From chapter 1, recall that the Grassmann manifold GF
n,k is the set of all k-dimensional
subspaces in n-dimensional Euclidean space Fn, where F represents either the real (R) or complex
(C) field. A related manifold formed by the set of all matrices in Fn×k with the k orthonormal
columns satisfying n ≥ k is called the Stiefel manifold VF
n,k. The study of these two manifolds
in the context of communications theory can be motivated in many ways. Using n to denote the block-length of a space-time code and k to denote the number of transmit antennas, [43] extended an argument of [123] to show that the appropriate coding spaces in a no-CSIT (Channel
State Information at Transmitter) system is a scaled version of either VF
n,k or GFn,k depending on
whether channel state information is available at the receiver or not. We have already argued earlier in this thesis that evaluating the performance of a n-transmit antenna multiple-input multiple- output (MIMO) system transmitting along k eigendirections of the channel matrix under finite-
rate feedback also leads one to geometric considerations on VF
n,k or GFn,k, depending on whether we
allocate unequal or equal powers along each eigenbeam. The observation that unitarily invariant problems on Euclidean spaces are naturally posed as unconstrained problems on these two manifolds [25,77] has encouraged their employment in tackling problems in areas such as signal processing [76] and computer vision [14, 114].
The applications of these manifolds in solving engineering problems requires one to analyze them as metric spaces, which in turn necessitates the delineation of a distance metric over them.
Based on different end-uses, researchers have suggested various metrics for GF
n,kand Vn,kF . For GFn,k,
[7] enlists six different metrics and concentrates on two of them, namely the geodesic metric arising from the “natural geometry of Grassmann manifold” and the chordal distance metric popularized
by a work of Conway et al. [16]. As [7] points out, Shannon [100] had analyzed GR
n,1 using the
geodesic metric (as early as in 1959) to demonstrate the existence of a code with minimum distance
θ over the manifold with its rate satisfying R ≥ − log sin θ − o(1). For the Stiefel manifold VF
n,k, two
points out the maximum likelihood receiver rule imposes the chordal distance metric on GF n,k and
the Euclidean metric on VF
n,k for the space-time coding system mentioned earlier. The study of
geodesic metrics by [7] and [41] arises from the realization that a detailed geometric analysis of these manifolds with a view to calculating geodesics or packings is greatly facilitated by the adoption of the geodesic metric. Here, invoking the local metric equivalence of these metrics [41] and a Riemannian geometric result from [36], we shall demonstrate that these distance metrics can be jointly analyzed under a single framework.
It turns out that the first step towards solving many theoretical questions involving either
VF
n,k or GFn,k lies in computing the volume of a ball over these manifolds. Despite the availability
of many geometric treatments of these manifolds as in [8] and [1], the precise volume has not been theoretically characterized so far. The earliest attempt to bound the volume was by Conway
et al. [16] who adopted a numerical simulation approach after isometrically embedding GR
n,k with the chordal distance metric into a Euclidean space of appropriate dimension. To improve this
rather loose bound, [7] (with a correction provided in [6]) studied GF
n,k in the domain of fixed k
and asymptotically increasing n for both the chordal and geodesic metrics. Under the chordal
distance metric, the first two terms in a series expansion for the ball volume expression over GF
n,k was calculated by Dai et al. in [20]. Using a volume measure of [2], they employed the Selberg’s generalization of the beta integral to obtain this answer. The calculations in [7] and [20] are not only tedious, but also specific to the Grassmann manifold due to their reliance on the notion of principal (or critical) angles in forming an integrable volume measure.
The collapse of GF
n,kto a single point at k = n motivates the study of Vn,nF in particular. After
an initial numerical study of packings and ball volume over the unitary group U (n) ≡ VC
n,n by [38],
an extension to arbitrary values of n and k was attempted by [41] who analyzed the manifold under both the geodesic and Euclidean metrics. Though the author recognized the importance of knowing the curvature of these manifolds, the reliance on numerical evaluation of the volume using bounds for the curvature terms led to conspicuous errors for many values of n and k (as seen in the first table on page 3448 of [41]).
In this chapter, we analyze GF
n,k and Vn,kF under the geodesic, chordal and Euclidean metrics.
We shall see that a little-known result on ball volumes by [36] enables one to improve upon the afore-mentioned results easily. We shall also demonstrate that under the geodesic metric, the precise series expansion for the volume of a ball can be given. This is accomplished by relating the curvature of these manifolds to the curvature of the orthogonal (for F = R) or unitary group (for F = C) through the notion of Riemannian submersions [26]. Further, the computations of ball
volume for GF
n,k and Vn,kF are unified by noting their near-identical geometric provenance from the
orthogonal or unitary groups.
While most of the afore-mentioned works utilize the ball volume expression to compute only
the Hamming upper and Gilbert-Varshamov lower bounds on rates of codes formed over GF
n,k, the
first term in this expansion plays a critical role in analyzing the performance of MIMO system with finite-rate feedback of channel information lying over these manifolds. Let H denote the channel
matrix. Let Vs denote the Nt× s matrix with its columns being the s right singular vectors of H
corresponding to its s largest singular values. If P denotes the diagonal power allocation matrix, one can define - recalling our discussion in Chapter 1 -
CCSIT, EHlog det(I + HVsP VsHHH),
and
CCSI−Fb, EHlog det(I + H ˆVsP ˆVsHHH).
Here, ˆVs is the quantized version of Vs. The aim of the analysis in this chapter, as before in Pn
analysis, is to characterize the variation of CCSIT−CCSI−Fbwith respect to Nf. For a general value of
P , we quantize Vsover the Stiefel manifold. If P = PonI, we quantize over the Grassmann manifold.
A Grassmannian quantization of Vs implies that ˆVs cannot be regarded as an approximation to
Vs, because dE(Vs, ˆVs) does not vanish as Nf → ∞. Hence, for the case of P = PonI analysis,
it is more accurate to regard Vs and ˆVs in the formulae above as representative bases for their
respective subspaces rather than as concrete matrices. This difference is not merely a matter of mathematical rigor as the final scaling in Grassmannian feedback for the above transmission
strategy is O
2−2NfN
which is twice as fast as the O
2−NfN
scaling obtained for Stiefel feedback. In this chapter, we also analyze a low complexity architecture called the GMD scheme, which derives its name from its use of the so-called Geometric Mean Decomposition of the channel matrix [49]. This scheme requires knowledge of the precise right singular vectors of the channel matrix, and hence requires quantization over the Stiefel manifold. Using the same geometric framework, we find
that CCSIT−CCSI−Fbvaries here as O
2−2NfN
, where N is the dimension of VC
Nt,rk(H). This scaling
is quicker than the O
2−NfN
stated earlier for the Stiefel feedback under the conventional SVD- based scheme, revealing that this low-complexity architecture is less susceptible to feedback errors than the SVD-based technique. The GMD scheme serves as an example of feedback information being used to reduce the complexity of the system implementation.
This chapter is organized into seven sections. Section 4.3 studies the geometry of GF
Nt,s
and VF
Nt,s, and provides the series expansion for the volume of a ball under the geodesic metric.
Section 4.4 makes use of the general geometric framework of Chapter 2 and mimics the calculation
technique used in Chapter 3 to compute the variation of CCSIT− CCSI−Fb difference as a function
of the feedback rate Nf for the conventional SVD-based scheme. Section 4.5 extends the analysis
to the case of the complexity-constrained GMD scheme. Section 4.6 concludes the chapter. The work in this chapter has been partially presented before in conferences [61] and [60].
The notation used in this chapter is similar to that of the previous chapter with some minor
additions. Ik is the k × k identity matrix. If the size k is clear from the context, then we drop the
subscript indicating the identity matrix as just I. On the Stiefel manifold, the identity element is denoted by Id.