Multipath Fading

Summary

Refractivity in the atmosphere (II)

Observed impairments in the Rx signal

Modeling multipath activity

Performance prediction

Countermeasures

HERALD Lab #5

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Summary

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.

Refractivity in the atmosphere (II)

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 a given 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_8

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_6

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.

Observed impairments in Rx signal

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.

Vettori

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).

recpow

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 shift d between two vector components is computed as a function of DL (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.

APD_D_1

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 (D₂).

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).

Modeling multipath activity

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 of minutes, 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 A₀ is given by :

where P₀ is called "multipath occurrence factor". (To be more precise, this is the Rayleigh "asymptotic" trend, derived for low probability and deep fade levels).

RAYLEIGH

An example of Rayleigh cumulative distribution, with P₀ = 1

Note that, if the reference fade depth A₀ 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.

Multipath Occurrence Factor

The Rayleigh model for multipath fade depth is described by a single parameter P₀.

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 P₀ value will apply in each case. So, the P₀ 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 P₀ factor when a radio hop is already working. However, the radio engineer needs prediction tools to estimate P₀ while a radio hop is at the design stage.

Several empirical formulas have been proposed, giving P₀ 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 :

where C (geoclimatic coefficient), Q (terrain profile coefficient), a (frequency exponent), and b (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, P₀ is proportional to frequency (the a exponent is equal, or close, to 1), while the b exponent 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 P₀ 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.

mult2A

Application of the Barnett-Vigants model:

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

mult1A

Application of the Barnett-Vigants model:

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

mult3A

Application of the Barnett-Vigants model:

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

mult4A

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.530-15. 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.530-15 (released September 2013) gives a model for the prediction of the Multipath Occurrence Factor P₀.

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 P₀. 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 P₀. 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 of a new experimental database, as already happened in recent years.

Performance prediction

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 P₀ is the multipath occurrence factor.

The outage time T_OUT during a given observation time T₀ (typically, one month), is finally given as T_OUT = T₀P_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 P₀, 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.530-15. 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 :

where :

is the Multipath Activity (directly related to the Multipath Occurrence Factor P₀);

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

B is the signature depth [dB];

t_r is the echo delay in the signature measurement.

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.530-15 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 P₀ is the multipath occurrence factor and h is the multipath activity, previously defined.

Notes :

1) If XPD > 35 dB, then put XPD = 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).

Countermeasures

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 A₁ and A₂ are the attenuations at the two diversity receivers, A₀ 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 A₀, 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.530-15:

The Improvement factor is a function of the reference attenuation A₀. F is the working frequency in GHz, D the path length in km, S the vertical spacing in m, P₀ 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 DF 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 A₀ (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 DF 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 DF_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 :

where h 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 Improvement I_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.530-15 as a function of k_NS (complete formulas are not added here).

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.530-15 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).

Depth2_1

Equipment signatures without and with an adaptive equalizer.