Announcement

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1

Review Ch.2

Power decays with distance

Path-loss

Shadowing

Multiplath

+ ?

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

Statistical Multipath Model

t

t

0

τ1 τ2 τ3

Delay spread = τ3- τ1

≈ 1/B

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

Narrowband Channel

t

t

0

τ1τ2τ3

Delay spread = τ3- τ1 << 1/B

≈ 1/B

Wideband Channel

t

t

0

τ1 τ2 τ3

Delay spread = τ3- τ1 >> 1/B

Text

≈ 1/B

! 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

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

t

t

0

τ1τ2τ3

Delay spread = τ3- τ1 << 1/B

≈ 1/B

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

t

t

0

τ1 τ2 τ3

Delay spread = τ3- τ1 >> 1/B

Text

≈ 1/B

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

c(⌧, t) =X

n

↵n(t)e�j�n(t)�(⌧)

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

J_0 (not the same as I_0!)! The zeroth-order Bessel function of the

first kind

25

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

Rayleigh Fading

27

pdf cdf

Rice Fading

28

pdf cdf

\nu = “mean of G”\sigma = “std of G”

Nakagami Fading

29

pdf cdf

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 Channel

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

Visual Representation

36

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.