Lecture Slides On Small Scale Fading.pdf
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FADING CHANNELS: SMALL SCALE FADING B. Sainath [email protected]
Department of Electrical and Electronics Engineering Birla Institute of Technology and Science, Pilani
March 18, 2019
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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OUTLINE 1
SMALL SCALE FADING
2
f -SELECTIVE VS FLAT FADING
3
SLOW VS FAST FADING
4
NARROWBAND FADING MODEL
5
RAYLEIGH & RICIAN FADING
6
NAKAGAMI FADING
7
OUTAGE PROBABILITY
8
REFERENCES
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Small Scale Fading Large scale fading
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Small Scale Fading Large scale fading Important for cell-site planning, link budget
Small scale fading ⇐ Focus lies here! Important for wireless communication system design
Wireless digital communication Employs high RF carrier frequency fc
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Small Scale Fading Large scale fading Important for cell-site planning, link budget
Small scale fading ⇐ Focus lies here! Important for wireless communication system design
Wireless digital communication Employs high RF carrier frequency fc e.g., fc = 900 MHz, fc = 1.9 GHz for mobile cellular communication
Multipath fading Due to constructive & destructive interference of transmitted EM waves
Figure: Source: Wireless communication by A. F. Molisch B. Sainath (BITS, PILANI)
SMALL SCALE FADING
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Multipath Fading Channel varies when mobile moves about carrier wavelength (λc ) e.g., 0.3 m for 900 MHz cellular
For vehicular speeds (e.g., 60 Kmph), Doppler shift of order of 100 Hz Driving force for diligent design of wireless communication systems!
In typical wireless applications, communication occurs in
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Multipath Fading Channel varies when mobile moves about carrier wavelength (λc ) e.g., 0.3 m for 900 MHz cellular
For vehicular speeds (e.g., 60 Kmph), Doppler shift of order of 100 Hz Driving force for diligent design of wireless communication systems!
In typical wireless applications, communication occurs in a passband [fc − W2 , fc + W2 ] Processing (coding/decoding, modulation/demodulation) takes place at baseband [− W2 , W2 ]
Figure: Passband and its baseband. Source: Tse & Viswanath Book B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Mulipath Resolution
Sampled baseband-equivalent channel model P ym = ` h` x[m − `] h` : `th complex channel filter tap at time m X h` ≈ aj ej2πfc τj j
" Sum is over all paths that fall in delay bin
# ` W
−
1 2W
,
` W
+
1 2W
Resolvable multipaths: up to delays of
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
5 / 33
Mulipath Resolution
Sampled baseband-equivalent channel model P ym = ` h` x[m − `] h` : `th complex channel filter tap at time m X h` ≈ aj ej2πfc τj j
" Sum is over all paths that fall in delay bin
Resolvable multipaths: up to delays of
# ` W
−
1 2W
,
` W
+
1 2W
1 W
Multipath fading: Frequency-flat & f −selective
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
5 / 33
Flat Fading Vs. f -Selective Fading (Multipath) Delay spread Td Important general parameter of wireless communication system Difference in propagation time between the longest & shortest paths with significant energy
Td = maxi,j τi − τj Coherence bandwidth Wc Minimum frequency separation for which the channel response is roughly independent Td of the channel dictates frequency coherence Wc ∝ T1 d
Td << Td >
B. Sainath (BITS, PILANI)
1 W
1 W
, Wc >> W ⇒ single tap, flat fading
, Wc < W ⇒ multiple taps, f -selective fading
SMALL SCALE FADING
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An Illustration of f -Selective & Flat Fading
Figure: (a) & (b): f -Selective fading & its spectral content; (c) & (d): flat fading & its spectral content. Source: Tse & Viswanath book B. Sainath (BITS, PILANI)
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Slow Fading Vs. Fast Fading Time variations of the channel How fast the filter taps hl [m] vary
Coherence time Tc ∝
1 Ds
Doppler spread Ds = maxi,j fc τi0 (t) − τj0 (t) Max. taken over all significant energy paths
Numerical example: v = 60 kmph, fc = 900 MHz Direct path has Doppler shift of
Figure: Source: Tse & Viswanath book
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Slow Fading Vs. Fast Fading Time variations of the channel How fast the filter taps hl [m] vary
Coherence time Tc ∝
1 Ds
Doppler spread Ds = maxi,j fc τi0 (t) − τj0 (t) Max. taken over all significant energy paths
Numerical example: v = 60 kmph, fc = 900 MHz Direct path has Doppler shift of fv = 50 Hz c Indirect path has shift of
Figure: Source: Tse & Viswanath book
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
8 / 33
Slow Fading Vs. Fast Fading Time variations of the channel How fast the filter taps hl [m] vary
Coherence time Tc ∝
1 Ds
Doppler spread Ds = maxi,j fc τi0 (t) − τj0 (t) Max. taken over all significant energy paths
Numerical example: v = 60 kmph, fc = 900 MHz Direct path has Doppler shift of fv = 50 Hz c Indirect path has shift of − fvc = −50 Hz Doppler spread Ds = 100 Hz
Figure: Source: Tse & Viswanath book
B. Sainath (BITS, PILANI)
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Fading Channel Parameters & Types
B. Sainath (BITS, PILANI)
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March 18, 2019
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Narrowband Fading Model (Td << Ts , Wc >> W ) Let s(t) denote transmitted signal, fc denote carrier frequency s(t) = Re{u(t)ej2πfc t }, where u(t) is equivalent low-pass signal with bandwidth W
s(t) propagates through wireless fading channel. Neglecting thermal noise, what is the expression for received signal r (t)? Sum of LOS path and resolvable1 multipath components (MPCs)
N(t): number of resolvable MPCs an (t) (or αn (t)): amplitude associated with pathloss and shadowing r (t) τn (t) , nc : path delay φDn (t): Doppler phase shift
an (t), τn (t) & φDn (t) = fDn = 1 Two
v cos θn , λ
R t
2πfDn (t 0 ) dt 0 ⇒ stationary & ergodic RPs
θn (t) denotes angle-of-arrival (AoA) relative to direction of motion
MPCs with delay τ1 and τ2 are resolvable if |τ1 − τ2 | >> W −1
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Canonical Form of Received Signal Consider the transmission of unmodulated carrier with phase shift φ0 s(t) = Re{ej2πfc t+φ0 } Received signal r (t) ⇐ stationary and ergodic random process r (t) = rI (t) cos (2πfc t) + rQ (t) sin (2πfc t) rI (t) =
PN(t) n=0
an (t) cos Φn (t), rQ (t) =
PN(t) n=0
an (t) sin Φn (t)
Φn (t) = 2πfc τn (t) − ΦDn − φ0
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Canonical Form of Received Signal Consider the transmission of unmodulated carrier with phase shift φ0 s(t) = Re{ej2πfc t+φ0 } Received signal r (t) ⇐ stationary and ergodic random process r (t) = rI (t) cos (2πfc t) + rQ (t) sin (2πfc t) rI (t) =
PN(t) n=0
an (t) cos Φn (t), rQ (t) =
PN(t) n=0
an (t) sin Φn (t)
Φn (t) = 2πfc τn (t) − ΦDn − φ0
From CLT, rI (t) and rQ (t) are jointly Gaussian if
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
11 / 33
Canonical Form of Received Signal Consider the transmission of unmodulated carrier with phase shift φ0 s(t) = Re{ej2πfc t+φ0 } Received signal r (t) ⇐ stationary and ergodic random process r (t) = rI (t) cos (2πfc t) + rQ (t) sin (2πfc t) rI (t) =
PN(t) n=0
an (t) cos Φn (t), rQ (t) =
PN(t) n=0
an (t) sin Φn (t)
Φn (t) = 2πfc τn (t) − ΦDn − φ0
From CLT, rI (t) and rQ (t) are jointly Gaussian if amplitudes an (t) are Rayleigh distributed & phases Φn (t) are uniformly distributed on [−π, π]
Quantities of interest:
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
11 / 33
Canonical Form of Received Signal Consider the transmission of unmodulated carrier with phase shift φ0 s(t) = Re{ej2πfc t+φ0 } Received signal r (t) ⇐ stationary and ergodic random process r (t) = rI (t) cos (2πfc t) + rQ (t) sin (2πfc t) rI (t) =
PN(t) n=0
an (t) cos Φn (t), rQ (t) =
PN(t) n=0
an (t) sin Φn (t)
Φn (t) = 2πfc τn (t) − ΦDn − φ0
From CLT, rI (t) and rQ (t) are jointly Gaussian if amplitudes an (t) are Rayleigh distributed & phases Φn (t) are uniformly distributed on [−π, π]
Quantities of interest: Autocorrelation, crosscorrelation, power spectral density (PSD), received power Note that performance measures, viz., outage probability, SEP, spectral efficiency, energy efficiency are functions of received SNR
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Narrowband Fading Model: Key Assumptions
Assumptions No dominant LOS component an (t), τn (t) & fDn (t) are constant over time intervals of interest an (t) ≈ an , τn (t) ≈ τn , and, fDn (t) ≈ fDn
Doppler phase shift at t = 0 is zero Phase offset will not affect analysis Phase of nth multipath component Φn (t) = 2πfc τn − 2πfDn t − φ0 is uniformly distributed Φn ∼ U [−π, π] ⇒ Mean value of Φn =?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Narrowband Fading Model: Key Assumptions
Assumptions No dominant LOS component an (t), τn (t) & fDn (t) are constant over time intervals of interest an (t) ≈ an , τn (t) ≈ τn , and, fDn (t) ≈ fDn
Doppler phase shift at t = 0 is zero Phase offset will not affect analysis Phase of nth multipath component Φn (t) = 2πfc τn − 2πfDn t − φ0 is uniformly distributed Φn ∼ U [−π, π] ⇒ Mean value of Φn =? Zero
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
12 / 33
Mean Values of rI (t), rQ (t)
Note that an and Φn are statistically independent MV of rI (t)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Mean Values of rI (t), rQ (t)
Note that an and Φn are statistically independent MV of rI (t) " # X E [rI (t)] = E an cos Φn = 0 n
MV of rQ (t)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
13 / 33
Mean Values of rI (t), rQ (t)
Note that an and Φn are statistically independent MV of rI (t) " # X E [rI (t)] = E an cos Φn = 0 n
MV of rQ (t) E [rQ (t)] = E
" X
# an sin Φn = 0
n
=⇒ r (t) = rI (t) cos(2πfc t) + rQ (t) sin(2πfc t) is
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
13 / 33
Mean Values of rI (t), rQ (t)
Note that an and Φn are statistically independent MV of rI (t) " # X E [rI (t)] = E an cos Φn = 0 n
MV of rQ (t) E [rQ (t)] = E
" X
# an sin Φn = 0
n
=⇒ r (t) = rI (t) cos(2πfc t) + rQ (t) sin(2πfc t) is zero-mean Gaussian random process
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
13 / 33
Crosscorrelation of rI (t), rQ (t)
E [rI (t)rQ (t)] = 0 (Prove this) rI (t) and rQ (t) are uncorrelated and also independent (?)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Crosscorrelation of rI (t), rQ (t)
E [rI (t)rQ (t)] = 0 (Prove this) rI (t) and rQ (t) are uncorrelated and also independent (?) (Ans. rI (t) and rQ (t) are Jointly Gaussian)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
14 / 33
Autocorrelation of rI (t)
Autocorrelation of rI (t) ArI (t, t + τ ) = E [rI (t)rI (t + τ )] Φn (t) = 2πfc τn − 2πfDn t − φ0 & Φn (t + τ ) = 2πfc τn − 2πfDn (t + τ ) − φ0 Relative to other phase terms, 2πfc τn changes rapidly Φn (t) uniformly distributed
Show that ArI (t, t + τ ) =
1 X 2 E an E [cos(2πfDn τ )] 2 n
θn fDn = v cos λ ArI (t, t + τ ) = ArI (τ ) ⇒ rI (t) is WSS random process
Similarly, show that rQ (t) is also WSS random process & ArI (τ ) = ArQ (τ )
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
15 / 33
Crosscorrelation of rI (t) & rQ (t)
Crosscorrelation of rI (t) and rQ (t) ArI ,rQ (t, t + τ ) = ArI ,rQ (τ ) = E [rI (t)rQ (t + τ )] For uniform scattering environment, ArI ,rQ (τ ) = 0
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Autocorrelation of Received Signal r (τ ) Ar (τ ) = E [r (t)r (t + τ )] = ArI (τ ) cos(2πfc τ ) + ArI ,rQ (τ ) sin(2πfc τ ) More assumptions on propagation channel Uniform scattering environment [Clarke, Jakes] Propagation channel
Figure: Uniform, isotropic scattering model. Dense scattering environment N multipath components (MPCs) with AoA θn = n∆θ, where ∆θ = Each MPC has same received power Pr
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
2π N
March 18, 2019
17 / 33
Autocorrelation & Received Power Autocorrelation ArI (τ ) = Pr =
1 2
Since
P
nE 2 E an
1 X 2 2πvτ E an E cos cos θn 2 n λ
2 an (How?) =
2Pr N
, we have
2πvτ Pr X ArI (τ ) = E cos cos n∆θ N n λ Pr X 2πvτ = cos n∆θ ∆θ E cos 2π n λ For very large N, ∆θ → 0. Express ArI (τ ) in integral form
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Autocorrelation Function (ACF) ArI (τ ) Autocorrelation in integral form Pr ArI (τ ) = 2π
Z
2π
cos
0
2πvτ cos θ λ
dθ,
= Pr J0 (2πfD τ ) where J0 (t) , the first kind
1 π
Rπ 0
exp (−jt cos θ) dθ ⇐ Bessel function of zeroth order of
Verify: 1 J0 (t) = 2π
Z
2π
cos (t cos θ) dθ 0
J0 (0) = 1
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Bessel Function & Observations
J0 (2πfD τ)
1
0.5
X: 0.38 Y: 0.008969
0
-0.5 0
0.5
1
1.5
2
2.5
3
fD τ
Observations:
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Bessel Function & Observations
J0 (2πfD τ)
1
0.5
X: 0.38 Y: 0.008969
0
-0.5 0
0.5
1
1.5
2
2.5
3
fD τ
Observations: ArI (τ ) = 0 for fD τ ≈ 0.4 For fD =
v , λ
vτ ≈ 0.4λ
Signal decorrelates over distance of approximately assumption
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
λ 2
under uniform Θn
March 18, 2019
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Bessel Function & Observations
J0 (2πfD τ)
1
0.5
X: 0.38 Y: 0.008969
0
-0.5 0
0.5
1
1.5
2
2.5
3
fD τ
Observations: ArI (τ ) = 0 for fD τ ≈ 0.4 For fD =
v , λ
vτ ≈ 0.4λ
Signal decorrelates over distance of approximately assumption For f = 900 MHz, λ = 0.3 m ⇒ vτ ≈ 13.33 cm
λ 2
under uniform Θn
MIMO: Antenna spacing ≈ 0.4λ for each antenna to receive independent fading path
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
20 / 33
Power Spectral Density (PSD)
Recall that ACF ArI (τ ) and PSD SrI (f ) form Fourier transform pair To determine PSD, use Laplace transform pair J0 (bt) ⇔ √ 1 b2 +s2
PSD: (Verify) SrI (f ) =
Pr r 2πfD
0,
1 2 , 1− f f
|f | ≤ fD ,
D
elsewhere.
Note that SrI (f ) = SrQ (f ) (Why?)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
21 / 33
Power Spectral Density (PSD)
Recall that ACF ArI (τ ) and PSD SrI (f ) form Fourier transform pair To determine PSD, use Laplace transform pair J0 (bt) ⇔ √ 1 b2 +s2
PSD: (Verify) SrI (f ) =
Pr r 2πfD
0,
1 2 , 1− f f
|f | ≤ fD ,
D
elsewhere.
Note that SrI (f ) = SrQ (f ) (Why?) (Ans. Since ACFs of rI (t) & rQ (t) are equal)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
21 / 33
PSD of r (t)
Exercise: Prove that Sr (f ) =
Pr r 4πfD
0,
1 1−
|f ±fc | fD
2 ,
|f ± fc | ≤ fD , elsewhere.
Observations: While SrI (f ) goes to ∞ at f = ±fD , Sr (f ) goes to ∞ at f = ±fc ± fD Not true in practice since uniform scattering model is an approximation
For dense scattering environments, PSD will be maximized at frequencies close to maximum Doppler frequency In general, fD (Θ) =
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
22 / 33
PSD of r (t)
Exercise: Prove that Sr (f ) =
Pr r 4πfD
0,
1 1−
|f ±fc | fD
2 ,
|f ± fc | ≤ fD , elsewhere.
Observations: While SrI (f ) goes to ∞ at f = ±fD , Sr (f ) goes to ∞ at f = ±fc ± fD Not true in practice since uniform scattering model is an approximation
For dense scattering environments, PSD will be maximized at frequencies close to maximum Doppler frequency Θ In general, fD (Θ) = v cos λ
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
22 / 33
Statistical Channel Models: Continuous & Discrete
Probabilistic model provide better insights into wireless systems Continuous-time & discrete-time multipath fading channel X y (t) = aj (t)x(t − τj (t)) + w(t) j
y[m] =
X
h` [m]x[m − `] + w[m]
`
For frequency-flat fading-single channel tap: y[m] = h[m]x[m] + w[m]
Q. How to probabilistically model channel filter taps?
Statistical Channel Models: Continuous & Discrete
Probabilistic model provide better insights into wireless systems Continuous-time & discrete-time multipath fading channel X y (t) = aj (t)x(t − τj (t)) + w(t) j
y[m] =
X
h` [m]x[m − `] + w[m]
`
For frequency-flat fading-single channel tap: y[m] = h[m]x[m] + w[m]
Q. How to probabilistically model channel filter taps?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
23 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
Q. What is the distribution of W = R 2 ?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
`
March 18, 2019
24 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
`
2 Q. What is the distribution of W = R ? (Ans. Power is exponentially 1 distributed with pdf pW (w) = σ2 exp − σw2 , w ≥ 0) `
`
Rician fading: many scattered paths plus one LOS path q q K 1 h` [m] = K+1 σ` ejθ + K+1 CN (0, σ`2 )
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
24 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
`
2 Q. What is the distribution of W = R ? (Ans. Power is exponentially 1 distributed with pdf pW (w) = σ2 exp − σw2 , w ≥ 0) `
`
Rician fading: many scattered paths plus one LOS path q q K 1 h` [m] = K+1 σ` ejθ + K+1 CN (0, σ`2 ) First term: specular path arriving with uniform phase θ
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
24 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
`
2 Q. What is the distribution of W = R ? (Ans. Power is exponentially 1 distributed with pdf pW (w) = σ2 exp − σw2 , w ≥ 0) `
`
Rician fading: many scattered paths plus one LOS path q q K 1 h` [m] = K+1 σ` ejθ + K+1 CN (0, σ`2 ) First term: specular path arriving with uniform phase θ Second term: large number of reflected & scattered paths
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
24 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
`
2 Q. What is the distribution of W = R ? (Ans. Power is exponentially 1 distributed with pdf pW (w) = σ2 exp − σw2 , w ≥ 0) `
`
Rician fading: many scattered paths plus one LOS path q q K 1 h` [m] = K+1 σ` ejθ + K+1 CN (0, σ`2 ) First term: specular path arriving with uniform phase θ Second term: large number of reflected & scattered paths K : {specular path energy}/{scattered paths energy} ⇐ Fading factor (indicator of severity of fading)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
24 / 33
Rician Distribution & Marcum Q Function Q: Prove that Z = |h` [m]| is Rician distributed with pdf 2 z z + s2 zs pZ (z) = 2 exp − I0 ,z ≥ 0 σ 2σ 2 σ2 I0 (.) denotes modified Bessel function of zeroth order Z 2π 1 I0 (z) = exp (z cos ψ) dψ 2π 0 Total average received power Pr = s2 + 2σ 2 Z ∞ Pr = z 2 pZ (z) dz 0
Average power in LOS component: s2 Average power in NLOS MPCs: 2σ 2
Fading factor K =
s2 2σ 2
⇐ Measure of severity of fading
K=0⇒
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
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Rician Distribution & Marcum Q Function Q: Prove that Z = |h` [m]| is Rician distributed with pdf 2 z z + s2 zs pZ (z) = 2 exp − I0 ,z ≥ 0 σ 2σ 2 σ2 I0 (.) denotes modified Bessel function of zeroth order Z 2π 1 I0 (z) = exp (z cos ψ) dψ 2π 0 Total average received power Pr = s2 + 2σ 2 Z ∞ Pr = z 2 pZ (z) dz 0
Average power in LOS component: s2 Average power in NLOS MPCs: 2σ 2
Fading factor K =
s2 2σ 2
⇐ Measure of severity of fading
K = 0 ⇒ Rayleigh fading K=∞⇒
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
25 / 33
Rician Distribution & Marcum Q Function Q: Prove that Z = |h` [m]| is Rician distributed with pdf 2 z z + s2 zs pZ (z) = 2 exp − I0 ,z ≥ 0 σ 2σ 2 σ2 I0 (.) denotes modified Bessel function of zeroth order Z 2π 1 I0 (z) = exp (z cos ψ) dψ 2π 0 Total average received power Pr = s2 + 2σ 2 Z ∞ Pr = z 2 pZ (z) dz 0
Average power in LOS component: s2 Average power in NLOS MPCs: 2σ 2
Fading factor K =
s2 2σ 2
⇐ Measure of severity of fading
K = 0 ⇒ Rayleigh fading K = ∞ ⇒ Rician fading Larger value of K ⇒ stronger LOS component! B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
25 / 33
Marcum Q Function
Characterize error probability performance of digital signals communicated over Rician fading channels 2 Z ∞ M−1 x x + a2 IM−1 (ax) dx QM (a, b) = x exp − a 2 b 2 Z ∞ √ x + s2 Q1 (s, y) = √ x exp − I0 (sx) dx 2 y Computation in MATLAB: Q = marcumq(a,b,m) computes the generalized Marcum Q
Marcum Q Function
Characterize error probability performance of digital signals communicated over Rician fading channels 2 Z ∞ M−1 x x + a2 IM−1 (ax) dx QM (a, b) = x exp − a 2 b 2 Z ∞ √ x + s2 Q1 (s, y) = √ x exp − I0 (sx) dx 2 y Computation in MATLAB: Q = marcumq(a,b,m) computes the generalized Marcum Q
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
26 / 33
Nakagami Distribution More general distribution Need for Nakagami distribution Some experimental data- not fit well into either Rayleigh or Rician distribution
Nakagami−m pdf (m fading parameter) 2mm y 2m−1 my 2 pY (y ) = exp − , m ≥ 0.5 Pr Γ(m)Pr m Z ∞ Γ(m) = t m−1 e−t dt 0
Rayleigh fading when
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
27 / 33
Nakagami Distribution More general distribution Need for Nakagami distribution Some experimental data- not fit well into either Rayleigh or Rician distribution
Nakagami−m pdf (m fading parameter) 2mm y 2m−1 my 2 pY (y ) = exp − , m ≥ 0.5 Pr Γ(m)Pr m Z ∞ Γ(m) = t m−1 e−t dt 0
Rayleigh fading when m = 1 2 For m = (K+1) ⇒ approximately Rician fading 2K+1 m = ∞ ⇒ No fading
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
27 / 33
Power Distribution
Let Z = Y 2 , Y denotes Nakagami RV pdf of X pZ (z) =
m Pr
m
x m−1 mz exp − ,z ≥ 0 Γ(m) Pr
Verify that for m = 1, power is exponentially distributed Verify that pZ (z) is a valid pdf, i.e., Z 0
B. Sainath (BITS, PILANI)
∞
m Pr
m
x m−1 mz exp − dz = 1 Γ(m) Pr
SMALL SCALE FADING
March 18, 2019
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Outage Probability Important physical layer performance measures Outage probability, symbol error probability (SEP), spectral efficiency
Outage probability pout : Probability that instantaneous SNR (iSNR) Γ falls below SNR threshold γth
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
29 / 33
Outage Probability Important physical layer performance measures Outage probability, symbol error probability (SEP), spectral efficiency
Outage probability pout : Probability that instantaneous SNR (iSNR) Γ falls below SNR threshold γth Z γth pout = pΓ (γ) dγ 0
pΓ (γ) denotes pdf of iSNR
Q: What is pout for Rayleigh fading?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
29 / 33
Outage Probability Important physical layer performance measures Outage probability, symbol error probability (SEP), spectral efficiency
Outage probability pout : Probability that instantaneous SNR (iSNR) Γ falls below SNR threshold γth Z γth pout = pΓ (γ) dγ 0
pΓ (γ) denotes pdf of iSNR
Q: What is pout for Rayleigh fading?
pout
γth = 1 − exp − γ
γ denotes average SNR
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
29 / 33
Numerical Example
Q: Consider Rayleigh fading channel. Average received power Pr = 20 dBm. What is the probability that the received power is below 10 dBm?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
30 / 33
Numerical Example
Q: Consider Rayleigh fading channel. Average received power Pr = 20 dBm. What is the probability that the received power is below 10 dBm? Pr = 0.1 W pth == 0.01 W pout = 1 − exp (−0.1) = 0.0952
Q: Repeat the problem when pth = 0 dBm
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
30 / 33
Numerical Example
Q: Consider Rayleigh fading channel. Average received power Pr = 20 dBm. What is the probability that the received power is below 10 dBm? Pr = 0.1 W pth == 0.01 W pout = 1 − exp (−0.1) = 0.0952
Q: Repeat the problem when pth = 0 dBm (Ans:1%) Exercise: Express pth in terms of pout . If pout is too small, show that pth ≈ Pr pout
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
30 / 33
Amount of Fading (AoF) Let Γ denote path SNR per symbol AoF is defined as
2 E Γ2 − E [Γ] AoF = 2 E [Γ]
Exercise: Shadow fading can be modeled by a log-normal distribution for different indoor and outdoor environments. Instantaneous fading SNR per symbol has pdf given by ! 2 (10 log10 γ − µ) pΓ (γ) = √ exp − , γ > 0, 2πσγ 2σ 2 where = ln1010 , and µ (dB) and σ (dB) are the mean and the standard deviation of 10 log10 Γ, respectively Derive an expression for the AoF in terms of σ 2 and
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
31 / 33
Reading Exercise
Figure: Comparison of narrowband & wideband fading channels. Bu denotes signal bandwidth. Source: http://slideplayer.com/slide/3921678/#
Wideband fading channel model Andrea Goldsmith book, Section 3.3.
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
32 / 33
References
”Fundamentals of wireless communication” by Tse & Viswanath “Wireless communications” by Andrea Goldsmith www.eecs.berkeley.edu/˜dtse/taiwan_course.pdf http://slideplayer.com/slide/3921678/#
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
33 / 33
Department of Electrical and Electronics Engineering Birla Institute of Technology and Science, Pilani
March 18, 2019
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
1 / 33
OUTLINE 1
SMALL SCALE FADING
2
f -SELECTIVE VS FLAT FADING
3
SLOW VS FAST FADING
4
NARROWBAND FADING MODEL
5
RAYLEIGH & RICIAN FADING
6
NAKAGAMI FADING
7
OUTAGE PROBABILITY
8
REFERENCES
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
2 / 33
Small Scale Fading Large scale fading
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
3 / 33
Small Scale Fading Large scale fading Important for cell-site planning, link budget
Small scale fading ⇐ Focus lies here! Important for wireless communication system design
Wireless digital communication Employs high RF carrier frequency fc
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
3 / 33
Small Scale Fading Large scale fading Important for cell-site planning, link budget
Small scale fading ⇐ Focus lies here! Important for wireless communication system design
Wireless digital communication Employs high RF carrier frequency fc e.g., fc = 900 MHz, fc = 1.9 GHz for mobile cellular communication
Multipath fading Due to constructive & destructive interference of transmitted EM waves
Figure: Source: Wireless communication by A. F. Molisch B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
3 / 33
Multipath Fading Channel varies when mobile moves about carrier wavelength (λc ) e.g., 0.3 m for 900 MHz cellular
For vehicular speeds (e.g., 60 Kmph), Doppler shift of order of 100 Hz Driving force for diligent design of wireless communication systems!
In typical wireless applications, communication occurs in
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
4 / 33
Multipath Fading Channel varies when mobile moves about carrier wavelength (λc ) e.g., 0.3 m for 900 MHz cellular
For vehicular speeds (e.g., 60 Kmph), Doppler shift of order of 100 Hz Driving force for diligent design of wireless communication systems!
In typical wireless applications, communication occurs in a passband [fc − W2 , fc + W2 ] Processing (coding/decoding, modulation/demodulation) takes place at baseband [− W2 , W2 ]
Figure: Passband and its baseband. Source: Tse & Viswanath Book B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
4 / 33
Mulipath Resolution
Sampled baseband-equivalent channel model P ym = ` h` x[m − `] h` : `th complex channel filter tap at time m X h` ≈ aj ej2πfc τj j
" Sum is over all paths that fall in delay bin
# ` W
−
1 2W
,
` W
+
1 2W
Resolvable multipaths: up to delays of
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
5 / 33
Mulipath Resolution
Sampled baseband-equivalent channel model P ym = ` h` x[m − `] h` : `th complex channel filter tap at time m X h` ≈ aj ej2πfc τj j
" Sum is over all paths that fall in delay bin
Resolvable multipaths: up to delays of
# ` W
−
1 2W
,
` W
+
1 2W
1 W
Multipath fading: Frequency-flat & f −selective
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
5 / 33
Flat Fading Vs. f -Selective Fading (Multipath) Delay spread Td Important general parameter of wireless communication system Difference in propagation time between the longest & shortest paths with significant energy
Td = maxi,j τi − τj Coherence bandwidth Wc Minimum frequency separation for which the channel response is roughly independent Td of the channel dictates frequency coherence Wc ∝ T1 d
Td << Td >
B. Sainath (BITS, PILANI)
1 W
1 W
, Wc >> W ⇒ single tap, flat fading
, Wc < W ⇒ multiple taps, f -selective fading
SMALL SCALE FADING
March 18, 2019
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An Illustration of f -Selective & Flat Fading
Figure: (a) & (b): f -Selective fading & its spectral content; (c) & (d): flat fading & its spectral content. Source: Tse & Viswanath book B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
7 / 33
Slow Fading Vs. Fast Fading Time variations of the channel How fast the filter taps hl [m] vary
Coherence time Tc ∝
1 Ds
Doppler spread Ds = maxi,j fc τi0 (t) − τj0 (t) Max. taken over all significant energy paths
Numerical example: v = 60 kmph, fc = 900 MHz Direct path has Doppler shift of
Figure: Source: Tse & Viswanath book
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
8 / 33
Slow Fading Vs. Fast Fading Time variations of the channel How fast the filter taps hl [m] vary
Coherence time Tc ∝
1 Ds
Doppler spread Ds = maxi,j fc τi0 (t) − τj0 (t) Max. taken over all significant energy paths
Numerical example: v = 60 kmph, fc = 900 MHz Direct path has Doppler shift of fv = 50 Hz c Indirect path has shift of
Figure: Source: Tse & Viswanath book
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
8 / 33
Slow Fading Vs. Fast Fading Time variations of the channel How fast the filter taps hl [m] vary
Coherence time Tc ∝
1 Ds
Doppler spread Ds = maxi,j fc τi0 (t) − τj0 (t) Max. taken over all significant energy paths
Numerical example: v = 60 kmph, fc = 900 MHz Direct path has Doppler shift of fv = 50 Hz c Indirect path has shift of − fvc = −50 Hz Doppler spread Ds = 100 Hz
Figure: Source: Tse & Viswanath book
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
8 / 33
Fading Channel Parameters & Types
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
9 / 33
Narrowband Fading Model (Td << Ts , Wc >> W ) Let s(t) denote transmitted signal, fc denote carrier frequency s(t) = Re{u(t)ej2πfc t }, where u(t) is equivalent low-pass signal with bandwidth W
s(t) propagates through wireless fading channel. Neglecting thermal noise, what is the expression for received signal r (t)? Sum of LOS path and resolvable1 multipath components (MPCs)
N(t): number of resolvable MPCs an (t) (or αn (t)): amplitude associated with pathloss and shadowing r (t) τn (t) , nc : path delay φDn (t): Doppler phase shift
an (t), τn (t) & φDn (t) = fDn = 1 Two
v cos θn , λ
R t
2πfDn (t 0 ) dt 0 ⇒ stationary & ergodic RPs
θn (t) denotes angle-of-arrival (AoA) relative to direction of motion
MPCs with delay τ1 and τ2 are resolvable if |τ1 − τ2 | >> W −1
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
10 / 33
Canonical Form of Received Signal Consider the transmission of unmodulated carrier with phase shift φ0 s(t) = Re{ej2πfc t+φ0 } Received signal r (t) ⇐ stationary and ergodic random process r (t) = rI (t) cos (2πfc t) + rQ (t) sin (2πfc t) rI (t) =
PN(t) n=0
an (t) cos Φn (t), rQ (t) =
PN(t) n=0
an (t) sin Φn (t)
Φn (t) = 2πfc τn (t) − ΦDn − φ0
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
11 / 33
Canonical Form of Received Signal Consider the transmission of unmodulated carrier with phase shift φ0 s(t) = Re{ej2πfc t+φ0 } Received signal r (t) ⇐ stationary and ergodic random process r (t) = rI (t) cos (2πfc t) + rQ (t) sin (2πfc t) rI (t) =
PN(t) n=0
an (t) cos Φn (t), rQ (t) =
PN(t) n=0
an (t) sin Φn (t)
Φn (t) = 2πfc τn (t) − ΦDn − φ0
From CLT, rI (t) and rQ (t) are jointly Gaussian if
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
11 / 33
Canonical Form of Received Signal Consider the transmission of unmodulated carrier with phase shift φ0 s(t) = Re{ej2πfc t+φ0 } Received signal r (t) ⇐ stationary and ergodic random process r (t) = rI (t) cos (2πfc t) + rQ (t) sin (2πfc t) rI (t) =
PN(t) n=0
an (t) cos Φn (t), rQ (t) =
PN(t) n=0
an (t) sin Φn (t)
Φn (t) = 2πfc τn (t) − ΦDn − φ0
From CLT, rI (t) and rQ (t) are jointly Gaussian if amplitudes an (t) are Rayleigh distributed & phases Φn (t) are uniformly distributed on [−π, π]
Quantities of interest:
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
11 / 33
Canonical Form of Received Signal Consider the transmission of unmodulated carrier with phase shift φ0 s(t) = Re{ej2πfc t+φ0 } Received signal r (t) ⇐ stationary and ergodic random process r (t) = rI (t) cos (2πfc t) + rQ (t) sin (2πfc t) rI (t) =
PN(t) n=0
an (t) cos Φn (t), rQ (t) =
PN(t) n=0
an (t) sin Φn (t)
Φn (t) = 2πfc τn (t) − ΦDn − φ0
From CLT, rI (t) and rQ (t) are jointly Gaussian if amplitudes an (t) are Rayleigh distributed & phases Φn (t) are uniformly distributed on [−π, π]
Quantities of interest: Autocorrelation, crosscorrelation, power spectral density (PSD), received power Note that performance measures, viz., outage probability, SEP, spectral efficiency, energy efficiency are functions of received SNR
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
11 / 33
Narrowband Fading Model: Key Assumptions
Assumptions No dominant LOS component an (t), τn (t) & fDn (t) are constant over time intervals of interest an (t) ≈ an , τn (t) ≈ τn , and, fDn (t) ≈ fDn
Doppler phase shift at t = 0 is zero Phase offset will not affect analysis Phase of nth multipath component Φn (t) = 2πfc τn − 2πfDn t − φ0 is uniformly distributed Φn ∼ U [−π, π] ⇒ Mean value of Φn =?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
12 / 33
Narrowband Fading Model: Key Assumptions
Assumptions No dominant LOS component an (t), τn (t) & fDn (t) are constant over time intervals of interest an (t) ≈ an , τn (t) ≈ τn , and, fDn (t) ≈ fDn
Doppler phase shift at t = 0 is zero Phase offset will not affect analysis Phase of nth multipath component Φn (t) = 2πfc τn − 2πfDn t − φ0 is uniformly distributed Φn ∼ U [−π, π] ⇒ Mean value of Φn =? Zero
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
12 / 33
Mean Values of rI (t), rQ (t)
Note that an and Φn are statistically independent MV of rI (t)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
13 / 33
Mean Values of rI (t), rQ (t)
Note that an and Φn are statistically independent MV of rI (t) " # X E [rI (t)] = E an cos Φn = 0 n
MV of rQ (t)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
13 / 33
Mean Values of rI (t), rQ (t)
Note that an and Φn are statistically independent MV of rI (t) " # X E [rI (t)] = E an cos Φn = 0 n
MV of rQ (t) E [rQ (t)] = E
" X
# an sin Φn = 0
n
=⇒ r (t) = rI (t) cos(2πfc t) + rQ (t) sin(2πfc t) is
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
13 / 33
Mean Values of rI (t), rQ (t)
Note that an and Φn are statistically independent MV of rI (t) " # X E [rI (t)] = E an cos Φn = 0 n
MV of rQ (t) E [rQ (t)] = E
" X
# an sin Φn = 0
n
=⇒ r (t) = rI (t) cos(2πfc t) + rQ (t) sin(2πfc t) is zero-mean Gaussian random process
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
13 / 33
Crosscorrelation of rI (t), rQ (t)
E [rI (t)rQ (t)] = 0 (Prove this) rI (t) and rQ (t) are uncorrelated and also independent (?)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
14 / 33
Crosscorrelation of rI (t), rQ (t)
E [rI (t)rQ (t)] = 0 (Prove this) rI (t) and rQ (t) are uncorrelated and also independent (?) (Ans. rI (t) and rQ (t) are Jointly Gaussian)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
14 / 33
Autocorrelation of rI (t)
Autocorrelation of rI (t) ArI (t, t + τ ) = E [rI (t)rI (t + τ )] Φn (t) = 2πfc τn − 2πfDn t − φ0 & Φn (t + τ ) = 2πfc τn − 2πfDn (t + τ ) − φ0 Relative to other phase terms, 2πfc τn changes rapidly Φn (t) uniformly distributed
Show that ArI (t, t + τ ) =
1 X 2 E an E [cos(2πfDn τ )] 2 n
θn fDn = v cos λ ArI (t, t + τ ) = ArI (τ ) ⇒ rI (t) is WSS random process
Similarly, show that rQ (t) is also WSS random process & ArI (τ ) = ArQ (τ )
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
15 / 33
Crosscorrelation of rI (t) & rQ (t)
Crosscorrelation of rI (t) and rQ (t) ArI ,rQ (t, t + τ ) = ArI ,rQ (τ ) = E [rI (t)rQ (t + τ )] For uniform scattering environment, ArI ,rQ (τ ) = 0
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
16 / 33
Autocorrelation of Received Signal r (τ ) Ar (τ ) = E [r (t)r (t + τ )] = ArI (τ ) cos(2πfc τ ) + ArI ,rQ (τ ) sin(2πfc τ ) More assumptions on propagation channel Uniform scattering environment [Clarke, Jakes] Propagation channel
Figure: Uniform, isotropic scattering model. Dense scattering environment N multipath components (MPCs) with AoA θn = n∆θ, where ∆θ = Each MPC has same received power Pr
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
2π N
March 18, 2019
17 / 33
Autocorrelation & Received Power Autocorrelation ArI (τ ) = Pr =
1 2
Since
P
nE 2 E an
1 X 2 2πvτ E an E cos cos θn 2 n λ
2 an (How?) =
2Pr N
, we have
2πvτ Pr X ArI (τ ) = E cos cos n∆θ N n λ Pr X 2πvτ = cos n∆θ ∆θ E cos 2π n λ For very large N, ∆θ → 0. Express ArI (τ ) in integral form
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
18 / 33
Autocorrelation Function (ACF) ArI (τ ) Autocorrelation in integral form Pr ArI (τ ) = 2π
Z
2π
cos
0
2πvτ cos θ λ
dθ,
= Pr J0 (2πfD τ ) where J0 (t) , the first kind
1 π
Rπ 0
exp (−jt cos θ) dθ ⇐ Bessel function of zeroth order of
Verify: 1 J0 (t) = 2π
Z
2π
cos (t cos θ) dθ 0
J0 (0) = 1
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
19 / 33
Bessel Function & Observations
J0 (2πfD τ)
1
0.5
X: 0.38 Y: 0.008969
0
-0.5 0
0.5
1
1.5
2
2.5
3
fD τ
Observations:
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
20 / 33
Bessel Function & Observations
J0 (2πfD τ)
1
0.5
X: 0.38 Y: 0.008969
0
-0.5 0
0.5
1
1.5
2
2.5
3
fD τ
Observations: ArI (τ ) = 0 for fD τ ≈ 0.4 For fD =
v , λ
vτ ≈ 0.4λ
Signal decorrelates over distance of approximately assumption
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
λ 2
under uniform Θn
March 18, 2019
20 / 33
Bessel Function & Observations
J0 (2πfD τ)
1
0.5
X: 0.38 Y: 0.008969
0
-0.5 0
0.5
1
1.5
2
2.5
3
fD τ
Observations: ArI (τ ) = 0 for fD τ ≈ 0.4 For fD =
v , λ
vτ ≈ 0.4λ
Signal decorrelates over distance of approximately assumption For f = 900 MHz, λ = 0.3 m ⇒ vτ ≈ 13.33 cm
λ 2
under uniform Θn
MIMO: Antenna spacing ≈ 0.4λ for each antenna to receive independent fading path
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
20 / 33
Power Spectral Density (PSD)
Recall that ACF ArI (τ ) and PSD SrI (f ) form Fourier transform pair To determine PSD, use Laplace transform pair J0 (bt) ⇔ √ 1 b2 +s2
PSD: (Verify) SrI (f ) =
Pr r 2πfD
0,
1 2 , 1− f f
|f | ≤ fD ,
D
elsewhere.
Note that SrI (f ) = SrQ (f ) (Why?)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
21 / 33
Power Spectral Density (PSD)
Recall that ACF ArI (τ ) and PSD SrI (f ) form Fourier transform pair To determine PSD, use Laplace transform pair J0 (bt) ⇔ √ 1 b2 +s2
PSD: (Verify) SrI (f ) =
Pr r 2πfD
0,
1 2 , 1− f f
|f | ≤ fD ,
D
elsewhere.
Note that SrI (f ) = SrQ (f ) (Why?) (Ans. Since ACFs of rI (t) & rQ (t) are equal)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
21 / 33
PSD of r (t)
Exercise: Prove that Sr (f ) =
Pr r 4πfD
0,
1 1−
|f ±fc | fD
2 ,
|f ± fc | ≤ fD , elsewhere.
Observations: While SrI (f ) goes to ∞ at f = ±fD , Sr (f ) goes to ∞ at f = ±fc ± fD Not true in practice since uniform scattering model is an approximation
For dense scattering environments, PSD will be maximized at frequencies close to maximum Doppler frequency In general, fD (Θ) =
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
22 / 33
PSD of r (t)
Exercise: Prove that Sr (f ) =
Pr r 4πfD
0,
1 1−
|f ±fc | fD
2 ,
|f ± fc | ≤ fD , elsewhere.
Observations: While SrI (f ) goes to ∞ at f = ±fD , Sr (f ) goes to ∞ at f = ±fc ± fD Not true in practice since uniform scattering model is an approximation
For dense scattering environments, PSD will be maximized at frequencies close to maximum Doppler frequency Θ In general, fD (Θ) = v cos λ
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
22 / 33
Statistical Channel Models: Continuous & Discrete
Probabilistic model provide better insights into wireless systems Continuous-time & discrete-time multipath fading channel X y (t) = aj (t)x(t − τj (t)) + w(t) j
y[m] =
X
h` [m]x[m − `] + w[m]
`
For frequency-flat fading-single channel tap: y[m] = h[m]x[m] + w[m]
Q. How to probabilistically model channel filter taps?
Statistical Channel Models: Continuous & Discrete
Probabilistic model provide better insights into wireless systems Continuous-time & discrete-time multipath fading channel X y (t) = aj (t)x(t − τj (t)) + w(t) j
y[m] =
X
h` [m]x[m − `] + w[m]
`
For frequency-flat fading-single channel tap: y[m] = h[m]x[m] + w[m]
Q. How to probabilistically model channel filter taps?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
23 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
Q. What is the distribution of W = R 2 ?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
`
March 18, 2019
24 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
`
2 Q. What is the distribution of W = R ? (Ans. Power is exponentially 1 distributed with pdf pW (w) = σ2 exp − σw2 , w ≥ 0) `
`
Rician fading: many scattered paths plus one LOS path q q K 1 h` [m] = K+1 σ` ejθ + K+1 CN (0, σ`2 )
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
24 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
`
2 Q. What is the distribution of W = R ? (Ans. Power is exponentially 1 distributed with pdf pW (w) = σ2 exp − σw2 , w ≥ 0) `
`
Rician fading: many scattered paths plus one LOS path q q K 1 h` [m] = K+1 σ` ejθ + K+1 CN (0, σ`2 ) First term: specular path arriving with uniform phase θ
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
24 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
`
2 Q. What is the distribution of W = R ? (Ans. Power is exponentially 1 distributed with pdf pW (w) = σ2 exp − σw2 , w ≥ 0) `
`
Rician fading: many scattered paths plus one LOS path q q K 1 h` [m] = K+1 σ` ejθ + K+1 CN (0, σ`2 ) First term: specular path arriving with uniform phase θ Second term: large number of reflected & scattered paths
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
24 / 33
Rayleigh Fading & Rician Fading Rayleigh frequency-flat fading: many scattered paths h` [m] ∼ N (0,
σ`2 2 )
+ jN (0,
σ`2 2 )
∼ CN (0, σ`2 )
‘CN ’ means circularly symmetric complex Gaussian (CSCG) r2 ,r ≥ 0 Let R = |h` [m]|. pdf of R: pR (r ) = σr2 exp − 2σ 2 `
`
2 Q. What is the distribution of W = R ? (Ans. Power is exponentially 1 distributed with pdf pW (w) = σ2 exp − σw2 , w ≥ 0) `
`
Rician fading: many scattered paths plus one LOS path q q K 1 h` [m] = K+1 σ` ejθ + K+1 CN (0, σ`2 ) First term: specular path arriving with uniform phase θ Second term: large number of reflected & scattered paths K : {specular path energy}/{scattered paths energy} ⇐ Fading factor (indicator of severity of fading)
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
24 / 33
Rician Distribution & Marcum Q Function Q: Prove that Z = |h` [m]| is Rician distributed with pdf 2 z z + s2 zs pZ (z) = 2 exp − I0 ,z ≥ 0 σ 2σ 2 σ2 I0 (.) denotes modified Bessel function of zeroth order Z 2π 1 I0 (z) = exp (z cos ψ) dψ 2π 0 Total average received power Pr = s2 + 2σ 2 Z ∞ Pr = z 2 pZ (z) dz 0
Average power in LOS component: s2 Average power in NLOS MPCs: 2σ 2
Fading factor K =
s2 2σ 2
⇐ Measure of severity of fading
K=0⇒
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
25 / 33
Rician Distribution & Marcum Q Function Q: Prove that Z = |h` [m]| is Rician distributed with pdf 2 z z + s2 zs pZ (z) = 2 exp − I0 ,z ≥ 0 σ 2σ 2 σ2 I0 (.) denotes modified Bessel function of zeroth order Z 2π 1 I0 (z) = exp (z cos ψ) dψ 2π 0 Total average received power Pr = s2 + 2σ 2 Z ∞ Pr = z 2 pZ (z) dz 0
Average power in LOS component: s2 Average power in NLOS MPCs: 2σ 2
Fading factor K =
s2 2σ 2
⇐ Measure of severity of fading
K = 0 ⇒ Rayleigh fading K=∞⇒
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
25 / 33
Rician Distribution & Marcum Q Function Q: Prove that Z = |h` [m]| is Rician distributed with pdf 2 z z + s2 zs pZ (z) = 2 exp − I0 ,z ≥ 0 σ 2σ 2 σ2 I0 (.) denotes modified Bessel function of zeroth order Z 2π 1 I0 (z) = exp (z cos ψ) dψ 2π 0 Total average received power Pr = s2 + 2σ 2 Z ∞ Pr = z 2 pZ (z) dz 0
Average power in LOS component: s2 Average power in NLOS MPCs: 2σ 2
Fading factor K =
s2 2σ 2
⇐ Measure of severity of fading
K = 0 ⇒ Rayleigh fading K = ∞ ⇒ Rician fading Larger value of K ⇒ stronger LOS component! B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
25 / 33
Marcum Q Function
Characterize error probability performance of digital signals communicated over Rician fading channels 2 Z ∞ M−1 x x + a2 IM−1 (ax) dx QM (a, b) = x exp − a 2 b 2 Z ∞ √ x + s2 Q1 (s, y) = √ x exp − I0 (sx) dx 2 y Computation in MATLAB: Q = marcumq(a,b,m) computes the generalized Marcum Q
Marcum Q Function
Characterize error probability performance of digital signals communicated over Rician fading channels 2 Z ∞ M−1 x x + a2 IM−1 (ax) dx QM (a, b) = x exp − a 2 b 2 Z ∞ √ x + s2 Q1 (s, y) = √ x exp − I0 (sx) dx 2 y Computation in MATLAB: Q = marcumq(a,b,m) computes the generalized Marcum Q
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
26 / 33
Nakagami Distribution More general distribution Need for Nakagami distribution Some experimental data- not fit well into either Rayleigh or Rician distribution
Nakagami−m pdf (m fading parameter) 2mm y 2m−1 my 2 pY (y ) = exp − , m ≥ 0.5 Pr Γ(m)Pr m Z ∞ Γ(m) = t m−1 e−t dt 0
Rayleigh fading when
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
27 / 33
Nakagami Distribution More general distribution Need for Nakagami distribution Some experimental data- not fit well into either Rayleigh or Rician distribution
Nakagami−m pdf (m fading parameter) 2mm y 2m−1 my 2 pY (y ) = exp − , m ≥ 0.5 Pr Γ(m)Pr m Z ∞ Γ(m) = t m−1 e−t dt 0
Rayleigh fading when m = 1 2 For m = (K+1) ⇒ approximately Rician fading 2K+1 m = ∞ ⇒ No fading
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
27 / 33
Power Distribution
Let Z = Y 2 , Y denotes Nakagami RV pdf of X pZ (z) =
m Pr
m
x m−1 mz exp − ,z ≥ 0 Γ(m) Pr
Verify that for m = 1, power is exponentially distributed Verify that pZ (z) is a valid pdf, i.e., Z 0
B. Sainath (BITS, PILANI)
∞
m Pr
m
x m−1 mz exp − dz = 1 Γ(m) Pr
SMALL SCALE FADING
March 18, 2019
28 / 33
Outage Probability Important physical layer performance measures Outage probability, symbol error probability (SEP), spectral efficiency
Outage probability pout : Probability that instantaneous SNR (iSNR) Γ falls below SNR threshold γth
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
29 / 33
Outage Probability Important physical layer performance measures Outage probability, symbol error probability (SEP), spectral efficiency
Outage probability pout : Probability that instantaneous SNR (iSNR) Γ falls below SNR threshold γth Z γth pout = pΓ (γ) dγ 0
pΓ (γ) denotes pdf of iSNR
Q: What is pout for Rayleigh fading?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
29 / 33
Outage Probability Important physical layer performance measures Outage probability, symbol error probability (SEP), spectral efficiency
Outage probability pout : Probability that instantaneous SNR (iSNR) Γ falls below SNR threshold γth Z γth pout = pΓ (γ) dγ 0
pΓ (γ) denotes pdf of iSNR
Q: What is pout for Rayleigh fading?
pout
γth = 1 − exp − γ
γ denotes average SNR
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
29 / 33
Numerical Example
Q: Consider Rayleigh fading channel. Average received power Pr = 20 dBm. What is the probability that the received power is below 10 dBm?
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
30 / 33
Numerical Example
Q: Consider Rayleigh fading channel. Average received power Pr = 20 dBm. What is the probability that the received power is below 10 dBm? Pr = 0.1 W pth == 0.01 W pout = 1 − exp (−0.1) = 0.0952
Q: Repeat the problem when pth = 0 dBm
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
30 / 33
Numerical Example
Q: Consider Rayleigh fading channel. Average received power Pr = 20 dBm. What is the probability that the received power is below 10 dBm? Pr = 0.1 W pth == 0.01 W pout = 1 − exp (−0.1) = 0.0952
Q: Repeat the problem when pth = 0 dBm (Ans:1%) Exercise: Express pth in terms of pout . If pout is too small, show that pth ≈ Pr pout
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
30 / 33
Amount of Fading (AoF) Let Γ denote path SNR per symbol AoF is defined as
2 E Γ2 − E [Γ] AoF = 2 E [Γ]
Exercise: Shadow fading can be modeled by a log-normal distribution for different indoor and outdoor environments. Instantaneous fading SNR per symbol has pdf given by ! 2 (10 log10 γ − µ) pΓ (γ) = √ exp − , γ > 0, 2πσγ 2σ 2 where = ln1010 , and µ (dB) and σ (dB) are the mean and the standard deviation of 10 log10 Γ, respectively Derive an expression for the AoF in terms of σ 2 and
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
31 / 33
Reading Exercise
Figure: Comparison of narrowband & wideband fading channels. Bu denotes signal bandwidth. Source: http://slideplayer.com/slide/3921678/#
Wideband fading channel model Andrea Goldsmith book, Section 3.3.
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
32 / 33
References
”Fundamentals of wireless communication” by Tse & Viswanath “Wireless communications” by Andrea Goldsmith www.eecs.berkeley.edu/˜dtse/taiwan_course.pdf http://slideplayer.com/slide/3921678/#
B. Sainath (BITS, PILANI)
SMALL SCALE FADING
March 18, 2019
33 / 33
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