Multipath Interference Characterization in Wireless Communication - - PowerPoint PPT Presentation

9/29/00 BYU Wireless Communications 1

Multipath Interference Characterization in Wireless Communication Systems

Michael Rice BYU Wireless Communications Lab

166

9/29/00 BYU Wireless Communications 2

Multipath Propagation

• Multiple paths between transmitter and receiver
• Constructive/destructive interference
• Dramatic changes in received signal amplitude and phase as a result of

small changes (λ/2) in the spatial separation between a receiver and transmitter.

• For Mobile radio (cellular, PCS, etc) the channel is time-variant

because motion between the transmitter and receiver results in propagation path changes.

• Terms: Rayleigh Fading, Rice Fading, Flat Fading, Frequency

Selective Fading, Slow Fading, Fast Fading ….

• What do all these mean?

167

9/29/00 BYU Wireless Communications 3

LTI System Model

h(t)

) (t s ) (t r

( ) ( ) ( )

∑ ∑

− = − =

− + = − =

1 1 1

) (

N k k j k N k k j k

t e a t a t e a t h

k k

τ δ δ τ δ

θ θ

line-of-sight propagation multipath propagation

( )

∑

− =

− + =

1 1

) ( ) (

N k k j k

t s e a t s a t r

k

τ

θ

line-of-sight component multipath component

168

9/29/00 BYU Wireless Communications 4

Some Important Special Cases

All the delays are so small and we approximate k

k

all for ≈ τ

( ) ( )

) ( ) ( ) ( ) (

1 1 1 1 1 1

t s e a a t s e a t s a t s e a t s a t r

N k j k N k j k N k k j k

k k k

      + = + ≈ − + =

∑ ∑ ∑

− = − = − = θ θ θ

τ

• sum of complex random numbers (random amplitudes

and phases)

• if N is large enough, this sum is well approximated by

complex Gaussian pdf

α

( ) ( )

2 2

, ~ , ~

a a I a a R I R

m N m N j σ α σ α α α α + =

{ } { } {

}

[ ]

π π θ α

θ θ

, ~ when

1 1 1 1

− = =       = =

∑ ∑

− = − =

U e E a E e a E E m

k N k j k N k j k a

k k

169

9/29/00 BYU Wireless Communications 5

Some Important Special Cases

All the delays are so small and we approximate k

k

all for ≈ τ

( ) ( ) [ ] [ ] ( ) ( )

) ( ) ( ) ( ) ( ) ( ) ( ) (

2 2 1 1 1 1 1 1

t s e a t s j a t s a t s e a a t s e a t s a t s e a t s a t r

j I R I R N k j k N k j k N k k j k

k k k

φ θ θ θ

α α α α α τ + + = + + = + =       + = + ≈ − + =

∑ ∑ ∑

− = − = − =

( ) ( )

) ( ) ( ) (

2 2 2 1 2 2

t s e X X t s e a t r

j j I R φ φ

α α + = + + =

( )

2 1

, ~

a

a N X σ

( )

2 2

, ~

a

N X σ

170

9/29/00 BYU Wireless Communications 6

Important PDF’s

( ) ( )

( )

        = + =         = + =

+ − + − 2 2 2 2 2 2 1 2 2 2 2 2 2 1 2 2 2 1

2 2 2 2 2

2 1 ) ( , ~ , ~

a u a a U a w a a W a a

ua I e u u p X X U a w I e w p X X W N X a N X

a a

σ σ σ σ σ σ

σ σ

Non-central Chi-square pdf Rice pdf

( ) ( )

( )

2 2 2

2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 1

2 1 ) ( , ~ , ~

a a

u a U w a W a a

e u u p X X U e w p X X W N X N X

σ σ

σ σ σ σ

− −

= + = = + = Chi-square pdf Rayleigh pdf

171

9/29/00 BYU Wireless Communications 7

Back to Some Important Special Cases

All the delays are so small and we approximate k

k

all for ≈ τ

( ) ( ) ( ) ( )

) ( ) ( ) ( ) (

2 2 1 1 1 1

t s e a t s e a t s a t s e a t s a t r

j I R N k j k N k k j k

k k

φ θ θ

α α τ + + = + ≈ − + =

∑ ∑

− = − =

Rice pdf “Ricean fading”

0 >

a

( ) ( ) ( ) ( )

) ( ) (

2 2 1 1 1 1

t s e t s e a t s e a t r

j I R N k j k N k k j k

k k

φ θ θ

α α τ + = ≈ − =

∑ ∑

− = − =

Rayleigh pdf “Rayleigh fading”

0 =

a

172

9/29/00 BYU Wireless Communications 8

Some Important Special Cases

All the delays are small and we approximate k

k

all for τ τ ≈

( ) ( ) ( ) ( ) ( )

τ α α τ τ

φ θ θ

− + + = − + ≈ − + =

∑ ∑

− = − =

t s e t s a t s e a t s a t s e a t s a t r

j I R N k j k N k k j k

k k

2 2 1 1 1 1

) ( ) ( ) ( ) ( Rayleigh pdf “Line-of-sight with Rayleigh Fading”

0 >

a

( ) ( ) ( ) ( ) ( )

τ α α τ τ

φ θ θ

− + = − ≈ − =

∑ ∑

− = − =

t s e t s e a t s e a t r

j I R N k j k N k k j k

k k

2 2 1 1 1 1

) ( Rayleigh pdf “Rayleigh fading”

0 =

a

173

9/29/00 BYU Wireless Communications 9

Multiplicative Fading

In the past two examples, the received signal was of the form ) ( ) ( t s Fe t r

jφ

= The fading takes the form of a random attenuation: the transmitted signal is multiplied by a random value whose envelope is described by the Rice or Rayleigh pdf. This is sometimes called multiplicative fading for the obvious

• reason. It is also called flat fading since all spectral components in

s(t) are attenuated by the same value.

174

9/29/00 BYU Wireless Communications 10

An Example

τ 1 θ f

2

) ( f H

( )

2

1 a +

( )

2

1 a −

( ) ( ) ( ) ( ) ( )

θ τ π τ δ δ

τ π θ θ

− + + = + = − + =

−

f a a f H ae f H t ae t t h

f j j

2 cos 2 1 1 ) (

2 2 ) 2 (

τ 1 θ f (dB) ) (

2

f H

( )

2 10 1

log 10 a +

( )

2 10 1

log 10 a −

h(t)

) (t s ) (t r

( )

f S

( )

f R

( )

f S f W W −

( )

f R f W W −

? ? ? ? ? ? ? ?

175

9/29/00 BYU Wireless Communications 11

Example (continued)

f

2

) ( f H

( )

f S f W W −

( )

f R f W W −

h(t)

) (t s ) (t r

( )

f S

( )

f R

τ is very small

f

2

) ( f H

( )

f S f W W −

( )

f R f W W −

τ is very large attenuation is even across the signal band (i.e. channel transfer function is “flat” in the signal band) attenuation is uneven across the signal band -- this causes “frequency selective fading”

176

9/29/00 BYU Wireless Communications 12

Another important special case

The delays are all different:

( )

∑

− =

− + =

1 1

) ( ) (

N k k j k

t s e a t s a t r

k

τ

θ 1 2 1 −

< < <

N

τ τ τ

• intersymbol interference

if the delays are “long enough”, the multipath reflections are resolvable.

177

9/29/00 BYU Wireless Communications 13

Two common models for non-multiplicative fading

∆ ∆ ∆

• ×

× × × × + + + +

1

α

2

α

3

α

2 − N

α

1 − N

α

) (t s ) (t r

Taped delay-line with random weights Additive complex Gaussian random process

( )

) ( ) ( ) ( ) (

1 1

t t s a t s e a t s a t r

N k k j k

k

ξ τ

θ

+ ≈ − + =

∑

− =

central limit theorem: approximately a Gaussian RP

178

9/29/00 BYU Wireless Communications 14

Multipath Intensity Profile

The characterization of multipath fading as either flat (multiplicative) or frequency selective (non-multiplicative) is governed by the delays: small delays ⇒ flat fading (multiplicative fading) large delays ⇒ frequency selective fading (non-multiplicative fading) The values of the delay are quantified by the multipath intensity profile S(τ) ( )

τ S τ

2

τ

1

τ

1 − N

τ

• 1. “maximum excess delay” or “multipath spread”

1 −

=

N m

T τ

∑

− =

− =

1 1

1 1

N k k

N τ τ

∑ ∑

− = − =

=

1 1 1 1 N k k N k k k

a a τ τ

• 2. average delay
• r
• 3. delay spread

2 1 1 2

1 1 τ τ στ − − =

∑

− = N k k

N

2 1 1 2 1 1 2 2

τ τ στ − =

∑ ∑

− = − = N k k N k k k

a a

• r

power

( ) ( ) ( )

{ }

( ) ( ) ( ) ( )

{ }

2 2 1 1 2 1 * 2 1,

τ τ τ τ δ τ τ τ τ τ h E S S h h E Rhh = − = =

uncorrelated scattering (US) assumption

179

9/29/00 BYU Wireless Communications 15

Characterization using the multipath intensity profile

( )

τ S τ

2

τ

1

τ

1 − N

τ

• 1. “maximum excess delay” or “multipath spread”

1 −

=

N m

T τ

∑

− =

− =

1 1

1 1

N k k

N τ τ

∑ ∑

− = − =

=

1 1 1 1 N k k N k k k

a a τ τ

• 2. average delay
• r
• 3. delay spread

2 1 1 2

1 1 τ τ στ − − =

∑

− = N k k

N

2 1 1 2 1 1 2 2

τ τ στ − =

∑ ∑

− = − = N k k N k k k

a a

• r

Compare multipath spread Tm with symbol time Ts: Tm < Ts ⇒ flat fading (frequency non- selective fading) Tm > Ts ⇒ frequency selective fading Compare multipath spread Tm with symbol time Ts: Tm < Ts ⇒ flat fading (frequency non- selective fading) Tm > Ts ⇒ frequency selective fading

power

180

9/29/00 BYU Wireless Communications 16

Spaced Frequency Correlation Function

( )

τ S τ

2

τ

1

τ

1 − N

τ

( )

f R ∆ f f Fourier Xform power

R(∆f) is the “correlation between the channel response to two signals as a function of the frequency difference between the two signals.” “What is the correlation between received signals that are spaced in frequency ∆f = f1-f2?” Coherence bandwidth f0 = a statistical measure

• f the range of frequencies over which the

channel passes all spectral components with approximately equal gain and linear phase. Compare coherence bandwidth f0 with transmitted signal bandwidth W: f0 > W ⇒ flat fading (frequency non- selective fading) f0 < W ⇒ frequency selective fading Compare coherence bandwidth f0 with transmitted signal bandwidth W: f0 > W ⇒ flat fading (frequency non- selective fading) f0 < W ⇒ frequency selective fading

equations (8) - (13) are commonly used relationships between delay spread and coherence bandwidth

181

9/29/00 BYU Wireless Communications 17

Time Variations

Important Assumption Multipath interference is spatial phenomenon. Spatial geometry is assumed fixed. All scatterers making up the channel are stationary -- whenever motion ceases, the amplitude and phase of the receive signal remains constant (the channel appears to be time-invariant). Changes in multipath propagation occur due to changes in the spatial location x of the transmitter and/or receiver. The faster the transmitter and/or receiver change spatial location, the faster the time variations in the multipath propagation properties. Important Assumption Multipath interference is spatial phenomenon. Spatial geometry is assumed fixed. All scatterers making up the channel are stationary -- whenever motion ceases, the amplitude and phase of the receive signal remains constant (the channel appears to be time-invariant). Changes in multipath propagation occur due to changes in the spatial location x of the transmitter and/or receiver. The faster the transmitter and/or receiver change spatial location, the faster the time variations in the multipath propagation properties.

( ) ( ) ( )

∑ ∑

− = − =

− + = − =

1 1 ) ( 1 ) (

) ( ) ( ) ( ) ( ) ( ) ; (

N k k x j k N k k x j k

x t e x a t x a x t e x a x t h

k k

τ δ δ τ δ

θ θ

line-of-sight propagation multipath propagation complex gains and phase shifts are a function of spatial location x.

182

9/29/00 BYU Wireless Communications 18

Spatially Varying Channel Impulse Response

τ x

) ; ( x t h

• channel impulse response changes

with spatial location x

• generalize impulse response to

include spatial information

• Transmitter/receiver motion cause

change in spatial location x

• The larger , the faster the rate of

change in the channel.

• Assuming a constant velocity v, the

position axis x could be changed to a time axis t using t = x/v.

) ; ( ) ( x t h t h → x

• 183

9/29/00 BYU Wireless Communications 19

Generalize the Multipath Intensity Profile

( ) ( ) ( )

{ }

( ) ( ) ( ) ( )

{ }

2 2 1 1 2 1 * 2 1,

τ τ τ τ δ τ τ τ τ τ h E S S h h E Rhh = − = =

( ) ( ) ( )

{ }

( ) ( ) ( ) ( ) ( )

{ }

( ) ( ) ( )

{ }

( ) ( ) ( )

{ }

t t h t h E t S t h t h E t t S x h x h E x x S x x S x h x h E x x Rhh ∆ + = ∆ = = − = = ; ; ; ; ; , ; ; ; , ; , ; ; ; , ; ,

* 2 1 * 2 1 2 1 * 2 1 2 1 2 1 1 2 2 1 1 * 2 1 2 1

τ τ τ τ τ τ τ τ τ τ τ δ τ τ τ τ τ From before… The generalization … US assumption US assumption v x t / = WSS assumption this function is the key to the WSSUS channel

184

9/29/00 BYU Wireless Communications 20

A look at S(τ;∆t)

τ x ∆

) ; ( x S ∆ τ

τ t ∆

) ; ( t S ∆ τ

( ) ( )

; τ τ S S = S(τ) S(τ)

185

9/29/00 BYU Wireless Communications 21

Time Variations of the Channel: The Spaced-Time Correlation Function

t ∆ ) ; ( t S ∆ τ

integrate along delay axis

t ∆ ) ( t R ∆

( ) ∫

∆ = ∆ τ τ d t S t R ) ; (

186

9/29/00 BYU Wireless Communications 22

Time Variations of the Channel: The Spaced-Time Correlation Function

T t ∆ ) ( t R ∆

R(∆t) specifies the extent to which there is correlation between the channel response to a sinusoid sent at time t and the response to a similar sinusoid at time t+∆t. Coherence Time T0 is a measure of the expected time duration over which the channel response is essentially invariant. Slowly varying channels have a large T0 and rapidly varying channels have a small T0.

187

9/29/00 BYU Wireless Communications 23

Re-examination of special cases

( ) ( ) [ ]

) ( ) ( ) ( ) ( ) (

1 1 1 1 1 1

t s a t s e a a t s e a t s a t s e a t s a t r

N k j k N k j k N k k j k

k k k

α τ

θ θ θ

+ =       + = + ≈ − + =

∑ ∑ ∑

− = − = − =

( ) ( ) [ ] [ ]

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

1 1 ) ( 1 1 ) ( 1 1 ) (

t s t a t s x a t s e x a a t s e x a t s a x t s e x a t s a t r

N k x j k N k x j k N k k x j k

k k k

α α τ

θ θ θ

+ = + =       + = + ≈ − + =

∑ ∑ ∑

− = − = − =

complex Gaussian RV

v x t / =

complex Gaussian Random Process with autocorrelation

( ) ( ) ( )

{ }

( )

t R t t t E t R ∆ = ∆ + = ∆ α α

α *

From before… The generalization …

188

9/29/00 BYU Wireless Communications 24

Commonly Used Spaced-Time Correlation Functions

( )

∞ < ∆ < ∞ − = ∆ t t R

a 2

2σ

( )

t v ae

t R

∆ −

= ∆

λ π

σ

2 2

2

( )

      ∆ = ∆ t v J t R

a

λ π σ 2 2

2

( )

2

2

2

      ∆ −

= ∆

t v ae

t R

λ π

σ

( )

t v t v t R

a

∆       ∆ = ∆ λ π λ π σ 2 2 sin 2

2

Time Invariant Land Mobile (Jakes) Exponential Gaussian “Rectangular”

t ∆

( )

t R ∆ t ∆

( )

t R ∆ t ∆

( )

t R ∆ t ∆

( )

t R ∆ t ∆

( )

t R ∆

189

9/29/00 BYU Wireless Communications 25

Characterization of time variations using the spaced-time correlation function

T t ∆ ) ( t R ∆

• Fast Fading

– T0 < Ts – correlated channel behavior lasts less than a symbol ⇒ fading characteristics change multiple times during a symbol ⇒ pulse shape distortion

• Slow Fading

– T0 > Ts – correlated channel behavior lasts more than a symbol ⇒ fading characteristics constant during a symbol ⇒ no pulse shape distortion ⇒ error bursts…

190

9/29/00 BYU Wireless Communications 26

Doppler Power Spectrum Frequency Domain View of Time-Variations

T t ∆ ) ( t R ∆

Fourier Xform

( )

ν S ν

d

f Time variations on the channel are evidenced as a Doppler broadening and perhaps, in addition as a Doppler shift of a spectral line. Doppler power spectrum S(ν) yields knowledge about the spectral spreading of a sinusoid (impulse in frequency) in the Doppler shift domain. It also allows us to glean how much spectral broadening is imposed on the transmitted signal as a function of the rate of change in the channel state. Doppler Spread of the channel fd is the range of values of ν over which the Doppler power spectrum is essentially non zero.

191

9/29/00 BYU Wireless Communications 27

Doppler Power Spectrum and Doppler Spread

( )

ν S ν

d

f Compare Doppler Spread fd with transmitted signal bandwidth W: fd > W ⇒ fast fading fd < W ⇒ slow fading Compare Doppler Spread fd with transmitted signal bandwidth W: fd > W ⇒ fast fading fd < W ⇒ slow fading

equations (18) - (21) are commonly used relationships between Doppler spread and coherence time

192

9/29/00 BYU Wireless Communications 28

Common Doppler Power Spectra

Time Invariant Land Mobile (Jakes) Exponential (1st order Butterworth) Gaussian “Rectangular”

ν

( )

ν S

( ) ( )

ν δ σ ν

2

2

a

S =

( ) ( )

2 2 2

/ 2 λ ν σ ν v S

a

− =

( ) ( )

( )

2 2 2

/ / 2 λ ν π λ σ ν v v S

a

+ =

( ) ( )

( )2

2

/ 2 2

/ 2

λ ν

λ π σ ν

v a

e v S

−

=

( )

     < < − =

• therwise

/ / /

2

λ ν λ λ σ ν v v v S

a

ν

( )

ν S ν

( )

ν S ν

( )

ν S ν

( )

ν S

193

9/29/00 BYU Wireless Communications 29

Putting it all together…

( )

f R ∆ f f

( )

τ S τ

2

τ

1

τ

1 − N

τ power T t ∆ ) ( t R ∆

( )

ν S ν

d

f

) ; ( t f S ∆ ∆

= ∆f = ∆t

Fourier transform Fourier transform

) ; ( ν τ S

∫

ν ν τ d S ) ; (

∫

τ ν τ d S ) ; (

) ; ( t S ∆ τ

Fourier transform

f ∆ ↔ τ

spaced-frequency correlation function multipath intensity profile spaced-time correlation function Doppler power spectrum spaced-frequency, spaced-time correlation function scattering function

194

9/29/00 BYU Wireless Communications 30

Scattering Function

delay τ frequency ν

) ; ( ν τ S

195

9/29/00 BYU Wireless Communications 31

References

• John Proakis, Digital Communications, Third Edition. McGraw-Hill. Chapter 10.
• William Jakes, Editor, Microwave Mobile Communications. John Wiley & Sons.

Chapter 1.

• William Y. C. Lee, Mobile Cellular Communications, McGraw-Hill.
• Parsons, J. D., The Mobile Radio Propagation Channel, John Wiley & Sons.
• Bernard Sklar, “Rayleigh Fading Channels in Digital Communication systems Part I:

Characterization,” IEEE Communication Magazine, July 1997, pp. 90 - 100.

• Peter Bello, “Characterization of Randomly Time Variant Linear Channels,” IEEE

Transactions on Communication Systems, vol. 11, no. 4, December 1963, pp. 360-393.

• R. H. Clarke, “A Statistical Theory of Mobile Radio Reception,” Bell Systems Technical

Journal, vol. 47, no. 6, July-August 1968, pp. 957-1000.

• M. J. Gans, “A Power-Spectral Theory of Propagatio9in in the Mobile Radio

Environment,” IEEE Transactions on Vehicular Technology, vol. VT 21, February 1972, pp. 27-38.,

196