arg max

1mMRe

T

0r

(t

)vm,

(t

)dt

= arg max1mM

Re Lk=1

T

0r(t)ck(t)sm, t k

Wdt

Um

Di it l C i ti Ch t 13 V 2010.09

.06 P Ni Ch 98 / 110

Discussions on assumptions:We assume:s(t) is band-limited to W.c is causal and time-limited to Tm and, at the same time,b d li it d t W


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( )band-limited to W.

W (f)c = 1Tm (i.e., L WTm 1) as stated in page 879in textbook.

The definition of Um requires T Tm (See page 871 in

textbook) such that the longest delayed version

s(t LW) = s(t WTmW) = s(t Tm)is still well-confined within the integration range

[0,T

). As a

result, the signal bandwidth is much larger than 1T; RAKEis used in the demodulation of spread-spectrum signals!WT WTm 1 W

1T

.

Di it l C i ti Ch t 13 V 2010.

09.

06 P Ni Ch 99 / 110

M = 2 case


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The receiver collectsthe signal energy fromall the received paths,which is somewhat anal-ogous to the gardenrake that is used to

gather together leaves,hays, etc. Consequently,the name RAKE re-ceiver has been coinedfor this receiver struc-

ture by Price and Green(1958). (I use sm,, but the textuses s,m.)

Di it l C i ti Ch t 13 V 2010.

09.

06 P Ni Ch 100 / 110

Alternative realization of RAKE receiver

The previous structure requires M delay lines.


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We can reduce the number of the delay lines to one by thefollowing derivation.

Let u= t tW

.

Um = Re Lk=1

T

0r(t)ck(t)sm, t kWdt

= Re L

k=1

T

0 r u+k

W ck u+k

W sm, (u)dtDi i l C i i Ch 13 V 2010

.

09.

06 P Ni Ch 101 / 110


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Di i l C i i Ch 13 V 2010.

09.

06 P Ni Ch 102 / 110

Performance of RAKE receiver

Suppose ck t = ck. Then,


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( )Um = Re Lk=1

T

0r(t)ck sm, t k

Wdt

= Re

L

k=

1

T

0 L

n=

1

cns1,

t

n

W+ zW

(t

)ck s

m,

t

k

Wdt

= Re L

k=1

L

n=1

cnc

k T

0s1, t k

W sm, t k

Wdt

+Re

L

k=1

T

0

zW

(t

)ck s

m, t

k

Wdt


Assumption (Add-and-delay property)The transmitted signal is orthogonal to its shifted

counterparts.

U d T TT

t k t k dt i l t


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Under T Tm,

T

0s

1, tk

W s

m, tk

Wdt is almostindependent of k,and nk = ReT0 zW(t)sm, t kWdtL

k=1Gaussian

white because

{s

m,

t k

W

}L

k=1 orthogonal;

hence, with k = ck,Um = Re L

k=1

ck2T0

s1, (t) sm, (t)dt+

Re L

k=1 c

k T

0 zW(t)sm, tk

Wdt=

L

k=1

2k

Re s1, t , sm, t +L

k=1

knk.


Therefore, the performance of RAKE is the same as theL-diversity maximal ratio combiner if{k}Lk=1 i.i.d.However, {k = ck}Lk=1 may not be identically distributed.In such case we can still obtain the pdf of b =

L k (if


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In such case, we can still obtain the pdf ofb =

k=1 k (if

M= 2) from

characteristic function ofk k( ) = 11 k

characteristic function ofb =

L

k=1 k L

k=1 k( ) =L

k=11

1 k

The pdf of is then given by the Fourier transform ofcharacteristic function:

f() =L

k=1k

k e

k

where k =L

i=1,ik

k

k i.



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Then,

Pe =

12 Lk=1 k

1

k

1+k

2L1

L

Lk=1 14k , BPSK, RAKE

12 Lk=1 k

1

k

2+k

2L1

L Lk=1 12k , BFSK, RAKE


Estimation of ckFor orthogonal signaling, we can estimate cn via

T

r t +n

s t + + s t dt


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0r t+ Ws1,(t) ++ sM,(t)dt

=L

k=1

ckT

0sm, t+ n

W

k

Ws1,(t) ++ sM,(t)dt

+

T

0 zt+n

Ws

1,(t) ++ s

M,(t)dt=

L

k=1

ckT

0sm, t+ n

W

k

W s

m,(t)dt+

T

0 zt+n

Ws1,(t) ++ sM,(t)dt (Orthogonality)= cn

T

0sm, (t) 2dt+ noise term (Add-and-delay)


M = 2 case


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Decision-feedback estimatorThis previous estimator only works for orthogonal signaling.For, e.g., PAM signal with


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s(t) = I g(t) where I {1,3, . . . ,(M 1)},we can estimate cn via

T

0

r t+n

Wg

(t)=

T

0 L

k=1

ck I gt+ nW

k

W + zt+ n

Wg(t)

=

L

k=1 ck

I

T

0 gt+n

W

k

Wg

(t)dt+ noise term= cn I

T

0g(t) 2dt+ noise term (Add-and-delay)


Final notes


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Usually it requires (t)cT > 100 in order to have an accurateestimate of{cn}Ln=1.Note that for DPSK and FSK with square-law combiner, it is

unnecessary to estimate {cn}Ln=1.So, they have no further performance loss due to aninaccurate estimate of

{cn

}Ln=1.


Chap13 Fading Channels I

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