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X

Y

BS

MS

d

Figure 2: Multipath channel seen at location (d, ) for one Tx&Rx antenna pair

s(t) y(t)hd,()

Figure 3: Causal LTI system representing multipath profile at location (d, )

write the output y(t) in terms of the input s(t) as

y(t) =

nn(d, ) e

jn(d,) s(t n(d, ))

which implies that the impulse response is

hd,() =n

n(d, ) ejn(d,) ( n(d, ))

Scales of Variation

As the MS moves, (d, ) change with time and the linear system associated with the channelbecomes LTV. There are two scales of variation:

The first is a small-scale variation due to rapid changes in the phase n as the mobile moves over

distances of the order of a wavelength of the carrier c = c/fc, where c is the velocity of light.This is because movements in space of the order of a wavelength cause changes in n of the orderof 1/fc, which in turn cause changes in n of the order of 2. (Note that for a 900 MHz carrier,c 1/3 m.)Modeling the phases n as independent Uniform[0, 2] random variables, we can see that theaverage power gain in the vicinity of (d, ) is given by

n

2n(d, ). We denote this average

power gain by G(d, ).

cV.V. Veeravalli, 2006 2

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The second is a large-scale variation due to changes in {n(d, )} and {n(d, )} both in thenumber of paths and their strengths. These changes happen on the scale of the distance betweenobjects in the environment.

To study these two scales of variation separately, we redraw Figure 3 in terms of two components

as shown in Figure 4.

x

s(t) y(t)

G(d, )

cd,()

Figure 4: Small-scale and large-scale variation components of channel

Here cd, is normalized so that the average power gain introduced by cd, is 1, i.e.

cd,() =n

n(d, )ejn(d,)( n(d, ))

where {n(d, )} is normalized so that

n 2n(d, ) = 1. The large-scale variations in (average)

amplitude gain are then lumped into the multiplicative term

G(d, ).

The goal of wireless channel modeling is to find useful analytical models for the variations in thechannel. Models for the large scale variations are useful in cellular capacity-coverage optimizationand analysis, and in radio resource management (handoff, admission control, and power control).Models for the small scale variations are more useful in the design of digital modulation anddemodulation schemes (that are robust to these variations). We hence focus on the small scale

variations in this class.

Small-scale Variations in Gain

s(t) y(t)

G

cd,()

Figure 5: Small-scale variations in the channel (with large-scale variations treated as constant).

Recall that the small scale variations in the channel are captured in a linear system with response

cd,() =n

n(d, )ejn(d,)( n(d, )) ,


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where the {n(d, )} are normalized so that

n 2n(d, ) = 1. As (d, ) changes with t, the channel

corresponding to the small-scale variations becomes time-varying and we get:

c(t; ) := cd(t),(t)() =n

n(t) ejn(t) ( n(t)) .

Treating the large scale variations

G(d, ) as roughly constant (see Figure 5), we obtain:

y(t) =

G

0

c(t; )s(t )d .

Finally, we may absorb the scaling factor

G into the signal s(t), with the understanding that thepower of s(t) is the received signal power after passage through the channel. Then

y(t) =

0

c(t; )s(t )d .

Doppler shifts in phase

For movements of the order of a few wavelengths, {n(t)} and {n(t)} are roughly constant, andthe time variations in c(t; ) are mainly due to changes in {n(t)}, i.e.,

c(t; ) n

n ejn(t) ( n) .

From this equation it is clear that the magnitude of the impulse response |c(t; )| is roughly inde-pendent of t. A typical plot of |c(t; )| is shown in Figure 6. The width of the delay profile (delayspread) is of the order of tens of microseconds for outdoor channels, and of the order of hundredsof nanoseconds for indoor channels. Note that the paths typically appear in clusters in the delay

profile (why?).

To study the phase variations n(t) in more detail, consider a mobile that is traveling with velocityv and suppose that the n-th path has an angle of arrival n(t) with respect to the velocity vectoras shown in Figure 7. (Note that we may assume that n is roughly constant over the time horizoncorresponding to a few wavelengths.) Then for small t,

n(t + t) n(t) 2fcvt cos nc

=2vt

ccos n ,

where c is the carrier wavelength and c is the velocity of light. The frequency shift introduced bythe movement of the mobile is hence given by

limt0

n(t + t) n(t)2t

=v

ccos n = fmax cos n ,

where fmax = v/c is called the maximum Doppler frequency. We will use this model for thevariations in n(t) to characterize small-scale variations statistically in the following section.

Definition 1. The quantity ds = max n(d, ) min n(d, ) is called the delay spread of thechannel.


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0 ds

|c(t; )|

0 (LOS)

n

n

Figure 6: Typical delay profile of channel with LOS path having delay 0.

Without loss of generality, we may assume that the delay corresponding to the first path arrivingat the receiver is 0. Then min n(d, ) = 0, ds = max n(d, ), and:

y(t) =

ds0

c(t; )s(t )d .

vvt

n

vt cos n

Figure 7: Doppler Shifts


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Frequency Nonselective (Flat) Fading

If the bandwidth of transmitted signal s(t) is much smaller than 1/ds, then s(t) does not changemuch over time intervals of the order of ds. Thus

y(t) s(t) ds0

c(t, ) = s(t)n

nejn(t) .

This implies that the multipath channel simply scales the transmitted signal without introducingsignificant frequency distortion. The variations with time of this scale factor are referred to asfrequency nonselective, or flat, fading.

Note that the distortions introduced by the channel depend on the relationship between the delayspread of the channel and the bandwidth of the signal. The same channel may be frequency selectiveor flat, depending on the bandwidth of the input signal. With a delay spread of 10 s correspondingto a typical outdoor urban environment, an AMPS signal (30 kHz) undergoes flat fading, whereasan IS-95 CDMA signal (1.25 MHz) undergoes frequency selective fading.

For flat fading, the channel model simplifies to

y(t) = V(t)s(t)

where

V(t) =

ds0

c(t, )d =n

nejn(t) .

Purely Diffuse Scattering - Rayleigh Fading

Our goal is to model

{V(t)

}statistically, but before we do that we distinguish between the cases

where the multipath does or does not have a line-of-sight (LOS) component. In the latter case,the multipath is produced only from reflections from objects in the environment. This form ofscattering is purely diffuse and can be assumed to form a continuum of paths, with no one pathdominating the others in strength. When there is a LOS component, it usually dominates all thediffuse components in signal strength.

To model {V(t)} statistically, we first fit a stochastic model to the phases {n(t)}n=1,2,....Assumption 1. The phases {n(t)}n=1,2,... are well modeled as independent stochastic processes,with n(t) being uniformly distributed on [0, 2] for each t and n.

Using this assumption, we get the following results:

{V(t)} is a zero-mean process. This is because

E [V(t)] =n

nE

ejn(t)

= 0

The process {vn(t)} defined by vn(t) = nejn(t) is a proper complex random process.


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Proof. We need to show that the pseudocovariance function of {vn(t)} equals zero.

C(t + , t) = E[vn(t)vn(t + )]

= E[(nejn(t))(ne

jn(t+))]

= 2

nE[ej(n(t)+n(t+))]

2nE[ej(2n(t)+2fmaxcos n)] = 0where the approximation on the last line follows from the Doppler equation.

If the number of paths is large, we may apply the Central Limit Theorem to conclude that{V(t)} is a proper complex Gaussian (PCG) random process.

First order statistics of {V(t)} for purely diffuse scattering

For fixed t, V(t) = VI(t) +jVQ(t) is PCG random variable with

E

|V(t)|2

=n

2n = 1 .

Since V(t) is proper, VI(t) and VQ(t) are uncorrelated and have the same variance, which equalshalf the variance of V(t). Since V(t) is also Gaussian, VI(t) and VQ(t) are independent as well.Thus VI(t) and VQ(t) are independent N(0, 1/2) random variables.Envelope and Phase Processes

The envelope process {(t)} and the phase process {(t)} are defined by

(t) = |V(t)| = V2I (t) + V2Q(t) , and (t) = tan1VQ(t)

VI(t)

.

We can write x(t) = V(t)s(t) in terms of (t) and (t) as:

x(t) = (t)ej(t)s(t) .

This means that for flat fading, the channel is seen as a single path with gain (t) and phaseshift (t). Note that and vary much more rapidly than the gain and phase of the individualpaths n and n (why?).

For fixed t, using the fact that VI(t) and VQ(t) are independentN(0, 1/2) random variables, it is easy to show that (t) and (t) are independent random variableswith (t) having a Rayleigh pdf and (t) being uniform on [0, 2]. The pdf of(t) is given by

p(x) = 2xex211{x0} .

It is easy to show that E[] =

/4 and E[2] = E|V(t)|2

= 1.

Since the envelope has a Rayleigh pdf, purely diffuse fading is referred to as Rayleigh fading.


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0 1 2 30

0.2

0.4

0.6

0.8

1

Figure 8: Rayleigh pdf

Scattering with a LOS component Ricean Fading

If there is a LOS (specular) path with parameters 0, 0 and 0(t) in addition to the diffusecomponents, then

V(t) = 0 ej0(t) +

1 20 V(t)

where {V(t)} is a zero mean PCG, Rayleigh fading process with variance E |V(t)|2 = 1.Note: {V(t)} is also a zero-mean process, but it is not Gaussian since the LOS component{0 ej0(t)} dominates the diffuse components in power. However, conditioned on {0(t)}, {V(t)}is a PCG process with mean {0 ej0(t)}.

Rice Factor: The Rice factor is defined by

=power in the LOS component

total power in diffuse components=

201 20

.

From the definition of it follows that

0 =

+ 1, and 1 20 =

1

( + 1).

For fixed t, the pdf of the envelope (t) can be found by first computing the joint pdf of (t) and(t), conditioned on 0(t). This is straightforward since, conditioned on 0(t), V(t) is a PCGrandom variable with mean 0e

j0(t).

We can then show that the pdf of (t) conditioned on 0(t) is not a function of 0(t), and weget:

p|0(x) =2x

1 2

0

I02x0

1 2

0 exp

x2 + 20

1 2

0 11{x0} = p(x)

This pdf is called a Ricean pdf and it can be rewritten in terms of as:

p(x) = 2x( + 1) I0

2x

( + 1)

expx2( + 1) 11{x0}

where I0() is the zeroth order modified Bessel function of 1st kind, i.e.,

I0(y) =1

2

exp(y cos )d .


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It is easy to see that when = 0, p(x) reduces to a Rayleigh pdf.

The pdf of 2(t) is easily computed as:

p2(x) =p(

x)

2x= ( + 1) I0 2x( + 1) exp[

x( + 1)

] 11{x0} .

0 1 2 30

0.5

1

1.5

2

RayleighRicean =1Ricean =5

Ricean =10

Figure 9: Ricean pdf for various Rice factors.

Signaling Through Slow Flat Fading Channels

We assume that the long-term variations in the channel are absorbed into

Em. Then

Em represents

the average received symbol energy (for symbol m) over the time frame for which the multipathprofile may be assumed to be constant. Then the received signal is given by:

r(t) = V(t)s(t) + w(t) = (t)ej(t)s(t) + w(t)

where E[2(t)] = 1.

For slow fading, (t) and (t) may b e assumed to be constant over each symbol period. Thus,for memorlyless modulation and symbol-by-symbol demodulation, y(t) for demodulation oversymbol period [0, Ts] may be written as

r(t) = ejsm(t) + w(t) (conditioned on symbol m being transmitted)

Average probability of error for slow, flat fading

The error probability is a function of the received signal-to-noise ratio (SNR), i.e., the receivedsymbol energy divided by the noise power spectral density. We denote the symbol SNR by s,and the corresponding bit SNR by b, where b = / and = log2 M.


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For slow, flat fading, the received SNR is

s =2EsN0

.

The average SNR (averaging over 2) is given by

s = E[2]

EsN0

=EsN0

.

The corresponding bit SNRs are given by

b =

, and b =

EsN0

=EbN0

.

Suppose the symbol error probability with SNR s is denoted by Pe(s). Then the average errorprobability (averaged over the fading) is

Pe =0

Pe(x)ps(x)dx

where ps(x) is the pdf of s.

For Rayleigh fading, 2 is exponential with mean 1; hence s is exponential with mean s, i.e.,

ps(x) =1

sexp

x

s

11{x0} .

For Ricean fading, s has pdf

ps(x) = + 1

sI0

2

x( + 1)s

exp

x( + 1)

s

11{x0} .

Pe for Rayleigh Fading

Useful result (see problem 1 of HW#10):0

Q(

x )ex/

dx =

1

2

1

2 +

.

BPSK

Pb(b) = Q(

2b ) .

Thus we get

Pb =

0

Q(

2x )pb(x)dx =1

2

1

b

1 + b

1

4b(for largeb).


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Binary coherent orthogonal modulation (e.g. FSK)

Pb(b) = Q(

b ) .

Here Pb is the same as that for BPSK with b replaced by b/2, i.e.,

Pb =1

2

1

b

2 + b

1

2b(for large b).

Binary DPSKPb(b) =

1

2eb .

In this we case we may integrate directly to get

Pb =

0

1

2ex

ex/b

bdx =

1

2(1 + b) 1

2b(for large b).

Binary noncoherent orthogonal modulation (FSK)

Pb(b) =1

2eb/2 .

Here Pb is the same as that for DPSK with b replaced by b/2, i.e.,

Pb =1

2 + b 1

b(for large b).

Similar expressions may be derived for other M-ary modulation schemes. Note that withoutfading the error probabilities decrease exponentially with SNR, whereas with fading the

error probabilities decrease much more slowly with SNR (inverse linear in case of Rayleighfading).

Pe for Ricean Fading

Direct approach: Compute Pe by direct integration of Pe(s) w.r.t. ps . This is cumbersomeexcept in some special cases.

Binary DPSK.

Pb = + 1

2( + 1 + b)exp

b

+ 1 + b

.

as done in class. Again, it is easy to check that we get the Rayleigh result when = 0

and the AWGN result as . Binary noncoherent FSK. Same as DPSK with b replaced by b/2. There are other approaches for computing expressions for the error probability for Ricean

fading channels, which are beyond the scope of this class.


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Diversity Techniques for Flat Fading Channels

Performance with fading is considerably worse than without fading, especially when the fadingis Rayleigh.

Performance may be improved by sending the same information on many (independently) fadingchannels

For signaling on L channels, the received signal on the -th channel is:r(t) = e

jsm,(t) + w(t), = 1, 2, . . . , L, m = 1, 2, . . . , M .

where the noise w(t) is assumed to be independent across channels.

If{ej}L=1 are independent, we get maximum diversity against fading.

How do we guarantee independence of channels? By separating them either in time, frequencyor space.

frequency separation must be 1ds , where ds is the delay spread time separation must be 1fmax , where fmax is the maximum Doppler frequency spatial separation must be > c2 , where c is the carrier wavelength.

Memoryless linear modulation with diversity

When symbol m is sent on the channelsr(t) = e

j

Es, am ejm g(t) + w(t), = 1, 2, . . . , L, m = 1, 2, . . . , M ,

where g(t) is a unit energy shaping function on channel , Es, is the average symbol energy onchannel , and the ams are normalized so that

1M

m a

2m = 1. We assume that the fading and

noise are independent across channels. Note that {w(t)} are independent PCG processes withPSD N0.

Optimum receiver: If we assume that the phases {} and the amplitudes {} are estimatedperfectly at the receiver, the optimum test statistic is formed by Maximal Ratio Combining(MRC) as

R =L=1

Es, ej r(t), g .

We proved that his was optimum in class; also see Problem 4 of HW#10.

The sufficient statistic R may be rewritten as

R =L=1

2Es, am ejm + L

=1

Es, ejW ,

where {W} are independent CN(0, N0) (note that the ej term multiplying W does not mattersince W is circularly symmetric).


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The MPE (ML) decision rule is the same as without diversity except that the constellation isscaled in amplitude based on the fading on the channels.

Special Case: BPSK with diversity

The sufficient statistic in this case takes the form

R = L=1

2Eb, + W ,

where W =L

=1 Eb, W is PCG with

E[|W|2] = N0L=1

Eb, 2 .

The MPE decision rule for equal priors (or the ML decision rule)is to decide +1 (symbol 1) if

RI > 0, and 1 (symbol 2) if RI < 0.

For fixed {},

Pb = P({RI > 0} | {symbol 2 sent}) = P

WI >

L=1

2Eb,

= Q

2 L

=1

2Eb,N0

= Q

2 L

=1

b,

= Q2b

where b, is the received bit SNR on the -th channel, and b = L=1 b, is the total receivedbit SNR.

The average BER is given by

Pb =

0

Q

2x

pb(x)dx .

Thus, we may evaluate Pb by first finding the pdf pb(x).

Special case: Rayleigh fading. If the fading is Rayleigh on all channels, then

Case 1: b,s are distinct for = 1, 2, . . . , L.

Pb =L=1

C2

1

b,

1 + b,

where

C =i=

b,b, b,i

.


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Case 2: b,s are identical, i.e. b, = b/L for all .

Pb =

A

bL

L L1=0

L 1 +

1 A

bL

with

A

bL

=

1

2

1

b

L + b

.

For large b,

A

bL

L

4band 1 A

bL

1 .

Thus

Pb

L

4b

L L=1

L 1 +

=

L

4b

L 2L 1L

.

Note that with diversity Pb decreases at (b)L

which is a significant improvement over theinverse linear performance obtained without diversity. (See Figure 10.)

5 10 15 20 25 30 35 40 4510

6

105

104

103

102

101

Average Bit SNR

Average

ProbabilityofBitError

Performance of BPSK with diversity on Rayleigh fading channel

L=1

L=2

L=4L=8

Figure 10: BPSK with diversity on Rayleigh fading channels.


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