A window having maximal energy concentration in the main lobe is given by the digital prolate spheroidal sequence (DPSS) of order 0 [256,136]. It is obtained by using all degrees of freedom (sample values) in an -point window to obtain a window transform which maximizes the energy in the main lobe of the window relative to total energy:
A prolate spheroidal wave function is defined as an eigenfunction of the integral equation
where is a rectangular windowing operation which zeros outside the interval .
Satisfying (3.37) means that window transform is an eigenfunction of this sequence of operations; that is, it can be zeroed outside the interval , inverse Fourier transformed, zeroed outside the interval , and forward Fourier transformed to yield the original Window transform multiplied by some scale factor (the eigenvalue of the overall operation). We may say that is the bandlimited extrapolation of its main lobe.
The sinc function in (3.37) can be regarded as a symmetric Toeplitz operator kernel), and the integral of multiplied by this kernel can be called a symmetric Toeplitz operator. This is a special case of a Hermitian operator, and by the general theory of Hermitian operators, there exists an infinite set of mutually orthogonal functions , each associated with a real eigenvalues .4.9 If denotes the largest such eigenvalue of (3.37), then its corresponding eigenfunction, , is what we want as our Slepian window, or prolate spheroidal window in the continuous-time case. It is optimal in the sense of having maximum main-lobe energy as a fraction of total energy.
The discrete-time counterpart is Digital Prolate Spheroidal Sequences (DPSS), which may be defined as the eigenvectors of the following symmetric Toeplitz matrix constructed from a sampled sinc function :