1. Statistical Multipath Channel Models
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
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
n = 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)

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

6. Reflector cluster
Two multipath components with delay  1 and  2 are "resolvable" if their delay difference considerably exceeds the inverse signal bandwidth
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. Let
 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
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
Set T = - t
c () is the standard time – invariant channel impulse response, the response at time '' to an impulse at time zero

11. Comparing these two expressions:
Substituting back:

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

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

17. If the channel is time – invariant, the time – varying parameters are constant
for channels with discrete multipath components
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 3.1
Consider 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

19. For typical carrier frequencies,
Where this is the case, a small change in  n (t) can result in a large phase change
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. 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
If it syncs to a multipath component with delay equal to the delay spread will be given by
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 Models
Assume delay spread is small compared to the bandwidth
Delay of the i th multipath component
original transmitted signal, s (t)
scale factor

24. independent of s (t) and u (t)
Let
narrowband for any T m

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. A measure of the degree to which two variables are related
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
cov = covariance =
E = Expected value
 = Mean value
 = Standard deviation

28. The main result of a correlation is called the correlation coefficient (or "r"). It ranges from -1.0 to 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. Accounts for sample size
Pearson Product - Moment Correlation Coefficient
Accounts for sample size
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.
= sample mean of x i
= 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
Problem: This formula may be unstable
Subtracting numbers in the denominator that may be very close to each other
Why?

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 .
where σy|x2  is the square of the error of a linear fit of yi  to x i  by the equation y = a + bx
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 

33. Interpretation of the size of a correlation Correlation Negative
Positive
Small to
0.10 to 0.29
Medium to
0.30 to 0.49
Large to
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
For continuous functions f (x) and g i the cross-correlation is defined as

36. Properties of Cross - Correlation
Similar in nature to the convolution of two functions
They are related by
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
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
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
For zero – centered signals (zero mean)

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
when f (t) is real
when f (t) is complex
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:
Similarly,
Therefore,
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
By the same process:
Conclusions:
r I (t) and r Q (t) are uncorrelated
They are independent

46. Autocorrelation of In – Phase Component
We can show that this expression is equal to
and
Where this is the case, we say that r I (t) and r Q (t) are wide – sense stationary (WSS) random processes

47. Cross – Correlate
The received signal
is also WSS with autocorrelation

48. Uniform Scattering Environment
Many scatterers densely packed wrt angle
Dense Scattering Environment

49. Assumptions:
N multipath components with angle of arrival
P r = Total received power

50. Substitute
Take the limit as N 
J 0 (x) is a Bessel function of the zeroth order

51. Similarly,
Autocorrelation is zero when f D   0.4
  = 0.4 
Independent at this point
Recorrelate later
The 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)

53. Recall:
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. for a relatively large range of  - values
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 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
is Rayleigh distributed with distribution
is the average received signal power based on path loss and shadowing

63. Use change of variables to integrate
Received signal power is exponentially distributed
Mean = 2  2
Equivalent lowpass signal for r (t)
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

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

66. Solution
Reflected
LOS