Insight into Frequency-Domain Extrapolations of Least-Squares-Based Curve Fitting Algorithms

Abstract

Many efforts have been made in order to identify state-space models of the black-box devices from their terminal admittance/impedance frequency responses. However, the band of the available admittance/impedance frequency responses is sometimes relatively narrow, from which some out-of-band critical modes of interest can thus not be intuitively observed. In order to overcome this limitation, the vector fitting (VF) algorithm is used in this paper to identify some out-of-band critical modes of an artificially created rational function. On its basis, the sensitivity index which depicts the effect of the partial fraction term on the in-band fitting error is derived in order to explain the VF's extrapolation behavior, and its effectiveness is verified on the cases where out-of-band modes of different numbers coexist.

Extrapolation of Band-Limited Frequency Responses for Out-of-Band Modal Synthesis

“…Since the in-band frequency responses are in essence the summation of all partial fraction terms (including both the in-band and out-of-band partial fraction terms), the information (e.g., the poles and residues) of the out-of-band partial fraction terms is indeed contained in the band-limited frequency responses. Therefore, the out-of-band partial fraction terms can in principle be identified from the in-band frequency responses with the help of the linear least squares methods using a "sufficiently high" fitting order, as validated in our recent work [18].…”

“…can be generated. Although all of the m out-of-band modes of f (s) can be identified if n is sufficiently high (i.e., n > m) as reported in [18], the exact number m and positions of the out-of-band modes of f (s) cannot be observed from the available band-limited frequency responses below f u . Therefore, whether n is "sufficiently high" or not cannot be intuitively determined, and n may be smaller than m. Furthermore, the n identified magnitude peaks of g(s) are not necessarily the same as the m actual magnitude peaks of f (s) as long as g(s) can give a sufficiently good fit to f (s) below f u , i.e.,…”

“…The sensitivity indexes ζ p1i (p 1i , r 1i ) and ζ r1i (p 1i , r 1i ) of a partial fraction term (e.g., r1i s−p1i ) of f (s) are introduced to describe how large the in-band fitting error (e.g., in the root-mean-square sense) within the fitting frequency interval [0, f u ] is induced by an unit identification error of p 1i and r 1i , respectively, which can be derived as [18]…”

“…2) Synthesis of Out-of-Band Modes of Different Numbers for Fig. 6: Based on ( 16), (18), and (19), the out-of-band modes of different numbers can be synthesized from Fig. 6 as plotted in Figs.…”

“…In this case, the identified out-of-band modes of the fitted reduced-order model can be different from the actual ones. In order to understand the extrapolation behavior of the reduced-order model in depth, the insight into the extrapolation of the VF on the band-limited frequency responses is provided in [18], which indicates that each identified out-of-band mode of the reduced-order model can represent the cumulative effect of the actual multiple out-of-band modes on the in-band frequency responses. As a reverse solution, it would be desirable to reproduce/identify the actual or synthesize the virtual out-of-band modes based on the reduced-order models.…”