Time Varying and Frequency Selective Radio Channel - KIT - IHE

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INSTITUTE OF RADIO FREQUENCY ENGINEERING AND ELECTRONICS

Time Varying and Frequency Selective Radio ChannelAdvanced Radio Communication IProf. Dr.-Ing. Marwan Younis

2 Institute of Radio Frequency Engineering and Electronics

Small-Scale Fading & Large-Scal Fading

small-scale fading: receiver moving through a spatial interference pattern.large-scale fading: slow changes in the propagation environment.


Multipath Effects

Multipath in the radio channel creates small-scale fading effects. This results in:

• rapid changes in signal strength over a small travel distanceor time interval

• frequency selectivity caused by multipath propagation delays• frequency modulation due to varying Doppler shifts on

different multipath signals


Spatial and Temporal Variations

Consider a moving mobile station, a fixed base station and a staticenvironment, i.e. only the mobile receiver is moving. Then, the spatial

variations of the signal are seen as temporal variations by the receiver as it moves through the interference pattern

time or position

5 Institute of Radio Frequency Engineering

and Electronics

Factors Influencing Small-Scale Fading

Channel Type:

• fixed-to-mobile

• line of sight (LOS) radio link

• satellite link

• stationary reception of TV/Radio

• …

Physical Factors:

• multipath propagation

• speed of the receiver

• speed of the surrounding objects

• signal bandwidth


Advanced Radio Communication I

Small-Scale Fading


Small-Scale Fading f(t)Larg-Scale Fading m(t)

The Components of Fading


2-Dimensional Gaussian distribution for ℜ(#$) and ℑ(#$)

'ℑ(#$()

ℜ(#$()

#$)

*)

*$

#$

ℜ(#$)

'ℑ(#$)

Superposition of Multipath Signals

#$(

*(


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Ricean PDF with K=SRV=0 (Rayleigh)Ricean PDF with K=SRV=0.51Ricean PDF with K=SRV=5Ricean PDF with K=SRV=50 (≈Gauß)

prob

abilit

y de

nsity

func

tion

(PD

F) [1

/Vol

t]

Probability Density Funciton

magnitude of the open circuit voltage VR



Large-Scale Fading (Log-Normal Fading)


-230 -210 -190 -170 -150 -130 -110 -90 -70 -50

standard deviation in dB ≈ 26mean power in dBm ≈ -122

received power [dBm]

prob

abilit

y de

nsity

func

tion

(PD

F)

Log-Normal Fading

GSM1800 Coverage Simulation: Calculated Large-Scale PDF


Question 1:Multi-Path creates small scale fading. What are the effects of fading?Question 2:Which physical factors (multipath, speed of Rx, movements of objects) are relevant for a satellite TV downlink?Question 3:What is the probability density function of the sum of a large number of independent random variables? Question 4:Which are the condition(s) to obtain a Rayleigh distribution for the magnitude of the Rx signal in a small-scale fading environment? Question 5:What does the Ricean factor K describe? Question 6:What probability density function describes large scale fading characterized

Questions



Channel Transfer Function and Impulse Response



The channel, the input signal and the output signal are modeled as linear time variant. Then they are completely described in:• Time Domain (time variable t)• Frequency Domain (frequency variable f)

In the time domain the channel, the input signal and the output signal are realquantities. Further, there DC component must be 0.

r(τ ) = h(τ )∗ s(τ )

output channel input

R( f ) = H ( f ) ⋅S( f )


t

convolution


spectrum of equivalent baseband signal

real partimaginary part

C(Δf )

f

real partimaginary part

+f0-f0

spectrum of bandpass signal for positive frequencies

spectrum of bandpass signal for negative frequencies

H ( f ) for f > 0H ( f ) for f < 0

Bandpass and Equivalent Baseband Signals



The channel, the input signal and the output signal are modeled as linear time variant. Then they are completely described in:• Time Domain (time variable t)• Frequency Domain (frequency variable f)

Since all signals are band-limited (bandpass) the equivalent (complex) basebandrepresentation can be used (known as low-pass or complex envelope)

bandpass

equivalent baseband

€

r(τ) = h(τ )∗ s(τ)

v(τ) =12c(τ )∗ x(τ)

time domain


€

R( f ) = H( f ) ⋅ S( f )

V (Δf ) =12C(Δf ) ⋅ X(Δf )

frequency domain


t

t



H ( f ) R( f )S( f )

C(Δf ) V (Δf )X(Δf )

bandpass

equivalent baseband

h(τ ) = c(τ ) cos 2π f0τ +∠c(τ )( )

H ( f ) = 12C( f − f0 )+

12C*(− f − f0 )


time

dom

ain

sign

al

time t in millisecond

Bandpass Signal and its Complex Envelope


Characterization of the Frequency-Selective Channel

– Time Domain –

20 Institute of Radio Frequency Engineering

and Electronics

Band-Limited Impulse Response Function

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

50 52 54 56 58 60 62 64 66 68 70

not bandlimited

ideal filter of bandwidth B=500kHzm

agnitude o

f equiv

ale

nt baseband r

esponse

time delay parameter t [µs]

Gaussian filter, B(-20dB)=500kHz


Normalized Power Delay Profile

-35

-30

-25

-20

-15

-10

-5

0

50 52 54 56 58 60 62 64 66 68 70norm

aliz

ed p

ower

del

ay p

rofil

e [d

B]

time delay parameter ! [#$]


Frequency-Selective Channel

The radio channel can be characterized:• in the time domain by the impulse response• in the frequency domain by the channel transfer function

In the time domain, the characterization is based on the powerdelay profile (PDP) function which describes the relative receivedpower as a function of the delay.

In order to compare different channels, parameters whichquantify the channel are utilized. The mean excess delay and theRMS delay spread are parameters determined directly from thePDP


Characterization of the Frequency-Selective Channel

– Frequency Domain –


Frequency-Selective Channel

In the frequency domain, the characterization is based on thefrequency autocorrelation function (ACF) which describes overwhich frequencies the channel is flat.

In order to compare different channels, parameters whichquantify the channel are utilized. The coherence or correlationbandwidth is a parameters determined directly from thefrequency ACF


Relating the Channel to the Signal

normalized frequency ACFbaseband transfer function


Characterization of the Time-Variant Channel

– Time Domain –


Signal Envelope and Coherence Time

normalized temporal ACFtime-varying envelope



In the time domain, the characterization is based on the temporalautocorrelation function (ACF) which describes how fast thechannel changes in time.

In order to compare different channels, parameters whichquantify the channel are utilized. The coherence or correlationtime is a parameters determined directly from the temporal ACF



– Frequency Domain –


Power Spectral Density (power Doppler Spectrum)



In the frequency domain, the characterization is based on thePower Spectral Density (PSD) or power Doppler spectrum(function) which is the received power spectrum for a puresinusoidal transmitted signal.

In order to compare different channels, parameters whichquantify the channel are utilized. The Doppler spread is ameasure of the spectral broadening.


Signal Envelope and Coherence Time

normalized temporal ACFtime-varying envelope


Geometry for Multipath Wave Propagation

last scattering center formultipath signal i

αvR,i

receiver positionat time t

receiver position

at time t=0

vR t

Δs =| vR| cos(αvR,i) t


Jakes Doppler Spectrum

-202468

10121416

-303691215182124

-250-200-150-100 -50 0 50 100 150 200 250

Jakes spectrum (linear)Jakes spectrum (in dB)

norm

aliz

ed J

akes

spe

ctru

m (l

inea

r)norm

alized Jakes spectrum [dB]

doppler frequency [Hz]

Time Varying and Frequency Selective Radio Channel - KIT - IHE

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