Slepian or DPSS Window

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 $M$ degrees of freedom (sample values) in an $M$ -point window $w(n)$ to obtain a window transform $W(\omega)\approx\delta(\omega)$ which maximizes the energy in the main lobe of the window relative to total energy:

$$\max_w \left[ \frac{ \mbox{main lobe energy} } { \mbox{total energy} } \right] \protect$$ (4.36)

In the continuous-time case, i.e., when $W(\omega)$ is a continuous function of $\omega\in(-\infty,\infty)$ , the function $W(\omega)$ which maximize this ratio is the first prolate spheroidal wave function for the given main-lobe bandwidth $2\omega_c$ [101], [202, p. 205].^4.8

A prolate spheroidal wave function is defined as an eigenfunction of the integral equation

$$\int_{-\omega_c}^{\omega_c} W(\nu) \frac{\sin[\pi D\cdot(\omega-\nu)]}{\pi(\omega-\nu)}\, d\omega = \lambda W(\omega) \protect$$ (4.37)

where $D$ is the nonzero duration of $w(t)$ in seconds. This integral equation can be understood as ``cropping'' $W(\omega)$ to zero outside its main lobe (note that the integral goes from $-\omega _c$ to $\omega_c$ , followed by a convolution of $W(\omega)$ with a sinc function which ``time limits'' the window $w(t)$ to a duration of $D$ seconds centered at time 0 in the time domain. In operator notation,

$$\begin{eqnarray*} && [\hbox{\sc Chop}_{2\omega_c}(W)]*[D\,\mbox{sinc}(D\omega)]\\ &=& \hbox{\sc FT}(\hbox{\sc Chop}_D(\hbox{\sc IFT}(\hbox{\sc Chop}_{2\omega_c}(W)))) \eqsp \lambda W \end{eqnarray*}$$

where $\hbox{\sc Chop}_D(w)$ is a rectangular windowing operation which zeros $w(t)$ outside the interval $t\in[-D/2,D/2]$ .

Satisfying (3.37) means that window transform $W(\omega)$ is an eigenfunction of this sequence of operations; that is, it can be zeroed outside the interval $[-\omega_c,\omega_c]$ , inverse Fourier transformed, zeroed outside the interval $[-D/2,D/2]$ , and forward Fourier transformed to yield the original Window transform $W(\omega)$ multiplied by some scale factor $\lambda$ (the eigenvalue of the overall operation). We may say that $W$ 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 $W$ 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 $W_m(\omega)$ , each associated with a real eigenvalues $\lambda_m$ .^4.9 If $\lambda_0$ denotes the largest such eigenvalue of (3.37), then its corresponding eigenfunction, $W_0(\omega)\;\leftrightarrow\; w_0(t)$ , 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 [13]:

$$S[k,l] = \frac{\sin[\omega_c T(k-l)]}{k-l}, \quad k,l=0,1,2,\ldots,M-1$$ (4.38)

where $M$ denotes the desired window length in samples, $\omega_c$ is the desired main-lobe cut-off frequency in radians per second, and $T$ is the sampling period in seconds. The main-lobe bandwidth is thus $2\omega_c$ rad/sec, counting both positive and negative frequencies.) The digital Slepian window (or DPSS window) is then given by the eigenvector corresponding to the largest eigenvalue. A simple matlab program is given in §F.1.2 for computing these windows, and facilities in Matlab and Octave are summarized in the next subsection.