1

ECE 480

Wireless Systems

Lecture 8

Statistical Multipath Channel Models

8 Mar 2012

2

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

3

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

4

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

5

• 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

6

• 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

7

• 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)

8

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

9

• 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

2

• 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

10

• 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

11

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

12

• 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

13

The time-varying impulse response corresponding to t 1 is

njn n

nc ,t e

2

10

14

15

• 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

17

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

18

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

19

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

20

• 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

21

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

22

• 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

23

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

24

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

25

• 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

26

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

27

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

28

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).

29

• 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.

30

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

31

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

32

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

33

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

34

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

35

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

36

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

37

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

38

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.

39

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)

40

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

41

• 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

42

• 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

43

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 [- , ]

44

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

45

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

46

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

47

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

48

Uniform Scattering EnvironmentMany scatterers densely packed wrt angle

Dense Scattering Environment

49

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

50

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

51

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

52

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

53

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

54

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 ()

55

• 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)

56

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

57

• Path loss decreases as d

• Shadowing and path loss shows slow variations

• Multipath shows much more rapid variations

58

• Fluctuation in power vs. distance

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

59

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

60

• 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

61

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

62

• 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

63

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

64

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

65

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

66

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