ece437_ch3
Post on 10-Dec-2015
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Announcement
! The TA prefers to have the paper copy of the HW. From next time, please bring paper copy of your HW solution.
1
C = 1/2 + e2/b2
Q(2/b),
d :=
10� log10
(e)
�dB
,
Pmin,dB =
¯Prx,dB(R)
1� Pout
(d, Pmin
)
= Q
✓Pmin,dB � ¯P
rx,dB(d)
�dB
◆
¯Prx,dB(d) = P
tx,dB +KdB
� 10� log10
d
d0
PL+S =Outage Probability& Cell Coverage Area
Statistical Multipath Model
! Random number of multipath components, each with ! Random amplitude ! Random phase ! Random delay ! Random Doppler shift (which we’ll consider later on)
! These components change with time (can either add constructively or destructively)
! Leads to time-varying channel impulse response
Time Varying Channel
! Each path is associated with ergodic and stationary: ! attenuation α(t) (path-loss and shadowing),
! delay τ(t) = r(t)/c,
! phase φ(t) = 2 π [ fc τ(t) - ∫u<t v/λ cos(θ(u)) du ]
with: c = speed of light, r(t) = path length, fc = carrier frequency, λ = wavelength, v = velocity, θ(t) = angle of arrival.
Signal Model (Appendix A)! Bandpass real-valued signal
! Transmit signal: s(t) bandpass, u(t) baseband
! Channel: hl(t) baseband: ! Received signal:
! Baseband signals: u(t) sent, v(t) received: i.e., time varying linear system.
! How the transmitted signal u(t) is affected by the channel depends on how its bandwidth B compares with the channel delay spread. Two cases to consider: ! narrowband channel and ! wideband channel.
Time Varying Channel
! Response of channel at t to impulse at t-τ:
! t is time when impulse response is observed ! t-τ is time when impulse got into the channel ! τ is how long ago impulse was put into the
channel for the current observation ! φn(t) = 2 π fc τn - φDn ! fc τn can be quite large --> huge phase variations
over small time
Time Varying Channel
Delay Spread
! Delay spread := maxm `not=‘ n|τn(t)-τm(t)|
! Delay spread << 1/bandwith: narrowband fading, non-resolvable fading = no inter-symbol interference
! Delay spread >> 1/bandwith: wideband fading, or resolvable fading = inter-symbol interference
! Delay spread is a random variable --> needs statistical characterization
! If path attenuation is too low then that path component is buried in noise and should not be counted in the delay spread --> power delay profile
! Indoor: fraction of microseconds; Outdoor: several microseconds;
Narrowband Channel
! Delay spread = maxm,n|τn(t)-τm(t)|<<1/B,then u(t)≈u(t- τn(t)) for all n (non-resolvable multipath)
! Channel input response becomes
! No signal distortion/spreading in time ! Multipath = complex scale factor attenuation
! Multipath = random variable
c(⌧, t) =X
n
↵n(t)e�j�n(t)�(⌧)
Wideband Channel
! Delay spread = maxm,n|τn(t)-τm(t)|>>1/B, delays are a continuum (resolvable multipath)
! Channel response given by
! A continuum of multipath components ! Signal distortion/spreading in time ! Multipath = random process
Narrowband Channel
! Delay spread << 1/B u(t)=1 (unmodulated carrier)
in-phase component quadrature component
r(t) = <{u(t)ej2⇡fct
X
n
↵n(t)e�j�n(t)
!}
= rI(t) cos(2⇡fct)� rQ(t) sin(2⇡fct)
c(⌧, t) =X
n
↵n(t)e�j�n(t)�(⌧)
In-Phase and Quadrature under CLT Approximation
! In phase and quadrature signal components:
! Without a dominant LOS, for N(t) large, we model rI(t) and rQ(t) as jointly Gaussian (because sum of large number of random variables -- by CLT).
! We assume φn(t) uniform and independent of αn(t). ! The received signal is thus completely characterized
by its mean, autocorrelation, and cross correlation.
Auto and Cross Correlation
! Assume φn~U[0,2π]
! Recall that θn is the multipath arrival angle
! Autocorrelation of in-phase/quad signal is
! Cross-correlation of in-phase/quad signal isP = average rx power = 1/2 sum_{n} E[α^2_n].
Uniform AOAs
! Under uniform scattering, in phase and quad comps have no cross-correlation and auto-correlation is
! The PSD of received signal is
Decorrelates over roughly half a wavelength
fDUsed to generate simulation values 0
SrI(f)
-fD
Signal Envelope Distribution
! CLT approx. leads rI and rQ to be iid Gaussian; the the envelope / modulo of rI +j rQ has a Rayleigh distribution (power is exponential, phase is uniform, and they are independent)
! When LOS component present, Ricean distribution
! Measurements support Nakagami distribution in some environments ! Similar to Ricean, but models “worse than Rayleigh” ! Lends itself better to closed form BER expressions
Level crossing rate and Average Fade Duration
! LCR: rate at which the signal crosses a specific value
! AFD: How long a signal stays below target R/SNR ! Derived from LCR
! For Rayleigh fading
! Depends on ratio of target to average level (ρ) ! Inversely proportional to Doppler frequency
Rt1 t2 t3
Wideband Channels
! Individual multipath components resolvable ! True when time difference between
components exceeds signal bandwidth
τ τNarrowband Wideband
Scattering Function
! Typically characterize its statistics, since c(τ,t) is different in different environments
! Underlying process WSS and Gaussian, so only characterize mean (0) and correlation
! Autocorrelation is Ac(τ1,τ2,Δt)=Ac(τ,Δt) ! Statistical scattering function:
τ
ρS(τ,ρ)=FΔt[Ac(τ,Δt)]
Multipath Intensity Profile
! Defined as Ac(τ,Δt=0)= Ac(τ)
! Determines average (TM ) and rms (στ) delay spread ! Approximate max delay of significant m.p.
! Coherence bandwidth Bc=1/TM
! Maximum frequency over which AC(Δf)=F[Ac(τ)]>0
! AC(Δf)=0 implies signals separated in frequency by Δf will be uncorrelated after passing through channel
τ
Ac(τ)TM
τ fAc(f)
0 Bc
Doppler Power Spectrum
! Sc(ρ)=F[Ac(τ=0,Δt)]= F[Ac(Δt)]
! Doppler spread BD is maximum Doppler freq. for which Sc (ρ)>0.
! Coherence time Tc=1/BD
! Maximum time over which Ac(Δt)>0
! Ac(Δt)=0 implies signals separated in time by Δt will be uncorrelated after passing through channel
ρ
Sc(ρ)
Bd
Main Points: narrowband! Statistical multipath model leads to a time-varying channel
impulse response
! Two multipath models: narrowband and wideband
! Narrowband model and CLT lead to in-phase/quad components that are jointly stationary Gaussian processes ! Processes completely characterized by their mean, auto-
correlation, and cross-correlation. ! Zero-mean ! Auto and cross-correlation depends on AOAs of
multipath.
Main Points: narrowband
! Uniform scattering makes autocorrelation of in-phase and quad follow Bessel function ! Signal components decorrelate over half wavelength ca. ! If cross correlation is zero in-phase/quadrature are indep.
! The power spectral density of the received signal has a bowel shape centered at carrier frequency ! PSD useful in simulating fading channels
! Narrowband fading distribution depends on environment ! Rayleigh, Ricean, and Nakagami all common
Main Points: wideband
! Wideband fading ! Scattering function characterizes rms delay spread and
Doppler spread. Key parameters for system design. ! Delay spread defines maximum delay of significant
multipath components. Inverse is coherence bandwidth of channel.
! Doppler spread defines maximum nonzero Doppler, its inverse is coherence time.
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