0 5 10 15 20 25 30 35 40
Source at Radius in Wavelengths (S)
Truncation Order (N) R = 1λ R = 2λ R = 3λ R = 4λ ε = −20dB ε = −30dB ε = −40dB N = kR
Figure 2.3: Error contours for the truncation of a field generated by near-field sources. For an
observation region of radiusR = 1,2,3,5λ, sources are present from the abscissa radius S. The dimensionality approaches the asymptotic value within one wavelengthS−R > λ. The sensitivity to near-field sources increases slightly with increasing observation radius. The lines corresponding to the truncation orderN =kRare shown for reference.
In answer to the question posed for the buffer distance between the domain of interest and any sources, the dimensionality reaches its asymptotic value for a buffer distance less than one wavelength. Furthermore, this does not vary significantly as the region size is increased.
2.7
Summary and Contributions
This chapter has developed the framework and clarified some existing results regarding the dimensionality of a multipath field over a finite domain of interest. It has been shown that this problem is important in developing a means of modelling and representation of a multipath field as is required to develop fundamental limits regarding the performance of communica- tions systems using spatial diversity.
Central to this chapter is the result that the dimensionality of an arbitrary two-dimensional multipath field is related to the radial extent of the region of observation scaled by the wave-
length of the narrow-band field,
D= 2⌈kR⌉+ 1 ≈12.57R/λ. (2.49)
While a numerical investigation indicates this is asymptotically correct for large regions, it is inappropriate for small regions. While a bound exists, it is conservative for larger radii. This motivated an attempt to create a tighter formal bound. Whilst some progress was made, it remains incomplete and important fundamental difficulties were identified. Through fur- ther analysis and numerical investigation it was shown that these dimensionality results can be extended to include fields with sources near the domain of interest, with the influence of sources decaying to insignificance outside a few wavelengths distance from the region boundary.
The following specific contributions were made in this chapter: 1. Provided a comparison of two existing dimensionality results:
• A dimensionality of2kR+ 1, although not rigorously derived, appears to be the correct asymptotic expression asR → ∞.
• The bound on the dimensionality ofekR+ 1, valid for allR, is conservative by a multiplicative factor ofe/2≈1.35for largeR, but tighter at smallR. In practical MIMO applications, small radii are arguable more relevant.
2. Presented a numerical study to consider the effective dimensionality of regions over a wide range of radii. This supported the use of the bound for small radii whilst the asymptotic dimensionality was2kRasR → ∞.
3. Pursued the path of deriving a bound to obtain a tighter result for the dimensionality bound as R → ∞. This motivated the development of a conjectured bound on the Bessel functionJn(z)in the regionz < n. Several difficulties were highlighted in the attempt to use this in the development of a tighter dimensionality bound.
4. Considered the impact of near-field sources on the required dimensionality for the field representation. Analysis and numerical investigation demonstrated that the influ- ence of sources need only be considered when within a few wavelengths of the region boundary.
Chapter 3
Impact of Direction of Arrival on
Dimensionality
3.1
Introduction
The previous chapter developed the framework for understanding dimensionality of the mul- tipath field. This chapter investigates the effect of restricting the direction of arrival of the multipath field.
Often in wireless communications the directions of arrival are constrained in direction or only span a sector. This restriction on the field can be incorporated into a model to use a more appropriate basis function and more compact parameterisation of the field. It has been noted that the richness, dimensionality or degrees of freedom for a spatial field decrease as the angular diversity is reduced [47, 84, 156]. Whilst such results suggest the dimensionality increases linearly with the angular spread, this has not been rigorously proven for a general region. A formal expression of this relationship is an important tool in better understand- ing the impact of angular diversity on the upper limits of the capacity of a communications system operating in a finite domain of interest. Conventional works on the limits of the capacity of a multiple antenna communications system rely on specifics of the antenna ge- ometry or spatial correlation models. By capturing the inherent dimensionality of the spatial field, it is possible to show the existence of an upper limit without reference to any specific configuration, thus providing a guide and reference for optimal system design.
The existing results that relate dimensionality to angular diversity are not rigourous and are based on simulation and approximations. Our aim is to provide a tight foundation to these intuitive results. We have seen that the dimensionality varies linearly with the radius of
the domain of interestR. We now introduce a second variable, A, representing the angular diversity. This leads to a spatial analogy of the well known 2W T dimensionality of time- bandwidth constrained signals discussed in Section 2.2. In this work we consider the problem of a circular region with the source directions constrained to a single contiguous interval. The effect of different angular distributions is the subject of Chapter 4. The impact of discrete clusters of scatterers was considered in [93].
The work presented in this chapter provides a formal proof of the linear relationship between the dimensionality and angular spread. This key result has been published by the author [157].