Theory of Multipath Shape Factors for Small-Scale Fading Wireless Channels

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5.8 Theory of Multipath Shape Factors for Small-Scale Fading Wireless Channels

A new method of analysis has been developed that characterizes the flat, small-scale fading experienced

by a receiver in an arbitrary spatial-temporal channel [Dur00]. This method characterizes a

multipath channel with arbitrary spatial complexity to three shape factors that have simple, intuitive

geometrical interpretations. Furthermore, these shape factors describe the statistics of received

[+] Enlarge Image

Figure 5.27 Indoor wideband impulse responses simulated by SIRCIM at 1.3 GHz. Also shown

are the distributions of the rms delay spread and the narrowband signal power distribution. The

channel is simulated as being obstructed in an open-plan building, T–R separation is 25 m. The

rms delay spread is 137.7 ns. All multipath components and parameters are stored on disk [from

[Rap93a] © IEEE].

 

signal fluctuations in a fading multipath channel. Analytical expressions for level-crossing rate,

average fade duration, envelope autocovariance, and coherence distance are easily derived using the

new shape factor theory, and match all of the classical results in [Jak74].

5.8.1 Introduction to Shape Factors

The term small-scale fading describes the rapid fluctuations of received power level due to

small, sub-wavelength changes in receiver position. This effect is due to the constructive and

destructive interference of the numerous multipath waves that impinge upon a wireless receiver

[Jak74]. The resulting signal strength fluctuations affect, in some way, nearly every aspect of

receiver design: dynamic range, equalization, diversity, modulation scheme, and channel and

error-correction coding. These fluctuations are a function of direction of travel as related to the

angle of arrival of multipath delay.

Numerous researchers have measured and analyzed the first-order statistics of these processes,

which mostly involves the characterization of small-scale fading with a probability density

function (PDF) discussed in Section 5.6 [Ric48], [Suz77], [Cou98]. The autocorrelation statistics of

[+] Enlarge Image

Figure 5.28 Urban wideband impulse responses simulated by SMRCIM at 1.3 GHz. Also shown

are the distributions of the rms delay spread and the narrowband fading. T–R separation is

2.68 km. The rms delay spread is 3.8 μs. All multipath components and parameters are saved on

disk. [from [Rap93a] © IEEE].

fading processes, or second-order statistics, include measures of a process such as power spectral

density (PSD), level-crossing rate, and average fade duration, as discussed in Section 5.7

It turns out that second-order statistics are heavily dependent on the angles-of-arrival of

received multipath, yet they have traditionally been studied using an omnidirectional, azimuthal

propagation model as shown in Section 5.7 [Jak74]. That is, multipath waves usually are assumed

to arrive at the receiver with equal power from the horizon in every possible direction. Truthfully,

no realistic channel resembles this idealized model, but it is a reasonable multipath propagation

model for receivers operating in heavily shadowed regions with a dense concentration of scatterers,

and yields analytical results which resembled early field measurements [Cla68]. Clarke’s model

conveniently produces analytical statistics that are isotropic, unrealistically identical regardless of

the direction traveled by the mobile receiver. Unfortunately, recent measurements and models have

shown that the arriving multipath in a local area bears little resemblance to the omnidirectional

propagation assumption and, in fact, the direction of travel relative to the arriving multipath is a

key factor in producing the observed fading [Ros97], [Fuh97]. An approximately omnidirectional

channel model does not accurately describe fading statistics if directional or smart antenna systems

are employed at a receiver [Win98, Lib99, Mol01].

The new concept of multipath shape factors allows the quantitative analysis of any distribution

of non-omnidirectional multipath waves in a local area (where the local average signal

strength is assumed to be wide-sense stationary). Three principle shape factors—the angular

spread, the angular constriction, and the azimuthal direction of maximum fading—are defined

and exactly related to the average rate at which a received signal fades [Dur99]. Four of the basic

second-order small-scale fading statistics—level-crossing rate, average fade duration, autocovariance,

and coherence distance—may be described in terms of this multipath shape factor theory.

As shown subsequently, several classical propagation problems are analyzed easily using multipath

shape factors.

5.8.1.1 Multipath Shape Factors

Three multipath shape factors influence second-order fading statistics, and these may be derived

from the angular distribution of multipath power, p(θ), which is a general representation of fromthe-

horizon propagation in a local area [Gan72]. This representation of p(θ) includes antenna gains

and polarization mismatch effects [Dur98a]. The shape factors are based on the complex Fourier

coefficients of p(θ):

where F_n is the nth complex Fourier coefficient. The utility of these three shape factors becomes

clear in Section 5.8.1.2.

Angular Spread, Λ

The shape factor angular spread, Λ, is a measure of how multipath concentrates about a single

azimuthal direction of arrival. We define angular spread to be

where F₀ and F₁ are defined by Equation (5.90). There are several advantages to defining angular

spread by Equation (5.91). First, since angular spread is normalized by F₀ (the total amount of

local average received power), it is invariant under changes in transmitted power. Second, Λ is

invariant under any series of rotational or reflective transformations ofp (θ). Finally, this definition

is intuitive; angular spread ranges from 0 to 1, whereby 0 denotes the extreme case of a single

multipath component from a single direction and 1 denotes no clear bias in the angular distribution

of received power (e.g., Clarke’s model).

It should be noted that other definitions exist in the literature for angular spread. These

definitions involve either beamwidth or the RMS θ calculations and are often ill-suited for general

application to arbitrary channels or periodic functions such as p(θ) [Ebi91], [Nag96], [Ful98],

[Jen98].

Angular Constriction, 

The shape factor angular constriction, , is a measure of how multipath concentrates about two

azimuthal directions. We define angular constriction to be

where F₀, F₁, and F₂ are defined by Equation (5.90). Much like the definition of angular spread,

the measure for angular constriction is invariant under changes in transmitted power or any series

of rotational or reflective transformations of p(θ). The possible values of angular constriction, ,

range from 0 to 1, with 0 denoting no clear bias in two arrival directions and 1 denoting the

extreme case of exactly two multipath components arriving from different directions.

Azimuthal Direction of Maximum Fading, θ_max

A third shape factor, which may be thought of as a directional, or orientation, parameter, is the

azimuthal direction of maximum fading, θ_max. We define this parameter to be

The θ_max value corresponds to the direction in which a mobile user would move in order to

experience the maximum fading rate in the local area.

5.8.1.2 Fading Rate Variance Relationships

Complex received voltage, received power, and received envelope are the three basic stochastic

processes that are studied in small-scale fading. In order to understand how these stochastic

processes evolve over space, it is useful to study the position derivatives or rate-of-changes of

the three processes. Since the mean derivative of a stationary process is zero, the mean-squared

derivative is the simplest statistic that measures the fading rate of a channel. In fact, a meansquared

derivative of a stationary process is actually the variance of the rate-of-change. This

section, therefore, presents equations for describing the rate variance relationships of small-scale

received complex voltage, power, and envelope fluctuations. All of these relationships are

proven exactly in Appendix C.

Complex Received Voltage, 

The complex received voltage,   , is a baseband representation of the summation of numerous

multipath waves that have impinged upon the receiver antenna and have excited a complex voltage

component at the input of a receiver (see Section 5.2). Appendix C.1 derives the rate variance, 

for the complex voltage of a receiver traveling along the azimuthal direction θ:

where   is the wavelength of the carrier frequency, P_T is the local average mean-squared

received power (units of volts-squared). Note that the dependence on multipath angle-of-arrival

in Equation (5.94) may be reduced to the three basic shape factors: angular spread, angular constriction,

and the azimuthal direction of maximum fading. The physical significance of , is

that it describes the spatial selectivity of a channel in a local area and, by extension, the average

complex voltage fluctuations for a mobile receiver as it moves over a local area.

             

              Received Power, P(r)

Received power, P_r, is equal to the magnitude-squared of complex voltage,  . Note that this

definition of power yields units of volts-squared rather than watts, which would differ only by a

constant of proportionality related to the input impedance of the receiver; the volts-squared definition

is more general and independent of the receiver used.

The mathematical operation of taking the squared magnitude of a complex quantity is a

nonlinear operation, so in order to derive a rate variance relationship for received power, we will

assume that the channel is Rayleigh fading. This assumption, however, is unnecessary for the

derivation of Equation (5.94). Appendix C.2 derives the rate variance,  for the power of a

receiver traveling along the azimuthal direction θ:

Once again, the dependence on multipath angle-of-arrival in Equation (5.95) may be reduced

entirely to the three basic shape factors. The physical significance of   is that it describes the

average received power fluctuations in a local area where the signal envelope undergoes Rayleigh

flat fading.

Received Envelope, R(r)

Received envelope, R(r), is equal to the magnitude of complex voltage, . Once again, we

assume that the channel is Rayleigh fading to calculate the mean-squared fading rate. Appendix

C.3 shows how this assumption leads to the envelope rate variance,

Again, Equation (5.96) depends on Λ,, and θ_max. The physical significance of  is that it

describes the average envelope fluctuations in a local area where the signal envelope undergoes

Rayleigh flat fading.

5.8.1.3 Comparison to Omnidirectional Propagation

Applying the three shape factors, Λ,, and θ_max  to the classical omnidirectional propagation

model of Section 5.7, we find that there is not a bias in either one or two directions of angle-ofarrival,

leading to maximum angular spread (Λ = 1) and minimum angular constriction (  = 0).

The statistics of omnidirectional propagation are isotropic, exhibiting no dependence on the azimuthal

direction of receiver travel, θ.

If the rate variance relationships of Equations (5.94)–(5.96) are normalized against their

values for omnidirectional propagation, then they reduce to the following form:

where  is a normalized fading rate variance. Equation (5.97) provides a convenient way to

analyze the effects of the shape factors on the second-order statistics of small-scale fading.

First, notice that angular spread, Λ, describes the average fading rate within a local area. A

convenient way of viewing this effect is to consider the fading rate variance taken along two perpendicular

directions within the same local area. From Equation (5.97), the average of the two

fading rate variances, regardless of the orientation of the measurement, is always given by averaging

the variances observed over two perpendicular directions within the local area

Equation (5.98) clearly shows that the average fading rate within a local area decreases with

respect to omnidirectional propagation as multipath power becomes more and more concentrated

about a single azimuthal direction. A method for measuring multipath angular spread

based on this relationship has been presented in [Pat99].

Second, notice that angular constriction,  , does not affect the average fading rate within a

local area, but describes the variability of fading rates taken along different azimuthal directions,

θ. From Equation (5.97), fading rate variance   will change as a function of direction of

receiver travel θ, but will always fall within the following range:

The upper limit of Equation (5.99) corresponds to a receiver traveling in the azimuthal direction

of maximum fading (θ = θ_max ) while the lower limit corresponds to travel in a perpendicular

direction (θ = θ_max +  Equation (5.99) clearly shows that the variability of fading rates

within the same local area increases as the channel becomes more and more constricted.

It is interesting to note that the propagation mechanisms of a channel are not uniquely

described by the three shape factors Λ, , and θ_max. An infinitum of propagation mechanisms

exist which may have the same set of shape factors and, by extension, lead to channels which

exhibit nearly the same end-to-end performance. In fact, Equation (5.97) provides rigorous

mathematical criteria for a multipath channel that may be treated as “pseudo-omnidirectional”:

Under the condition of Equation (5.100), angular spread becomes approximately 1 and angular

constriction becomes approximately 0. Thus, the second-order statistics of the channel behave

nearly identical to the classical omnidirectional channel developed by Clarke and Gans.

5.8.2 Examples of Fading Behavior

This section presents four different analytical examples of non-omnidirectional propagation

channels that provide insight into the shape factor definitions and how they describe fading rates.

Consider the simplest small-scale fading situation where two coherent, constant-amplitude

multipath components, with individual powers defined by P₁ and P₂, arrive at a mobile receiver

separated by an azimuthal angle  We call this the two-wave channel model. Figure 5.29 illustrates

this angular distribution of power, which is mathematically defined as

where θ_o is an arbitrary offset angle and δ(·) is an impulse function. By applying Equations

(5.91)–(5.93), the expressions for Λ, , and θ_max for this distribution are

The angular constriction, γ, is always 1 because the two-wave model represents perfect clustering

about two directions. The limiting case of two multipath components arriving from the same

direction   results in an angular spread, Λ, of 0. An angular spread of 1 results only when

two multipath of identical powers (P₁ = P₂) are separated by   Figure 5.29 shows how

the fading behavior changes as multipath separation angle, increases for the case of two

equal-powered waves. Thus, increasing α changes a channel with low spatial selectivity into a

channel with high spatial selectivity that exhibits a strong dependence on the azimuthal direction

of receiver motion.

5.8.2.1 Sector Channel Model

Consider another theoretical situation where multipath power is arriving continuously and uniformly

over a range of azimuth angles. This model has been used to describe propagation for

directional receiver antennas with a distinct azimuthal beam [Gan72]. The function p(θ) will be

defined by

Figure 5.29 Fading properties of two multipath components of equal power [from [Dur00]

©IEEE].

The angle indicates the width of the sector (in radians) of arriving multipath power and the

angle θ_ο is an arbitrary offset angle, as illustrated by Figure 5.30. By applying Equations (5.91)–

(5.93), the expressions for Λ,, and θ_max for this distribution are

[+] Enlarge Image

The limiting cases of these parameters and Equation (5.95) provide deeper understanding of

angular spread and constriction.

Figure 5.30 Fading properties of a continuous sector of multipath components [from [Dur00]

©IEEE].

Figure 5.30 graphs the spatial channel parameters, Λ and, as a function of sector width,

  The limiting case of a single multipath arriving from precisely one direction corresponds to

, which results in the minimum angular spread of Λ = 0. The other limiting case of uniform

illumination in all directions corresponds to   (omnidirectional Clarke model), which

results in the maximum angular spread of Λ = 1. The angular constriction, , follows an opposite

trend. It is at a maximum ( = 1) when  and at a minimum ( = 0) when  The

graph in Figure 5.30 shows that as the multipath angles of arrival are condensed into a smaller

and smaller sector, the directional dependence of fading rates within the same local area

increases. Overall, however, fading rates decrease with decreasing sector size

Figure 5.31 Fading properties of double-sectored multipath components [from [Dur00] ©IEEE].

5.8.2.2 Double Sector Channel Model

Another example of angular constriction may be studied using the Double Sector model of

Figure 5.31. Diffuse multipath propagation over two equal and opposite sectors of azimuthal

angles characterize the incoming power. The equation that describes this angular distribution of

power is

The angle α is the sector width and the angle θ_o is an arbitrary offset angle. By applying

Equations (5.91)–(5.93), the expressions for Λ,  , and θ_max for this distribution are

Note that the value of angular spread, Λ, is always 1. Regardless of the value of an

equal amount of power arrives from opposite directions, producing no clear bias in the direction

of multipath arrival.

The limiting case of   (omnidirectional propagation) results in an angular constriction

of = 0. As α decreases, the angular distribution of power becomes more and more constricted.

In the limit of  , the value of angular constriction reaches its maximum, = 1. This

case corresponds to the above-mentioned instance of two-wave propagation. Figure 5.31 shows

how the fading behavior changes as sector width α increases, making the fading rate more and

more isotropic while the RMS average remains constant.

5.8.2.3 Ricean Channel Model

A Ricean channel model results from the addition of a single plane wave and numerous diffusely

scattered waves [Ric48]. If the power of the scattered waves is assumed to be evenly distributed

in azimuth, then the channel may be modeled by the following p(θ):

where K is the ratio of coherent to diffuse incoherent power, often referred to as the Ricean

K-factor. By applying Equations (5.91)–(5.93), the expressions for Λ,, and θ_max for this distribution

are

Figure 5.32 depicts the spatial channel parameters, Λ and, as a function of K-factor. For

very small K-factors, the channel appears to be omnidirectional (Λ = 1 and  = 0). As the K-factor

increases, the angular spread of the Ricean channel decreases and the angular constriction

increases. This indicates that the overall fading rate in the Ricean channel decreases and that the

differences between the minimum and maximum fading rate variances within the same local area

but different directions increases.

5.8.3 Second-Order Statistics Using Shape Factors [Dur00]

With an understanding of how shape factors describe fading rate variances, it is possible to re-derive

many of the basic second-order statistical measures of fading channels given in Section 5.7.3 in

terms of the three shape factors. Level-crossing rates, average fade duration, spatial autocovariance,

Figure 5.32 Fading properties of Ricean-model multipath components [from [Dur00] ©IEEE].

 

and coherence distance expressions that were originally derived under the assumption of omnidirectional

multipath propagation will now be cast in terms of the angular spread, the angular constriction,

and the azimuthal direction of maximum fading [Dur99], [Dur99a], [Dur99b].

The derivations focus on Rayleigh channels, since these types of channels are analytically

tractable. A Rayleigh fading signal is one whose envelope, R, follows a Rayleigh PDF, p_R(r),

given by Equation (5.49) which can be expressed as

where P_T is the average total power received in a local area (units of volts-squared).

5.8.3.1 Level-Crossing Rates and Average Fade Duration

The general expression for a level-crossing rate is given by (5.80) [Jak74]:

where R is the threshold level and  is the joint PDF of envelope and its time derivative.

For a Rayleigh-fading signal, the level-crossing rate of the envelope process is

The variable ρ is the normalized threshold level, such that ρ = R²/P_T [Jak74]. Note that  is

simply the time-derivative equivalent of , derived in Appendix C.1, which arises from a

mobile receiver traveling through space with a constant velocity in an otherwise static channel

(transmitter and scatterers are fixed).

By substituting Equation (5.94) into Equation (5.111), we arrive at an exact expression for

the level-crossing rate in a Rayleigh fading channel with any arbitrary spatial distribution of

multipath power and any direction of mobile receiver travel, θ:

The average fade duration,, is defined to be [Jak74], [Cla68]

Substitution of the Rayleigh PDF of Equations (5.109) and (5.112) into Equation (5.113)

yields

Equations (5.112) and (5.114) are useful tools for studying small-scale fading statistics in the

presence of non-omnidirectional multipath, and yield identical results to those given in Section 5.7.3

when an omnidirectional channel is assumed.

5.8.3.2 Spatial Autocovariance

Another important second-order statistic is the spatial autocovariance of received voltage envelope.

The autocovariance function determines the correlation of received voltage envelope as a function

of change in receiver position and is useful for studies in spatial diversity [Jak74], [Vau93].

Appendix D develops an approximate expression for the spatial autocovariance function of envelope

based on shape factors [Dur99a]. The approximation is given by

Equation (5.115) allows us to estimate the envelope correlation between two points in space separated

by a distance r along an azimuthal direction θ. The behavior of Equation (5.115) is benchmarked

in Section 5.8.5 against several known analytical solutions presented in [Jak74].

5.8.3.3 Coherence Distance

Coherence distance, D_c, is the separation distance in space over which a fading channel appears

to be unchanged. As shown in Chapter 7, coherence distance is important in the design of wireless

receivers that employ spatial diversity to combat spatial selectivity. For mobile receivers, a

similar parameter called coherence time, T_c, is the elapsed time over which a fading channel

appears to be constant (see Equation (5.40.b)). For the case of a static channel, the coherence

time of a mobile receiver may be calculated from the coherence distance (T_c = D_c /v, where v is

the speed of the mobile).

Definitions for coherence distance may be based on the envelope autocovariance function.

A convenient definition for the coherence distance, D_c, is the value that satisfies the equation

ρ(Dc) = 0.5 [Ste94]. The classical value for coherence distance in an omnidirectional Rayleigh

channel is given by

Using the generalized autocovariance function of Equation (5.115) leads to a new definition of

coherence distance:

For omnidirectional propagation, Equation (5.117) differs from Equation (5.116) by only –3.0%.

Furthermore, Equation (5.117) captures the behavior of non-omnidirectional multipath. As

angular spread, Λ, decreases, the coherence distance in a local area increases. As the angular

constriction,, increases, the coherence distance develops a strong dependence on orientation, θ.

5.8.4 Applying Shape Factors to Wideband Channels

The theory presented in Section 5.8 was originally developed for a flat fading small scale

assumption. By realizing that wideband channels may be modeled as discrete, resolvable multipath

components in time delay, one can readily see how the theory may be applied to each

resolvable time delay bin, as shown in Figure 5.4. The shape factor theory allows the fading statistics

of individual multipath to be studied [Pat99].

5.8.5 Revisiting Classical Channel Models with Shape Factors

As a point of comparison, we now analyze three well-known cases of propagation that have analytical

solutions as described in Sections 5.7.1–5.7.3 [Jak74]. The cases are analyzed using the shape

factor approach as outlined in Section 5.7.4 for mobile receivers with speed v. This approach is

shown to produce quick, comprehensive, and—most importantly—accurate solutions.

The first case corresponds to a narrowband receiver operating in a local area with multipath

arriving from all directions, such that the angular distribution of power, p(θ), is a constant.

The receiver antenna is assumed to be an omnidirectional whip, oriented perpendicular to the

ground. Due to the vertical electric-field polarization of the whip antenna, this propagation scenario

is referred to as the E_z-case [Cla68].

The second two cases correspond to the same narrowband receiver in the same omnidirectional

multipath channel, but with a small loop antenna mounted atop the receiver such that the

plane of the loop is perpendicular to the ground. The antenna pattern of the small loop antenna

attenuates the arriving multipath such that the angular distribution of power becomes

where A is some arbitrary gain constant. Unlike the omnidirectional E_z-case, the statistics of this

propagation scenario will depend on the direction of travel by the receiver. The H_x-case will refer to

a receiver traveling in a direction perpendicular to the main lobes of the loop antenna pattern (θ = 0).

The H_y-case will refer to a receiver traveling in a direction parallel to the main lobes .

Figure 5.33 illustrates the E_z, H_x, and H_y cases for the modeled receiver antennas.

The first step is to calculate the three spatial parameters from the angular distribution of

power, p(θ), using Equations (5.91)–(5.93). The spatial parameters for the E_z-case are Λ,, and

θ_max = 0. Since this case is omnidirectional, the angular spread is at a maximum (Λ = 1) and the

angular constriction is at a minimum ( = 0). For the H_x- and H_y-cases, the spatial parameters are

Figure 5.33 Three different multipath-induced mobile-fading scenarios [from [Dur00] ©IEEE].

Λ = 1,  = 1/2, and θ_max   Since the impinging multipath have no clear bias in one direction,

the angular spread is at a maximum just like the E_z-case. However, there is clearly a bias in two

directions, resulting in an increased angular constriction of   = 1/2.

After substitution of these parameters into Equation (5.112) along with the appropriate

direction of mobile travel, the level-crossing rates for the three cases become

The corresponding average fade durations are

These expressions exactly match the original solutions presented by Clarke in [Cla68].

Now substitute the channel shape factors into the approximate spatial autocovariance

functions in Equation (5.115). The results for the three cases are

These three functions are compared to their more rigorous analytical solutions in Figure 5.34

through Figure 5.36. Note that all three model the spatial autocovariance function consistent

with the approximation made in the derivation of Equation (5.115). The behavior is nearly exact

for values of r equal to or less than a correlation distance.

Figure 5.34 Comparison between Clarke theoretical and the shape theory approximation for

envelope autocovariance functions for E_z-case [from [Dur00] ©IEEE].

Figure 5.35 Comparison between Clarke theoretical and the shape theory approximation for

envelope autocovariance functions for H_x-case [from [Dur00] ©IEEE].

Figure 5.36 Comparison between Clarke theoretical and approximate envelope autocovariance

functions for H_y-case [from [Dur00] ©IEEE].

 

The shape factor technique for finding fading statistics is an intuitive way to relate the

physical channel characteristics to the fading behavior. In the previous examples, the spatial

parameters may be calculated analytically or even estimated intuitively by simply looking at the

distributions of multipath power in Figure 5.33. The use of spatial parameters to find level crossing

rate, average fade duration, and spatial autocovariance is quite simple when compared to the

full analytical solutions of the E_z, H_x, and H_y-cases presented in [Jak74]. The proposed solution

is also more comprehensive. For example, once the shape factors have been found, Equations

(5.112), (5.114), and (5.115) provide statistics for all directions of travel for the H_x- and H_y-cases,

and not just specific directions such as θ = 0 or   Thus, specific fading behaviors

for various directions of receiver motion are modeled easily.

The solution form of Equations (5.112), (5.114), and (5.115) reveals an interesting property

about statistics in Rayleigh-fading channels. Since the three shape factors only depend on

low-order Fourier coefficients, many of the second-order statistics of Rayleigh-fading channels

are insensitive to the higher-order multipath structure. The general biases of angular spread and

angular constriction truly dominate the space and time evolution of these fading processes.

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