1 sysc4607 – lecture 5 outline announcements: tutorial important: review of probability theory and...
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SYSC4607 – Lecture 5 Outline
Announcements:Tutorial important: Review of Probability
Theory and Random Processes
Review of Last Lecture Narrowband Fading Model
In-Phase and Quad Signal Components
Auto and Crosscorrelation
Signal Envelope Distributions
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Review of Last Lecture
Model Parameters from Empirical DataObtain path loss model from MMSE curve fitObtain STD of data relative to path loss
model (for log-normal shadowing)
Random Multipath Model
Time Varying Channel Impulse Response))(()(),(
)(
0
)( tettc n
tN
n
tjn
n
-Random (large) number of components-Random delays/gains/doppler on each
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Narrowband Model
Assume delay spread maxm,n|n(t)-m(t)|<<1/B
(nonresolvable multipath components) Then u(t)u(t-). Received signal given by
No signal distortion (spreading in time) Multipath affects complex scale factor in
brackets. Characterize scale factor by setting u(t)=ej0
)(
0
)(2 )()(Re)(tN
n
tjn
tfj nc etetutr
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In-Phase and Quadratureunder CLT
Approximation In phase and quadrature signal components:
For N(t) large, rI(t) and rQ(t) jointly Gaussian by CLT (sum of large # of random variables).
Received signal is also Gaussian characterized by its mean and autocorrelation function.
If n(t) uniform, the in-phase/quad components are zero-mean, and jointly WSS with the same autocorrelation function (next slide).
),(cos)()()(
0
tttr n
tN
nnI
)(sin)()()(
0
tttr n
tN
nnQ
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Auto and Cross Correlation
Assume n~U[0,2]
Recall that n is the multipath arrival angleAutocorrelation of inphase/quad signal is
Cross Correlation of inphase/quad signal is
Autocorrelation of received signal is
)2sin()()2cos()()( , crrcrr fAfAAQII
/)cos()],2[cos()(5.0)()( 2nDD
nnrr vffEEAA
nnQI
)]2[sin()(5.0)()( 2,,
nIQQI Dn
nrrrr fEEAA
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Uniform AOAs
Under uniform scattering, in phase and quad components have no cross-correlation (independent) and autocorrelation is
The PSD of received signal is
)2()()( 0 Drrr fJPAAQI
Decorrelates over roughly half a wavelength
)]2([)(
)]()([5.0)(
0 Drr
crcrr
fJPfS
ffSffSfS
I
II
F
fc+fDUsed for simulating the fading process fc
Sr(f)
fc-fD
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Signal Envelope Distribution
CLT approx. leads to Rayleigh distribution for envelope (power is exponential)
When LOS component present, Rician distribution is used
Measurements support Nakagami distribution in some environmentsHas both Rayleigh and Rician as special casesAlso models “worse than Rayleigh” situations
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Main Points
Narrowband model has in-phase and quad. comps that are zero-mean stationary Gaussian processesAuto and cross correlation depends on AOAs of
multipath
Uniform scattering makes autocorrelation of inphase and quad follow Bessel functionSignal components decorrelated over half
wavelengthCross correlation is zero (in-phase/quadrature
indep.)
Fading distribution depends on environmentRayleigh, Rician, and Nakagami all common