## Small-Scale Fading II

## (and basics about random processes)

PROF. MICHAEL TSAI 2014/3/31

## Random processes

**2**

t

t
𝑥_{2}(𝑡)

𝑥_{1}(𝑡)

t
𝑥_{3}(𝑡)

𝑋(𝑡)

One realization of X(t)

………

𝑋(𝜏) is a random variable:

𝑋(𝜏)
𝑓_{𝑋 𝜏} 𝑋

## Joint CDF for a random process

**• If we sample X(t) at times 𝒕**_{𝟎}, … 𝒕_{𝒏}**, we can have a joint **
**cdf of samples at those times:**

**3**

𝑃_{𝑋 𝑡}_{0} _{,…,𝑋 𝑡}_{𝑛} 𝑥_{0}, … , 𝑥_{𝑛} = 𝑝 𝑋 𝑡_{0} ≤ 𝑥_{0}, 𝑋 𝑡_{1} ≤ 𝑥_{1}, … , 𝑋 𝑡_{𝑛} ≤ 𝑥_{𝑛}

t

X(t) 𝑋 𝑡_{0} 𝑋 𝑡_{1} … 𝑋 𝑡_{𝑛}

## Stationary Random

## Process (Strict-sense)

**• A random process X(t) is stationary if for all T, all n, **
**and all sets of sample times 𝒕**_{𝟎}, 𝒕_{𝟏}, … , 𝒕_{𝒏} **we have:**

**4**

𝑝 𝑋 𝑡_{0} ≤ 𝑥_{0}, 𝑋 𝑡_{1} ≤ 𝑥_{1}, … , 𝑋 𝑡_{𝑛} ≤ 𝑥_{𝑛} =

𝑝 𝑋 𝑡_{0} + 𝑇 ≤ 𝑥_{0}, 𝑋 𝑡_{1} + 𝑇 ≤ 𝑥_{1}, … , 𝑋 𝑡_{𝑛} + 𝑇 ≤ 𝑥_{𝑛}

If time shifts does not matter, then it is stationary

## Mean (First Moment)

**5**

t

t
𝑥_{2}(𝑡)

𝑥_{1}(𝑡)

t
𝑥_{3}(𝑡)

𝐸[𝑋 𝑡 ]

……… Averaging over all realizations

𝐸[𝑋 𝑡 ]

𝐸 𝑋 𝜏

## Autocorrelation (Second Moment)

**• “How similar a random process and a shifted version of itself is”**

**• Autocorrelation of a random process is defined as:**

**6**

𝐴_{𝑋} 𝑡, 𝑡 + 𝜏 ≜ 𝐸 𝑋 𝑡 𝑋 𝑡 + 𝜏

t

t
𝑥_{𝑗}(𝑡 + 𝜏)

𝑥_{𝑖}(𝑡)

Shifted by 𝜏

### ×

All possible combinations of realizations

𝜏 𝐴 𝑡, 𝑡 + 𝜏

For a particular t

### =

## For stationary random processes…

**• Mean**

**• Autocorrelation**

**7**

𝐸 𝑋 𝑡 = 𝐸 𝑋 𝑡 − 𝑡 = 𝐸 𝑋 0 = 𝜇_{𝑋}

Constant. Does not change with t.

𝐴_{𝑋} 𝑡, 𝑡 + 𝜏 = 𝐸 𝑋 𝑡 − 𝑡 𝑋 𝑡 + 𝜏 − 𝑡 = 𝐸[𝑋 0 𝑋 𝜏 ] ≜ 𝐴_{𝑋}(𝜏)

## 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 𝒕**_{𝟎}, … , 𝒕_{𝒏} **and **
𝒕_{𝟎}^{′} , … , 𝒕_{𝒎}^{′} **.**

**• Two random processes are independent if we have**

**8**

𝑝_{𝑋 𝑡}

0 ,…,𝑋 𝑡_{𝑛} 𝑌 𝑡_{0}^{′} ,…,𝑌 𝑡_{𝑚}^{′} 𝑥_{0}, … , 𝑥_{𝑛}, 𝑦_{0}, … , 𝑦_{𝑚} =

𝑝 𝑋 𝑡_{0} ≤ 𝑥_{0}, … , 𝑋 𝑡_{𝑛} ≤ 𝑥_{𝑛}, 𝑌 𝑡_{0}^{′} ≤ 𝑦_{0}, … , 𝑌 𝑡_{𝑚}^{′} ≤ 𝑦_{𝑚}

Similar to how you can define a joint cdf for two random variables

𝑝_{𝑋 𝑡}

0 ,…,𝑋 𝑡_{𝑛} 𝑌 𝑡_{0}^{′} ,…,𝑌 𝑡_{𝑚}^{′} 𝑥_{0}, … , 𝑥_{𝑛}, 𝑦_{0}, … , 𝑦_{𝑚} =

𝑝 𝑋 𝑡_{0} ≤ 𝑥_{0}, 𝑋 𝑡_{1} ≤ 𝑥_{1}, … , 𝑋 𝑡_{𝑛} ≤ 𝑥_{𝑛} 𝑝 𝑌 𝑡_{0}^{′} ≤ 𝑦_{0}, 𝑌 𝑡_{1}^{′} ≤ 𝑦_{1}, … , 𝑌 𝑡_{𝑛}^{′} ≤ 𝑦_{𝑛}

## 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 𝝉.**

**• If both random processes are stationary, we have**

**9**

𝐴_{𝑋𝑌} 𝑡, 𝑡 + 𝜏 ≜ 𝐸 𝑋 𝑡 𝑌 𝑡 + 𝜏

𝐸 𝑋 𝑡 𝑌 𝑡 + 𝜏 = 𝐸 𝑋 𝑡 𝐸 𝑌 𝑡 + 𝜏

𝐴_{𝑋𝑌} 𝑡, 𝑡 + 𝜏 = 𝐸 𝑋 𝑡 𝑌 𝑡 + 𝜏 = 𝐸 𝑋 0 𝑌 𝜏 = 𝐴_{XY} 𝜏

## Wide-Sense Stationary (WSS)

**• A process is wide-sense stationary if**

**and**

• 𝑨_{𝑿}**(𝝉) has its maximum value at 𝝉 = 𝟎.**

**10**

𝐸 𝑋 𝑡 = 𝜇_{𝑋}

𝐴_{𝑋} 𝑡, 𝑡 + 𝜏 = 𝐸 𝑋 𝑡 𝑋 𝑡 + 𝜏 = 𝐴_{𝑋} 𝜏

𝐴_{𝑋} 𝜏 ≤ 𝐴_{𝑋} 0 = 𝐸 𝑋^{2} 𝑡

A random process is always “the most similar” to the version of itself without shifting.

## Ergodicity

**11**

t

t
𝑥_{2}(𝑡)

𝑥_{1}(𝑡)

t
𝑥_{3}(𝑡)

𝑋(𝑡)

Expectation value over time is the same as expectation over all possible realizations
𝐸_{𝑡} .

𝐸_{𝑖} .

## Power Spectral Density

**• The power spectral density of a WSS process is **

**defined as the Fourier transform of its autocorrelation **
**function with respect to 𝝉:**

**• PSD takes its name from the fact that the expected **
**power of a random process X(t) is the integral of its **
**PSD:**

**12**

𝑆_{𝑋} 𝑓 =

−∞

∞

𝐴_{𝑋} 𝜏 exp −𝑗2𝜋𝑓𝜏 𝑑𝜏

𝐸 𝑋^{2} 𝑡 = 𝐴_{𝑋} 0 =

−∞

∞

𝑆_{𝑋} 𝑓 𝑑𝑓

## 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:**

**13**

𝑋_{𝑔} =

0 𝑇

𝑔 𝑡 𝑋 𝑡 𝑑𝑡

Linear combination of samples

𝐸 𝑋_{𝑔} =

0 𝑇

𝑔 𝑡 𝐸 𝑋 𝑡 𝑑𝑡

𝑉𝑎𝑟 𝑋_{𝑔} =

0 𝑇

0 𝑇

𝑔 𝑡 𝑔 𝑠 𝐸 𝑋 𝑡 𝑋 𝑠 𝑑𝑡 𝑑𝑠 − 𝐸^{2}[𝑋_{𝑔}]

## Gaussian random processes

**• Samples of a random process, 𝑿 𝒕**_{𝒊} **, 𝒊 = 𝟎, … , 𝒏, are **
**jointly Gaussian random variables, if we let 𝒈 𝒕 =**
𝜹 𝒕 − 𝒕_{𝒊} **.**

**14**

𝑋_{𝑔} =

0 𝑇

𝑔 𝑡 𝑋 𝑡 𝑑𝑡

𝑋_{𝑔} =

0 𝑇

𝛿 𝑡 − 𝑡_{𝑖} 𝑋 𝑡 𝑑𝑡 = 𝑋 𝑡_{𝑖}

## Recap: 2 important aspects (of channel time variation)

**15**

## Example: white noise

**• White noise is a zero-mean WSS random process with **
**a PSD that is constant over all frequencies.**

• 𝑁_{0} **is often called as one-sided white noise PSD. **

**• By inverse Fourier transform, the autocorrelation can **
**be obtained:**

**16**

𝐸 𝑋 𝑡 = 0 𝑆_{𝑋} 𝑓 = ^{𝑁}^{0}

2 for some constant 𝑁_{0}

𝐴_{𝑋} 𝜏 = 𝑁_{0}

2 𝛿 𝜏

White noise is not correlated with any shifted version of itself.

(Not similar at all after ANY time period)

**17**

t

t_{0}

_{0} _{1} _{2} _{3} _{4} _{5} _{6} _{(t}

0)

(t_{1})
t_{1}

t_{2}

(t_{2})
t_{3}

(t_{3})

h_{b}(t,)

### Two main aspects

### of the wireless

### channel

## Doppler Effect

**• Difference in path lengths 𝚫𝐥 = 𝒅 𝒄𝒐𝒔𝜽 = 𝒗𝚫𝐭 𝐜𝐨𝐬𝜽**

**• Phase change 𝚫𝝓 =** ^{𝟐𝝅𝚫𝐥}

𝝀 = ^{𝟐𝝅𝒗𝚫𝐭}

𝝀 𝐜𝐨𝐬𝜽

**• Frequency change, or Doppler shift, **

𝒇_{𝒅} = 𝟏
𝟐𝝅

𝚫𝝓

𝚫𝐭 = 𝒗

𝝀 𝒄𝒐𝒔𝜽

**18**

### 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=𝜆 = ^{𝑐}

𝑓_{𝑐} = ^{3×10}^{8}

1850×10^{6} = 0.162 (𝑚)

• Vehicle speed 𝑣 = 60𝑚𝑝ℎ = 26.82 ^{𝑚}

𝑠

1. 𝑓_{𝑑} = ^{26.82}

0.162cos 0 = 160 𝐻𝑧
2. 𝑓_{𝑑} = ^{26.82}

0.162cos 𝜋 = −160 (𝐻𝑧)
3. Since cos ^{𝜋}

2 = 0, there is no Doppler shift!

**19**

𝒇_{𝒅} = 𝟏
𝟐𝝅

𝚫𝝓

𝚫𝐭 = 𝒗

𝝀𝒄𝒐𝒔𝜽

## 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 𝜽 (incoming angle)**

**• Multi-path: all different angles**

**• Many Doppler shifts Doppler spread**

**20**

## Narrow-band Fading Model

**• Sending an unmodulated carrier wave with random **
**phase offset 𝝓**_{𝟎}**:**

**• Received signal becomes**

**21**

𝑠 𝑡 = 𝑅𝑒{exp 𝑗 2𝜋𝑓_{𝑐}𝑡 + 𝜙_{0} } = cos 2𝜋𝑓_{𝑐}𝑡 + 𝜙_{0}

𝑟 𝑡 = 𝑅𝑒

𝑛=1 𝑁 𝑡

𝛼_{𝑛} 𝑡 exp −𝑗𝜙_{𝑛} 𝑡 exp 𝑗2𝜋𝑓_{𝑐}𝑡

= 𝑟_{𝐼} 𝑡 cos(2𝜋𝑓_{𝑐}𝑡) − 𝑟_{𝑄} 𝑡 sin(2𝜋𝑓_{𝑐}𝑡)

Sum of many MPC Carrier with
frequency 𝑓_{𝑐}

**22**

𝑟_{𝐼} 𝑡 =

𝑛=1 𝑁 𝑡

𝛼_{𝑛} 𝑡 cos 𝜙_{𝑛} 𝑡 𝑟_{𝑄} 𝑡 =

𝑛=1 𝑁 𝑡

𝛼_{𝑛} 𝑡 sin 𝜙_{𝑛} 𝑡

𝜙_{𝑛} 𝑡 = 2𝜋𝑓_{𝑐}𝜏_{𝑛} 𝑡 − 𝜙_{𝐷}_{𝑛} − 𝜙_{0}

= 𝑟_{𝐼} 𝑡 cos(2𝜋𝑓_{𝑐}𝑡) − 𝑟_{𝑄} 𝑡 sin(2𝜋𝑓_{𝑐}𝑡)
𝑟 𝑡 = 𝑅𝑒

𝑛=1 𝑁 𝑡

𝛼_{𝑛} 𝑡 exp −𝑗𝜙_{𝑛} 𝑡 exp 𝑗2𝜋𝑓_{𝑐}𝑡

Doppler Shift Carrier phase shift (same for all MPC) Phase shift due to delay

Since N(t) is large & we assume 𝛼_{𝑛}(𝑡) and 𝜙_{𝑛}(𝑡) are independent for different MPC,
we can approximate 𝑟_{𝐼}(𝑡) and 𝑟_{𝑄}**(𝑡) as jointly Gaussian random processes. **

## Some assumptions

**• No dominant LOS component**

• 𝜶_{𝒏} 𝒕 , 𝒇_{𝑫}_{𝒏} 𝒕 , 𝒂𝒏𝒅 𝝉_{𝒏} **𝒕 change slowly over time**

• 𝟐𝝅𝒇_{𝒄}𝝉_{𝒏} **changes rapidly relative to all other phase **
**terms**

• 𝝓_{𝒏}**(𝒕) uniformly distributed on [−𝝅, 𝝅].**

• 𝜶_{𝒏} **and 𝝓**_{𝒏} **are independent of each other.**

**23**

𝜙_{𝑛} 𝑡 = 2𝜋𝑓_{𝑐}𝜏_{𝑛} 𝑡 − 𝜙_{𝐷}_{𝑛} − 𝜙_{0}

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.**

**24**

𝐸 𝑟_{𝐼} 𝑡 = 𝐸

𝑛

𝛼_{𝑛} cos 𝜙_{𝑛} 𝑡 =

𝑛

𝐸 𝛼_{𝑛} 𝐸[cos 𝜙_{𝑛}(𝑡)] = 0

𝐸 𝑟_{𝑄} 𝑡 = 0

## Un-correlated

**25**

𝐸 𝑟_{𝐼} 𝑡 𝑟_{𝑄} 𝑡 = 𝐸

𝑛

𝛼_{𝑛}𝑐𝑜𝑠𝜙_{𝑛} 𝑡

𝑚

𝛼_{𝑚} sin 𝛼_{𝑚} 𝑡

=

𝑛 𝑚

𝐸 𝛼_{𝑛}𝛼_{𝑚} 𝐸 cos 𝜙_{𝑛} 𝑡 sin 𝜙_{𝑚} 𝑡

=

𝑛

𝐸 𝛼_{𝑛}^{2} 𝐸 cos 𝜙_{𝑛} 𝑡 sin 𝜙_{𝑛} 𝑡 =

𝑛

𝐸 𝛼_{𝑛}^{2} 𝐸 sin 2𝜙_{𝑛} 𝑡

2 = 0

𝛼_{𝑛} and 𝜙_{𝑛} are not correlated.

=

𝑛,𝑚 𝑛≠𝑚

𝐸 𝛼_{𝑛} 𝐸 𝛼_{𝑚} 𝐸 cos 𝜙_{𝑛} 𝑡 𝐸 sin 𝜙_{𝑚} 𝑡 +

𝑛

𝐸 𝛼_{𝑛}^{2} 𝐸 cos 𝜙_{𝑛} 𝑡 sin 𝜙_{𝑛} 𝑡
Different MPC’s 𝛼_{𝑛} and 𝜙_{𝑛} are independent

Uniformly distributed over −𝜋, 𝜋 , so =0.

𝑟_{𝐼}(𝑡) and 𝑟_{𝑄} 𝑡 are uncorrelated, and they are Gaussian processes

they are independent.

## Autocorrelation

**26**

𝐴_{𝑟}_{𝐼} 𝑡, 𝑡 + 𝜏 = 𝐸 𝑟_{𝐼} 𝑡 𝑟_{𝐼} 𝑡 + 𝜏 =

𝑛

𝐸 𝛼_{𝑛}^{2} 𝐸 cos 𝜙_{𝑛} 𝑡 cos 𝜙_{𝑛} 𝑡 + 𝜏

= .5𝐸[cos(2𝜋𝑓_{𝐷}_{𝑛}𝜏)] + .5𝐸 cos 4𝜋𝑓_{𝑐}𝜏_{𝑛} − 4𝜋𝑓_{𝐷}_{𝑛}𝑡 − 2𝜋𝑓_{𝐷}_{𝑛}𝜏 − 2𝜙_{0}
𝐸 cos 𝜙_{𝑛} 𝑡 cos 𝜙_{𝑛} 𝑡 + 𝜏 =

= 𝐸[.5 cos 𝜙_{𝑛} 𝑡 + 𝜏 − 𝜙_{𝑛} 𝑡 + .5 cos 𝜙_{𝑛} 𝑡 + 𝜏 + 𝜙_{𝑛} 𝑡 ]

𝜙_{𝑛} 𝑡 + 𝜏 = 2𝜋𝑓_{𝑐}𝜏_{𝑛} − 2𝜋𝑓_{𝐷}_{𝑛} 𝑡 + 𝜏 − 𝜙_{0}
𝜙_{𝑛} 𝑡 = 2𝜋𝑓_{𝑐}𝜏_{𝑛} − 2𝜋𝑓_{𝐷}_{𝑛}𝑡 − 𝜙_{0}

Large and uniformly distributed over [−𝜋, 𝜋]

0

𝐴_{𝑟}_{𝐼} 𝑡, 𝑡 + 𝜏 = .5

𝑛

𝐸 𝛼_{𝑛}^{2} 𝐸[cos(2𝜋𝑓_{𝐷}_{𝑛}𝜏)] = .5

𝑛

𝐸 𝛼_{𝑛}^{2} 2𝜋𝑣𝜏 cos 𝜃_{𝑛}
𝜆

Only depends on 𝜏, so WSS!

## Autocorrelation

**• Finally, **

**27**

𝐴_{𝑟}_{𝐼}_{,𝑟}_{𝑄} 𝑡, 𝑡 + 𝜏 = 𝐴_{𝑟}_{𝐼}_{,𝑟}_{𝑄} 𝜏 = 𝐸 𝑟_{𝐼} 𝑡 𝑟_{𝑄} 𝑡 + 𝜏

= −.5

𝑛

𝐸 𝛼_{𝑛}^{2} sin 2𝜋𝑣𝜏 cos𝜃_{𝑛}

𝜆 = −𝐸 𝑟_{𝑄} 𝑡 𝑟_{𝐼} 𝑡 + 𝜏

𝑟 𝑡 = 𝑟_{𝐼} 𝑡 cos 2𝜋𝑓_{𝑐}𝑡 − 𝑟_{𝑄} 𝑡 sin 2𝜋𝑓_{𝑐}𝑡

𝐴_{𝑟} 𝜏 = 𝐸 𝑟 𝑡 𝑟 𝑡 + 𝜏 = 𝐴_{𝑟}_{𝐼} 𝜏 cos 2𝜋𝑓_{𝑐}𝜏 + 𝐴_{𝑟}_{𝐼}_{,𝑟}_{𝑄} 𝜏 sin(2𝜋𝑓_{𝑐}𝜏)
Also only depends on 𝜏, WSS!

The received signal, representing how the channel changes over time

## Amplitude distribution - Rayleigh

• 𝒛 𝒕 = 𝒓 𝒕 = 𝒓_{𝑰}^{𝟐} 𝒕 + 𝒓_{𝑸}^{𝟐} 𝒕

• 𝒓_{𝑰}**(𝒕) and 𝒓**_{𝑸}**(𝒕) 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):**

**28**

Channel path loss

t

𝑝_{𝑍} 𝑧 = 2𝑧

𝑃_{𝑟} exp −𝑧^{2}

𝑃_{𝑟} = 𝑧

𝜎^{2} exp − 𝑧^{2}

2𝜎^{2} , 𝑧 ≥ 0

This is the famous Rayleigh distribution!

## 2-variable joint

## Gaussian distribution

**• PDF for 2-variable joint Gaussian distribution:**

**• 𝝆: X and Y’s correlation (in our case, 0)**

• 𝝁_{𝑿} **and 𝝁**_{𝒀}**: X and Y’s mean**

• 𝝈_{𝑿}^{𝟐} **and 𝝈**_{𝒀}^{𝟐}**: X and Y’s variance (in our case, both are 𝝈**^{𝟐})

**29**

𝑓 𝑋, 𝑌 = 1

2𝜋𝜎_{𝑋}𝜎_{𝑌} 1 − 𝜌^{2} exp − 1
2 1 − 𝜌^{2}

𝑋 − 𝜇_{𝑋} ^{2}

𝜎_{𝑋}^{2} − 2𝜌(𝑋 − 𝜇_{𝑋})(𝑌 − 𝜇_{𝑌})

𝜎_{𝑋}𝜎_{𝑌} + 𝑌 − 𝜇_{𝑌} ^{2}
𝜎_{𝑌}^{2}

𝑓 𝑋, 𝑌 = 1

2𝜋𝜎^{2} exp −1
2

𝑋^{2} + 𝑌^{2}
𝜎^{2}
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 𝒛**^{𝟐} 𝒕 = 𝒓 𝒕 ^{𝟐} **to obtain**

**30**

𝑝_{𝑍}^{2} 𝑥 = 1

𝑃_{𝑟} exp − 𝑥

𝑃_{𝑟} = 1

2𝜎^{2} exp − 𝑥

2𝜎^{2} , 𝑥 ≥ 0

## Example: Rayleigh fading

**• Consider a channel with Rayleigh fading (no LOS!) **
**and average received power 𝑷**_{𝒓} **= 𝟐𝟎 dBm. Find the **
**probability that the received power is below 10 dBm.**

**• We want to find the probability that 𝒁**^{𝟐} < 𝟏𝟎 𝒅𝑩𝒎 =
**𝟏𝟎 𝒎𝑾.**

**31**

𝑝 𝑍^{2} < 10 =

0

10 1

100exp − 𝑥

100 𝑑𝑥 = 0.095

## With a LOS component – Ricean (or Rician)

**• If the channel has a fixed LOS component then 𝒓**_{𝑰}(𝒕)
**and 𝒓**_{𝑸}**(𝒕) are no longer zero-mean variables. **

**• The received signal becomes the superposition of a **
**complex Gaussian component and a LOS component.**

**32**

Rayleigh: lots of random nLOS components

Ricean: Rayleigh + one strong static component

(LOS or strong reflection nLOS)

## Ricean distribution

**• Amplitude distribution:**

• 𝟐𝝈^{𝟐} = _{𝒏,𝒏≠𝟎} 𝑬[𝜶_{𝒏}^{𝟐}**] is the average power in the nLOS**
**MPCs. **

• 𝒔^{𝟐} = 𝜶_{𝟎}^{𝟐} **is the power in the dominant strong **
**component.**

• 𝑰_{𝟎}**(𝒙): the modified Bessel function of zeroth order. **

**33**

𝑝_{𝑍} 𝑧 = 𝑧

𝜎^{2} exp −𝑧^{2} + 𝑠^{2}

2𝜎^{2} 𝐼_{0} 𝑧𝑠

𝜎^{2} , 𝑧 ≥ 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.

**34**

𝑃_{𝑟} =

0

∞

𝑧^{2}𝑝_{𝑍} 𝑧 𝑑𝑧 = 𝑠^{2} + 2𝜎^{2}

𝐾 = 𝑠^{2}
2𝜎^{2}

## Ricean and Rayleigh

**35**

36

**Example: Intra-car Wireless Channel Measurements**

Which distribution fit the empirical amplitude distribution function the best?

Lognormal, Nakagami, Rician, Rayleigh, and Weibull

Engine compartment Center

(Parked)

Engine compartment Under the engine

(Parked)

CDF CDF

**Parked: Weibull**

37

Engine compartment Trunk

(Driving)

Under the engine Engine Compartment

(Driving)

CDF CDF

**Driving: Rician/Nakagami**

38

Channel model

No Line-of-Sight component Rayleigh

Rician

## 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.**

**39**

𝑻_{𝒄,𝟎.𝟗} = 𝒂𝒓𝒈𝒎𝒊𝒏_{𝝉} 𝑨_{𝒓} 𝝉

𝑨_{𝒓} 𝟎 < 𝟎. 𝟗

The first time interval that normalized autocorrelation drops below the threshold.

40

10^{-3} 10^{-2} 10^{-1} 10^{0} 10^{1} 10^{2} 10^{3} 10^{4}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time spacing (seconds)

Normalized Correlation

Scenario 1 (Parked) Scenario 2 (Driving)

**50% correlation**

**~ 2.5 sec** **~ 60 sec**

Worst Channel

**Coherence time is always on the order of seconds.**

IE/UE Channel

## Fast and slow fading channel

**41**

𝜏

𝜏
𝐴_{𝑟} 𝜏

𝐴_{𝑟} 𝜏

𝑇_{𝑐}

𝑇_{𝑐}
𝑇_{𝑠} t

𝑇_{𝑠}: symbol period

Slow fading

Fast fading

f
𝐵_{𝑠}

𝐵_{𝑠}: signal bandwidth
𝐵_{𝐷}

𝐵_{𝐷}: Doppler Spread

f

𝑇_{𝑆} > 𝑇_{𝑐}

𝐵_{𝐷}

𝐵_{𝐷}: Doppler Spread

f
𝐵_{𝐷} > 𝐵_{𝑆}

𝐵_{𝐷} ≪ 𝐵_{𝑆}

𝑇_{𝐶} ≫ 𝑇_{𝑆}