Maximum Main-Lobe Energy Window: DPSS

Question: How do we use all $M$ degrees of freedom (sample values) in an $M$ -point window $w(n)$ to obtain $W(\omega)\approx\delta(\omega)$ in some optimal sense?

That is, we wish to perform the following optimization:

$$\max_w \left[ \frac{ \mbox{main lobe energy} } { \mbox{total energy} } \right]$$

In the continuous-time case [ $\omega\in(-\infty,\infty)$ ], this problem is solved by a prolate spheroidal wave function, an eigenfunction of the integral equation

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

where $D$ is the nonzero time-duration of $w(t)$ in seconds.

Interpretation:

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

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

$W$ is thus the bandlimited extrapolation of its main lobe ( $\omega\in[-\omega_c,\omega_c]$ )

The optimal window transform $W$ is an eigenfunction of this operation sequence corresponding to the largest eigenvalue.

The resulting optimal window $w$ has maximum main-lobe energy as a fraction of total energy.

It may be called the Slepian window, or prolate spheroidal window in the continuous-time case.

In discrete time, we need Discrete Prolate Spheroidal Sequences (DPSS), eigenvectors of the following symmetric Toeplitz matrix constructed from a sampled sinc function:

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

• $M =$ window length in samples
• $\omega_c =$ main-lobe cut-off frequency (rad/sec)
• $T =$ sampling period in seconds.

The DPSS window (digital Slepian window) is then given by the eigenvector corresponding to the largest eigenvalue.