Ofdm Channels GREAT

Channels and Channel Models

EIT 140, tom<AT>eit.lth.se

Channel types: single-user / multi-user

Depending on the topology of the channel, we distinguish single-user channelsand multi-user channels:

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Single-user channel

Multiple-access channel

Broadcast channel

Interference channel

Channel types: SISO, MIMO, SIMO, MISODepending on the number of ports of a user-to-user link, we distinguish

Single-input single-output (SISO) channel

Single-input multi-output (SIMO) channel

Multi-input single-output (MISO) channel

Multi-input multi-output (MIMO) channel

SISO

SIMO

MISO

MIMO

= port-to-port link

Examples:

Multi-antenna systems

Multi-pair cables

Channel types: information theoretic view

Sometimes, all the blocks like modulation, demodulation, up-conversion,(physical) channel, down-conversion, etc. are modelled as a single entity calleddigital channel:

Modulator DemodulatorChannel

Channel

Digital

Up-Converter Down-Converter A/D ConverterD/A Converter

“digital” refers to the quantisation in amplitude (the set of outputsymbols is finite)

digital channel is described by transition probabilities p(yk |xl ), i.e., theconditional probabilities that yk is detected given that xl was transmitted

Channel types: classification according to medium

Depending on the medium, we distinguish

guided channels

wire (e.g.: copper twisted-pairs in the access network)cable (e.g.: coax cables used in cable networks)fibre (e.g.: optical fibres in backbone networks)microwave guides (e.g.: feeder “pipes” for high-power RFtransmitters, radar)

unguided channels

wireless channelunderwater acoustic channel

Channel properties

The transmitted waveforms may experience effects like

reflection

absorption

attenuation (scaling in amplitude)

dispersion (spreading) in time

refraction (bending due to variation of the media’s refraction index)

diffraction (scattered re-radiation, caused by an edge or an object whosesize is in the order of the wave length)

Channel properties cont’d

The net effect of every channel can be described by

modification of the signal

addition of noise

sequence notation: r(n) = h(n)∗s(n) + w(n) matrix notation: r = Hs + w

Depending on the channel properties, a channel can be

linear / non-linear channels

time-invariant / time-variant (fading) channels

frequency-flat / frequency-selective (time-dispersive) channels

The additive noise can be

Gaussian / non-Gaussian

correlated in time/frequency, spatially (in MIMO system), over users (inmulti-user systems)

Wireline channel: physical mechanisms/effects

essentially time-invariant, frequency-selective attenuation, orequivalently, dispersion in time

crosstalk: electromagnetic coupling among wire pairs (also calledloops) in a cable

extrinsic noise/interference (impulse noise, radio frequencyinterference)

background noise (thermal noise, front-end noise)

Wireline channel: physical mechanisms/effects cont’d

1 12 2

......

K K

RFI impulses

FEXT NEXT

side A side B

Far-end crosstalk (FEXT)

Near-end crosstalk (NEXT)

Impulse noise

Radio frequency interference (RFI)

Wireline channel: modelling as LTI system

Assuming proper termination, the insertion loss can be modelled as LTIsystem:

Hloop(f , d)=e− d

d0(k1

√f +k2f )

e−j d

d0k3f , (47)

where

f is the frequency in Hz

d is the length of the loop in m

k1, k2, k3 are constants depending on the diameter of the wire;exemplary values for 0.5mm loop:k1 =3.8 · 10−3, k2 =−0.541 · 10−8, k3 =4.883 · 10−5

Wireline channel: modelling as LTI system

Assuming proper termination, the NEXT coupling and FEXT couplingcan be modelled via LTI systems:

HFEXT(f , d)=kff

f0

√

d

d0|Hloop(f , d)|, kf =10−45/20, f0 =1MHz, d0 =1 km

(48)

HNEXT(f , d)=kn

(f

f0

) 34 √

1− |Hloop(f , d)|4, kn =10−50/20, f0 =1MHz

(49)where

f is the frequency in Hz

d is the coupling length of the loops in m

Wireline channel: receive PSDs

frequency f in MHz

PSD

indB

m/H

z

Signal

−150

−130

−110

−90

−70

−50

0 5 10 15

NEXTFEXTAWGN500m1000m2000m

Transmit signal PSD: flat −60 dBm/Hz

Wireless channel: physical mechanisms/effects

Fixed terminals

Path lossBackground noise

Mobile terminal(s)

Path lossBackground noiseDoppler effectTime-varying impulse response

→ dispersion in frequency→ receive signal amplitude fluctuations (fading)

Dispersion in time, or equivalently, frequency selectivity

Obstacle-free transmission: path loss

The receive signal power is given by

Pr = PtGtGrLp. (50)

Pt is the transmit power

Gt is the transmit antenna gain (ratio of the received powercompared to the power an isotropic antenna would receive; for adish antenna with effective area A, the antenna gain is roughlyG ≈ 4πA/λ2),

Gr is the receive antenna gain and

Lp is the free-space path loss, given by

Lp =

(λ

4πd

)2

(51)

d distanceλ wavelength

Presence of obstacles: ray tracing

Simple two-ray model

d

hthr

Pr = PtGtGrh2t h2

r

d4(d2 ≫ hthr) (52)

ht height of transmit antenna

hr height of receive antenna

Simplified path loss model

Pr = PtGtGr Pr(d0)/Pt︸ ︷︷ ︸

K

(d0

d

)γ

(53)

d0 reference distance

K ratio of receive and transmit power for d0

path loss exponent γ, depends on wavelength and environment,typically in the range 2− 8 for 1GHz

Mobile terminal(s)

Most often, only one of the terminals is moving, which we call themobile terminal (MT). The other one, the fixed terminal (FT), doesnot move

Due to reciprocity, it does not matter whether we observe downlink(FT → MT) or uplink (MT → FT)

Mobile terminal(s): Doppler effect

When the FT transmits a signal with frequency f = c/λ, the MTreceives this signal at frequency f + ν = f + v ′/λ, where v ′ is therelative velocity of the MT with respect to the FT.

Note that v ′ is a signed quantity

FT

MT1

MT2

v1

v2

v ′1 = v1 cos α1

v ′2 = −v2 cos α2

α1

α2

ν is called the Doppler shift (example: ν ≈ 83 Hz for 100 km/h and900 MHz)

Mobile terminal(s): characterisation of effects

Dispersion in time

transmitted beam is reflected and scattered along the way →multi-path propagationoften, there is neither a direct beam from the FT to the MT nor astationary reflection (both of which are referred to as line of sight(LOS) components)if the beams arrive with different delays, time-dispersion of thetransmitted signal occurs

Dispersion in frequency

if either the MT or scatterers are moving, each received beam has adifferent relative velocity with respect to the MT →frequency-dispersion of the transmitted signal occursmotion is not the exclusive cause of frequency dispersion; moregenerally, frequency dispersion is caused by a time-varying channelimpulse response

Fluctuations in amplitude (fading)

Characterisation of amplitude fluctuations (fading)

path loss (dotted green curve)

large-scale (macroscopic) fading (dashed blue curve)

small-scale (microscopic) fading (solid red curve)

time (∝ distance)

amplit

ude

indB

Small-scale fading: Rayleigh distribution, Rice distribution

FT

MT v

ψ1

ψ2

many waves arrive from arbitrary directions

the amplitudes AI ∼ N (mI , σ2) and AQ ∼ N (mQ , σ2) of inphaseand quadrature receive component, respectively, are independentand Gaussian distributed (central limit theorem)

no LOS component: mI = mQ = 0 → amplitude U =√

A2I+ A2

Q

has Rayleigh distribution

with LOS component: mI 6= 0 and/or mQ 6= 0 → amplitude U hasRice distribution

Small-scale fading: Rayleigh/Rice distribution cont’d

Rayleigh distribution:

pU (u) =u

σ2e− u2

2σ2 , u ≥ 0. (54)

Rice distribution:

pU (u) =u

σ2e− u2+s2

2σ2 I0(us

σ2), u ≥ 0; s =

√

m2I

+ m2Q

= A0.

(55)

u

pU

(u)

A0 = 0

A0 = 1

A0 = 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5

Large-scale fading

Models the channel property changes caused by movement of theMT

Characterises the mean value of the small-scale fading model

The log-normal distribution has been found to yield a good matchwith measurements

The mean value in dB γdB is Gaussian distributed

pγdB(γdB) =

1√2πσdB

e− (γdB−mdB)2

2(σdB)2 , (56)

where σdB, the standard deviation of γdB, is typically in the range of6-12 dB.Then the distribution of γ = 10γdB/20 is given by

pγ(γ) =20

γ√

2πσdB ln 10e− (20 log10 γ−mdB)2

2(σdB)2 . (57)

Charaterisation of dispersion in time and frequency

Deterministic analysis

channel is modelled as linear time-variant (LTV) system, describedby a time-variant impulse response h(τ, t)

time-variant frequency response H(f , t) = Fτ{h(τ, t)}delay Doppler spreading function s(τ, ν) = Ft{h(τ, t)}output Doppler spreading function B(f , ν) = Ft{H(f , t)}

Charaterisation of dispersion in time and frequency

Stochastic analysis

auto-correlation function Rhh(τ1, τ2, t1, t2) = E {h∗(τ1, t1)h(τ2, t2)}of the impulse response

wide-sense stationarity (WSS) assumption: Rhh(τ1, τ2, t1, t2)depends only on the time difference ∆t = t2 − t1uncorrelated scattering (US) assumption: scatterers actindependently → it is sufficient to observe Rhh(τ, t1, t2)WSS + US → WSSUS assumption: it is sufficient to observe thedelay cross-power spectral density Rhh(τ, ∆t)

time-frequency correlation function

RHH (∆f , ∆t) = Fτ{Rhh(τ, ∆t)}scattering function Rs (τ, ν) = F∆t{Rhh(τ, ∆t)}Doppler cross-power spectral density

RB (∆f , ν) = F∆t{RHH (∆f , ∆t)}

Exemplary system functions

Summary of wireless channel characterisation measures

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h(τ, t)

Time-variantimpulse response

H(f , t)

Time-variantfrequency response

s(τ, ν)

Delay Dopplerspreading function

B(f , ν)

Output Dopplerspreading function

Rhh (τ, ∆t)

Delay cross-powerspectral density

RHH (∆f , ∆t)

Time-frequencycorrelation function

Rs (τ, ν)

Scattering function

RB (∆f , ν)

Doppler cross-powerspectral density

P(τ)

Delay powerdensity spectrum

RH (∆f )

Frequency correlation

RH (∆t)

Time correlation

P(ν)

Doppler powerdensity spectrumFτ (∆f )

Fτ (∆f )

Fτ (∆f )

Fτ(f

)F

τ(f

) Ft (ν)Ft (ν) F∆t (ν)F

∆t (ν) F∆t (ν)

∆f=

0∆f=

0

∆t =0∆t =0

Ct (∆t) Cf ,t (∆f , ∆t)

| · |2 Cf (∆f )

(·)−1

(·)−1

TcohCoherence

time

BdopDoppler

bandwidth

TmultiMulti-path

spread

BcohCoherencebandwidth

Charaterisation of dispersion in time and frequency

Two functions commonly used in practice:

1 delay power density spectrum (power delay profile)P(τ) = Rhh(τ, ∆t)|

∆t=0 specifies time-dispersion (or equivalently,frequency-selectivity) characteristic

2 Doppler power density spectrum (Doppler spectrum)P(ν) = RB (∆f , ν)|

∆f =0 specifies frequency dispersion, orequivalently, the correlation of realisations observed over time of agiven coefficient of the tapped delay line filter

Charaterisation of dispersion in time and frequency

Two scalars commonly used in practice:

1 The multi-path spread Tmulti specifies the approximate support ofthe power delay profile P(τ), or equivalently, the approximate lengthof the channel impulse response. Dual measure: coherence

bandwidth Bcoh ≈ 1/Tmulti.

2 The coherence time Tcoh specifies the approximate support of thetime correlation function RH (∆t), or equivalently, the time duringwhich the impulse response remains constant. Dual measure:Doppler bandwidth Bdop ≈ 1/Tcoh.

Assessment of wireless channels

The parameters Tcoh and Tmulti of a wireless channel have to be seen incontext with symbol period Tsym of the system.

replacementsBdop

Bcoh

Tmulti

Tcoh

frequency selectivefast fading

frequency flatfast fading

frequency selectiveslow fading

frequency flatslow fading

Ergodicity

useful description of a linear channel with additive noise:

r = Hs + n

s ∈ CS : channel input. r ∈ CR : channel output. H ∈ CR×S :channel matrix. n ∈ CR : additive noise.

Ergodic channelrn = Hnsn + nn.

Hn are realizations of a random process. Transmittedsymbol/codeword sn, n = 0, 1, . . . , N ; N ≫ “sees” all channelstates. Valid for fast fading channels.

Nonergodic channel: Consider the model

rn = Hsn + nn,

Here, H is constant over the symbol/codewordsn, n = 0, 1, . . . , N ; N ≫. Transmitted symbol/codeword “sees”only one state (H). Valid for slowly fading channels.

Block fading

Interleavers spreads out codewords in time and/or frequency.

Long interleaver can thus turn a nonergodic channel (where eachcodesymbol of a codeword sees one channel state only) into anergodic channel (where each codesymbol of a codeword sees adifferent channel state)

Block fading characterizes the situation in between those twoextremes. If the interleaver is not long enough, blocks ofcodesymbols see the same channel state.

Summary

1 Classification of channels

single-user, multi-userSISO, MIMO, MISO, SIMO“digital” channels (BSC, DMC)physical medium (copper, coax, fiber, air/space)

2 Wireline channel

essentially time-invariant, strongly frequency selective

3 Wireless channel

fixed terminals

static view on attenuation (link budget) is sufficient, limited bybackground noise

mobile terminal(s)

time-variant (frequency-dispersive), time-dispersive (frequencyselective)

ergodicity