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What is small scale fading?Small scale fading is used to describe the rapid fluctuation of the amplitude, phases, or multipath delays of a radio signal over a short period of time or travel distance.

. Factors influencing small scale fading

•Multi path propagation

•Speed of the mobile

•Speed of surrounding objects

•The Transmission Bandwidth of the Signal

SMALL SCALE FADING AND MULTIPATH

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Doppler shift:

/cos2/2 tVl Apparent change in frequency is given by

/cos2/ vtfd

The phase change in the received signal due to the difference in in path length

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Impulse response model of a multipath channel Mobile radio channel may be modeled as linear filter with time varying impulse response, consider the case where time variation is strictly due to receiver motion in space.

dtdhxtdhtxtdy ),()(),()(),(

t

dtvthxtvty ),()(),(

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Since v is constant y(vt,t) is just a function of t therefore

From above equation it is clear that mobile radio channel Can be modeled as linear time varying channel.As v is constant over short distances andX(t)-transmitted band pass waveformY(t)- the received waveformH(t , )- impulse response of time varying multiple radio Channel and it is function of both t and .- represents channel multi path delay for fixed value of tt- time variation due to motion

Therefore above equation can be expressed as

),()(),()()( thtxdthxty

),()(),()(),()(),( tdhtxtvthtxdtvthxtvtyt

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Since received signal in a multi path channel consists of series of attenuated , time delayed , phase shifted replicas of the transmitted signal and the base band impulse response of multi path channel can be expressed as

1

0

1

0

)()exp()(

))((.)],()(2exp[),(),(

N

i

iiib

N

i

icib

jah

tittfjtath

Where ai(t,T) and Ti(t) are real amplitudes and excess delaysRespectively of ith multi path component at time t.and the term under the exponent represents the phase shift

- Is the unit impulse function which determines the specificmulti path bins that have components at time t and excess delay i .

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Relationship between bandwidth and received power

In actual wireless communication systems the impulse response of a multi path channel is measured in the field using channel sounding techniques.

the received local ensemble average power of wideband and narrow band signals are equivalent

When the transmitted signal has bandwidth much greater than bandwidth of channel, then the multipath structure is completely resolved by he received signal at anytime and received power varies very little since the the individual multi path amplitudes do not change rapidly over local area.However if the transmitted signal has a very narrow bandwidth the the multi path is resolved by received signal And large signal fluctuations occur at the receiver due to the phase shift of the many unresolved multi path components.

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Small-scale multipath measurements Direct RF pulse measurements Spread spectrum sliding correlator measurements Swept frequency measurements

They are also called wideband channel sounding techniques

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Parameters of Mobile Multipath Channels

Time dispersion parameterCoherence bandwidthDoppler spread and Coherence time

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Time dispersion parameters

Mean excess delay, RMS delay and Excess delay spread (X dB) are multipath channel parameters that can be determined from a power delay profile

The Mean excess delay is the first moment of the power delay profile and is defined to be,

kk

kk

k

k

2k

kk

2k

)P(τ

τ)P(τ

a

τa = τ

The RMS delay spread is the square root of the second central moment of the power delay profile and is defined to be,

Where a and τ are the real amplitudes and excess delays

22 )( where

kk

2k

kk

k

2k

2k

k

2k

2

)P(τ

τ)P(τ

a

τa =

10

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Coherence Bandwidth

Coherence bandwidth is a range of frequencies over which the channel can be considered “flat” I.e., a channel which passes all spectral components with approximately equal gain and linear phase.

I.e., it is the range of frequencies over which two frequency components have a strong potential for amplitude correlation

For Example: If the coherence bandwidth is defined as the bandwidth over which the frequency correlation function is above 0.9, then the

coherence bandwidth is approximately,

BC ≈ 1/ 50 στ

If the definition is relaxed so that the frequency correlation function is above 0.5, then the coherence bandwidth is approximately,

BC ≈ 1/ 5 στ

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Doppler Spread & Coherence Time

Doppler spread BD is a measure of the spectral broadening caused by the time rate change of the mobile radio channel and is defined as the range of frequencies over which the received Doppler spectrum is essentially non-zero.

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Coherence time TC is the time domain dual of Doppler spread and is used to characterize the time varying nature of the frequency dispersiveness of the channel in the time domain.

The Doppler spread and Coherence time are inversely proportional to one another, I.e., TC ≈ 1 / fm eq. (2)

Coherence time is actually a statistical measure of the time duration over which the channel impulse response is essentially invariant and quantifies the similarity of the channel response at different times.

Coherence time is the duration over which two received signals have a strong potential for amplitude correlation

If the reciprocal bandwidth of the baseband signal is greater than the coherence time of the channel, then the channel will change during the transmission of the baseband message, thus causing distortion at the receiver

Doppler Spread & Coherence Time (2)

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If the coherence time is defined as the time over which the time correlation function is above 0.5, then the coherence time is approximately,

TC ≈ 9 / 16Π fm eq. (1)

Where fm is the maximum Doppler shift given by, fm = / A popular thumb rule for modern digital communications is to

define the coherence time as the geometric mean of the eq.(1) & (2). That is,

TC = √ 9 / 16Π fm = 0.423 / fm

Doppler Spread & Coherence Time (3)

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Small Scale Fading:

Different types of transmitted signals undergo different types of fading depending upon the relation between the

Signal Parameters: Bandwidth, Symbol Period

and

Channel Parameters: RMS Delay Spread,

Doppler Spread In any mobile radio channel a wave can be dispersed either in Time or in Frequency. These time and frequency dispersion mechanisms lead to four possible distinct effects which depend on the nature of transmitted signal, the channel and the velocity.

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Flat Fading: A received signal is said to have underwent Flat Fading if “The

Mobile Radio Channel has a constant gain and linear phase response over a Bandwidth which is greater than the Bandwidth of the transmitted Signal”

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Frequency Selective Fading:

The channel creates frequency selective fading on the received signal when the channel possesses a constant gain and linear phase response over a bandwidth, which is smaller than the bandwidth of the transmitted signal

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Hence a signal will undergo fast fading if

cs TT and Ds BB

Note: Fast fading only deals with the rate of change of the channel due to motion. fast fading occurs only for very low data rates.

Fast Fading:

In Fast Fading channel, the channel impulse response changes at a rate much faster than the transmitted baseband signal. This causes frequency dispersion due to Doppler spreading, which leads to signal distortion

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Hence a signal will undergo slow fading if

cs TT and Ds BB

Note: Fast and Slow Fading deal with the relationship between the time rate of change in the channel and the transmitted signal, and not with the propagation path loss models.

Slow Fading

In Slow Fading channel the channel impulse response changes at a rate much slower than the transmitted baseband signal.

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Rayleigh Fading Distribution: Rayleigh Fading Distribution in mobile radio channels is commonly used to describe the statistical time varying nature of the received envelope of a flat fading signal or the envelope of an individual multipath component.The pdf of a Rayleigh distribution:

0 (r < 0)

)0(2

exp)(2

2

2

r

rrrp

CDF

2

2

0 2exp1)()()(

r

drrpRrPRPR

r

22

The variance of the Rayleigh distribution is given byr

The rms value of the envelope is 2

 

2)(][][

2

0

2222

drrprrErEr

22 4292.02

2

The mean value of the Rayleigh distribution is given by meanr

2533.12

)(][0

drrrprErmean

The median value of r is found by solving

177.1medianr

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Ricean Fading Distribution: When there is a dominant stationary (nonfading) signal

component present, such as a line-of-sight propagation path, the small scale-scale fading envelope distribution is Ricean.

•The Ricean distribution is often defined in terms of a parameter K called the Ricean Factor

2

2

2A

K

dBA

dBK2

2

2log10)(

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Statistical Models for Multipath Fading Channels

Clarke’s Model for Flat Fading Two-ray Rayleigh Fading Model Saleh and Valenzuela Indoor Statistical

Model SIRCIM and SMRCIM Indoor and Outdoor

Statistical Models Level Crossing and Fading Statistics

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Clarke’s Model

the random received signal envelope r has a Rayleigh distribution:

2/

)68.4(

00

02

exp

20

2

2

2

2

E

where

r

rrr

rp

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Level Crossing and Fading Statistics

)80.4(2,2

0

efrdrRprN mR

Two important statistics: 1. level crossing rate 2. average fade duration

What is level crossing rate (LCR)?

The expected rate at which the Rayleigh fading envelope, normalized to the local rms signal level, crosses a specified level in a positive-going direction.

The number of level crossings per second is:

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)81.4(Pr1

RrNR

)82.4(1

Pr i

iTRr

)83.4(exp1Pr 2

0

drrpRrR

What is average fade duration?

The average period of time for which the received signal is below a specified level R.

For a Rayleigh fading signal, this is given by

where is the probability that the received signal r is less than R and is given by

Rr Pr

where is the duration of the fade and T is the observation interval of the fading signal.

i

For a Rayleigh distribution,

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)84.4(2

12

mf

e

where p(r) is the pdf of a Rayleigh distribution.

Using equations (4.80), (4.81), (4.83), the average fade duration can be expressed as

The average fade duration helps determine the most likely number of signaling bits that may be lost during a fade.

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Two-ray Rayleigh Fading ModelClarke’s model and the statistics for Rayleigh fading are for flat fading conditions, and do not consider multipath time delay.

A commonly used multipath model is an independent Rayleigh fading two-ray model

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.

Saleh and Valenzuela Indoor Statistical Model

It is a simple multipath model for indoor channels based on measurement results.

SIRCIM and SMRCIM Indoor and Outdoor Statistical Models

Rappaport and Seidel developed an elaborate, empirically derived statistical model and wrote a computer program called SIRCIM.

SIRCIM: Simulation of Indoor Radio Channel Impulse-response Models.

The model is based on the discrete impulse response channel model.

Huang produced a similar program named SMRCIM.

SMRCIM: Simulation of Mobile Radio Channel Impulse-response Models.

The program generates small-scale urban cellular and microcellular channel impulse responses.