## 1. Introduction

[2] In electromagnetic analysis, field quantities can be solved in the frequency domain by utilizing the Method of Moments (MoM), in which the problem of interest is typically solved at a set of discrete frequency points [*Harrington*, 1982]. This phenomenology has been implemented in many commercial software packages such as TIDES [*Zhang et al.*, 2008]. In MoM, the object being analyzed is meshed into small triangular or quadrilateral patches normally less than one wavelength. For high-frequency domain calculation, finer meshes are required therefore the number of unknowns increases dramatically along with the frequency of interest. This requires longer computation time at each frequency sample point, typically including the low-frequency samples. A similar time consuming process arises with the need to measure the experimental frequency response of a system at a large number of points. It is desirable to reduce the number of sample points and apply interpolation techniques to reconstruct the high-resolution frequency response accurately from a sparsely sampled data set. The Pade approximation, based on the Taylor expansion of the parameter of interest as a function of frequency, may be used to interpolate the frequency domain data [*Brezinski*, 1980]. Another more versatile approach is the Cauchy method developed by *Cauchy* [1821]. In the Cauchy method, the system frequency response is modeled by a ratio of two polynomials and has been applied to the solution of electromagnetic field problems [*Kottapalli et al.*, 1991; *Adve and Sarkar*, 1993; *Adve et al.*, 1997; *Yang and Sarkar*, 2007]. Once the coefficients of the polynomials are determined from the given sparsely sampled data, the response of interest can be interpolated/extrapolated at other neighboring frequencies. In the work of *Yang and Sarkar* [2007], the Cauchy method is applied to amplitude-only sparse and incomplete data and the phase information is also reconstructed as a byproduct.

[3] The real problem is that in many electromagnetic systems, the system response appears to oscillate in the high-frequency range. According to the sampling theorem, the minimum number of uniform samples depends on the period of the oscillation and cannot be further reduced if the limitation is reached, which still means a huge computational load even when an interpolation technique is applied. In this paper, a new approach is proposed to smooth the phase function of the system response. Although the magnitude of the field quantity is unchanged, the period of both the real and the imaginary parts of the system response are increased, therefore less frequency sample points are required to accurately reproduce the frequency response in the high-frequency band.

[4] Instead of using the Cauchy method, the matrix pencil technique is applied for data interpolation. In matrix pencil, the frequency function is modeled as a sum of complex exponentials, thus it is more suitable for periodic data set. The matrix pencil method has been widely used in many ways such as modeling the transient response of electromagnetic systems [*Hua and Sarkar*, 1989, 1990; *Sarkar and Pereira-Filho*, 1995], computing the input impedance of electrically wide slot antennas [*Kahrizi et al.*, 1997], analyzing the complex modes in lossless closed conducting structures [*Sarkar and Salazar*, 1994], and in diverse applications, such as inverse synthetic-aperture radar [*Hua et al.*, 1993] and radio-direction finding [*Ouibrahim et al.*, 1988; *Himed*, 1990]. Compared to the Cauchy method, the matrix pencil method is computationally more efficient and more robust to noise.

[5] Another interpolation technique called Z-matrix interpolation [*Newman*, 1993] can also be applied in MoM. Instead of interpolating the frequency function, this technique interpolates the impedance matrix, which changes smoothly over the frequency. Therefore, the simple linear or low-order polynomial interpolations can be utilized. However, the matrix inversion still needs to be performed at each frequency point. Also the Z-matrix interpolation cannot be used in the measurement of RCS data.

[6] In section 2 we provide a brief review of matrix pencil, combined with total least squares (TLS) implementation. The phase-smoothing method is introduced in section 3. Two numerical examples are presented in section 4, followed by the conclusion in section 5.