Multipath Interference Characterization in Wireless Communication - PowerPoint PPT Presentation

Multipath Interference Characterization in Wireless Communication Systems Michael Rice BYU Wireless Communications Lab 9/29/00 BYU Wireless Communications 1 166

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? 9/29/00 BYU Wireless Communications 2 167

LTI System Model − 1 N ( ) ∑ ( ) θ = j δ − τ h t a e t k k k 0 ( t ) ( t ) k = h ( t ) s r 1 N − ( ) ( ) ∑ = δ + j θ δ − τ a t a e t k 0 k k = 1 k line-of-sight multipath propagation propagation 1 N − ∑ ( ) ( ) = ( ) + j θ − τ r t a s t a e s t k 0 k k k = 1 line-of-sight multipath component component 9/29/00 BYU Wireless Communications 3 168

Some Important Special Cases τ ≈ 0 for all All the delays are so small and we approximate k k 1 N − ( ) ∑ ( ) ( ) θ = + j − τ r t a s t a e s t k 0 k k k = 1 1 N − ( ) ∑ ≈ ( ) + j θ a s t a e s t k 0 k = 1 k  N − 1  ∑ • sum of complex random numbers (random amplitudes j θ ( ) = a + a e s t  k  0 k   and phases) = 1 k • if N is large enough, this sum is well approximated by α complex Gaussian pdf  − 1  N { } ∑ α = α + α j = α = j θ m E E a e   k R I a k   ( ) 1 k = 2 α ~ , σ N m { } { } − 1 N ∑ R a a = j θ E a E e ( ) k k ~ , 2 α σ N m k = 1 I a a [ ] = 0 when θ ~ − π , π U k 9/29/00 BYU Wireless Communications 4 169

Some Important Special Cases τ ≈ 0 for all All the delays are so small and we approximate k k ( ) ( ) 2 2 ( ) ( ) = + α + α j φ r t a e s t − 1 N ( ) ∑ 0 R I ( ) ( ) θ = + j − τ r t a s t a e s t k 0 k k 2 2 ( ) = + j φ X X e s t 1 k = 1 2 1 N − ( ) ∑ ≈ ( ) + j θ a s t a e s t k 0 k k = 1 ( ) ( ) ~ , 2 ~ 0 , 2 X N a σ X N σ  N − 1  ∑ 1 0 2 a a j θ ( ) = a + a e s t  k  0 k   = 1 k [ ] ( ) = + α a s t 0 [ ] ( ) = + α + α a j s t 0 R I ( ) ( ) 2 2 φ ( ) = a + α + α e j s t 0 R I 9/29/00 BYU Wireless Communications 5 170

Important PDF’s ( ) ~ , σ 2 ( ) X N a ~ 0 , 2 σ 1 0 X N a ( ) 1 a 2 ~ 0 , σ ( ) X N ~ 0 , 2 σ 2 a X N 2 a 2 2 W = X + X 2 2 = + 1 2 W X X 1 2 2 a + w 1   0 − a Non-central w 1 2 − 2 σ   ( ) = 0 Chi-square pdf p w e I w a 2 2   ( ) = σ p w e 0 W Chi-square pdf 2 σ 2 σ 2 a   W 2 2 σ a a a = 2 + 2 U X X = 2 + 2 U X X 1 2 1 2 2 2 a + u   2 0 u − ua u ( ) u − 2 2 σ   Rice pdf = 0 ( ) p u e I 2 a 2 σ = Rayleigh pdf   p u e 0 a U 2 2 σ σ   U 2 σ a a a 9/29/00 BYU Wireless Communications 6 171

Back to Some Important Special Cases τ ≈ 0 for all All the delays are so small and we approximate k k 0 0 0 > a 0 = a 1 1 N − N − ∑ ( ) ∑ ( ) ( ) = ( ) + j θ − τ ( ) = j θ − τ r t a s t a e s t r t a e s t k k 0 k k k k = 1 = 1 k k N − 1 N − 1 ( ) ( ) ∑ ∑ ( ) j θ j θ ≈ a s t + a e s t ≈ a e s t k k 0 k k = 1 = 1 k k ( ) ( ) ( ) ( ) 2 2 2 2 φ ( ) φ ( ) = a + α + α e j s t = α + α e j s t 0 R I R I Rice pdf Rayleigh pdf “Ricean fading” “Rayleigh fading” 9/29/00 BYU Wireless Communications 7 172

Some Important Special Cases τ ≈ τ for all All the delays are small and we approximate k k 0 0 0 > a 0 = a 1 1 N − N − ∑ ( ) ∑ ( ) ( ) = ( ) + j θ − τ ( ) = j θ − τ r t a s t a e s t r t a e s t k k 0 k k k k = 1 = 1 k k N − 1 N − 1 ∑ ( ) ∑ ( ) ( ) j θ j θ ≈ a s t + a e s t − τ ≈ a e s t − τ k k 0 k k = 1 = 1 k k ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 ( ) φ φ = a s t + α + α e j s t − τ = α + α e j s t − τ 0 R I R I Rayleigh pdf Rayleigh pdf “Line-of-sight with Rayleigh Fading” “Rayleigh fading” 9/29/00 BYU Wireless Communications 8 173

Multiplicative Fading In the past two examples, the received signal was of the form ( ) ( ) = j φ r t Fe s t 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. 9/29/00 BYU Wireless Communications 9 174

An Example ( ) ( ) 2 ( ) θ h t = δ t + ae j δ t − τ ( f ) H ( ) 1 ( 2 ) = + j θ − π f τ H f ae ( ) ( ) ( ) 2 2 1 1 2 2 cos 2 + = + + π τ − θ a H f a a f ( ) 2 1 − a f ( t ) ( t ) h ( t ) s r 1 θ τ ( ) ( ) S f R f 2 ( ) (dB) H f ( ) 2 10 log 10 1 + a ( ) ( ) S f R f ? ? ? ? ( ) 2 10 log 10 1 ? ? − a ? ? f f f − − W W W W 1 θ τ 9/29/00 BYU Wireless Communications 10 175

Example (continued) attenuation is even across the signal band (i.e. channel ( t ) ( t ) h ( t ) s r transfer function is “flat” in the ( ) ( ) signal band) S f R f τ is very small 2 ( ) ( f ) ( ) H S f R f f f f − − W W W W attenuation is uneven across the signal band -- this causes τ is very large “frequency selective fading” 2 ( ) ( f ) ( ) H S f R f f f f − − W W W W 9/29/00 BYU Wireless Communications 11 176

Another important special case τ < τ < < τ The delays are all different: � 1 2 N − 1 1 N − ∑ ( ) ( ) = ( ) + j θ − τ r t a s t a e s t k 0 k k = 1 k intersymbol interference if the delays are “long enough”, the multipath reflections are resolvable . 9/29/00 BYU Wireless Communications 12 177

Two common models for non-multiplicative fading ( t ) s ∆ ∆ ∆ � Taped delay-line with × × × × × α α α α α random weights 1 2 3 − 2 − 1 N N ( t ) + + + + r central limit theorem: approximately a Gaussian RP Additive complex 1 Gaussian random process N − ∑ ( ) ( ) = ( ) + j θ − τ r t a s t a e s t k 0 k k = 1 k ( ) ( ) ≈ a s t + ξ t 0 9/29/00 BYU Wireless Communications 13 178

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 ) 1. “maximum excess delay” or “multipath spread” { } ( ) ( ) ( ) τ 1 , τ = * τ τ R hh E h h 2 1 2 = τ T ( ) ( ) 1 m N − = τ δ τ − τ S uncorrelated 1 1 2 2. average delay scattering (US) − 1 N ∑ { } a τ assumption ( ) ( ) 1 1 N − k k 2 τ = τ ∑ S E h or τ = τ τ = = 1 k k − 1 − 1 N N ∑ 1 k = a ( ) k τ S 1 k = 3. delay spread − 1 N ∑ 2 2 τ a power 1 1 N − k k ∑ 2 2 σ τ = τ − τ or 2 σ τ = = 1 − τ k k − 1 N − 1 N ∑ τ 2 = 1 k a τ τ τ k 1 2 N − 1 = 1 k 9/29/00 BYU Wireless Communications 14 179

Characterization using the multipath intensity profile ( ) τ S power Compare multipath spread T m with symbol Compare multipath spread T m with symbol time T s : time T s : τ τ τ τ T m < T s ⇒ flat fading (frequency non- 1 2 N − 1 T m < T s ⇒ flat fading (frequency non- selective fading) selective fading) 1. “maximum excess delay” or “multipath spread” T m > T s ⇒ frequency selective fading = τ T T m > T s ⇒ frequency selective fading 1 m N − 2. average delay − 1 N ∑ a τ 1 1 N − k k ∑ or τ = τ τ = = 1 k k − 1 − 1 N N ∑ 1 k = a k 1 k = 3. delay spread − 1 N ∑ 2 2 τ a 1 1 N − k k ∑ 2 2 σ τ = τ − τ or 2 σ τ = = 1 − τ k k − 1 N − 1 N ∑ 2 = 1 k a k = 1 k 9/29/00 BYU Wireless Communications 15 180