s-Expanded Transfer Function using UaL Decomposition with Convolution & Deconvolution

This article provides two new features that in combinations facilitate the construction of a transfer function in RC and RL circuits, and in s-expanded polynomials. First is the introduction of tuple-based array manipulations, where multiplications and divisions are replaced with convolutions and deconvolutions, respectively. Second, it introduces a new UaL decomposition for tuple-based matrices. The UaL decomposition is proven to be computationally more efficient compared to the traditional LU factorization. It is shown that the growth of the entries in both U and L matrices are gradual and follow a geometrical pattern, as the decomposition progresses. Another important feature of the UaL decomposition is that it does not generate nonzero remainders, or simply, it is “cancellation free”. The method is applied to the MNA representation of an RC (or RL) circuit. Finally, it is shown that working in the tuple-based format puts the U and L matrix entries into s-expanded form, such as P(s) = a
n
s
ⁿ
+ a
n-1
s
^n-1
+…+ a
0
, which in turn produces s-expanded transfer functions.