Lecture 15-2003p [compatibility Mode]

Wireless and Mobile  Networking Dr. Faramarz Hendessi I f h U i fT h Isfahan Univ. of Tech. Spring 2009

Lecture 15: Lecture 15:

Basic Principles

Fading Distribution and  Models

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Types of Small‐scale Fading Small-scale Fading (Based on Multipath Tİme Delay Spread)

Flat Fading

Frequency Selective Fading

1. BW Signal < BW of Channel 2. Delay Spread < Symbol Period

1. BW Signal > Bw of Channel 2. Delay Spread > Symbol Period

Small-scale Fading (Based on Doppler Spread)

Fast Fading 1. High Doppler Spread 2. Coherence Time < Symbol Period 3. Channel variations faster than baseband signal variations

Slow Fading 1. Low Doppler Spread 2. Coherence Time > Symbol Period 3. Channel variations smaller than baseband signal variations

Fading  Distributions •

• •

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Describes how the received signal amplitude changes with time.  – Remember that the received signal is combination of multiple  signals arriving from different directions, phases and amplitudes.  – With the received signal we mean the baseband signal, namely the  With the received signal we mean the baseband signal namely the envelope of the received signal (i.e. r(t)).  It is a statistical characterization of the multipath fading.  Two distributions – Rayleigh Fading – Ricean Fading

Rayleigh Distributions •

Describes the received signal envelope distribution for channels,  where all the components are non‐LOS:  – i.e. there is no line‐of–sight (LOS) component.

Ricean Distributions •

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Describes the received signal envelope distribution for channels where  one of the multipath components is LOS component.  – i.e. there is one LOS component.

Rayleigh Fading

Rayleigh Fading

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Rayleigh Fading Distribution • The Rayleigh distribution is commonly used to  describe the statistical time varying nature of  the received envelope of a flat fading signal, or  the envelope of an individual multipath the envelope of an individual multipath  component. • The envelope of the sum of two quadrature  Gaussian noise signals obeys a Rayleigh  distribution. ⎧ r r 2

⎪ exp(− 2 ) p(r ) = ⎨σ 2 2σ ⎪0 r <0 ⎩

0≤ r ≤ ∞

• σ is the rms value of the received voltage  before envelope detection, and σ2 is the time‐ average power of the received signal before  envelope detection.

Rayleigh Fading Distribution • The probability that the envelope of the   received signal does not exceed a specified  value of R is given by the CDF:  R

−

P(R) = Pr (r ≤ R) = ∫ p(r)dr =1− e ∞

0

rmean = E[ r ] = ∫ rp ( r ) dr = σ 0

rmedian

π

2

2σ 2

= 1.2533σ

1 = 1.177σ found by solving = 2

rrms = 2σ σ r2 = E [ r 2 ]− E 2 [ r ] =

∫

∞

0

r 2 p ( r ) dr −

• rpeak=σ and p(σ)=0.6065/σ

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R2

σ 2π 2

rmedian

∫ p (r )dr 0

= 0. 4292σ 2

Rayleigh PDF 0.7

0.6065/σ 0.6

mean = 1.2533σ median = 1.177σ

0.5

variance = 0.4292σ2

0.4

0.3

0.2

0.1

0 0

1 σ

22 σ

33 σ

4 4σ

5 5σ

A typical Rayleigh fading envelope  at 900MHz.

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Ricean Distribution • •

When there is a stationary (non‐fading)  LOS signal present, then the  envelope distribution is Ricean.   The Ricean distribution degenerates to Rayleigh when the dominant  component fades away. p y

Ricean Fading Distribution •

•

When there is a dominant stationary signal component present, the  small‐scale fading envelope distribution is Ricean.  The effect of a  dominant signal arriving with many weaker multipath signals gives rise  to the Ricean distribution. The Ricean distribution degenerates to a Rayleigh distribution when  th d i the dominant component fades away. t tf d ⎧ r ( r 2 + A2 ) Ar exp[ − ]I 0 ( 2 ) ⎪ p ( r ) = ⎨σ 2 2σ 2 σ ⎪0 r <0 ⎩

A≥0

•

The Ricean distribution is often described in terms of a parameter K which is defined as the ratio between the deterministic signal power and the variance of the multipath.

• •

K is known as the Ricean factor As A→0, A→0 K → ‐∞ dB, dB Ricean distribution degenerates to Rayleigh distribution. 2 K =

٧

0 ≤ r ≤ ∞,

A

2σ 2

CDF  •

Cumulative distribution for three small‐scale fading measurements  and their fit to Rayleigh, Ricean, and log‐normal distributions.

PDF • Probability density function of Ricean distributions: K=‐∞dB (Rayleigh) and K=6dB. For  K>>1, the Ricean pdf is approximately Gaussian  about the mean.

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Rice time series

Nakagami Model • Nakagami Model p(r ) =

2m m r 2 m−1 exp( p(− Γ(m)Ω m

m 2 r ) Ω

• r: envelope amplitude • Ω=: time‐averaged power of received signal • m: the inverse of normalized variance of r2

– Get Rayleigh when m=1 G tR l i h h 1

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Small‐scale fading mechanism

• Assume signals arrive  from all angles in the  horizontal plane  0<α<360 • Signal amplitudes are  equal, independent of  α • Assume further that  there is no multipath  delay: (flat fading  assumption) • Doppler shifts fn =

v

λ

cos a n

Small‐scale fading: effect of Doppler  in a multipath environment • fm, the largest Doppler shift

S bbEz ( f ) =

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⎛ f ⎞ 1 ⎟⎟ k 1 − ⎜⎜ 8πf m ⎝ 2 fm ⎠

2

Carrier Doppler spectrum

• Spectrum Empirical investigations show  results that deviate from this model Power Model Power goes to infinity at fc+/‐fm

Baseband Spectrum Doppler Faded Signal

• Cause baseband spectrum has a maximum  frequency of 2fm

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Simulating Doppler/Small‐scale fading

Simulating Doppler fading • Procedure in page 222

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Level Crossing Rate (LCR) Threshold (R)

LCR is defined as the expected rate at which the Rayleigh fading envelope, normalized to the local rms signal level, crosses a specified threshold level R in a positive going direction. It is given by:

NR = 2π fmρe−ρ

2

where

ρ = R / rrms

(specfied envelopevaluenormalized torms)

NR : crossingspersecond

Average Fade Duration Defined as the average period of time for which the received signal is below a specified level R. For Rayleigh distributed fading signal, it is given by:

τ=

(

2 1 1 Pr[r ≤ R] = 1− e−ρ NR NR

eρ −1 R τ= , ρ= rrms ρfm 2π 2

Example 5.7, 5.8, 5.9

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)

Fading Model: Gilbert‐Elliot Model Fade Period Signal Amplitude Threshold

Time t

Bad

Good

(Fade)

(Non-fade)

Gilbert‐Elliot Model 1/AFD Bad

Good (Non-fade) (Non fade)

((Fade))

1/ANFD

The channel is modeled as a Two-State Markov Chain. Each state duration is memory-less and exponentially distributed. The rate going from Good to Bad state is: 1/AFD (AFD: Avg Fade Duration The rate going from Bad to Good state is: 1/ANFD (ANFD: Avg Non-Fade Duration)

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Simulating 2‐ray multipath

• a1 and a2 are independent Rayleigh fading • φ1 and φ2 are uniformly distributed over  [0 2π) [0,2π)

Simulating multipath with Doppler‐induced Rayleigh fading

EE 542/452 Spring 2008

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Saleh and Valenzuela Indoor Model •

•

Measured same‐floor indoor characteristics – Found that, with a fixed receiver, indoor channel is very slowly  time‐varying – RMS delay spread: mean 25ns, max 50ns – Maximal delay spread 100ns‐200ns – With no LOS, path loss varied over 60dB range and obeyed log  distance power law, 3 > n > 4 Model assumes a structure and models correlated multipath  components.

• Multipath model – Multipath components arrive in clusters, follow Poisson  distribution. Clusters relate to building structures.   g – Within cluster, individual components also follow Poisson  distribution.  Cluster components relate to reflecting objects near  the TX or RX. – Amplitudes of components are independent Rayleigh variables,  decay exponentially with cluster delay and with intra‐cluster delay

SIRCIM and SMRCIM  indoor/outdoor Models •

These models were developed by Rappaport and seidel SIRCIM is a  computer program , that generates small scale indoor channel  response measurements.

•

The most salient feature of the model is that it produces multipath channel  conditions that are very realistic since they are based on real world  measurements and may thus be used for meaningful system design in  factories and office buildings

•

These programs are very useful and poplar and are used in over 100  institutions. Model can measure individual multipath fading and small scale  receiver spacing. Multipath delay inside the building was found to be 40ns to 800ns. Mean multipath delay ranged from 30‐300 ns. Arriving multipath component has a Gaussian distribution. Average number of multipath components range from 9 to 36

• • • • •

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SIRCIM and SMRCIM  indoor/outdoor Models

• SIRCIM Model

– Based on measurements at 1300MHz in 5  factory and other buildings factory and other buildings – Model power‐delay profile as a piecewise  function ⎧ TK TK < 110ns ⎪ 1− 367 ⎪⎪ T −110 PR (TK , S1 ) = ⎨0.65− K 110ns < TK < 200ns 360 ⎪ ⎪0.22- TK − 200 200ns < T < 500ns K ⎪⎩ 1360

T ⎧ 0.55 + K TK < 100ns ⎪ 667 PR (TK , S2 ) = ⎨ T −100 ⎪0.08+ 0.62exp( K ) 100ns < TK < 500ns 75 ⎩

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