Small-Scale Fading II
(and basics about random processes)
PROF. MICHAEL TSAI 2015/4/24
Random processes
2
t
t
()
()
t
()
()
One realization of X(t)
………
( ) is a random variable:
( )
Joint CDF for a random process
• If we sample X(t) at times , … , we can have a joint cdf of samples at those times:
3
,…, , … , = ≤ , ≤ , … , ≤
t
X(t) …
Stationary Random
Process (Strict-sense)
• A random process X(t) is stationary if for all T, all n, and all sets of sample times , , … , we have:
4
≤ , ≤ , … , ≤ =
+ ≤ , + ≤ , … , + ≤
If time shifts does not matter, then it is stationary
Mean (First Moment)
5
t
t
()
()
t
()
[ ]
……… Averaging over all realizations
[ ]
Autocorrelation (Second Moment)
• “How similar a random process and a shifted version of itself is”
• Autocorrelation of a random process is defined as:
6
! , + ≜ +
t
t
#( + )
$()
Shifted by
×
All possible combinations of realizations
! , + For a particular t
=
For stationary random processes…
• Mean
• Autocorrelation
7
= − = 0 = (
Constant. Does not change with t.
! , + = − + − = [ 0 ] ≜ ! ( )
Two random processes
• Two random processes X(t) and Y(t) defined on the same underlying probability space have a joint cdf:
for all possible sets of sample times , … , and ) , … , *) .
• Two random processes are independent if we have
8
,…, + , ,…,+ -, , … , , ., … , ./ =
≤ , … , ≤ , 0 ) ≤ ., … , 0 /) ≤ ./
Similar to how you can define a joint cdf for two random variables
,…, + , ,…,+ -, , … , , ., … , ./ =
≤ , ≤ , … , ≤ 0 ) ≤ ., 0 ) ≤ ., … , 0 ) ≤ .
Cross-correlation
• The cross-correlation between two random processes X(t) and Y(t) is defined as
• Two random processes are uncorrelated if
for all t and 1.
• If both random processes are stationary, we have
9
! + , + ≜ 0 +
0 + = 0 +
! + , + = 0 + = 0 0 = !23
Wide-Sense Stationary (WSS)
• A process is wide-sense stationary if
and
• 45(1) has its maximum value at 1 = .
10
= (
! , + = + = !
! ≤ ! 0 =
A random process is always “the most similar” to the version of itself without shifting.
Ergodicity
11
t
t
()
()
t
()
()
Expectation value over time is the same as expectation over all possible realizations
.
$ .
Power Spectral Density
• The power spectral density of a WSS process is
defined as the Fourier transform of its autocorrelation function with respect to 1:
• PSD takes its name from the fact that the expected power of a random process X(t) is the integral of its PSD:
12
8 = 9 !A exp −=2? @
BA
= ! 0 = 9 8A
BA
Gaussian random processes
• A random process X(t) is a Gaussian process if, for all values of T and all functions g(t), the random variable
has a Gaussian distribution.
• We usually use this to model the noise for a communication receiver.
• Mean & variance:
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C = 9 D @E
Linear combination of samples
C = 9 D @E
FGH C = 9 9 D D I I @ @IE
− [C]
E
Gaussian random processes
• Samples of a random process, 5 J , J = , … , , are jointly Gaussian random variables, if we let K = L − J .
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C = 9 D @E
C = 9 M − E $ @
= $
Example: white noise
• White noise is a zero-mean WSS random process with a PSD that is constant over all frequencies.
• N is often called as one-sided white noise PSD.
• By inverse Fourier transform, the autocorrelation can be obtained:
15
= 0 8 = O for some constant N
! = N
2 M
White noise is not correlated with any shifted version of itself.
(Not similar at all after ANY time period)
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t
t0
τ0 τ
1 τ
τ 3
2 τ
4 τ
5 τ
6 τ(t0) τ(t1) t1
t2
τ(t2) t3
τ(t3)
hb(t,τ)
Two main aspects
of the wireless
channel
Doppler Effect
• Difference in path lengths PQ = R STUV = WPX YZ[V
• Phase change P\ = ]^PQ_ = ]^WPX_ YZ[V
• Frequency change, or Doppler shift,
`R =
]^ P\
PX = W
_ STUV
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Example
• Consider a transmitter which radiates a sinusoidal carrier frequency of 1850 MHz. For a vehicle moving 60 mph, compute the received carrier frequency if the mobile is moving
1. directly toward the transmitter.
2. directly away from the transmitter
3. in a direction which is perpendicular to the direction of arrival of the transmitted signal.
• Ans:
• Wavelength=a = cb
d = fg××e h = 0.162 (k)
• Vehicle speed l = 60km = 26.82 /o 1. p = q.f.qcos 0 = 160 uv
2. p = q.f.qcos ? = −160 (uv)
3. Since cos w = 0, there is no Doppler shift!
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`R =
]^ P\
PX = W
_ STUV
Doppler Effect
• If the car (mobile) is moving toward the direction of the arriving wave, the Doppler shift is positive
• Different Doppler shifts if different V (incoming angle)
• Multi-path: all different angles
• Many Doppler shifts Doppler spread
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Narrow-band Fading Model
• Sending an unmodulated carrier wave with random phase offset \:
• Received signal becomes
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I = xy{exp = 2?b + { } = cos 2?b + {
H = xy } ~ exp −={
O
exp =2?b
= H cos (2?b) − H sin(2?b)
Sum of many MPC Carrier with frequency b
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H = } ~ cos {
O
H = } ~ sin {
O
{ = 2?b − { − {
= H cos (2?b) − H sin(2?b) H = xy } ~ exp −={
O
exp =2?b
Doppler Shift Carrier phase shift (same for all MPC) Phase shift due to delay
Since N(t) is large & we assume ~() and {() are independent for different MPC, we can approximate H() and H() as jointly Gaussian random processes.
Some assumptions
• No dominant LOS component
• , ` , R 1 change slowly over time
• ]^`S1 changes rapidly relative to all other phase terms
• \( ) uniformly distributed on [−^, ^].
• and \ are independent of each other.
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{ = 2?b − { − {
Very large
Zero-mean
• Similarly,
• So, E[r(t)]=0, and it is a zero-mean Gaussian process.
• If there is a dominant LOS component, then this is no longer true.
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H = } ~
cos { = } ~ [cos {()]
= 0
H = 0
Un-correlated
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H H = } ~I{
} ~/sin ~/
/
= } } ~~/ cos { sin {/
/
= } ~ cos { sin {
= } ~ sin 2{
2
= 0
~ and { are not correlated.
= } ~ ~/ cos { sin {/
/,/
+ } ~ cos { sin {
Different MPC’s ~ and { are independent
Uniformly distributed over −?, ? , so =0.
H() and H are uncorrelated, and they are Gaussian processes
they are independent.
Autocorrelation
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! , + = H H + = } ~ cos { cos { +
= .5 [cos(2? )] + .5 cos 4?b − 4? − 2? − 2{
cos { cos { + =
= [.5 cos { + − { + .5 cos { + + { ]
{ + = 2?b − 2? + − { { = 2?b − 2? − {
Large and uniformly distributed over [−?, ?]
0
! , + = .5 } ~ [cos(2? )]
= .5 } ~
2?l cos a
Only depends on , so WSS!
Autocorrelation
• Finally,
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!, , + = !, = H H +
= −.5 } ~ sin 2?l cos
a
= − H H +
H = H cos 2?b − H sin 2?b
! = H H + = ! cos 2?b + !, sin (2?b )
Also only depends on , WSS!
The received signal, representing how the channel changes over time
Amplitude distribution - Rayleigh
• = = ] + ]
• ( ) and ( ) are both zero-mean Gaussian random process (so at a given time, two Gaussian random
variables).
• z(t)’s distribution - the amplitude distribution of r(t):
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Channel path loss
t
v = 2v
exp − v
= v
exp − v
2 , v ≥ 0
This is the famous Rayleigh distribution!
2-variable joint
Gaussian distribution
• PDF for 2-variable joint Gaussian distribution:
• : X and Y’s correlation (in our case, 0)
• 5 and : X and Y’s mean
• 5] and ]: X and Y’s variance (in our case, both are ])
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, 0 =
, 0 = 1
2? exp −1
2 + 0
The rest of the derivation can be found here:
http://www.dsplog.com/2008/07/17/derive-pdf-rayleigh-random-variable/
Power distribution:
Rayleigh
• We can obtain the power distribution by making the change of variables ] = ] to obtain
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= 1
exp −
= 1
2 exp −
2 , ≥ 0
Example: Rayleigh fading
• Consider a channel with Rayleigh fading (no LOS!) and average received power = ] dBm. Find the probability that the received power is below 10 dBm.
• We want to find the probability that ] < R¢* =
*£.
30
¤ < 10 = 9 1
100 exp −
100 @
= 0.095
With a LOS component – Ricean (or Rician)
• If the channel has a fixed LOS component then ( ) and ( ) are no longer zero-mean variables.
• The received signal becomes the superposition of a complex Gaussian component and a LOS component.
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Rayleigh: lots of random nLOS components
Ricean: Rayleigh + one strong static component
(LOS or strong reflection nLOS)
Ricean distribution
• Amplitude distribution:
• ]] = ∑, §[ ]] is the average power in the nLOS MPCs.
• U] = ] is the power in the dominant strong component.
• (¨): the modified Bessel function of zero-th order.
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v = v
exp −v + I
2 © vI
, v ≥ 0
Ricean distribution
• The average power in the Ricean fading is
• The Ricean distribution is often described in terms of a fading parameter K, defined by
• K is the ratio of the power in the dominant component to the power in the other random MPCs.
• K=0, then Ricean degenerates to Rayleigh
• K=∞, then Ricean becomes a non-fading LOS channel.
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= 9 v A v @v = I + 2
« = I 2
Coherence Time
• Coherence Time:
Coherence time is a statistical measure of the range of time over which the channel can be considered “static”.
• 90% coherence time:
• We can define 50% coherence time in a similar way too.
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¬S,. = K*J1 4 1
4 < .
The first time interval that normalized autocorrelation drops below the threshold.
Fast and slow fading channel
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!
!
b
b
o t
o: symbol period
Slow fading
Fast fading
f
®o
®o: signal bandwidth
®
®: Doppler Spread
f
¯ > b
®
®: Doppler Spread
f
® > ®¯
® ≪ ®¯
² ≫ ¯
