Adaptive Filters

CHAPTER 36 Hyperbolic Rotations

It is sometimes necessary, especially when deriving fast least-squares algorithms (as we shall discuss in the next chapter), to employ J–unitary (also called hyperbolic) transfor mations, as opposed to unitary transformations, in order to annihilate certain entries in a pre-array of numbers. A J–unitary transformation Θ is one that satisfies

for some signature matrix J, i.e., a diagonal matrix with ±1 entries. The special case J = I corresponds to unitary transformations and was studied in Chapter 34. In this chapter, we extend the results to the J–unitary case, starting with Givens rotations and followed by Householder transformations.

36.1 HYPERBOLIC GIVENS ROTATIONS

As in our discussions in Chapter 34, we again distinguish between the cases of real data and complex data.

Real Data

Thus, consider a 1 × 2 real-valued vector z = [ a b ], and assume that we wish to determine a 2 × 2 matrix Θ that transforms it to the form:

(36.1)

for some nonzero real number α to be determined, and where Θ is required to be hyper bolic, i.e., it should satisfy

Unfortunately, and in contrast to the case of orthogonal Givens transformations in Sec. 34.1, the transformation ...