In this paper, a novel phase-smoothing method is proposed to efficiently interpolate the fine resolution frequency data from sparse samples in the high-frequency range. This is achieved by taking the effect of the complex exponential term e−jkr in the electromagnetic field into account, which causes the oscillation in the system response especially in the high-frequency domain. By phase-smoothing, even though the magnitude of the field quantity is unchanged, both real and imaginary parts of the frequency response become smoother. The interpolation is performed separately for both real and imaginary parts so that the sample rate required for accurate reconstruction is significantly reduced. The interpolation is carried out by the matrix pencil method, the coefficients of which are calculated by using the total least squares implementation to improve accuracy. Several numerical examples are presented to illustrate the applicability of this unique phase-smoothing method in ultra-high-frequency bands.
 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 . 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 , the Cauchy method is applied to amplitude-only sparse and incomplete data and the phase information is also reconstructed as a byproduct.
 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.
 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.
 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.
 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.
2. Total Least Squares Matrix Pencil
 In the matrix pencil method, we model the frequency domain function as a sum of complex exponentials,
where f is the frequency interval of the sample points. The problem is to find the best estimates for Ri, zi and M given H(fi). Now let us construct two known data matrices with the same dimensions:
where P referred to as the pencil parameter. Then (1) can be written in matrix forms as:
Now consider the matrix pencil:
where I is the M × M identity matrix. In general when M ≤ P ≤ N − M the rank of H2 − λH1 will be M. However, if λ = zi, i = 1, 2, …, M, the ith colomn of H2 − λH1 is zero, and the rank of the matrix reduced to M − 1. This implies that zi's are the generalized eigenvalues of matrix pair (H1, H2). Therefore zi's can obtained by finding the eigenvalues of H2, where is the Moore-Penrose pseudo-inverse of H1.
 In the presence of noise, the total least squares (TLS) matrix pencil can be applied to combat noise. In this implementation, the new data matrix is constructed as:
Note that the matrices H1 and H2 are matrix H with last row and first row deleted respectively. To reduce the variance of zi, L is normally chosen between N/3 and N/2. Next, the singular value decomposition (SVD) of H gives:
where U and V are unitary matrices whose columns are the eigenvectors of HH* and H * H, and Σ is the diagonal matrix consists of the singular values of H, i.e., the square root of the eigenvalues of H * H, in descending order. The choice of M is given by the dominant singular values in the range
where d is the number of significant decimal digits in the data. This is equivalent to filtering the noise presented in the data. In practice overestimates of M does not affect the solution severely however underestimates of M results in lack-of-fit which leads to large errors; therefore it is always preferable to use a larger M value.
 Once M is fixed, the filtered version of SVD can be expressed as:
where and are the first M rows of U and V respectively, and is M × M upper left square portion of Σ. Furthermore, the filtered version of H1 and H2 are given by:
where 1 and 2 are equal to without the last and the first row respectively. The parameters zi are given by the nonzero eigenvalues of U2. Once the parameters M and zi are all known, the residuals Ri can be solved from the following least squares matrix equation:
3. Phase-Smoothing Method
 In electromagnetic systems, the field quantity is calculated by utilizing the Green's function, given by:
where f is the frequency and r is the distance from the source. At a certain distance r, both real and imaginary parts of G are a sinusoidal function of f. Furthermore the magnitude of G is always equal to one but the phase is a linear function of f, and the slope of this linear function decides the period (here the frequency interval) of the real and imaginary parts. This periodic variation in the Green's function will be reflected in the electromagnetic field. Therefore if we can divide the frequency response by e−j2πfr/c, the magnitude of the frequency response will not be affected but the phase response will be altered by a linear phase difference. It is equivalent to viewing the frequency response at a different time origin, since for linear time invariant (LTI) system, the linear frequency difference corresponds to a shift in time domain. When the distance r is correctly selected, the overall trend of phase response will be flat thus both real and imaginary parts will be smoothed. Consequently, the number of samples for both real and imaginary parts can be reduced. The key factor is how to find the correct distant r, if it is not known.
 If the data being processed are near-field data from simulation or are measurement data for the electromagnetic field, the exact distance may be known and can be used directly. However, sometimes the distance information of the far-field is not available or different kinds of frequency domain data is processed, and the r is unknown. In this case the phase-smoothing method can still be applied. A small band of high-resolution data will be required as training data to get the phase function over this small band. Keep in mind that here the phase function is referred to as the continuous phase function, not the principle phase. Then the phase function is fitted with a straight line by linear regression, the suitable r value can be obtained from the slope of that straight line.
 The phase-smoothing method does not work for the magnitude-only data. For measurement data sometimes the phase information is not easy to obtain. In this case a two step process is performed: the phase response has to be retrieved from the magnitude response at first [Yang and Sarkar, 2007], then the phase smoothing method is applied to reduce the number of samples required for accurate frequency response interpolation.
 Once the smoothed frequency response is obtained from phase-smoothing, the TLS matrix pencil is applied to this new data set to get the wideband high-resolution frequency response, which will be multiplied by the same e−j2πfr/c to restore the original actual data set. Here we want to point out that even though the real and imaginary parts of the data should be interpolated independently, it is not necessary to separate them apart from the complex data. Due to the nature of matrix pencil, the complex quantity containing both real and imaginary parts can be fit into matrix pencil directly and the correct parameters for both of them can be estimated simultaneously. So comparing with applying matrix pencil on the original data set, there is no extra computation cost.
4. Numerical Examples
 In this section, numerical examples illustrate the application of the phase-smoothing method. Two examples are presented using data simulated with the commercially available TIDES parallel integral equation solver using the Method of Moments [Zhang et al., 2008]. Both examples are scattering scenes with two Impulse Radiating Antennas (IRAs) [Taylor and Sarkar, 2005] and a perfect electrical conducting (PEC) sphere. Each of the two IRAs has a 6 foot diameter paraboloidal reflector and two perpendicular feed arms at 45° from the centerline of the reflector. The PEC sphere scatterer is situated 1.2 m above the origin with diameter 20 inches. In the first scattering scene [Taylor and Sarkar, 2006a], the two IRAs are placed 15m from the origin. The transmit (Tx) IRA is located along the x axis and tilted upward toward the front and center of the scatterer. The receive (Rx) IRA is located 0.915m above the Tx IRA and it is tilted downward toward the front and center of the sphere. No ground plane is used in this simulation. Figure 1 depicts the dimensions and orientation of the components assembled in the scene.
 In this example, the frequency interval of the data is 1 MHz, the response is calculated at 2000 frequency sample points from 1 MHz to 2 GHz using a computer simulation code. Figure 2 shows that the response of the receive IRA antenna is pretty smooth at low frequency but becomes oscillatory at higher frequencies. We focus our analysis from 1 to 2 GHz, and set the distance r = 16 m. The original and smoothed phase functions are plotted together in Figure 3, and it is clear the slope of the overall phase trend is flattened.
Figures 4a and 4b show the close-up look of original and smoothed real and imaginary parts respectively. It indicates that the period of the smoothed response is approximately twice of the original period of original response, and this is true for the whole frequency band from 1 to 2 GHz. Therefore, the required number of sample points can be reduced by half.
 Next, the original data is downsampled and the matrix pencil method is applied to the downsampled data directly. The error in the approximation is calculated by comparing the interpolated data with the original data. The percentage error rate is defined by:
Figures 5 and 6plot the interpolation result and the associated error rate when the downsample rate is 4, i.e., the frequency interval of the input data is 4 MHz. The interpolated data agree with the original data very well and the frequency spectrum is successfully reconstructed. When the downsample rate increases to 5, which is shown in Figures 7 and 8, the error becomes large. Therefore, the maximum frequency interval is 4 MHz if matrix pencil is applied without phase-smoothing.
 Next, phase-smoothing is applied before matrix pencil method and the data is recovered back after interpolation. Figures 9 and 10 illustrate the final interpolation result with phase-smoothing and the error rate. It is clear that the phase-smoothing method successfully extends the minimum frequency interval from 4 MHz to 8 MHz while the same level of error is maintained. It also agrees with our expectation when looking at Figure 3, where the oscillating periods of both real and imaginary responses are doubled. Therefore, by performing phase-smoothing, the total computation time can be reduced by half.
 The second scattering scene has two IRAs situated on a PEC ground in the xy-plane [Taylor and Sarkar, 2006b]. The Rx IRA is located along the x axis at a distance 35 m in front of the sphere and the Rx IRA is tilted upward toward the front and center of the sphere. The Tx IRA is located at an angle of 45° and 15 m from the x axis, and is also tilted upward toward the sphere. Figure 11 depicts the dimensions and orientation of the components assembled in the scattering scene with ground plane. In this example, the frequency interval of the data is still 1 MHz and the response is calculated at 2000 frequency sample points from 1 MHz to 2 GHz. The frequency spectrum of Rx IRA is plotted in Figure 12. Again, the analysis is focused from 1 to 2 GHz. In this example, the optimal distance is found to be r = 40.5 m. Both the original and smoothed phase functions are plotted in Figure 13, which shows that after phase-smoothing, the slope of the overall phase trend is horizontal.
Figures 14a and 14b give the close-up look of original and smoothed real and imaginary parts respectively. It is seen that the period of real and imaginary parts are significantly enlarged, and they are much smoother compared to the original functions.
 Now we compare the performance of matrix pencil interpolation with and without phase-smoothing. It can be easily verified that for this scenario, the maximum frequency interval required for accurate interpolation is 2 MHz when phase-smoothing is not used. Figures 15 and 16 show the result when the frequency interval is 3 MHz, for which the error is very obvious. However, when phase-smoothing is applied, the maximum frequency interval can be relaxed to up to 10 MHz, the interpolation result and the error percentage rate of which are plotted in Figures 17 and 18. The error is less than 1% for over most of the frequency band. Therefore for this particular problem, the phase-smoothing method will save 80% of computation comparing to standard TLS matrix pencil method, and it will save 90% of total computation time computation comparing to the direct calculation from MoM without using any interpolation technique.
 In this paper, a novel phase-smoothing method to smooth the frequency response of an electromagnetic system is proposed. The phase function is flattened by dividing the system function by normalized Green's function e−jkr, therefore both real and imaginary parts of the response are smoothed. Compared to ordinary TLS matrix pencil, the new matrix pencil with phase-smoothing provides accurate interpolating at much lower sampling rate in the frequency domain, therefore computation time can be significantly reduced. The benefit of phase-smoothing depends on the actual problem and is case by case, however it is always desirable to apply this method since no extra cost is introduced in the process. The phase-smoothing method is particularly attractive for ultra-high-frequency range wide-band interpolation. Combining this with phase reconstruction from amplitude-only data can be very useful in fast computation of frequency domain system response.