Utilization of Laplace-Domain Frequency Scaling Circuit Techniques to Enable Numerical Conversion of Transfer Functions from Pole-Residue Form to Pole-Zero Factored Form
Publication Date: 2014-Jan-13
The IP.com Prior Art Database
Disclosed is a technique for applying filter frequency scaling methods to a rational function approximation of the system transfer function of a signal integrity interconnect model, so that the function can be more easily converted from partial fraction expansion form to pole-zero factored form. This enables subsequent pole-zero map analysis in the complex analog frequency s-plane or complex discrete frequency z-plane.
Page 01 of 5
Utilization of Laplace -
-Domain Frequency Scaling Circuit Techniques to Enable
Domain Frequency Scaling Circuit Techniques to Enable
Numerical Conversion of Transfer Functions from Pole -
-Residue Form to Pole
Residue Form to Pole - -Zero
This disclosure address the application of filter scaling theory, a set of well-known techniques for the analysis and design of analog filters, to the analysis of high-speed interconnect models in signal integrity engineering. Specifically, it involves applying frequency scaling methods to a rational function approximation of the system transfer function, so that the function can be converted from partial fraction expansion form to pole-zero factored form. This enables subsequent analysis of the system either in the complex analog frequency s-domain (or s-plane) or, alternatively, in the complex discrete frequency z-domain (or z-plane) once the matched-Zs- to z-domain transform has been applied.
Because of the prevalence of vector fitting macromodeling processes in signal integrity engineering, the transfer function for the interconnect is most likely to be encountered in
the partial fraction expansion form of its rational function approximation :
Ignoring the direct term, D, which is usually not present for a physically realizable passive system, note that the partial fraction expansion in the form of (1.1) is a sum of 1st-order partial fraction sections of the form
Note also that, in the s-plane, this 1st-order function would have a pole at and that
thus can also be said to have a zero at infinity. The latter demonstrates one of the
pragmatic limitations of s-plane analysis; namely, the frequent presence of poles and zeros at distances very far from the axes' origin. As such, it is advantageous to look at the transfer function in the z-domain, in lieu of in the s-domain. To do so, it is necessary to
apply an s- to z- transform to obtain .
Known s- to z- transform methods that are commonly utilized to create are the
impulse invariance method (IIM), the bilinear transform (BLT), and the matched-Z transform
-. All of these transforms have known limitations. Specifically, the IIM maps the zeros of incorrectly in the frequency domain. The BLT displays frequency warping at high
frequencies, and thus obtains incorrect mappings of poles and zeros except at a single frequency point, where the magnitudes of and can be matched. The matched-Z transform
Page 02 of 5
does not work directly on the partial fraction expansion form of (1.1), and thus must be
first converted to a factored pole-zero form; this conversion can be difficult in high-frequency bands.
This last problem is significant to s-domain pole-zero map analysis as well. This is because, even with the options for performing pole-zero analysis in the z-domain and the practical limitations of the s-plane for high-frequency pole-zero positions, it is still useful to be able to examine pole-zero maps di...