(and basics about random processes)

Small-Scale Fading II

(and basics about random processes)

PROF. MICHAEL TSAI 2014/3/31

Random processes

2

t

t 𝑥₂(𝑡)

𝑥₁(𝑡)

t 𝑥₃(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:

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𝑃_{𝑋 𝑡0 ,…,𝑋 𝑡𝑛} 𝑥₀, … , 𝑥_𝑛 = 𝑝 𝑋 𝑡₀ ≤ 𝑥₀, 𝑋 𝑡₁ ≤ 𝑥₁, … , 𝑋 𝑡_𝑛 ≤ 𝑥_𝑛

t

X(t) 𝑋 𝑡₀ 𝑋 𝑡₁ … 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:

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𝑝 𝑋 𝑡₀ ≤ 𝑥₀, 𝑋 𝑡₁ ≤ 𝑥₁, … , 𝑋 𝑡_𝑛 ≤ 𝑥_𝑛 =

𝑝 𝑋 𝑡₀ + 𝑇 ≤ 𝑥₀, 𝑋 𝑡₁ + 𝑇 ≤ 𝑥₁, … , 𝑋 𝑡_𝑛 + 𝑇 ≤ 𝑥_𝑛

If time shifts does not matter, then it is stationary

Mean (First Moment)

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t

t 𝑥₂(𝑡)

𝑥₁(𝑡)

t 𝑥₃(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:

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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

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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

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𝑝_{𝑋 𝑡}

0 ,…,𝑋 𝑡_𝑛 𝑌 𝑡₀^′ ,…,𝑌 𝑡_𝑚^′ 𝑥₀, … , 𝑥_𝑛, 𝑦₀, … , 𝑦_𝑚 =

𝑝 𝑋 𝑡₀ ≤ 𝑥₀, … , 𝑋 𝑡_𝑛 ≤ 𝑥_𝑛, 𝑌 𝑡₀^′ ≤ 𝑦₀, … , 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 𝑡₀^′ ,…,𝑌 𝑡_𝑚^′ 𝑥₀, … , 𝑥_𝑛, 𝑦₀, … , 𝑦_𝑚 =

𝑝 𝑋 𝑡₀ ≤ 𝑥₀, 𝑋 𝑡₁ ≤ 𝑥₁, … , 𝑋 𝑡_𝑛 ≤ 𝑥_𝑛 𝑝 𝑌 𝑡₀^′ ≤ 𝑦₀, 𝑌 𝑡₁^′ ≤ 𝑦₁, … , 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

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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.

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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 𝑋² 𝑡

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

Ergodicity

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t

t 𝑥₂(𝑡)

𝑥₁(𝑡)

t 𝑥₃(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:

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𝑆_𝑋 𝑓 =

−∞

∞

𝐴_𝑋 𝜏 exp −𝑗2𝜋𝑓𝜏 𝑑𝜏

𝐸 𝑋² 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:

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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 − 𝐸²[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 .

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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)

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Example: white noise

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

• 𝑁₀ is often called as one-sided white noise PSD.

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

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𝐸 𝑋 𝑡 = 0 𝑆_𝑋 𝑓 = ^𝑁0

2 for some constant 𝑁₀

𝐴_𝑋 𝜏 = 𝑁₀

2 𝛿 𝜏

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

₀ ₁ ₂ ₃ ₄ ₅ _{6 (t}

0)

(t₁) t₁

t₂

(t₂) t₃

(t₃)

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

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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=𝜆 = ^𝑐

𝑓_𝑐 = ^3×108

1850×10⁶ = 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!

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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

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Narrow-band Fading Model

• Sending an unmodulated carrier wave with random phase offset 𝝓_𝟎:

• Received signal becomes

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𝑠 𝑡 = 𝑅𝑒{exp 𝑗 2𝜋𝑓_𝑐𝑡 + 𝜙₀ } = cos 2𝜋𝑓_𝑐𝑡 + 𝜙₀

𝑟 𝑡 = 𝑅𝑒

𝑛=1 𝑁 𝑡

𝛼_𝑛 𝑡 exp −𝑗𝜙_𝑛 𝑡 exp 𝑗2𝜋𝑓_𝑐𝑡

= 𝑟_𝐼 𝑡 cos(2𝜋𝑓_𝑐𝑡) − 𝑟_𝑄 𝑡 sin(2𝜋𝑓_𝑐𝑡)

Sum of many MPC Carrier with frequency 𝑓_𝑐

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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_𝑐𝜏_𝑛 𝑡 − 𝜙_𝐷𝑛 − 𝜙₀

= 𝑟_𝐼 𝑡 cos(2𝜋𝑓_𝑐𝑡) − 𝑟_𝑄 𝑡 sin(2𝜋𝑓_𝑐𝑡) 𝑟 𝑡 = 𝑅𝑒

𝑛=1 𝑁 𝑡

𝛼_𝑛 𝑡 exp −𝑗𝜙_𝑛 𝑡 exp 𝑗2𝜋𝑓_𝑐𝑡

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_𝑫𝒏 𝒕 , 𝒂𝒏𝒅 𝝉_𝒏 𝒕 change slowly over time

• 𝟐𝝅𝒇_𝒄𝝉_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.

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𝜙_𝑛 𝑡 = 2𝜋𝑓_𝑐𝜏_𝑛 𝑡 − 𝜙_𝐷𝑛 − 𝜙₀

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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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

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𝐸 𝑟_𝐼 𝑡 𝑟_𝑄 𝑡 = 𝐸

𝑛

𝛼_𝑛𝑐𝑜𝑠𝜙_𝑛 𝑡

𝑚

𝛼_𝑚 sin 𝛼_𝑚 𝑡

=

𝑛 𝑚

𝐸 𝛼_𝑛𝛼_𝑚 𝐸 cos 𝜙_𝑛 𝑡 sin 𝜙_𝑚 𝑡

=

𝑛

𝐸 𝛼_𝑛² 𝐸 cos 𝜙_𝑛 𝑡 sin 𝜙_𝑛 𝑡 =

𝑛

𝐸 𝛼_𝑛² 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_𝑛² 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

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𝐴_𝑟𝐼 𝑡, 𝑡 + 𝜏 = 𝐸 𝑟_𝐼 𝑡 𝑟_𝐼 𝑡 + 𝜏 =

𝑛

𝐸 𝛼_𝑛² 𝐸 cos 𝜙_𝑛 𝑡 cos 𝜙_𝑛 𝑡 + 𝜏

= .5𝐸[cos(2𝜋𝑓_𝐷𝑛𝜏)] + .5𝐸 cos 4𝜋𝑓_𝑐𝜏_𝑛 − 4𝜋𝑓_𝐷𝑛𝑡 − 2𝜋𝑓_𝐷𝑛𝜏 − 2𝜙₀ 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_𝑐𝜏_𝑛 − 2𝜋𝑓_𝐷𝑛 𝑡 + 𝜏 − 𝜙₀ 𝜙_𝑛 𝑡 = 2𝜋𝑓_𝑐𝜏_𝑛 − 2𝜋𝑓_𝐷𝑛𝑡 − 𝜙₀

Large and uniformly distributed over [−𝜋, 𝜋]

0

𝐴_𝑟𝐼 𝑡, 𝑡 + 𝜏 = .5

𝑛

𝐸 𝛼_𝑛² 𝐸[cos(2𝜋𝑓_𝐷𝑛𝜏)] = .5

𝑛

𝐸 𝛼_𝑛² 2𝜋𝑣𝜏 cos 𝜃_𝑛 𝜆

Only depends on 𝜏, so WSS!

Autocorrelation

• Finally,

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𝐴_{𝑟𝐼,𝑟𝑄} 𝑡, 𝑡 + 𝜏 = 𝐴_{𝑟𝐼,𝑟𝑄} 𝜏 = 𝐸 𝑟_𝐼 𝑡 𝑟_𝑄 𝑡 + 𝜏

= −.5

𝑛

𝐸 𝛼_𝑛² sin 2𝜋𝑣𝜏 cos𝜃_𝑛

𝜆 = −𝐸 𝑟_𝑄 𝑡 𝑟_𝐼 𝑡 + 𝜏

𝑟 𝑡 = 𝑟_𝐼 𝑡 cos 2𝜋𝑓_𝑐𝑡 − 𝑟_𝑄 𝑡 sin 2𝜋𝑓_𝑐𝑡

𝐴_𝑟 𝜏 = 𝐸 𝑟 𝑡 𝑟 𝑡 + 𝜏 = 𝐴_𝑟𝐼 𝜏 cos 2𝜋𝑓_𝑐𝜏 + 𝐴_{𝑟𝐼,𝑟𝑄} 𝜏 sin(2𝜋𝑓_𝑐𝜏) 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):

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Channel path loss

t

𝑝_𝑍 𝑧 = 2𝑧

𝑃_𝑟 exp −𝑧²

𝑃_𝑟 = 𝑧

𝜎² exp − 𝑧²

2𝜎² , 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)

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𝑓 𝑋, 𝑌 = 1

2𝜋𝜎_𝑋𝜎_𝑌 1 − 𝜌² exp − 1 2 1 − 𝜌²

𝑋 − 𝜇_𝑋 ²

𝜎_𝑋² − 2𝜌(𝑋 − 𝜇_𝑋)(𝑌 − 𝜇_𝑌)

𝜎_𝑋𝜎_𝑌 + 𝑌 − 𝜇_𝑌 ² 𝜎_𝑌²

𝑓 𝑋, 𝑌 = 1

2𝜋𝜎² exp −1 2

𝑋² + 𝑌² 𝜎² 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_𝑍² 𝑥 = 1

𝑃_𝑟 exp − 𝑥

𝑃_𝑟 = 1

2𝜎² exp − 𝑥

2𝜎² , 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.

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𝑝 𝑍² < 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.

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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:

• 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.

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𝑝_𝑍 𝑧 = 𝑧

𝜎² exp −𝑧² + 𝑠²

2𝜎² 𝐼₀ 𝑧𝑠

𝜎² , 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.

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𝑃_𝑟 =

0

∞

𝑧²𝑝_𝑍 𝑧 𝑑𝑧 = 𝑠² + 2𝜎²

𝐾 = 𝑠² 2𝜎²

Ricean and Rayleigh

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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

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Engine compartment  Trunk

(Driving)

Under the engine  Engine Compartment

(Driving)

CDF CDF

Driving: Rician/Nakagami

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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.

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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.

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10^-3 10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴

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

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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