## Multipath Fading |
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Refractivity in the atmosphere (II)

Observed impairments in the Rx signal

©
2001-2016, Apus Cloud Project e Luigi Moreno

_______________________________________________________________

In this
Session multipath propagation is considered. First, refractivity conditions are
discussed and the received signal impairments are presented (signal attenuation
and distortion). Multipath activity
statistics are described, according to the Rayleigh model, and the multipath
occurrence factor is defined. These models are applied for outage prediction,
for both narrow-band and wide-band systems. Finally, multipath countermeasures,
space and frequency diversity, are considered.

A general
introduction to the effect of the atmosphere refractive index on radio
propagation and specifically of a vertical refractivity gradient has been given
in a
previous
Session
.

In that
context, we mainly considered constant gradient conditions, and we defined the
"standard atmosphere" as the condition with vertical refractivity
gradient G = - 40 N/km (k-factor =
1.33). Still under the assumption of a
constant refractivity gradient, other conditions are the "sub-refractive
atmosphere" (G
.>.-.40 N/km; k.<.
1.33)
and the "super-refractive atmosphere" (G
.<.-.40 N/km; k.>.1.33).

A
constant vertical refractivity gradient means that the ray trajectory suffers the same curvature, at any
elevation in the atmosphere. Under this
condition, a direct ray trajectory is identified, from the Tx antenna to the Rx
antenna, with launching angle
agiven by:

_{
}

where_{
}

_{
}

R_{E}
is the equivalent earth radius (8500 km with standard k-factor = 1.33), H_{T}
and H_{R} are the antenna heights at the transmitter and receiver,
respectively, and D is the path length.

*
Ray trajectories in "constant gradient"
atmosphere
*

More generally, the vertical
refractivity gradient may deviate from a constant-gradient model.It may be assumed as constant within
atmospheric layers of
limited
height (stratified atmosphere).
In the real case,
the transition from one layer to another is smoothed in some measure.

A stratified
atmosphere model is useful in explaining the
different bending of ray trajectories, when
they travel at different elevations in the atmosphere.

In these conditions, the "gradient profile" may be
such that
not only a
direct ray, but multiple rays, with different launching angles, reach the
receiver antenna through several spatially disjointed paths. This is called
"multipath propagation".

*
Ray trajectories under multipath propagation
conditions
*

As a result, the received signal is made by several
components (signal echoes), adding together with random amplitude, delay, and
relative phase shift.

**
Signal attenuation
**

Using a vectorial representation of signals, the received
signal, under multipath propagation, can be viewed as the addition of multiple
vectors.

The
component vectors may interfere each other, at a given time instant, in a
constructive or destructive way, depending on the relative phase shifts.

*
Addition of multiple signal echoes, represented by
vectors,
*

*
at two subsequent time instant
*

The relative phase of component vectors depend on the
difference in the path length traveled by each signal component.Note that the wavelength is of the order of
centimeters and even
small
movements in atmospheric layers may
significantly modify the path distances and the
relative vector phases.

So, at different time instants, variations in the component
vector phases may produce sudden variations in the resultant vector
amplitude; the received signal power
may be almost cancelled, for short
periods (fraction of a second, or few seconds).

*
An example of received signal power vs. time,
*

*
during a multipath propagation event
*

The
above figure can be compared with graphical definition of received signal thresholds and
margins, as given in a previous Session.

Clearly,
during multipath events, the received signal power may fade below the hop
threshold, so that a system outage is observed.This will be discussed in a subsequent section.

**
Signal
distortion
**

The phase shiftd
between two vector
components is computed as a function of
D
L (length
difference in the paths traveled by the two rays) and of the signal wavelength
l:

_{
}

The above formula shows that the relative phase of component
vectors depend on the signal frequency (or wavelength).The
pictures
above
can be thought as valid for a given frequency, but slightly different
phase patterns are applicable to adjacent frequencies.

This means that multipath fading is "frequency
selective".

While a deep fading condition is observed at a given
frequency F1, the signal at a different frequency F2 (some MHz apart) is
probably received with lower attenuation.

Because
of the fast variability of multipath events, this condition could be reversed
in a very short time (a deep fading at frequency F2 and a higher Rx power at
frequency F1).

We
recall that, for undistorted transmission, the transmission channel must have a
"flat" amplitude response in the whole signal bandwidth. A similar
requirement applies to group-delay response.

During
multipath events, it has been observed that the transmission channel cannot be
considered as a "flat response" channel if the monitored bandwidth
exceeds some 10 -12 MHz.

Therefore,
"narrowband" signals (approximately below 10 MHz bandwidth) do not
suffer the frequency selective effect of multipath propagation.

On the
other hand, distortion caused by frequency selectivity represents a further
impairment (in addition to signal attenuation) for "wideband" signals
(approximately above 15 MHz bandwidth).

Amplitude
and Group-Delay distortions produce Intersymbol Interference on digital
signals, thus worsening the receiver performance for a given signal-to-noise
ratio (Rx power).

**
Degradation
of Cross-pol discrimination
**

An additional impairment due
to multipath fading is a degradation of the receiver cross-polar
discrimination. Such discrimination is
required when multiple RF channels are transmitted in a radio hop and both
polarization are used (co-channel
or interleaved
channel arrangements).

Under non-fading conditions,
the hop performance are determined by the
antenna cross-polar
discrimination
(XPD), both at the transmitter and at the receiver.

During multipath events, as
far as the signal attenuation is moderate, the cross-polar signal is usually
well correlated to the co-polar one and the XPD performance is maintained.

On the other hand, when
signal attenuation becomes deeper, the XPD appears to be degraded, mainly
because of the antenna response to multipath components.

The mechanism can be
clarified by considering the co-pol and cross-pol antenna patterns. While the
co-pol pattern usually shows a rather flat maximum in the pointing direction,
the cross-pol pattern has a very narrow minimum in the same direction.

*
Antenna response to two
rays, with slightly different arrival angles:
*

*
the two co-pol components
are almost equal, while
*

*
the difference between the
cross-pol components is large (
**
D
**
_{
2
}
*

The two co-pol components
may almost cancel (if with opposite phase), while the dominant cross-pol
component is large in any case. So a significant degradation may affect the
overall XPD.

A second mechanism may be
involved in the XPD degradation during multipath events, when some multipath
components are produced by reflection or terrain scattering.In that case, the signal polarization of the
reflected or scattered signal is rotated (in some measure) and the cross-pol
signal is increased.

Performance prediction
models usually assume that, as far as the signal attenuation is within some
10-15 dB, the XPD is determined by the antenna measured performance.On the other hand, for deeper fadings, some
XPD degradation is expected (up to 1 dB additional degradation for 1 dB
additional signal attenuation).

Multipath
events are observed with daily and seasonal cycles, when suitable refractive
gradient profiles are more often observed.
A multipath activity period can last tens ofminutes, or even one or several hours.

A
prediction model of multipath activity is implemented by correlating
significant radio link and environmental parameters with statistical
observation of multipath events.

**
Radio
and environmental parameters
**

Radio link parameters which have been
recognized as affecting multipath events are :

·
Working frequency;

·
Path length;

·
Path inclination.

Environmental
conditions which are likely to produce multipath events are :

· flat terrain;

·
strong evaporation (high
temperature and humidity);

· absence of wind.

It is
often useful to identify climatic regions with specific characteristics, so
that multipath activity can, in some measure, be correlated with regional
parameters. Particularly in tropical climates, long multipath events can be
observed.

**
Statistical
observation of multipath events
**

By
monitoring a radio hop during multipath events, a number of recordings, similar
to the above figure, can be
collected. This enables to build up
statistical data about the time periods with fade depth below given thresholds.

A large amount of similar experiments have
shown that fade depth statistics are well approximated by a Rayleigh
distribution (at least for fade depth greater than about 15
.
dB).According to that distribution, the probability that the signal
fade depth A (in dB) is deeper than a given value A_{0} is given by :

_{
}

where P_{0}
is called "multipath occurrence factor".(To be more precise, this is the Rayleigh "asymptotic"
trend, derived for low probability and deep fade levels).

*
An example of Rayleigh cumulative distribution, with
P _{0} = 1
*

Note
that, if the reference fade depth A_{0} increases 10 dB, then the
corresponding probability is lower by a factor 10 (the diagram slope is 10 dB /
decade).

This
experimental
result
is in good agreement with mathematical analysis, applied to the
random vector model, previously mentioned.
It can be shown that, if we add a large number of vectors, with random
amplitudes and phases, then the resultant vector amplitude is a random variable
with Rayleigh distribution.

The
Rayleigh model for multipath fade depth is described by a single parameter P_{0}.

We can
imagine to collect fade depth statistics on a given radio hop in different time
periods, or on radio hops with different length, working frequency, and/or in
different climates. We expect that, in
some measure, the experimental results approximate the Rayleigh formula given
above, even if a different P_{0} value will apply in each case.So, the P_{0} parameter gives a
measure of the "multipath activity" on a given hop and within a given
time period.

The
above example suggests an experimental means to estimate the P_{0}
factor when a radio hop is already working.
However, the radio engineer needs prediction tools to estimate P_{0}
while a radio hop is at the design stage.

Several
empirical formulas have been proposed, giving P_{0} as a function of
radio hop parameters and of environmental conditions. The relevant factors are
those mentioned in a previous section.

Most of
these formulas have the following structure :

_{
}

whereC (geoclimatic coefficient),Q (terrain profile coefficient),
a(frequency exponent), andb
(path length exponent) are
empirical parameters. They are usually
estimated by processing large amounts of experimental data, or can derive from
more complex formulas, again related to the results of field measurements.

Generally,
P_{0} is proportional to frequency (the
a
exponent
is equal, or close, to 1), while the
bexponent is in the range is 3.-.
3.6 (the multipath occurrence increases about
ten times when the hop length is doubled).

Probably, the most popular model for P_{0}
prediction is the Bell Labs formula (reported in papers by W.T. Barnett and
A. Vigants, in the early
70's). The general formula mentioned above is applied (frequency in
GHz, distance in km), with the following parameters:

· a= 1;

· b= 3;

· C = 1×
10^{-5}for dry mountainous regions;

· C = 2.1×
10^{-5}for continental temperate regions;

· C = 3.1×
10^{-5}for maritime temperate regions;

· C = 4.1×
10^{-5}for maritime sub-tropical, high humidity and
temperature regions;

· Q = 1 /s^{
1.3
}

· s =
profile roughness, measured in
meters as the standard deviation of terrain elevations at 1 km intervals (in
any case,
s
must be in
the range 6 m to 42 m).

Examples
of the Barnett-Vigants model are given below.

*
Application of the Barnett-Vigants model:
*

*
High dry mountainous regions; high roughness
terrain (
**
s
**
= 42 m)
*

*
Application of the Barnett-Vigants model:
*

*
Temperate continental regions; average rolling
terrain (
**
s
**
= 24 m)
*

*
Application of the Barnett-Vigants model:
*

*
Temperate maritime regions; low roughness terrain (
**
s
**
= 12 m)
*

*
Application of the Barnett-Vigants model:
*

*
Sub-tropical, high humidity regions; flat
terrain (
**
s
**
= 6 m)
*

An
alternative model is proposed by ITU-R Rec. P.530-9. The model structure is
slightly different and more complex with respect to the general formula mentioned above. This model has been
frequently revised in recent ITU-R meetings and probably it is not yet at a
final version.

**
ITU-R
Multipath occurrence model
**

ITU-R Rec. P.530-9 (released
June 2001) gives a model for the prediction of the Multipath Occurrence Factor
P_{0}.

The model provides two
different formulas, to be applied for detailed link design or for preliminary
planning, respectively. The main
difference in the two approaches is that the detailed design makes use of data
on terrain roughness around the radio path.

(Note :Rec. P.530 gives the Rayleigh formula in
%; a 0.01 factor is added in the
P_{0}.
expressions
given below to take account of this).

__
Detailed link design
__:

_{
}

where :K (= geoclimatic factor) is given by :

_{
}

e_{
p
}
= path
inclination in milliradians;

HL = elevation of the lower antenna in meters;

dN1 = refractivity gradient in the lowest 65 m of the
atmosphere, not exceeded for 1% of an average year;

s_{A} = area roughness around the radio path.

The refractivity gradient
dN1 is
provided on a 1.5° grid in latitude and longitude in ITU-R
Rec. P.453.

The area
roughness is defined as the standard deviation of terrain heights (m) within a
110 km x 110 km area with a 30 s resolution.

__
Preliminary planning
__:

_{
}

where :K (= geoclimatic factor) is given by :

_{
}

and the other symbols are
already defined above.

__
Comment
__

The ITU-R model derives from
the processing of a significant amount of
P_{0}.
estimates, at several
frequencies (up to 37 GHz) and with various path lengths in different climatic
environments.

The mathematical approach is
mainly based on minimizing the standard deviation between empirical data and
prediction formulas by means of multiple regressions.The positive aspect is that the model is well related to
observations in real links. It is
stated that the
overall standard
deviations of error using the proposed models is of the order of 5 dB
(including the contribution from year-to-year variability).

On the other hand, a physical model underlying formula
structure and parameter choice is not clearly defined, so that it appears that
the proposed approach could be revised on the basis ofa new experimental database, as already happened
in recent years.

In a previous Session, general
concepts about fade margin and outage prediction have been briefly
discussed. In particular, it was found
convenient to distinguish between two outage conditions :

· when the outage is only caused by insufficient Rx power (received signal level below the hop threshold);

· when distortion in the Rx signal is expected to contribute to the outage, even when the Rx power is still above the hop threshold.

In the context of multipath propagation, the first condition applies to "narrowband" signals, since it is assumed that they do not suffer any distortion during multipath events.On the other hand, the second condition applies to "wideband" signals, which may be severely distorted by frequency selective multipath.

**
Outage
prediction in Narrowband systems
**

Outage events are observed when the Rx power is below the hop threshold.

Taking account of the multipath fading Rayleigh distribution,
the outage probability P_{OUT}, can be predicted as :

_{
}

where A is the signal attenuation caused by
multipath propagation, FM is the hop Fade Margin, and P_{0} is the multipath occurrence factor.

The outage time T_{OUT} during a
given observation time T_{0} (typically,one month), is finally given as
T_{OUT} =T_{0 }P_{OUT}.

In conclusion, two parameters are required for outage time prediction :

· the hop Fade Margin, given by the Link Budget computation;

·
the multipath
occurrence factor P_{0}, given by some model for multipath activity, as
the Barnett-Vigants one, presented
above.

In this
context, the Fade Margin is often referred as the Flat Fade Margin, since it is
used to compensate for non-selective (flat) attenuation.

**
Outage prediction in Wideband
systems
**

The prediction of Outage Time
in Wideband systems takes account that outage events may be caused by the
combined effect of signal attenuation and distortion.As a result, the outage condition may be observed even if the Rx
power is still above the receiver power threshold.

Reference will be made to
the prediction model reported in ITU-R Rec. P.530-9.Using a simplified approach, the model deals separately with the
two impairments (signal attenuation and distortion), so that the general formula
for outage probability prediction is :

_{
}

where P_{NS} is the
outage probability due to signal attenuation (non-selective outage component),
which is given by the same
outage
formula derived for narrowband systems
, while P_{S} is the outage
probability due to signal distortion (selective outage).

The selective component P_{S}
depends on the receiver sensitivity to signal distortion. The Signature Measurement
is the tool used to characterize a radio equipment under this aspect.P_{S} is given by :

_{
}

_{
}
is the Multipath
Activity (directly related to the Multipath Occurrence Factor
P_{0});

_{
}
is the mean time
delay [ns] of multipath echo components, which is a function of the hop length
D (in km);

W is the signature width [GHz];

B is the signature depth [dB];

t_{
r
}
is the echo
delay in the signature measurement.

Subscript "M" indicates
that the signature was measured with a Minimum -Phase channel, while subscript "NM"
refers to a Non-Minimum-Phase channel.

**
Outage contribution from X-pol
interference
**

Since multipath events have
an impact in
reducing discrimination
between cross-polarized signals
, multipath outage is increased by the
effect of cross-polar interference.

The Rec. P.530-9 prediction
model assumes that cross-polar interference contributes to the outage
probability with an additive term P_{XP}.

_{
}

where :

(C/I)_{0} is the threshold Carrier-to-Interference ratio;

XPD is the minimum cross-pol discrimination of the Tx and Rx
antennas;

_{
}
is an empirical
parameter, where P_{0} is the multipath occurrence factor and
h
is the multipath activity, previously
defined.

Notes :

1) IfXPD > 35 dB, then putXPD = 35 dB in the P_{XP} formula;

2) If a Cross-Pol Interference Canceller
(XPIC) is used, then the threshold C/I must be reduced by an amount equal to
the XPIC gain;

3) if two separate antennas
are used to transmit the cross-polarized signals, then the Q definition is
revised, by replacing the 0.7 factor with the K factor below :

_{
}

(s = vertical antenna spacing,
l= signal wavelength).

Several
techniques have been devised to reduce the impairments caused by multipath
propagation.

**
Space Diversity
**

As with reflection paths, two Rx antennas, with a suitable vertical
spacing, receive the multipath component signals with different phase patterns.

So, in a well arranged space diversity configuration, the Rx signals at the two antennas will exhibit a low correlation and the probability of deep fading at the same time can be significantly lowered. Typical spacing is of the order of 150 - 200 wavelengths.

A diversity improvement
factor I_{SD} is defined as :

_{
}

where A_{1} and A_{2} are the attenuations at
the two diversity receivers, A_{0} is a reference attenuation and
Prob
{X , Y} means
probability that events X and Y are true at the same instant (joint
probability).

The Barnett-Vigants model is extended to space diversity
reception, giving :

_{
}

where F is the working frequency in GHz, D the path length in
km, S the vertical spacing in m, and V is the difference of the two antenna
gains in dB.
Note that the improvement factor is a function of the
reference attenuation A
_{
0
}
,
so at different fade levels a different improvement is predicted.

The Outage Time
prediction, for a Narrowband system, is derived from the Single Rx prediction and the
definition of
diversity
improvement
:

_{
}

**
ITU-R model for Space Diversity
improvement
**

An alternative formula to
predict the space diversity improvement is given by ITU-R Rec. P.530-9:

_{
}

The Improvement factor isa function of the reference attenuation A_{
0
}.
F is the working
frequency in GHz, D the path length in km, S the vertical spacing in m,
P_{0}is the Multipath Occurrence Factor
and V is the
difference of the two antenna gains in dB (if any).

(Note :coefficients have been revised in comparison
with ITU-R formula because Rec. P.530 gives the Multipath Occurrence factor in
%).

**
1+1 Frequency Diversity
**

Again,
we refer to
general
concepts on diversity techniques
.

In this
case, we exploit the
frequency
selective
nature of multipath fading, so that two RF channels with suitable
frequency spacing exhibit the low
correlation property, which guarantees a low probability of deep fading in the
two channels at the same time.

Since a
protection channel is often required in multi-channel radio-relay systems in
case of equipment failure, it can be convenient that the same protection
channel be used also as a frequency diversity countermeasure to multipath
fading.

For
effective multipath protection, fast quality detector and switching circuits
are required.

In a 1+1 configuration, one working channel is continuously protected
by one spare channel. Similarly to
Space
diversity
, a Frequency Diversity Improvement Factor I_{FD} can be
defined. According to the
Barnett-Vigants model, also applied in ITU-R Rec. P.530, it can be estimated as
:

_{
}

where
F is the average
working frequency and
D
F is the channel spacing (both in GHz), D is the path
length in km. Also in this case,
the
improvement factor is a function of the reference attenuation A
_{
0
}(in dB).

**
N
+ 1 Frequency Diversity
**

The frequency diversity arrangement can be extended
from the 1+1 configuration, as assumed above, to N+1 configurations, where one
RF channel is used as a protection for N working channels

In N+1 systems it is
expected that the frequency diversity effectiveness is reduced in some
measure.

If, in the unprotected
condition, M channels are in the outage state, then using frequency protection
the number of outage channels is reduced to M-1.A fairly complex probability and combinatorial problem must be
solved to estimate the outage time reduction given by N+1 frequency diversity.

With good approximation, a
simplified solution is obtained by defining an "equivalent channel
spacing". By this approach, the
Frequency Diversity improvement in N+1 systems with channel spacing
D
F is equal to the
improvement in an "equivalent" 1+1 diversity system with channel
spacing
DF_{EQ}given by:

_{
}

So, we can use again the
previous (1+1) improvement formula, with
D
F_{EQ}instead of
DF_{.}

**
Outage in Wideband systems with
Diversity
**

In rather general terms, it
can be stated that the outage probability in a diversity system (P_{OUT, DIV})
is related to the outage probability with single reception (P_{OUT, SINGLE})
through the formula :

_{
}

whereh
is the
(previously defined)
multipath
activity
(that is the fraction of time with multipath events) and k is the
correlation factor between the two diversity signals.

In the case of the
non-selective outage probability, the Diversity ImprovementI_{DIV} = (P_{OUT,SINGLE} /
P_{OUT,DIV} ) is given by empirical formulas, for both Space and Frequency Diversity.
Then, the above formula can be
reversed to derive the non-selective correlation factor k_{NS}:

_{
}

On the other hand, the
selective correlation factor k_{S} is given by Rec. P.530-9 as a
function of k_{NS},

Once the (non-selective and
selective) correlation factors are known, the outage probabilities can be
computed using the general formula
reported above, for both the non-selective outage component (P_{NS,DIV})
and the selective one (P_{S,DIV}).

Finally, the two
outage components are combined to give the overall outage probability:

_{
}

Note :The outage prediction model reported by
ITU-R Rec. P.530-9 gives different formulas to combine the non-selective and
selective outage components in the single
and diversity conditions.

**
Adaptive
equalizers
**

Adaptive equalization is
part of the demodulation process. The
equalizer is implemented as a self-adjusting circuit (at the IF or baseband
stage), which is able to partially compensate for multipath distortion in
wideband digital systems.

The objective is to reduce
the selective outage component,
so that (with an ideal equalizer) outage should be observed only when the
received power fades below the Rx threshold.

The IF equalizer is usually
described in the frequency domain, as a circuit whose transfer function is
complementary to the multipath channel transfer function. The overall transfer
function (transmission channel plus equalizer) should approximate an ideal
non-distorting channel.

The BB equalizer is usually
described in the time domain, as a transversal filter (or decision feedback
filter), which cancels undesired tails in the transmission channel impulse
response, so reducing intersymbol interference.In some radio equipment, the BB equalizer and the
Cross-pol Interference Canceller
(XPIC)
are implemented in a single circuit.

The receiver signature
gives a measure of the effectiveness of an adaptive equalizer.By comparing the signature with and without
equalizer, the improvement (outage reduction) given by the equalizer can be
estimated (see the
selective outage
prediction formula
, based on signature parameters).

*
Equipment signatures without
and with an adaptive equalizer.
*

Further Readings

Rummler W.D. et al., "Multipath fading channel models for microwave digital radio", IEEE Comm. Magazine, vol. 24, n. 11, November 1986, pp. 30-42.

Greenstein M.J. and Shafi M., "Outage calculation methods for microwave digital radio", IEEE Comm. Magazine, vol. 25, n. 2, February 1987, pp. 30-39.

Martin A.L., "Dispersion signatures; some results of laboratory and field measurements", European Conf. on Radio Relay, Munich, 1986.

Greenstein L.J. and Yeh Y.S., "A simulation study of space diversity and adaptive equalization in microwave digital radio", AT&T Bell Lab Tech. J., vol. 64, n. 4, April 1985, pp. 907-935.

Sebald G. et al., "Advanced time- and frequency-domain adaptive equalization in multilevel QAM digital radio systems", IEEE Journal on Selected Areas in Communications, vol. JSAC-5, n. 3, April 1987.

Giger A.J. and Barnett W.T., "Effects of Multipath Propagation on Digital radio", IEEE Trans. on Communications, vol. 29, n. 9, Sept. 1981, pp. 1345-52.

Barnett W.T., "Multipath propagation at 4, 6 and 11 GHz", BSTJ, vol. 51, n. 2, February 1972, pp.321-361.

Vigants A., "Space diversity engineering", BSTJ, vol. 54, n. 1, January 1975, pp.103-142.

Vigants A. and Pursley M.V., "Transmission unavailability of frequency-diversity protected microwave systems caused by multipath fading", BSTJ, vol. 58, n. 8, October 1979, pp.1779-96.

**
End of
Session #5
**

_______________________________________________________________

©
2001-2016, Apus Cloud Project e Luigi Moreno