*Local reflections cause multipathEach path has a random gain, with random magnitude and random phaseEach gain is represented in baseband asReceiver, and/or reflectors, may be movingSmall Scale Fading

Building 2

Building 1

v

*Assume a group of paths with small relative delayNet effect is one path with random gain and phase According to Central Limit Theorem, the net gain Rejf is complex Gaussian with zero meanThe envelope R is Rayleigh distributed and the phase f is uniform [0, 2p] Small Scale Rayleigh Fading

Building 2

Building 1

v

*The net gain is the sum of all closely delayed paths:Each of greal and gimag is the sum of many independent random variablesHence greal and gimag are independent and Gaussian with zero mean and variance 2 eachFading gain g = greal + j gimag is complex Gaussian with zero mean and variance 22 (sum of two variances)Rayleigh Fading

Rayleigh DistributionFrom probability theory we know:

Received amplitude follows Rayleigh distribution

Received power follows Exponential distribution

Received phase follows Uniform distribution *

Amplitude, Rayleigh Distribution

*

Power, Exponential Distribution*

Fading of 16 QAM signalSignal has a higher probability of being weekFor example, to receive the 16-QAM signal we must estimate and compensate for the amplitude and phase *No fadingFaded signals with random amplitude and phase

Effect of MobilityFading gain changes with timeg(t)=greal(t) + j gimag(t)Fading change rate depends on the maximum Doppler frequencyCoherence time

Fading Example for R*

Statistical Properties 1Complex fading gain g(t)

The two parts greal(t) and gimag(t) are zero mean

The two parts greal(t) and gimag(t) are statistically independent*

Statistical Properties 2Fading gain is correlated over timeUsually Jakes model is used in mobile comm.Autocorrelation function given by

Jo() is the Bessel function of order zeroAg(Dt) indicates how much the gain is correlated with itself after delay DtPower spectral density of fading is the FT of the autocorrelation function

*

Statistical Properties 3Usually the fading gain is normalized to unity power, i.e, s2=1/2

*fD Dtf/fD

Rician Fading ChannelIf the channel also includes a LOS component we get Rician fadingFading gain is now

greal is Gaussian with mean S and variance s2The envelope R is Rician distributed (see Proakis chapter 2)

Rician Fading ChannelThe channel amplitude R is Rician

I0 = modified Bessel function of order zeroNow, when s(t) is transmitted

Power ratio K=LOS/faded=S2/(2s2)If K=0 we are back to Rayleigh fadingAs K increases, more power to LOS *

Rician PDF*K=1, 2, 3

*Channel may consists of groups of delays (echoes)Each group is composed of many closely delayed pathsMaximum Delay Spread: Delay between first and lastTypically few microseconds outdoor and less than hundred of nanoseconds indoorChannel with large delay spread is an FIR filter:Large Delays Effect

t

t

t

t

S

Channel input

Channel output

Power Delay ProfilePower of the multipath decay as delay increases according to power delay profileEach path gl has a varianceExample, exponential profileExample, uniform profileTypically, fading is normalizedMean delay spread:RMS delay spread

*

*Time and Delay PictureChannel may have many resolvable pathsEach path at a certain delayEach path changes with time, t, and has its delay, t

Autocorrelation function:Scattering function: twice Fourier Transform of the Autocorrelation function, over Dt and Dt

*Simulating Classical Fading ModelJakes model

Building 2

House

Public house

House

*Simulating Classical Fading ModelAssume a mobile station in the middle of 4N reflectorsReflections with equal amplitude but different DopplerDoppler from path with incident angle an is fn=fM cos(an) , fM is the maximum Doppler Reflectors have different propagation delay around the circle

*Classical Fading ModelAfter some mathematical manipulations, the gain of the path hk(t):

*Classical Fading ModelWith L resolvable multipath, the channel model is given byThe gains vl select the desire delay profileThey are normalize the total channel power to 1

t

t

t

t

S

Channel input

Channel output

*Walch Codes of length 16nk

11111111111111111-11-11-11-11-11-11-11-111-1-111-1-111-1-111-1-11-1-111-1-111-1-111-1-111111-1-1-1-11111-1-1-1-11-11-1-11-111-11-1-11-1111-1-1-1-11111-1-1-1-1111-1-11-111-11-1-11-111-111111111-1-1-1-1-1-1-1-11-11-11-11-1-11-11-11-1111-1-111-1-1-1-111-1-1111-1-111-1-11-111-1-111-11111-1-1-1-1-1-1-1-111111-11-1-11-11-11-111-11-111-1-1-1-111-1-11111-1-11-1-11-111-1-111-11-1-11

*Fading ReferencesClassical Model: W. C. Jakes, editor, Microwave Mobile Communications, New York, Wiley 1974Modifications: P. Dent, G. E. Bottomley, and T. Croft, Jakes fading model revisited, Electronic letters, vol. 29, pp. 1162-1163, June 1993Good reference: Chapter on Fading channels in Digital Communications by Bernard Sklar

*Fast: Channel changes within symbol. TcTsSelective: Delay Spread > symbol time TsNon-Selective: Delay Spread < symbol time TsEffects on Signal

DefinitionsCoherence time = 1/max doppler = 1/fDCoherence bandwidth = 1/max delay spreadSlow fading: Symbol time < coherence timeNon-selective fading: Signal bandwidth < coherence bandwidthFast fading and selective fading are the opposite*

*Fast Fading:Due to high speed High distortion to the received signal Slow Fading:Terminal may fall in a fading null for long timeWorse performanceEffects on Signal, cont.

time

time

g(t)

s(t)

time

g(t)

Fast Fading

Slow Fading

*Effects on Signal, cont.

frequency

Signal spectrum

frequency

Channel gain

frequency

Channel gain

Selective

Non-Selective

*Receiver Antenna DiversityTransmitter Antenna DiversityTransmitter and Receiver Antenna Diversity (MIMO Systems)Rake ReceiverChannel EqualizationChannel Coding

Fading Counter Measures

*Receiver may have two or more antennasTwo main types:Antenna Selection: Select stronger antenna signal. Best for slow, non-selective fadingAntenna Combining:Optimally combine signal of antennas (MRC)More complexity & better performanceReceiver Antenna Diversity

*Maximal Ratio Combining

ho

h1

so

Maximal Ratio Combining Receiver Diversity

so ho

ho*

|h0|2 so

so h1

h1*

|h1|2 so

+

|h0|2 so + |h1|2 so

*Two antennas are used in TxTwo successive symbols are pre-coded as shownNeed two orthogonal sources for two channels estimation

Transmit Diversity

*Same as Tx Diversity, but with two RxsWe have 4 channels, h0, h1, h2 and h3Each receiver combines as beforeThe two receivers are then combinedTx & Rx Diversity (MIMO)

ho

h1

s0 then -s1*

s1 then s0*

h2

h3

Combine

so ( |h0|2 + |h1|2 + |h2|2 + |h3|2 )

s1 ( |h0|2 + |h1|2 + |h2|2 + |h3|2 )

Combine

+

+

*Used for Direct Sequence Spread Spectrum SystemsMultipath diversity = multipath is advantageousOne finger (correlator) per pathEach finger synchronized to one pathFinger outputs combined (MRC)Rake Receiver

*Need to estimate channel gain for each pathRake Receiver performs Maximal Ratio CombiningNumber of fingers = number of paths (ideally)Small inter-path interferenceRake Receiver, Cont.

*Equalizers attempt to compensate for channel fading effectsLinear Equalizer: FIR filter with adaptive tap weightsAdaptation to minimize some criteriaMost famous: Least Mean Square (LMS)Other criteria: Recursive Least Squares, Kalman Filter, etc.Channel Equalization

*LMS: wj(n+1)=wj(n) e*(n) yj(n) Linear Equalizer

Z-1

X

Z-1

X

Z-1

X

Z-1

X

+

Threshold

e

+

-

w0

w1

w2

wN-1

data

X

wN-2

y0

y1

y2

yN-1

*SummaryFading Types:Large Scale: Distance + ShadowingSmall Scale: Fast or Slow & Flat or SelectiveCounter Measures:Diversity TypesRakeEqualization

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