Characteristic Basis Functions of the Energy Radiation Pattern of a Sparse True Time Delay Array

Abstract

Abstract-A set of characteristic basis functions of the energy radiation pattern for a true-time-delay array of equi-spaced elements radiating a pulsed/transient wave-field was derived. This set is determined by the array layout and by the set of excitation waveforms that can be used to expand the actual excitation pulse. It is established that the characteristic basis function set spans the mapping of the square amplitudes of the discrete Fourier transform of the excitation coefficients to the energy radiation pattern. This mapping is further used to analyze array performance and re-examine the term array sparsity. Additional use of this set can be found in synthesizing an array radiation pattern to meet prescribed requirements.

Characterization of the energy radiation pattern of a transient antenna array: Eigen-analysis and bounds

“…This simplification is due to the suppression of the temporal coordinates in both the excitation and the far-field domains, which results in the use of the excitation signals' auto-and crosscorrelation functions [1], [5]. Though time independent, the ERP still captures the kinematic properties of the TTD array as encapsulated by the beam-radiation-skeleton (BS).…”

“…Thus, employing this property of circulant matrixes, given a radiation direction the ERP is reformulated in an alternative "spectral" bilinear form with a diagonal matrix whose entries (eigen-values) are the DFT vector of the first row of the circulant matrix. The corresponding ("excitation") vector is a spectral excitation vector, which is the DFT of the zero padded original excitation vector [5]. It is readily noted that since the circulant structure is maintained for different radiation directions, with a change of only the correlation matrix entries, only the entries of the diagonal "spectral" matrix are changed (according to the direction).…”

“…of entire domain or sub-domain classes. The weights are related to the DFT elements of the excitation vector [5]. Furthermore, it follows that the set of spectral contributions serves as the basic building block of the ERP and as such can be identified as the characteristic basis functions (CBFs) set for the corresponding array layout (geometrical and excitation pulse).…”

“…This reformulation as a vector-matrix-vector product enables a further decomposition in terms of spectral contributions by a matrix eigen-decomposition. To that end, the cross-correlation Toeplitz-structured matrix is embedded into a larger circulant matrix, with approximately twice the size of the Toeplitz matrix, [4] while the excitation vector is zero padded [5]. Recall that all circulant matrixes with the same size have the same eigen-vectors that are the discrete Fourier transform (DFT) basis vectors [3], [4].…”

Characteristic Basis Function set of the far field of a transient array antenna

“…The resulting eigen-set of the mapping is identified as the Characteristic Basis function (CBFs) set of the array. These CBFs are the "natural" building blocks of the transient radiation pattern (see, e.g., [4]). Moreover The analysis of the CBFs is expected to yield a greater understanding of the connection between the array's pulsed excitation and the corresponding transient radiation.…”