cross correlation - stanford university · 3 main points lnarrowband model has in-phase and quad....
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EE359 – Lecture 5 Outlinel Announcements:
l HW posted, due Friday 4pml Background on random processes in Appendix B
l Review of Last Lecture: Narrowband Fadingl Auto and Cross Correlation of In-Phase and
Quadrature Signal Componentsl Correlation and PSD in uniform scatteringl Signal Envelope Distributionsl Wideband Channels and their Characterization
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Review of Last Lecturel Model Parameters from Measurementsl Random Multipath Modell Channel Impulse Response
l Many multipath components, Amplitudes change slowly, Phases change rapidly
l For delay spread max|tn(t)-tm(t)|<<1/B, u(t)»u(t-t).l Received signal given by
))(()(),(1
)( tettc n
N
n
tjn
n ttdat j -=å=
-
þýü
îíì
úû
ùêë
éÂ= å
=
-)(
0
)(2 )()()(tN
n
tjn
tfj nc etetutr fp a- No signal distortion in time - Multipath yields complex
scale factor in brackets
ii i
i
ii
i
2
l For u(t)=ejf0,l In-phase and quadrature signal components:
l For N(t) large, rI(t) & rQ(t) jointly Gaussian by CLTl Received signal characterized by its mean,
autocorrelation, and cross correlation. Let
l If ji(t) uniform, in-phase/quad components are mean zero, independent, and stationary (WSS)
0)()(2)( fftpf --= ttftnDncn
)2sin()()2cos()()( 00 fpfp +-+= tftrtftrtr cQcI
Review Continued:Narrowband Model
),2cos()()()(
0
)( tfettr c
tN
n
tjnI
n pa få=
-= )2sin()()()(
0
)( tfettr c
tN
n
tjnQ
n pa få=
-=
fi~U[0,2p]
ii
ii ii
i ii
Eqi
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Cross Correlation
l Cross Correlation of inphase/quad signal is
l Thus, , so rI(t) and rQ(t) independent
l Autocorrelation of received signal is
l Thus, r(t) is stationary (WSS)
)2sin()()2cos()()( , tpttptt crrcrr fAfAAQII
-=
),2cos()()()(
0
)( tfettr c
tN
n
tjnI
n pa få=
-= )2sin()()()(
0
)( tfettr c
tN
n
tjnQ
n pa få=
-= , fi~U[0,2p]
0)0(, =QI rr
A
i ii i ii
Eqi
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Uniform Scatteringl Multipath comes uniformly from all directions
l Power in each component is the same:
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Autocorrelation and PSD under uniform scattering
l Under uniform scattering, in phase and quad comps have no cross correlation and autocorrelation is
l The PSD of received signal is
)2()()( 0 tptt Drrr fJPAAQI
==
Decorrelates over roughly half a wavelength
)]2([)(
)]()([25.)(
0 tp Drr
crcrr
fJPfS
ffSffSfS
I
II
F=
++-=
fc+fDUsed to generate simulation values fc
Sr(f)
fc-fD
.4l
vt=d
1 fD=v/l
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Signal Envelope Distribution
l CLT approx. leads to Rayleigh distribution (power is exponential)
l When LOS component present, Ricean distribution 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
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Wideband Channelsl Individual multipath components resolvablel True when time difference between components
exceeds signal bandwidthl Requires statistical characterization of c(t,t)
l Assume CLT, stationarity and uncorrelated scatteringl Leads to simplification of its autocorrelation function
uB/1<<Dt uB/1>>Dt
t t1tD 2tD
Narrowband Wideband
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Main Pointsl Narrowband model has in-phase and quad. comps
that are zero-mean stationary Gaussian processesl Auto and cross correlation depends on AOAs of multipath
l Uniform scattering makes autocorrelation of inphaseand quad comps of RX signal follow Bessel functionl Signal components decorrelate over half wavelengthl The PSD has a bowel shape centered at carrier frequency
l Fading distribution depends on environmentl Rayleigh, Ricean, and Nakagami all common
l Wideband channels have resolvable multipathl Will statistically characterize c(t,t) for WSSUS model
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