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4.5 Types of Small Scale Fading

fading depends on • channel characteristics (rms delay spread, Doppler spread)• transmitted signal (symbol bandwidth, symbol period)

Two independent propagation mechanisms

(1) multipath delay spread leads to • time dispersion • frequency selective fading or flat fading (BS << BC)

(2) Doppler spread leads to • frequency dispersion • time selective fading

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Small Scale Fading

(2) Doppler spreadBD = Doppler spreadTC = coherence time

BS = signal bandwidth, TS = symbol period = BS-1

(1) multipath time-delay spread

Flat FadeBS < BC

TS >

= rms delay spreadBC = coherence bandwidth

Frequency Selective Fade

BS > BC

TS <

High Doppler Spread

Fast FadeTS > TC

BS < BD

Slow FadeTS < TC

BS > BD

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4.5.1 Fading Due to Multipath Time Delay Spread

time dispersion from multipath causes either • flat fading • frequency selective fading

4.5.1.1 Flat Fading (FF) channel occurs when BC >> BS & TS >>

• channel response has constant gain & linear phase

• channel’s multipath structure preserves signal’s spectral characteristics

• time varying received signal strength due to gain fluctuations MPCs

- amplitude changes in received signal, r(t) due to gain fluctuations

- spectrum of r(t) transmission is preserved

• typically cause deep fades compared to non-fading channel- in a deep fade, may require 20dB-30dB transmit power for low BER

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Channels Complex Envelope, hb(t,) is approximated as having no excess delay

• a single delta function with = 0

Distribution of instantaneous gain for FF channels is an important RF design issue

• Rayleigh Distribution is the most common amplitude distribution

• Rayleigh Flat Fading Model assumes channel induces time varying amplitude according to Rayleigh Distribution

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time response

s(t)

0 TS

h(t,)

0

frequency response

channel

r(t)

0 TS+,

S(f)

fc

H(f)

fc

R(f)

fc

Flat Fading Time & Frequency Response

assume << Ts

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4.5.1.2 Frequency Selective Fading (FSF)

if BS >> FF bandwidth s(t) experiences FSF

• channel impulse response has multipath delay spread > TS

• r(t) is distorted & includes multiple versions of transmitted waveform that are

- attenuated (faded)- delayed in time

FSF is due to time-dispersion of transmitted symbols within channel• inter-symbol interference (ISI) induced by channel• different frequency components of R(f) have different gains

FF Bandwidth = Channel Bandwidth over which signals experience • constant gain & linear phase• statistical variation in amplitude

s(t) r(t) Channel

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• difficult to model each multipath signal must be modeled

• channel must be considered to be a linear filter

• wideband multipath measurements are used to develop multipath models

FSF also known as wideband channels BS > bandwidth of channel’s impulse response

Small Scale FSF Models used for analysis of mobile communications

1. Statistical Impulse Response Models • 2-ray Rayleigh Fading model

- models impulse response as 2 delta functions ()

- ’s independently fade and have sufficient time delay between them to induce FSF

2. Computer Generated Impulse Responses

3. Measured Impulse Responses

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Frequency Selective Fading Time & Frequency Response

frequency response

S(f)

fc fc

H(f) R(f)

fc

time responses(t)

0 TS

h(t,)

0

r(t)

0 TS TS+ channel

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FSF channel - different gain for different spectral components of S(f)• S(f) = spectrum of transmitted signal s(t)

• BS = bandwidth of transmitted signal s(t)

• FSF caused by MPC delays which exceed TS = BS-1

• time varying channel gain & phase over spectrum of s(t)• results in distorted signal r(t)

Common Rules of thumb

• Channel is FF if TS 10

• Channel is FSF if TS < 10

• actual values depend on modulation technique• time-delay spread has significant impact on BER

BS > BC and TS < FSF fading occurs when

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channel impulse response becomes

L

lllth

1)(~),(

~

channel response is time invariant but frequency dependent

FSF Channel model for large scale effects in complex phasor notation

• a signal’s MPCs may have different path lengths

• complex gain assumed constant practically, true if receiver & transmitter are stationary

= complex gains l~l = relative delay of lth path

L

lll tstx

1)(~~)(~

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e.g. assume 2-ray FSF Channel with impulse response:)(~)(),(

~22 th

• 2 = relative delay of 2nd path

• simulate channel digitally with sample period = Ts = 2 (fs = sampling

frequency)

• channels frequency response will vary for different values of α22.42.2

2.01.81.61.41.21.00.8

0.60.4

0.20.0

-0.4 -0.2 0.0 0.2 0.4frequency (sample rates)

ampl

itud

e re

spon

se

2 = -j2 = 0.52 = j/2α2 channel effects

0.5 attenuation at edges of channelj0.5 attenuation is on left side -j null exists at 0.25fs

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4.5.2 Fading Effects Due to Doppler Spread

• fading is affected by s(t)’s rate of change vs channel’s rate of change

(i) fast fading channel: channel impulse response changes rapidly during symbol period TS

(ii) slow fading channel: channel impulse response changes slowly during TS

4.5.2.1 Fast Fading - results in distorted r(t)

Fast Fading occurs when

(i) symbol period is longer than coherence time: TS > TC and

(ii) symbol bandwidth is less than Doppler Spread: BS < BD

• time-selective fading: BD causes frequency dispersion

• signal distortion from fast fading increases with larger BD

Fast Fading assumes all reflected paths have equal lengths & delays

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4.5.2.2 Slow Fading

Fast fading or Slow fading determined by relationship of TS to

(i) velocity of mobile terminals and

(ii) velocity of objects in the channel

• deals with time rate of change of channel & signal (small scale fading)

• does not deal with path loss models (large scale fading)

Slow Fading occurs when

(i) symbol period is much less than coherence time: TS << TC and

(ii) symbol bandwidth is much greater than Doppler Spread: BS >> BD

• channel assumed static over one or more symbol periods, TS

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Power Spectrum of received signal = power spectra of transmitted signal convolved with power spectra of fading process

• multiplication in time domain = convolution in frequency domain

Time-selective Channel

Sr(f) = S(f) Ss(f)

• Sr(f) = power spectrum of received signal

• S(f) = power spectrum of fading process

• Ss(f) = power spectrum of transmitted signal

• channel impulse response is time varying

h(t,τ) = α(t)δ(t)

• α(t) is time varying received signal strength is also time varying

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Frequency H(z)

1.0

0.8

0.6

0.4

0.2

0.0

Fre

quen

cy R

espo

nse

-2000 -1000 0 1000 2000

doppler spread spectrumoriginal spectrum

Nominal transmitted spectrum power spectrum of fading process original spectrum vs doppler spread spectrum

Assume doppler spectrum 20% of signal bandwidth B3dB 5fm

• fm = maximum doppler shift

• B3dB = 3dB bandwidth of nominal spectrum

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N

nn tsttx

1)()(~Re)(~ )(~)(~ tst

Flat Fading Channel Response: time selective & frequency flat

time varying channel impulse response )()(~),(~

ttth

l~

L

lllth

1)(~),(

~

time invariant but frequency dependent channel response

Frequency Selective Channel: large scale effects

= complex gains

l = relative delay of lth path

L

lll tstx

1)(~~)(~

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TS = Transmitted Symbol Period

BS = Transmitted Baseband Signal BW

TS

Frequency SelectiveFast Fade

Frequency SelectiveSlow Fade

FlatSlow Fade

FlatFast Fade

TS

TC

BS

Frequency SelectiveFast Fade

Frequency SelectiveSlow Fade

FlatSlow Fade

FlatFast Fade

BS

BD

BC

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4.5.3 General Channels

(i) flat-flat channel: neither frequency or time varying

(ii) time selective and frequency selective channel• MPCs of signal have different lengths frequency selective fading• interaction of MPCs generated locally time-varying fading

received signal given as

L

lll ttsttx

1))((~)(~)(~

time varying channel impulse response is

L

lll tttth

1))(()(~),(

~

• channel model includes large and small scale effects• h(t,) is for finite number of signal paths

- represents continuum of MPC with arbitrarily small differences- both time selective & frequency selective

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received signal = channel response transmitted signal• with either discrete or continuum number of MPCs

dtsthtx )(),(

~)(~

• time varying frequency response (transfer function) given by

H(t,f) = F[ h(t,τ)]

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e.g. time varying impulse response

consider )()(~)()(~),(~

221 ttth

• where {αi(t)} are independent Rayleigh Processes

• at time t = t0 Fourier Transform reveals power spectrum

• different frequencies with nulls and strong responses vary with time

400 300 200 100 0 -100 -200 -300 -400 kHz

2520151050

0.350.300.250.200.150.100.050.00

spec

trum

time (m

s)

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4.6 Rayleigh & Ricean Distributions

4.6.1 Rayleigh Fading Distributions • in mobile RF channels commonly used to model describe statistical time varying nature of

(i) received envelope for FF signal

(ii) envelope of a single MPC

• envelope sum of 2 quadrature Gaussian noise signals obeys Rayleigh distribution

Rx Speed ≈ 120km/hrsi

gnal

leve

l (d

B a

bout

rm

s)

0 50 100 (ms)

1050

-5-10-15-20-25

Rayleigh distributed signal vs time

λ/2

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∆x ∆0.10

m5.03 rad (288o)

0.11 m

5.53 rad (317o)

0.12 m

6.03 rad (345o)

Multipath Reflections due to local objects arrive at stationary receiver

• nth path has electric field strength En & relative phase θn

• complex phasor of N signal reflections given by

Ẽ =

N

nnn jE

1exp

• Ẽ is a RV representing effects of multipath channel

e.g. fc = 2.4GHz ( 0.125m) ∆ = 50.26 ∆x

∆ = x2 ∆x = difference in path length

∆ = change in phase

small changes in path length large changes in phase

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Model for complex channel envelope is Σ iid complex RVs

• Central Limit Theorem: for large N Ẽ becomes Gaussian distribution

= Zr +jZi

N

nnnn

N

jE1

)sin()cos(lim

where Zr and Zi are real Gaussian random variables

thus E[Ẽ] = 0 and Ẽ is a 0-mean Gaussian RV

Consider mean of nth component of Ẽ , denoted as Enexp(jθn)

E[Enexp(jθn)] = E[En] E[exp(jθn)]

2

0

2

0

sincos nnnn djd= E[En]

= E[En] (0+0) = 0

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• θn- θm is the difference of 2 random phases = random phase

• by symmetry power is equally distributed between real & imaginary parts of Ẽ

• P0 = average receive power

Consider variance (power) in Ẽ given by mean square value

E[|Ẽ|2] =

N

mmm

N

nnn jEjE

11

expexp E

N

nmn

N

mmn jEE

1 1

exp = E

01

2 PEN

nn

= E

(equals 0 for all n ≠ m)

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Complex Envelop, Ẽ has 0-mean Zr & Zi have Gaussian PDF

2

2

2exp

2

1)(

r

rzz

zfr

2 = P0/2 for Zr and Zi

Define the amplitude of the complex envelope, Ẽ as

R = 22ri ZZ

PDF of amplitude is determined by changing variables of )( rz zfr

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= rms value of received voltage before envelope detection

2 = time average power of received signal before envelope detection

p(r) = (4.49)

0

2exp 2

2

2 σ

rr

0 ≤ r ≤

r < 0

Received Signal Envelope Voltage

0 2 3 4 5 r

p(r)

0.6065p(r) = Pr[r = ] =

Rayleigh PDF given by

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Rayleigh CDF

• probability that received signal’s envelope doesn’t exceed specific value, R

P(R) = Pr[r R] =

R Rdrrp

02

2

2exp1)(

(4.50)

rmean = E[r] =

0

2533.12

)( drrrp (4.51)

Mean Value of Rayleigh Distributed Signal

Variance of Rayleigh Distribution represents ac power in signal envelope

(4.52)

2r = E[r2] – E2[r] =

0

22

2)(

drrpr

22 4292.02

2

=

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Median Value of r is found by

2

1)(

0

medianr

drrp (4.53)

(4.54)rmedian = 1.177

Envelope’s rms Value is the square root of mean square or

= rms voltage of original signal prior to envelope detection

22 2 =

r = ][ 2rE

0

2 )( drrpr=

= 1.414

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-40 -30 -20 -10 0 10amplitude 20log10(R/Rrms)

Pr

(r <

R)

1

0.1

0.01

10-3

10-4

Rayleigh Distribution Graph

Conceptually, each location corresponds to different set of {n}

• deep fades >20dB (R < 0.1Rrms) occur only about 1% • wide variation in received signal strength due to local reflections

• if received signal were measured from a number of stationary locations power measurements would show Rayleigh distribution

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Rayleigh Fading Signal

• mean & median differ by only 0.07635 (0.55dB)

• in practice - median is often used

- fading data usually consists of field measurements & a particular

distribution cannot be assumed

- use of median allows easy comparison of different fading distributions with widely varying means

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4.6.2 Ricean Fading Distributions

if signal has dominant stationary component (LOS path) small scale fading envelope has Ricean Distribution

• random MPCs arrive at different angles & are superimposed on dominant signal• envelope detector output yields DC component + random MPCs

Ricean distribution is result of dominant signal arriving with weaker MPCs

• as dominant component fades Ricean distribution degenerates to Rayleigh distribution

• composite signal resembles noise signal with envelope that is Rayleigh

• same as in the case of sine wave detection in thermal noise

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A = peak amplitude of dominant signal

0() = modified Bessel function of the 1st kind & zero order

p(r) = (4.55)

I0

0

2exp 22

22

2 σ

ArArr

A ≥ 0 , r ≥ 0

r < 0

Ricean PDF

Ricean Distribution often described in terms of Ricean Factor K

• K = ratio of deterministic signal power to multipath variance

K = 2

2

2A

dBA

2

2

2log10

K (dB) =

(4.56)

• as dominate path attenuates A grows small • Ricean distribution degenerates to Rayleigh

received signal envelope voltage

K = - dB

K = 6 dB

p(r)

r

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-40 -30 -20 -10 0 10amplitude 20log10(R/Rrms)

Pr (r < R)1

0.1

0.01

10-3

10-4

K = 0 (Rayleigh)K=5dBK=10dBK=13dB

probability of deep fades reduced as K grows

Ricean Fading