SMALL SCALE FADING AND
MULTIPATH
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
1
Doppler shift:
The phase change in the received signal due to the
difference in in path length
  2l /   2Vt cos / 
Apparent change in frequency is given by
fd   / 2t  v cos / 
2
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.

y (d , t )  x(t )  h(d , t ) 
 x( )h(d , t   )d

t
y(vt, t )   x( )h(vt, t   )d

3
Since v is constant y(vt,t) is just a function of t
therefore
t
y(vt, t )   x( )h(vt, t   )d  x(t )  h(vt, t )  x(t )  h(d , t )

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 and
X(t)-transmitted band pass waveform
Y(t)- the received waveform
H(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 t
t- time variation due to motion
Therefore above equation can be expressed as

y(t )   x( )h(t , )d  x(t )  h(t , )

4
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
N 1
hb(t , )   ai (t , ) exp[ j 2fci (t )  (t , )]. (  i(t ))
i 0
N 1
hb( )   ai exp( ji ) (  i )
i 0
Where ai(t,T) and Ti(t) are real amplitudes and excess delays
Respectively 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 specific
multi path bins that have components at time t and excess
.
delay i

5
narrow band signals are equivalent
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.
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.
6
Small-scale multipath measurements
 Direct RF pulse measurements
 Spread spectrum sliding correlator measurements
 Swept frequency measurements
They are also called wideband channel sounding techniques
7
Parameters of Mobile Multipath Channels
Time dispersion parameter
Coherence bandwidth
Doppler spread and Coherence time
8
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,
a2τ
P(τ )τ

τ=
a


 P(τ
k k
k
k
k
k
2
k
k
k
)
k
Where a and τ are the real amplitudes and excess delays
The RMS delay spread is the square root of the second central
moment of the power delay profile and is defined to be,
     ( )
2
2
where
2 =
2 2
a
 kτk
k
a
k
2
k

2
P(τ
)
τ
 k k
k
 P(τ
k
)
k
9
10
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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Doppler Spread & Coherence Time (2)
 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
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Doppler Spread & Coherence Time (3)
 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
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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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16
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”
17
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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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
Hence a signal will undergo fast fading if
Ts  Tc
and
Bs  BD
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.
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Slow Fading
In Slow Fading channel the channel impulse response
changes at a rate much slower than the transmitted
baseband signal.
Hence a signal will undergo slow fading if
Ts  Tc
and
Bs  BD
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.
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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:
 r
 r2 
( 0  r   )
p( r )   2 exp 
2 
 2 

0
(r < 0)
CDF
 r2 

P( R)  Pr (r  R)   p(r )dr  1  exp 
2 
 2 
0
R
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The mean value rmean of the Rayleigh distribution is given by


rmean  E[r ]   rp (r )dr  
2
0
 1.2533
The variance  r of the Rayleigh distribution is given by

 r  E[r ]  E [r ]   r p(r )dr 
2
2
2
2
0
 2
2


   2    0.4292 2
2

2
The rms value of the envelope is 2
The median value of r is found by solving
rmedian  1.177
22
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
A2
K
2 2
A2
K (dB)  10log 2 dB
2
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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:
 r
 r2 
 exp  2 
p r    2
 2 

0

where
0r 
(4.68)
r0
 2  E02 / 2
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Level Crossing and Fading Statistics
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:

N R   rpR, rdr 
2 f m e
 2
(4.80)
0
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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

1
Prr  R
NR
(4.81)
where Prr  R is the probability that the received signal r is less
than R and is given by
Prr  R 
1
i

T i
(4.82)
where  i is the duration of the fade and T is the observation interval
of the fading signal.
For a Rayleigh distribution,
R

Prr  R   pr dr  1  exp   2
0

(4.83)
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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
e 1

f m 2
2
(4.84)
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 Model
Clarke’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.
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