1 EE359 – Lecture 6 Outline l Announcements: l HW due tomorrow 4pm l Review of Last Lecture l Wideband Multipath Channels l Scattering Function l Multipath Intensity Profile l Doppler Power Spectrum 1 Review of Last Lecture l For f n ~U[0,2p], r I (t),r Q (t) zero mean, WSS, with l Uniform AoAs in Narrowband Model l In-phase/quad comps have zero cross correlation and l PSD maximum at the maximum Doppler frequency l PSD used to generate simulation values ) 2 ( ) ( ) ( 0 t p t t D r r r f J P A A Q I = = Decorrelates over roughly half a wavelength l q t t p t q / cos ), ( ] 2 [cos ) ( n D r D r r v f A f E P A n Q n n I = = = ) ( ] 2 [sin ) ( , , t t p t q Q I n n Q I r r D r r r A f E P A - = = .4l vt=d fc+fD Sr(f) fc fc-fD 1 fD=v/l 2 Review Continued: Signal Envelope Distribution l CLT approx. leads to Rayleigh distribution (power is exponential * ) l When LOS component present, Rician distribution is used l Measurements support Nakagami distribution in some environments l Similar to Ricean, but models “worse than Rayleigh” l Lends itself better to closed form BER expressions To cover today *Correct in lecture 5 handout; Reader corrections on next slide Reader correction 3 Reader correction: Rayleigh Distribution (Section 3.2.2, pp. 87-88, correct in 1 st Ed.) l X and Y zero-mean Gaussian, variance s 2 , define Z: l Signal envelope z(t)=|r(t)|; r(t) has power P r =2 s 2 l Envelope: Z, z(t), and|r(t)| are Rayleigh distributed l Power: Z 2 , z 2 (t), and |r(t)| 2 are exponentially distributed 4
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1
EE359 – Lecture 6 Outline
l Announcements:l HW due tomorrow 4pm
l Review of Last Lecturel Wideband Multipath Channelsl Scattering Functionl Multipath Intensity Profilel Doppler Power Spectrum
1
Review of Last Lecturel For fn~U[0,2p], rI(t),rQ(t) zero mean, WSS, with
l Uniform AoAs in Narrowband Modell In-phase/quad comps have zero cross correlation and
l PSD maximum at the maximum Doppler frequencyl PSD used to generate simulation values
)2()()( 0 tptt Drrr fJPAAQI
==Decorrelates over roughly half a wavelength
lqttpt q /cos),(]2[cos)( nDrDrr vfAfEPAnQnnI===
)(]2[sin)( ,, ttpt q QInnQI rrDrrr AfEPA -==
.4l
vt=d
fc+fD
Sr(f)
fcfc-fD
1 fD=v/l
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Review Continued:Signal Envelope Distribution
l CLT approx. leads to Rayleigh distribution (power is exponential*)
l When LOS component present, Riciandistribution is used
l Measurements support Nakagamidistribution in some environmentsl Similar to Ricean, but models “worse than
Rayleigh”l Lends itself better to closed form BER
expressions
To cover today
*Correct in lecture 5 handout; Reader corrections on next slide
Reader correction
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Reader correction: Rayleigh Distribution (Section 3.2.2, pp. 87-88, correct in 1st Ed.)
l X and Y zero-mean Gaussian, variance s2, define Z:
l Signal envelope z(t)=|r(t)|; r(t) has power Pr=2s2
l Envelope: Z, z(t), and|r(t)| are Rayleigh distributed
l Power: Z2, z2(t), and |r(t)|2 are exponentially distributed
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Wideband Channelsl Individual multipath components resolvablel True when time difference between
components exceeds signal bandwidthl High-speed wireless systems are wideband for
most environments
uB/1<<Dt uB/1>>Dt
t t1tD 2tD
Narrowband Wideband
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Scattering Functionl Typically characterize c(t,t) by its statistics,
since it is a random processl Underlying process WSS and Gaussian, so
only characterize mean (0) and correlationl Autocorrelation is Ac(t1,t2,Dt)=Ac(t,Dt)
l Correlation for single mp delay/time differencel Statistical scattering function:
l Average power for given mp delay and doppler
t
rs(t,r)=FDt[Ac(t,Dt)]
Easy to measure
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Multipath Intensity Profile
l Defined as Ac(t,Dt=0)= Ac(t)l Determines average (µTm ) and rms (sTm) delay spread l Approximates maximum delay of significant multipath
l Coherence bandwidth Bc=1/sTm
l Maximum frequency over which Ac(Df)=F[Ac(t)]>0l Ac(Df)=0 implies signals separated in freq. by Df will be
uncorrelated after going through channel: freq. distortion
t
Ac(t)TM
uT Bm
/1>>s
t1tD 2tD f
cu BB >>
Ac(f)0 Bc
Wideband signal distorted in time and in frequency
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Doppler Power Spectrum
l Doppler Power Spectrum: Sc(r)=FDt [Ac(Df=0,Dt)≜Ac(Dt)]
l Power of multipath at given Dopplerl Doppler spread Bd: Max. doppler for which Sc (r)=>0.
l Coherence time Tc=1/Bd: Max time over which Ac(Dt)>0l Ac(Dt)=0Þ signals separated in time by Dt uncorrelated after passing through channel
l Why do we look at Doppler w.r.t. Ac(Df=0,Dt)?l Captures Doppler associated with a narrowband signall Autocorrelation over a narrow range of frequenciesl Fully captures time-variations, multipath angles of arrival
Ac(Df,Dt)=Ft[Ac(t,Dt)]
r
Sc(r)
Bd
Scattering Function: s(t,r)=FDt[Ac(t,Dt)]
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Main Pointsl Wideband channels have resolvable multipath
l Statistically characterize c(t,t) for WSSUS modell Scattering function characterizes rms delay and Doppler
spread. Key parameters for system design.
l Delay spread defines maximum delay of significant multipath components. Inverse is coherence BWl Signal distortion in time/freq. when delay spread exceeds
inverse signal BW (signal BW exceeds coherence BW)
l Doppler spread defines maximum nonzero doppler, its inverse is coherence timel Channel decorrelates over channel coherence time
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