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Utilization of Laplace-Domain Frequency Scaling Circuit Techniques to Enable Numerical Conversion of Transfer Functions from Pole-Residue Form to Pole-Zero Factored Form

IP.com Disclosure Number: IPCOM000234122D
Publication Date: 2014-Jan-13
Document File: 5 page(s) / 173K

Publishing Venue

The IP.com Prior Art Database

Abstract

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.

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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

Zero

Factored Form

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 [1]:


(1.1)

    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


(1.2)

Note also that, in the s-plane, this 1st-order function would have a pole at and that

(1.3)

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

[2]-[5]. 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

1


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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...