Multipath Interference Characterization in Wireless Communication Systems
Michael Rice BYU Wireless Communications Lab
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• 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?
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167 LTI System Model
δ θ τ N −1 = j k ()− h(t) ∑δak e t k = s(t) r(t) k 0 h(t) N −1 ()θ δ ( ) = + j k −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
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168 Some Important Special Cases
τ ≈ All the delays are so small and we approximate k 0 for all k
N −1θ = + 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 α N −1 αα = α + α m = E{}= E a θe j k R j I a ∑ k k =1 2 ()σ N −1 {} α R ~ N ma , a θ = {}j k ()∑ E ak E e 2 θ ~ N m ,σ k =1 I a a = []−π π 0 when k ~ U ,
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169 Some Important Special Cases
τ ≈ All the delays are so small and we approximate k 0 for all k
θ α 2 2 N −1 = ()()+ + α j r(t) a0 R I eφ s(t) = + τ j k ()− r(t) a0s(t) ∑ ak e s t k 2 2 jφ θk=1 = + X1 X 2 e s(t) N −1 () ≈ + j k a0s(t) ∑ ak e s t θk=1 N −1 ()σ 2 ()σ 2 X1 ~ N a0 , a X 2 ~ N 0, a = + j k a0 ∑ ak e s(t) α k =1 = []+ a0 α s(t) []α = a + + j s(t) 0 Rα I = ()()+ 2 + α 2 jφ a0 R I e s(t)
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170 Important PDF’s
σ σ ()2 σ X1 ~ N a0 , a ()2 () X1 ~ N 0, a 2 σ () X 2 ~ N 0, a 2 X 2 ~ N 0, a
= 2 + 2 W X1 X 2 = 2 + 2 W X1 X 2 σ a2 +w σ − 0 2 w 1 2 a0 Non-central σ − = a 1 2 2 pW (w) 2 e I0 w 2 = a Chi-square pdf 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 σ − 0 u2 u σ 2 ua − 2 a 0 σ 2 ()= Rice pdf u 2 a pU u 2 e I0 2 ()= σ pU u 2 e Rayleigh pdf a a σ a
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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 θ θ N −1 N −1 = + j k ()− = j k τ()− r(t) a0s(t) ∑ak e sτ t k r(t) ∑ ak e s t k k=1 k=1 θ θ N −1 () N −1 () ≈ + j k ≈ j k a0s(t) ∑ak e s t ∑ ak e s t = = ()()αk 1 ()()k 1α = + 2 + α 2 jφ = 2 + α 2 jφ a0 R I e s(t) R I e s(t)
Rice pdf Rayleigh pdf
“Ricean fading” “Rayleigh fading”
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172 Some Important Special Cases
τ ≈τ All the delays are small and we approximate k for all k
> = a0 0 a0 0 θ τ θ τ N −1 N −1 = + j k ()− = j k ()− r(t) a0s(t) ∑ ak e s t k r(t) ∑ak e s t k k=1 θ τ k=1 θ τ N −1 N −1 ≈ + j k ()− ≈ j k ()− a0s(t) ∑αak e s t ∑αak e s t k=1 α k=1 α ()() ()() = + 2 + 2 jφ ()−τ = 2 + 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”
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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.
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174 An Example δ δ θ τ h(t) = ()t + ae j t − ( ) H ( f ) 2 θ () − π τ H f =1+ ae j( 2 f ) () 2 π τ 2 H f =1+ a2 + 2a cos()2 f −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
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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
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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
if the delays are “long enough”, the multipath reflections are resolvable.
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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 ()− r(t) a0s(t) ∑ ak e s t k k=1 τ ≈ +ξ a0s(t) (t)
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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) τ τ τ 1. “maximum excess delay” or “multipath spread” R (), = E{}h* τ h ()() hh 1 2 1 2 =τ τ δ Tm N −1 = S()(τ − ) uncorrelated 1 1τ 2 scattering (US) 2. average delay N −1 τ τ τ assumption () () − τ ∑ ak k 2 1 N 1 S = E{}h τ = τ or = k =1 ∑ k N −1 N −1 = k 1 ∑ a ()τ k S k =1 3. σdelay spread σ N −1 τ 2 2 − ∑ a 1 N 1τ k k = 2 −τ 2 = 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
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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 τ τ ∑ a τ 1 N −1 k k = ∑τ or = k =1 − k N −1 N 1 k =1 ∑ ak k =1 3. σdelay spread σ N −1 τ a 2 2 N −1τ ∑ k k 1 2 2 k =1 2 τ = ∑ −τ or = −τ − k τ N −1 N 1 k =1 2 ∑ ak k =1
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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.
equations (8) - (13) are commonly used relationships between delay spread and coherence bandwidth
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181 Time Variations
ImportantImportant Assumption Assumption
MultipathMultipath interference interference is is spatial spatial phenomenon. phenomenon. Spatial Spatial geometry geometry is is assumed assumed fixed. fixed. All All scatterersscatterers making making up up the the channel channel are are stationary stationary -- -- whenever whenever motion motion ceases, ceases, the the amplitude amplitude and and phasephase of of the the receive receive signal signal remains remains constant constant (the (the channel channel appears appears to to be be time-invariant). time-invariant). ChangesChanges in in multipath multipath propagation propagation occur occur due due to to changes changes in in the the spatial spatial location location x x of of the the transmittertransmitter and/or and/or receiver. receiver. The The faster faster the the transmitter transmitter and/or and/or receiver receiver change change spatial spatial location, location, the faster the time variations in the multipath propagation properties. the faster the time variations in the multipathδ propagation properties. θ τ N −1 = δ j k (x) ()− h(t; x) ∑ ak (x)e t k (x) k =0 N −1 ()θ δ ( ) = + j k (x) −τ a0 (x) t ∑ ak (x)e t k (x) complex gains and k =1 phase shifts are a function of spatial location x. line-of-sight multipath propagation propagation
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182 Spatially Varying Channel Impulse Response
• channel impulse response changes with spatial location x • generalize impulse response to include spatial information
h(t; x) h(t) → h(t; x)
x • Transmitter/receiver motion cause change in spatial location x
• The larger x & , the faster the rate of τ 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.
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183 Generalize the Multipath Intensity Profile
τ τ From before… The generalization … τ τ τ τ τ τ R (), = E{}h* τ h ()() R ()()(), ; x , x = E{}h* ; x h ; x US hh 1 2 τ 1 2 US hh 1 2 1 2 δ τ1 1 2 2 assumption = ()(δ τ − ) τ = ()()τ − S 1 1τ 2 assumption S 1; x1, x2 1 2 τ τ () () ()()()τ 2 = {}* S = E{}h τ Sτ ; x1, x2 E h ; x1 h ; x2 t = x / v τ ()= {}* τ ()() S ;t1,t2 E h ;t1 h ;t2 τ WSS τ assumption S();∆t = E{}h* ;t h ()()τ;t + ∆t
this function is the key to the WSSUS channel
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184 A look at S(τ;∆t)
τ ∆ S( ; x) S(τ;∆t)
S(τ) S(τ) ∆ x ∆t
τ τ
S()τ = S τ; (0 )
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185 Time Variations of the Channel: The Spaced-Time Correlation Function
τ ∆ S( ; t) R(∆t) ∆t
∆t
τ R()∆t = ∫ S( ;∆t)dτ
integrate along delay axis
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186 Time Variations of the Channel: The Spaced-Time Correlation Function
R(∆t)
∆t 0 T0
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.
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187 Re-examination of special cases
From before… The generalization …
N −1 N −1 θ θ = + j k (x) ()− j k r(t) a s(t) a (x)e s t (x) r(t) = a s(t) + ∑ a e s()t − 0 ∑ k k 0 k k k =1 k=1 τ τ N −1 () N −1 () θ j (x) θ k j k ≈ a s(t) + a (x)e s t ≈ a s(t) + ∑ a e s t 0 ∑ k 0 k k =1 k=1 N −1 N −1 jθ ( x) θ = + k j k a ∑a (x)e s(t) = a + ∑ a e s(t) 0 k 0 k k=1 k =1 = []α+ = []+α a0 (x) s(t) a0 s(t) t = x / v [] = +α complex Gaussian RV a0 (t) s(t)
complex Gaussian Random Process with autocorrelation
α * Rα ()∆t = E{}t α ()(t + ∆t ) = R()∆t
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188 Commonly Used Spaced-Time Correlation Functions
R()∆t
Time Invariant ()∆ = σ 2 − ∞ < ∆ < ∞ R t 2 a t ∆t R()∆t σ Land Mobile π v R()∆t = 2 2 J 2 ∆t (Jakes) a 0 λ ∆t R()∆t
π v −2 ∆t ()∆ = σ 2 λ Exponential R t 2 a e ∆t R()∆t 2 π v − ∆t ()∆ = σ 2 λ R t 2 a e Gaussian ∆t R()∆t π v σ ∆ sin 2λ t 2 R()∆t = 2 ∆ “Rectangular” a π v t 2 ∆t λ
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189 Characterization of time variations using the spaced-time correlation function
R(∆t)
∆t 0 T0
• 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…
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190 Doppler Power Spectrum Frequency Domain View of Time-Variations
R(∆t) S()ν
Fourier Xform
∆t ν 0 T0 fd
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.
ν Doppler Spread of the channel fd is the range of values of over which the Doppler power spectrum is essentially non zero.
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191 Doppler Power Spectrum and Doppler Spread
CompareCompare Doppler Doppler Spread Spread f df with with S()ν d transmittedtransmitted signal signal bandwidth bandwidth W W: : ⇒ fd > W ⇒ fast fading fd > W fast fading ⇒ fdf < < W W ⇒ slow slow fading fading ν d fd
equations (18) - (21) are commonly used relationships between Doppler spread and coherence time
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192 Common Doppler Power Spectra
ν S()ν σ ()= 2δ ν () Time Invariant S 2 a ν ν σ S()ν Land Mobile 2 2 S()= ν a (Jakes) 2 − v / λ()2 ν ()ν ν σ S π 2 2v λ/ Exponential S()= ν a ()2 + λ()2 (1st order Butterworth) v / ν S()ν
ν ν 2 σ 2 − Gaussian ()= 2 a ()v / λ 2 S π e ν ν σ v /()λ 2 S()ν λ λ 2 ν a − < < λ S()= v / v / “Rectangular” v / ν 0 otherwise
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193 Putting it all together…
Fourier transform S(τ;∆t) S(∆f ;∆t) spaced-frequency, spaced-time correlation function τ ↔ ∆f
R(∆t) ∆t = 0 ∆f = 0 R()∆f spaced-frequency spaced-time correlation function correlation function f ∆t 0 T0 f0
Fourier transform Fourier transform
S()τ S()ν multipath intensity Doppler power profile spectrum power τ τ τ τ ν 1 2 N −1 f τ ν τ ν d ∫ S( ; )dν ∫ S( ; )dτ
S(τ ;ν ) scattering function
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194 Scattering Function
S(τ ;ν )
delay τ
frequency ν
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195 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.,
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