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 N −1 θ = j k δ ()−τ h(t) ∑ ak e t k = s(t) r(t) k 0 h(t) N −1 θ = δ ()+ j k δ (−τ ) a0 t ∑ ak e t k k =1 line-of-sight multipath propagation propagation N −1 θ = + j k ()−τ r(t) a0s(t) ∑ ak e s t k k =1 line-of-sight multipath component component 9/29/00 BYU Wireless Communications 3 168 Some Important Special Cases τ ≈ All the delays are so small and we approximate k 0 for all k N −1 θ = + j k ()−τ r(t) a0s(t) ∑ ak e s t k k=1 N −1 θ ≈ + j k () a0s(t) ∑ ak e s t k=1 N −1 θ = + j k a0 ∑ ak e s(t) • sum of complex random numbers (random amplitudes = k 1 and phases) α • if N is large enough, this sum is well approximated by complex Gaussian pdf N −1 θ α = α + α m = E{}α = E a e j k R j I a ∑ k k =1 2 α ()σ N −1 R ~ N ma , a θ = ∑ E{}a E{}e j k 2 k α ~ N()m ,σ k =1 I a a = θ []−π π 0 when k ~ U , 9/29/00 BYU Wireless Communications 4 169 Some Important Special Cases τ ≈ All the delays are so small and we approximate k 0 for all k 2 2 φ N −1 = ()()+α + α j θ r(t) a0 R I e s(t) = + j k ()−τ r(t) a0s(t) ∑ ak e s t k 2 2 jφ k=1 = + X1 X 2 e s(t) N −1 θ ≈ + j k () a0s(t) ∑ ak e s t k=1 N −1 ()σ 2 ()σ 2 θ X1 ~ N a0 , a X 2 ~ N 0, a = + j k a0 ∑ ak e s(t) k =1 = []+α a0 s(t) = []+α + α a0 R j I s(t) = ()()+α 2 + α 2 jφ a0 R I e s(t) 9/29/00 BYU Wireless Communications 5 170 Important PDF’s ()σ 2 X1 ~ N a0 , a ()σ 2 X1 ~ N 0, a ()σ 2 X 2 ~ N 0, a ()σ 2 X 2 ~ N 0, a = 2 + 2 W X1 X 2 = 2 + 2 W X1 X 2 2 + − a0 w 2 w 1 2σ a0 Non-central − = a 1 2σ 2 pW (w) 2 e I0 w 2 = a Chi-square pdf σ σ pW (w) e 2 a a Chi-square pdf σ 2 2 a 2 2 = + 2 2 U X1 X 2 = + U X1 X 2 a2 +u 2 − 0 u2 u σ 2 ua − 2 a 0 σ 2 ()= Rice pdf u 2 a pU u 2 e I0 2 ()= σ σ pU u 2 e Rayleigh pdf a a σ a 9/29/00 BYU Wireless Communications 6 171 Back to Some Important Special Cases τ ≈ All the delays are so small and we approximate k 0 for all k > = a0 0 a0 0 N −1 N −1 θ θ = + j k ()−τ = j k ()−τ r(t) a0s(t) ∑ak e s t k r(t) ∑ ak e s t k k=1 k=1 N −1 N −1 θ θ ≈ + j k () ≈ j k () a0s(t) ∑ak e s t ∑ ak e s t k=1 k=1 = ()()+α 2 + α 2 jφ = ()()α 2 + α 2 jφ a0 R I e s(t) R I e s(t) Rice pdf Rayleigh pdf “Ricean fading” “Rayleigh fading” 9/29/00 BYU Wireless Communications 7 172 Some Important Special Cases τ ≈τ All the delays are small and we approximate k for all k > = a0 0 a0 0 N −1 N −1 θ θ = + j k ()−τ = j k ()−τ r(t) a0s(t) ∑ ak e s t k r(t) ∑ak e s t k k=1 k=1 N −1 N −1 θ θ ≈ + j k ()−τ ≈ j k ()−τ a0s(t) ∑ ak e s t ∑ak e s t k=1 k=1 = + ()()α 2 + α 2 jφ ()−τ = ()()α 2 + α 2 jφ ()−τ a0s(t) R I e s t R I e s t 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 r(t) = Fe jφ 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 θ 2 h(t) = δ ()t + ae j δ (t −τ ) H ( f ) H ()f =1+ ae j(θ −2πfτ ) 2 H ()f 2 =1+ a2 + 2a cos()2πfτ −θ ()1+ a ()1− a 2 f s(t) h(t) r(t) θ 1 τ S()f R()f H ( f ) 2 (dB) ()+ 2 10log10 1 a S()f R()f ? ? ? ? ()− 2 ? ? 10log10 1 a ? ? f f f −W W −W W θ 1 τ 9/29/00 BYU Wireless Communications 10 175 Example (continued) attenuation is even across the s(t) h(t) r(t) signal band (i.e. channel transfer function is “flat” in the () () τ is very small S f R f signal band) 2 S()f H ( 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 S()f H ( 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 L N −1 N −1 θ = + j k ()−τ r(t) a0s(t) ∑ ak e s t k k=1 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 s(t) ∆ ∆ L ∆ Taped delay-line with α × α × α × α × α × random weights 1 2 3 N −2 N −1 + + + + r(t) central limit theorem: approximately a Gaussian RP Additive complex N −1 Gaussian random process θ = + j k ()−τ r(t) a0s(t) ∑ ak e s t k k=1 ≈ +ξ a0s(t) (t) 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 τ = {}* ()()τ τ Rhh 1, 2 E h 1 h 2 T =τ − = S()(τ δ τ −τ ) m N 1 uncorrelated 1 1 2 scattering (US) 2. average delay N −1 τ assumption − ∑ ak k 2 1 N 1 S()τ = E{}h ()τ τ = ∑τ or τ = k =1 − k N −1 N 1 k =1 ∑ ak S()τ = 3. delay spread k 1 N −1 2 2 − ∑ a τ 1 N 1 k k σ = τ 2 −τ 2 σ = k =1 −τ 2 power τ ∑ or − k τ N −1 τ N 1 k =1 2 τ τ τ ∑ ak 1 2 N −1 k =1 9/29/00 BYU Wireless Communications 14 179 Characterization using the multipath intensity profile S()τ Compare multipath spread Tm with symbol Compare multipath spread Tm with symbol power time Ts: τ time Ts: τ τ τ 1 2 N −1 ⇒ Tm < Ts ⇒ flat fading (frequency non- Tm < Ts flat fading (frequency non- selectiveselective fading) fading) 1. “maximum excess delay” or “multipath spread” ⇒ =τ Tm > Ts ⇒ frequency selective fading Tm N −1 Tm > Ts frequency selective fading 2. average delay N −1 ∑ a τ 1 N −1 k k τ = ∑τ or τ = k =1 − k N −1 N 1 k =1 ∑ ak = 3. delay spread k 1 N −1 2τ 2 N −1 ∑ ak k 1 2 2 k =1 2 σ τ = ∑τ −τ or σ = −τ − k τ N −1 N 1 k =1 2 ∑ ak k =1 9/29/00 BYU Wireless Communications 15 180 Spaced Frequency Correlation Function S()τ R()∆f Fourier Xform power τ f τ τ τ 1 2 N −1 f0 R(∆f) is the “correlation between the channel response to two signals as a function of the Compare coherence bandwidth f0 with frequency difference between the two signals.” Compare coherence bandwidth f0 with transmittedtransmitted signal signal bandwidth bandwidth W W: : “What is the correlation between received ⇒ f0 > W ⇒ flat fading (frequency non- signals that are spaced in frequency ∆f = f -f ?” f0 > W flat fading (frequency non- 1 2 selectiveselective fading) fading) Coherence bandwidth f = a statistical measure ⇒ 0 f0 < W ⇒ frequency selective fading of the range of frequencies over which the f0 < W frequency selective fading channel passes all spectral components with approximately equal gain and linear phase.