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EE359 – Lecture 5 Outline
Announcements:HW posted, due Thursday 5pmBackground on random processes in
Appendix BLecture notes: No need to take notes unless
you want to! Most important aspects of lecture are in the
handouts Everything is in the book My lecture notes available by request
Review of Last LectureNarrowband Fading ModelIn-Phase and Quad Signal
ComponentsCrosscorrelation of RX Signal in
NB FadingCorrelation and PSD in uniform
scatteringSignal Envelope Distributions
Review of Last Lecture
Model Parameters from Measurements Random Multipath Model Channel Impulse Response
Received signal characteristicsMany multipath componentsAmplitudes change slowlyPhases change rapidly
))(()(),(1
)( tettc n
N
n
tjn
n
Narrowband Model Assume delay spread maxm,n|tn(t)-tm(t)|
<<1/B Then u(t)u(t-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)=ejf0, yielding
)(
0
)(2 )()()(tN
n
tjn
tfj nc etetutr
)2sin()()2cos()()( 00 tftrtftrtr cQcI
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 vars).
Received signal characterized by its mean, autocorrelation, and cross correlation.
If jn(t) uniform, the in-phase/quad components are mean zero, indep., and stationary.
),(cos)()()(
0
tttr n
tN
nnI
)(sin)()()(
0
tttr n
tN
nnQ
0)()(2)( ttft
nDncn
Auto and Cross Correlation
Recall that qn is the multipath arrival angle
Autocorrelation of inphase/quad signal is
Cross Correlation of
inphase/quad signal is
Autocorrelation of received signal
is
/cos],2[cos)()( nDDrr vffPEAAnnnQI
)(]2[sin)( ,, QInnQI rrDrr AfPEA
)2sin()()2cos()()( , crrcrr fAfAAQII
),2cos()()()(
0
)( tfettr c
tN
n
tjnI
n
)2sin()()()(
0
)( tfettr c
tN
n
tjnQ
n
, fn~U[0,2p]
Uniform AOAs
Under uniform scattering, in phase and quad comps have no cross correlation and autocorrelation is
The PSD of received signal is
)2()()( 0 Drr fPJAAQI
Decorrelates over roughly half a wavelength
)]2([)(
)]()([25.)(
0 Dr
crcrr
fPJfS
ffSffSfS
I
II
F
fc+fDUsed to generate simulation values fc
Sr(f)
fc-fD
Signal Envelope Distribution
CLT approx. leads to Rayleigh distribution (power is exponential)
When LOS component present, Ricean distribution is used
Measurements support Nakagami distribution in some environmentsSimilar to Ricean, but models “worse than
Rayleigh”Lends itself better to closed form BER
expressions
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 comps of RX signal follow Bessel functionSignal components decorrelate over half
wavelength The PSD has a bowel shape centered at carrier
frequency
Fading distribution depends on environmentRayleigh, Ricean, and Nakagami all
common
