A convenient expression for the scattering matrix of a junction of
normalized waveguides is obtained in terms of
(78) |
The Householder structure (80) can be exploited to speed up
the matrix-vector multiplication. Define
The unnormalized Householder reflection is structurally a unitary transformation in the sense that remains a constant diagonal matrix after quantization of the vector [110]. Hence, if all the incoming waves are scaled by , normalized junctions implemented in this way conserve their losslessness even under coefficient quantization. To be conservative, it is sufficient to scale the incoming waves by a constant slightly less than , in such a way that the junction has a small loss. Preferably, however, we may scale the so that they sum to for some integer , in which case becomes a power of implementable as a simple shift in fixed-point binary arithmetic.
Similarly, the unnormalized junction (38) can be interpreted as an ``oblique Householder'' reflection, in the sense that the sum of the vectors and is colinear with the vector which lies on the diagonal of the parallelogram whose edges are given by the incoming and outgoing wave vectors. As we have already noticed in section 8, even in this case the matrix can be implemented as a structurally lossless transformation, and therefore it is well suited as a building block for large networks using fixed-point computations.
Computations (64) and (82) do not lend themselves to highly parallel implementations because of the scalar product needed to compute or . A scalar product needs additions to be computed on processors [55].