ece 480 wireless systems lecture 8 statistical multipath channel models 8 mar 2012

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ECE 480

Wireless Systems

Lecture 8

Statistical Multipath Channel Models

8 Mar 2012

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Fading Models for Multipath Channels

• Time delay spread

• Consider a pulse transmitted over a multipath channel (10 – ray trace)

• Received signal will be a pulse train, each delayed by a random amount due to scattering

• Result can be a distorted signal

• Time - varying nature

• Mobility results in a multipath channel due to reflections

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Time – Varying Channel Impulse Response

c

c c

j f t

s t Re u t cos f t Im u t sin f t

Re u t e

2

2 2

Transmitted signal

u (t) is the equivalent lowpass signal for s (t) with bandwidth B u with a carrier frequency, f c

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Neglecting noise

c n D n

N tj f t t

n nn

r t Re t u t t e

2

0

n = 0 LOS path

Unknowns:

• N (t) = number of resolvable multipath components

• For each path (including LOS):

• Path length = r n (t)

• Delay

• Doppler phase shift

• Amplitude n (t)

nn

r tt

c

nD t

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• The nth resolvable multipath component may result from a single reflector or with multiple reflectors

• Single reflector n (t) is a function of the single reflector

• is the phase shift

• is the Doppler shift

• is the Doppler phase shift

c nj fn t e 2

n

nD

v cos tf

n nD Dt

f t d t 2

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• Reflector cluster

• Two multipath components with delay 1 and 2 are "resolvable" if their delay difference considerably exceeds the inverse signal bandwidth

uB 1 2 1

1

• If u (t - 1) ~ u (t - 2) the two components cannot be separated at the receiver and are "unresolvable"

• Unresolvable signals are usually combined into a single term with delay

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• The amplitude of unresolvable signals will typically undergo fast variations due to the constructive and destructive combining

• Typically, wideband channels will be resolvable while narrowband channels may not

• Since n (t), n (t), and Dn (t) change with time they are characterized as random processes

• The received signal is also stationary and ergodic (can be characterized from a sample)

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c n D n

N tj f t t

n nn

r t Re t u t t e

2

0

Let nn c n Dt f t 2

n c

N tj t j f

n nn

r t Re t e u t t e

2

0

n (t) is a function of path loss and shadowing

n (t) is a function of delay and Doppler

• They may be assumed to be independent

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• The received signal can be obtained by convolving the equivalent lowpass time – varying channel response c ( , t) and upconverting it to the carrier frequency

cj f tr t Re c ,t u t d e

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• The time 't' is when the impulse response is observed at the receiver

• The time 't - ' is when the impulse is launched into the channel relative to 't'

• If there is no physical reflector in the channel, c ( , t) = 0

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• For time – invariant channels

c ,t c , t T • Set T = - t

c ,t c • c () is the standard time – invariant channel

impulse response, the response at time '' to an impulse at time zero

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n c

N tj t j f

n nn

r t Re t e u t t e

2

0

cj f tr t Re c ,t u t d e

2

Comparing these two expressions:

n

N tj t

n nn

c t , t e t

0

Substituting back:

n c

N tj t j f

n nn

r t Re t e u t t e

2

0

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• Consider the system in the figure where each multipath component corresponds to a single reflector

• At time t 1 there are 3 multipath components

• Impulses launched into the channel at time t 1 - i with i = 1, 2, 3 will all be received at time t 1

• Impulses launched at any other time will not be received

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The time-varying impulse response corresponding to t 1 is

njn n

nc ,t e

2

10

14

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• At time t 2 there are two multipath components

• Impulses launched at time t 2 - 'i (i = 1, 2) will be received at time t 2

• The time – varying impulse response is

nj 'n n

nc ,t ' e '

1

20

16

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nN

jn n

nc e

0

If the channel is time – invariant, the time – varying parameters are constant

for channels with discrete multipath components

jnc e

for channels with a continuum of multipath components

For stationary channels, the response to an impulse at time t 1 is just a shifted version of its response to an impulse at time t 2 t 1

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Example 3.1Consider a wireless LAN operating in a factory near a conveyor belt. The transmitter and receiver have a LOS path between them with gain 0 , phase 0 , and delay 0. Every T 0 seconds, a metal item comes down the conveyor belt, creating an additional reflected signal path with gain 1 , phase 1 , and delay 1. Find the time – varying impulse response, c ( , t) of this channel. Solution

For t n T 0 the channel response is LOS. For t = n T 0 , the response will include both the LOS and the reflected path

j

j j

c ,t e t n T

e e t n T

0

0 1

0 0 0

0 0 1 1 0

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For typical carrier frequencies,

c nf t 1

Where this is the case, a small change in n (t) can result in a large phase change

nn c n Dt f t 02

This phenomenon, called “fading”, causes rapid variation in the signal strength vs. distance

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• The impact of multipath on the received signal is a function of whether the time delay spread is large or small wrt the inverse signal bandwidth

• If the delay is small, the LOS and multipath components are typically unresolvable

• If the delay spread is large, they are typically resolvable into some number of discrete components

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nN

jn n

nc e

0

Time – invariant channel model

• The demodulator may sync to the LOS component or to one of the other components

• If it syncs to the LOS component (smallest delay 0), the delay spread is a constant

m n nT max 0

• If it syncs to a multipath component with delay equal to the delay spread will be given by

m n nT max

• In time – varying channels, T m becomes a random variable

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• Some components have much lower power than others

• If the power is below the noise floor, it will not contribute significantly to the delay spread

• May be characterized by two factors determined from the power delay profile

• Average delay spread

• RMS delay spread (most common)

• Range of delay spread

• Indoors: 10 – 1000 ns

• Suburbs: 200 – 2000 ns

• Urban: 1 – 30 s

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Narrowband Fading ModelsAssume delay spread is small compared to the bandwidth

m n mT t t t tB

1

iu t u t

Delay of the i th multipath component

c nj f t j tn

nr t Re u t e t e

2

cj f tu t e 2

nj tn

nt e

original transmitted signal, s (t)

scale factor

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nj tn

nt e independent of s (t) and u (t)

Let cj f tcs t Re e cos f t 02

02

narrowband for any T m

c n

N tj f t j t

nn

c Q c

r t Re e t e

r t cos f t r t sin f t

2

0

2 2I

N t

n nn

r t t cos t

0

I

N t

Q n nn

r t t sin t

0

nn c n Dt f t 02

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• If N (t) is large, the Central Limit Theorem applies n (t) and n (t) are independent

• r I (t) and r Q (t) can be approximated as Gaussian

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Correlation

• Correlation (correlation coefficient) indicates the strength and direction of a linear relationship between two random variables.

• In general statistical usage, correlation refers to the departure of two variables from independence.

A measure of the degree to which two variables are related

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The correlation ρ X, Y between two random variables X and Y with expected values μ X and μ Y and standard deviations σ X and σ Y is defined as

X Yx,y

X Y X Y

E X Ycov X ,Y

cov = covariance = X YE X Y

E = Expected value

= Mean value

= Standard deviation

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X Y

X

Y

E X E Y

E X E X

E Y E Y

2 2 2

2 2 2

X ,Y

E X,Y E X E Y

E X E X E Y E Y

2 2 2 2

• The main result of a correlation is called the correlation coefficient (or "r"). It ranges from -1.0 to +1.0. The closer r is to +1 or -1, the more closely the two variables are related.

• If r is close to 0, it means there is no relationship between the variables. If r is positive, it means that as one variable gets larger the other gets larger. If r is negative it means that as one gets larger, the other gets smaller ("inverse" correlation).

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• If the variables are independent the correlation is 0

• The converse is not true because the correlation coefficient detects only linear dependencies between two variables.

• Example: Suppose the random variable X is uniformly distributed on the interval from −1 to 1, and Y = X 2. Then Y is completely determined by X, so that X and Y are dependent, but their correlation is zero; they are uncorrelated.

• However, in the special case when X and Y are jointly normal independence is equivalent to uncorrelatedness.

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Pearson Product - Moment Correlation Coefficient

Suppose we have a series of n  measurements of X  and Y  written as x i  and y i  where i = 1, 2, ..., n and that X  and Y  are both normally distributed.

Accounts for sample size

n

i i

x ,yx y

x x y yr

n s s

1

1

x = sample mean of x i

y = sample mean of y i

s x = sample mean of x i

s y = sample mean of y i

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We can use the same basic formula for the sample as for the entire population

n n n

i i i i

x yn n n n

i i i i

n x y x yr

n x x n y y

1 1 12 2

2 2

1 1 1 1

Problem: This formula may be unstable

Why? Subtracting numbers in the denominator that may be very close to each other

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The sample correlation coefficient is the fraction of the variance in y i  that is accounted for by a linear fit of x i  to y i .

x yx y

y

r

22

21

where σy|x2  is the square of the error of a linear fit of

yi  to x i  by the equation y = a + bx

n

i iy x y a b x 22

1

n

y iy y 22

1

Since the sample correlation coefficient is symmetric in x i  and y i , we will get the same value for a fit of x i  to y i 

y xx y

x

r

22

21

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Interpretation of the size of a correlation

Correlation Negative PositiveSmall - 0.29 to - 0.10 0.10 to 0.29

Medium - 0.49 to - 0.30 0.30 to 0.49

Large - 0.50 to - 1.00 0.50 to 1.00

These criteria are somewhat arbitrary

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Cross - Correlation

• In signal processing, the cross-correlation is a measure of similarity of two signals

• Used to find features in an unknown signal by comparing it to a known one

• It is a function of the relative time between the signals

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For discrete functions f i and g i the cross-correlation is defined as

j i jij

f g f * g

For continuous functions f (x) and g i the cross-correlation is defined as

f g x f * t g x t d t

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Properties of Cross - Correlation• Similar in nature to the convolution of two

functions

• They are related by f t g t f * t g t

if f (t) or g (t) is an even function

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Autocorrelation

• Autocorrelation is the cross-correlation of a signal with itself

• Autocorrelation is useful for finding repeating patterns in a signal

• Determining the presence of a periodic signal which has been buried under noise

• Identifying the fundamental frequency of a signal which doesn't actually contain that frequency component, but implies it with many harmonic frequencies

• Different definitions in statistics and signal processing

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Statistics

• The autocorrelation of a discrete time series or a process X t is simply the correlation of the process against a time-shifted version of itself

• If X t is second-order stationary with mean μ then the definition is

i i kE X X

R k

2

• E is the expected value and k is the time shift being considered (usually referred to as the lag).

• This function has the property of being in the range [−1, 1] with 1 indicating perfect correlation (the signals exactly overlap when time shifted by k) and −1 indicating perfect anti-correlation.

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Signal processing

• Given a signal f(t), the continuous autocorrelation R f () is the continuous cross-correlation of f (t) with itself, at lag , and is defined as

fR f * f f t f * t d t f t f * t d t

Basically, autocorrelation is the convolution of a signal with itself

Note that, for a real function, f (t) = f * (t)

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Formally, the discrete autocorrelation R at lag j for signal x n is

n n jn

R j x m x m

For zero – centered signals (zero mean)

n n jn

R j x x

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• A fundamental property of the autocorrelation function is symmetry, R(i) = R(− i)

• In the continuous case, R f (t) is an even function

f fR R when f (t) is real

f fR R * when f (t) is complex

f fR R 0

• The continuous autocorrelation function reaches its peak at the origin, where it takes a real value

The same result holds in the discrete case

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• The autocorrelation of a periodic function is, itself, periodic with the very same period

• The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all τ) is the sum of the autocorrelations of each function separately

• Since autocorrelation is a specific type of cross-correlation, it maintains all the properties of cross-correlation

• The autocorrelation of a white noise signal will have a strong peak at  = 0 and will be close to 0 for all other

• A sampled instance of a white noise signal is not statistically correlated to a sample instance of the same white noise signal at another time

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Autocorrelation, Cross – Correlation, and Power Spectral Density

• Assumptions:

• No dominant LOS component

• Each of the multipath components is associated with a single reflector n (t) n = constant

n (t) n = constant

• f Dn (t) f Dn = constant

Dn (t) 2 f Dn t

n (t) 2 f c n + 2 f Dn t - 0

• 2 f c n changes more rapidly than the others

n (t) is uniformly distributed on [- , ]

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Under these Assumptions:

n n n nn n

E r t E cos t E E cos t 0I

QE r t 0Similarly,

E r t 0Therefore,

Zero – mean Gaussian process

If there is a dominant LOS product, the assumption of a random uniform phase no longer holds

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Correlation of In – Phase and Quadrature Components

QE r t r t 0IBy the same process:

• Conclusions:

• r I (t) and r Q (t) are uncorrelated

• They are independent

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Autocorrelation of In – Phase Component

Ir

A t , t E r t ,r t I I

We can show that this expression is equal to

I nr n Dn

nn

n

A t , t . E E cos f

. E E cos cos

2

2

0 5 2

0 5 2

n

nDf cos constant

I Ir rA t , t A

Where this is the case, we say that r I (t) and r Q (t) are wide – sense stationary (WSS) random processes

Q Qr rA t , t A and

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Cross – Correlate I Qr rA t , t and A t , t

I Q I Qr ,r r ,r Q

nn

n

Q

A t , t A E r t r t

. E sin cos

E r t r t

20 5 2

I

I

The received signal

c Q cr t r t cos f t r t sin f t 2 2I

is also WSS with autocorrelation

Qr r c r cA E r t r t A cos f A sin f 2 2I

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Uniform Scattering EnvironmentMany scatterers densely packed wrt angle

Dense Scattering Environment

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Assumptions:

• N multipath components with angle of arrival

n n

NN

2 2

• P r = Total received power

Nr

rn

P nA cos cosN

1

2I

rn

PE

N

2 2

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N

2

Substitute

Nr

rn

P nA cos cos

1

22I

Take the limit as N

rr r D

PA cos cos d P J f

2

00

2 22I

j x cosJ x e d

00

1

J 0 (x) is a Bessel function of the zeroth order

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rr Q

PA sin cos d

2

0

2 02

Similarly,

Autocorrelation is zero when f D 0.4

= 0.4

Independent at this point

Recorrelate laterThe expression = 0.4 turns out to be very significant and dictates such actions as antenna spacing

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Power Spectral Densities

• Take Fourier Transforms of the autocorrelation functions of r I (t) and r Q (t)

rr r Q r D

D

D

PS f S f A f f

f ff

elsewhere

2

2 1

1

0

I I

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Qr r c r cA E r t r t A cos f A sin f 2 2I

Recall:

r r r c r c

rc D

Dc

D

S f A . S f f S f f

Pf f f

f f ff

elsewhere

2

0 25

12

1

0

I I

PSD of the received signal r (t) under uniform scattering

This expression integrates to P r as required

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Goes to infinity at f = f D

Model is not valid in these ranges

However, PSD is maximized near these areas

PSD corresponds to the power density function (pdf) of the random Doppler frequency f D ()

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• Uniform scattering assumption is based on many scattered paths arriving uniformly from all angles with the same average power

can be considered a uniform random variable on [0, 2 ]

• By definition, p f (f) is proportional to the density of scatters at the Doppler frequency, f

• S r I is also proportional to this density

• We can characterize the PSD form the pdf p f (f)

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cos 1 for a relatively large range of - values

D Df f in this range

Power associated with all ot these multipath components will add together in the PSD at f f D

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• Path loss decreases as d

• Shadowing and path loss shows slow variations

• Multipath shows much more rapid variations

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• Fluctuation in power vs. distance

• A vehicle traveling at fixed velocity would experience variations over time similar to this figure

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Envelope and Power Distributions

• For any two Gaussian random variables X and Y

• Means are both zero X = Y

• We can show that the quantity

is Rayleigh distributed

Z X Y 2 2

• Z 2 is exponentially distributed

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• The Rayleigh distribution is a continuous probability distribution

• It is used when a two dimensional vector (e.g. wind velocity) has its two orthogonal components normally and independently distributed

• The absolute value (e.g. wind speed) will then have a Rayleigh distribution.

normalized distribution

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where erfi (z) is the complex error function Probability density function                                        

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• For n (t) normally distributed

• r I and r Q are both zero – mean Gaussian random variables

• Assume that each has a variance of 2

• The signal envelope

Qr

zz t r t r rP

2 22I

is Rayleigh distributed with distribution

zr r

z z z zp t exp exp xP P

2 2

2 2

2 02

r nn

P E 2 22 is the average received signal power based on path loss and shadowing

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Use change of variables to integrate

r

x xP

zr

p x e e xP

2

22

2

1 1 02

• Received signal power is exponentially distributed

• Mean = 2 2

• Equivalent lowpass signal for r (t)

LP Q

Q

r t r t j r t

r tarctan

r t

I

I

• If r I (t) and r Q (t) are uncorrelated Gaussian variables:

is uniformly distributed and independent

• r (t) has a Rayleigh distribution and is independent of

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Example 3.2

Consider a channel with Rayleigh fading and average received power P r = 20 dBm. Find the probability that the received power is below 10 dBm.

Solution

P r = 20 dBm = 100 mW.

We want the probability that Z 2 < 10 dBm = 10 mW

x

p Z e d x .

10

2 100

0

110 0 095100

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Problem 3.1

Consider-ray channel consisting of a direct ray plus a ground – reflected ray where the transmitter is a fixed base station at height h and the receiver is mounted on a truck (also at a height, h. The truck starts next to the base station and moves away at velocity . Assume that signal attenuation on each path follows a free – space path – loss model. Find the time – varying channel impulse at the receiver tot transmitter – receiver separation d = t sufficiently large for the length of the reflected ray to be approximated by

hr r ' dd

22

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n

N tj t

n nn

c t , t e t

0

Solution

j t j tc ,t t e t t e t 0 10 0 1 1

LOS Reflected