In Chapter 3 and Chapter 4 we considered the dimensionality for the representation of a multipath field in a circular region. The field was considered both with a uniform angular power spectrum and a restricted range of angles. For the isotropic case the basis functions are the complex exponentials. For the restricted angular range, the basis functions can be approximated by the prolate spheroidal wave functions. Further perturbations were shown to introduce greater complexity into the basis functions, suggesting that convenient solutions for such cases are unlikely.
Of interest in the general case is the essential dimensionality, or number of significant terms, that could be utilised if the correct basis was determined for a particular scenario. This provides a measure of the sub optimality of using the basis obtained from the isotropic case. This is a property of the eigenvalues from (4.37). In particular, we can consider the number of terms required for the expected residual error in a finite representation to fall below a set threshold.
Definition 4.8 Dimensionality of a Multipath Field.
For any set of eigenvalues from (4.37), givenε >0there exists some integerD(ε)such that,
D(ε) = arg min n P m≥nλm P mλm < ε . (4.85)
Of general interest is the value ofN =D(0.01)for which ourN term finite representation (4.36) will capture 99% of the multipath energy. The modelling error in using such a rep- resentation would be equivalent to a 20dB signal to noise ratio. We use this threshold as the definition of the essential dimensionality to analyse the eigenvalues obtained from the integral equation.
Strictly, D(ε)can only take on integer values. Of particular interest are the situations for whichD(ε)will be fairly small such as a communications system with a small number of antenna in a confined spatial region. The impact of changes to the region shape and field angular power distribution will be obscured by the coarse quantisation. To facilitate the
4.7 Dimensionality of Optimal Representation 0 0 6 12 −90 0 90 (a) Restricted to±90◦ 0 0 6 12 10−2 10−1 100 −45 0 45 (b) Restricted to±45◦ 0 0 6 12 10−2 10−1 100 −22.5 0 22.5 (c) Restricted to±22.5◦
Figure 4.4: Eigenvalues and first four angular basis functions for a circular region (R = λ) and uniform restricted angular spectrum. The top plot in each column provides a schematic of the region and source distribution. The second plot shows the restricted (◦) and eigenvalues with those for the uniform spectrum (×) also plotted for comparison. Restricting the angular range lowers the number of significant eigenvalues. The remaining four plots in each column show the basis functions, with the equivalent prolate spheroidal functions shown as a dashed line. The angular basis functions are zero beyond the domain shown in the figures.
0
0 6 12
−90 0 90
(a) Gaussian withσ= 52◦
0 0 6 12 10−2 10−1 100 −90 0 90 (b) Gaussian withσ= 26◦ 0 0 6 12 10−2 10−1 100 −90 0 90 (c) Gaussian withσ= 13◦
Figure 4.5: Eigenvalues and first four angular basis functions for a circular region (R = λ) and a truncated Gaussian angular source spectrum. The top plot in each column provides a schematic of the region and source distribution. The second plot shows the Gaussian spectrum (◦) eigenvalues with the uniform spectrum (×) eigenvalues also plotted for comparison. The remaining four plots in each columns show the basis functions for the Gaussian distribution. For comparison, the dashed line depicts the basis function for a uniform angular spread with the same angular variance.
4.7 Dimensionality of Optimal Representation
0
0 6 12
−180 −90 0 90 180
(a) Ellipse Axes2λandλ
0 0 6 12 10−2 10−1 100 −180 −90 0 90 180
(b) Ellipse Axes2λandλ/2
0 0 6 12 10−2 10−1 100 −180 −90 0 90 180
(c) Uniform Linear Array Figure 4.6: Eigenvalues and first four angular basis functions for an elliptical region with an unre-
stricted uniform angular source distribution. The top plot in each column shows a schematic of the region geometry and source distribution. The second plot shows the eigenvalues for the elliptical re- gion (◦) with the eigenvalues of a circular region (×) shown for comparison. As the region becomes more elliptical, the eigenvalues converge towards the limiting case of a line array which is shown for comparison in the third column. For the elongated domain of interest and uniform linear array, the
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 Effective Dimensionality Radius (λ) Isotropic Restricted ±90° Restricted ±45° Restricted ±22.5° 2kR+1 kR+1 kR/2+1 kR/4+1
Figure 4.7: Essential dimension of a region with restricted uniform angular power spectrum. Scat-
tered points are obtained from the numerical eigenvalues obtained from (4.37). Lines are plotted to show the empirical relationship D= 2kRA/π+ 1which shows an excellent correspondence. This figure demonstrates that dimensionality varies linearly with the radius and angular range.
analysis, we use an exponential interpolation to obtain a fractional dimensionality. The pro- cedure is detailed in Appendix A. The fractional definition of essential dimension provides smooth curves which aid in analysing the impact of configuration changes in the following examples.
Figure 4.7 show the essential dimension as the radius is varied for four different angular power spectra. The power spectra are uniformly distributed across |θ| < Afor A = 180◦, 90◦, 45◦ and 22.5◦. The values for D(0.01) obtained from the eigenvalues of (4.37) are
shown to be approximated by,
D(0.01) ≈2kRA
π + 1. (4.86)
Whilst the scattered points are obtained from the eigenequation results, the lines are direct plots of the simple relationship (4.86) and have not been in any way fitted to the data. The general relationship is to be expected, as discussed in Chapter 2, Section 3. The match between the scatter points and the lines will depend on the selected threshold value. A lower
4.7 Dimensionality of Optimal Representation 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 Effective Dimensionality
Major Axis Radius (λ) Circular
Elliptical 2:1 Elliptical 10:1 Elliptical 20:1 Uniform Linear Array 2kR+1
2kR/π+1
Figure 4.8: Essential dimension of an elliptical region with uniform angular power spectrum. Scat-
tered points are obtained from the numerical eigenvalues obtained from (4.37). Lines show the em- pirical relationship for a circle (D= 2kR+ 1) and uniform linear array (D= 2kR/π+ 1). Between these limits the variation of dimensionality with radius is dependent on the shape of the region.
threshold value would shift all the scatter points towards a higher effective dimensionality. In this case, the threshold of0.01provides an excellent match for the simple dimensionality expression (4.86).
For the isotropic case, an upper bound on the dimensionality of a multipath field as N = 2⌈ekR/2⌉+ 1 has been rigorously proven [41, 42]. From the plots in Figure 4.7 it can be seen that the lower value from (4.86) is a better match for the dimensionality as defined in Definition 4.8.
The next area of investigation is to study the impact of changes to the region shape. An elliptical region is selected as a simple perturbation of the circular region. Keeping the length of the major axis the same, the region is contracted by decreasing the minor axis. Figure 4.8 shows the effect of the overall region size and ratio of major to minor axis on the dimensionality. The field is isotropic from all angles. The dimensionality is decreased as the region becomes more elliptical. The limiting case of the uniform linear array with
D(0.01) ≈2kR1
0 45 90 135 180 0 2 4 6 8 10 12 Effective Dimensionality
Halfwidth of Uniform Angular Spread (A) Circular
Ellipse 4:1 Broadside Ellipse 4:1 Endfire Line Array Broadside Line Array Endfire 2kRA/π+1 2kRA/π2+1
Figure 4.9: The effect of increasing angular spread on the dimensionality. The incident field is
constrained to|θ|< AwhereAis the halfwidth of the uniform angular spread. The region lies within a radius ofλgiving a major axis or linear array of length2λ. For the asymmetric regions (elliptical and line array) the points are shown for the mean direction of arrival being aligned with the end or broad side of the array. The relative orientation has a significant impact on the dimensionality.
This is also a lower bound on the dimensionality. The scatter points approach this lower bound as the minor axis becomes insignificant compared with the wavelength (≪ λ/20). For such a region there is little diversity obtained from the width of the region since the field cannot change significantly over this distance.
Further analysis can introduce both a perturbation to the region shape and a restriction to the directions of arrival. For a circular region, the growth in dimensionality with the angular spread is linear. With an elliptical region, the growth in dimensionality is dependent on the relative orientation of the region shape and the mean direction of arrival. Figure 4.9 demonstrates the change in essential dimensionality with the angular spread. An increase in angular spread has more impact on the dimensionality when the mean direction of arrival is to the broad side of the region. The effect is minimised when the direction of arrival is collinear with the major axis of the region.
Finally consider the effect of a small angular range as it is moved around the array. The contribution to dimensionality will be maximal when aligned with the broad side of the region. The geometry for the region and angular power spectrum is shown in Figure 4.10.
4.7 Dimensionality of Optimal Representation
0
Major 2λ
Minor λ
Offset
(a) Ellipse with 2:1 axis ratio shown for −135◦ offset of mean
angle
0
Major 2λ
Minor λ/2
(b) Ellipse with 4:1 axis ratio shown for0◦offset of mean angle
0
ULA Length 2λ
Offset
(c) Uniform linear array shown for135◦offset of mean angle
Figure 4.10: Schematic of the geometries for the regions and offset in the mean angle. The three
figures show the three cases for which the data points are plotted in Figure 4.11. The angular range is
±12.25◦around the varied offset angle. A different offset angle is shown in each of the three figures.
The effect of the shift in mean angle offset on the dimensionality of the field is shown in Figure 4.11. The angular range is12.25◦ orA=π/8radians.
Consider the kernel, (4.67), for a uniform linear array. Using trigonometric identities it follows that λngn(θ) =P(θ) Z Ω gn(φ)sinc 2kRcos(θ+φ 2 ) sin( θ−φ 2 ) dφ ≈P(θ) Z θ′ +A θ′ −A gn(φ)sinc (kRcos(θ′)(θ−φ))dφ (4.88) with θ′ = (θ +φ)/2 and for small θ −φ. This is the much studied bandlimited kernel
with asymptotic dimensionality of2kRcos(θ′)A/π [142]. For a small angular spread and
elongated region, it is proposed that the dimensionality will vary as
D(0.01) = 2kRA|cos(θ)|
π + 1. (4.89)
This line is plotted in Figure 4.11 and provides a good match for the limiting case of the uniform linear array.
−1800 −135 −90 −45 0 45 90 135 180 0.5 1 1.5 2 2.5 3 Effective Dimensionality
Offset of Mean Angle of Arrival
Ellipse 2:1 Ellipse 4:1
Uniform Linear Array kR|cos(θ)|/4+1
Figure 4.11: The effect of relative orientation between the array and the direction of arrival. The
source distribution is restricted to a range of±12.25◦ and offset from the broadside of the elongated region by the abscissa angle. The dimensionality is maximised when the source distribution is aligned with the broad side of the array at0◦ and 180◦. The variation is more severe as the region becomes increasingly elliptical. Also shown is the theoretical curve that would be expected for a small angular range incident on a uniform linear array.
4.8
Summary and Contributions
This chapter presented a theoretical framework for the representation of a random multipath field in the angular domain. The framework leads to an integral equation whose solutions are the optimal basis function for such a representation. In analysing this equation it is clear that the most efficient representation depends directly on the scattering environment, through the angular power spectrumP(θ), and also the way in which the field is measured or observed, through the defined domain of interestΛ. Whilst the integral equation only leads to analytical solutions in the simplest cases, it was shown that it can be accurately solved numerically, providing a means to obtain the optimal angular representation.
4.8 Summary and Contributions The following specific contributions were made in this chapter:
1. Developed a framework for the representation of a random multipath field in the angu- lar domain. The angular domain representation implicitly captures the wave equation constraint. In addition, in comparison with a direct representation of the multipath field in space, the angular domain has one less dimension. The angular power spec- tra for a multipath field corresponds with the physical and engineering intuition of a spatial channel model.
2. Derived an integral eigenequation to determine the optimal set of deterministic angular basis functions for representing a random multipath field. The solution of this integral equation is dependent on both the angular power spectrum and the spatial domain of interest for the field. How the field is observed or measured has a direct bearing on the optimal representation in the angular domain.
3. Demonstrated that the integral equation in the angular framework has a direct corre- spondence to the Karhunen-Lo´eve expansion in the spatial domain. This result vali- dated the consistency and optimality of the proposed framework.
4. Derived the closed form solutions to the eigenequation for the cases of an isotropic field and singular direction of arrival with a circular region. Investigated and concluded that the eigenequation is not easily soluble in closed form for general configurations. 5. Detailed two suitable numerical techniques to accurately approximate the solution of
the integral equation using a discrete matrix equation. Demonstrated that the size of the matrix is determined by the extent of the domain of interest with a matrix dimension of2⌈kR⌉+ 1required for a two-dimensional region with maximum radiusRand field wavenumberk = 2π/λ.
6. Demonstrated that the high resolution details in the power spectrum,P(θb), beyond a certain point are largely irrelevant in determining the optimal representation. For the two-dimensional case it was shown that only the low order Fourier coefficients ofP(θ)
are significant. The critical characteristics of the system for determining the optimal basis set are the physical extent of the region of interest and the low frequency content of the angular power spectrum.
7. Presented examples to characterise the way in which the shape and distribution of the region and the power spectrum interact and effect the number of significant solutions of the eigenequation. This directly relates to the optimal number of terms needed to represent the random field. These examples demonstrated the macroscopic aspects of the interaction between the angular power spectrum and the domain of interest.
8. Introduced the essential dimensionality as the number of terms required to capture 99% of the energy of the random multipath field. This represents a 20dB signal to truncation error ratio. Analysis of the eigenvalue results demonstrated that this thresh- old is consistent with previously stated expressions for dimensionality. In particular, a circular region with a uniform restricted angular power spectrum has dimensionality
D = 2kR A/π + 1, where the range of incident angles is restricted to±A about the mean angle.