What window functions allow for controlling the location of nulls?

Question

Window functions are commonly used to control spectral sidelobes. Typically there is a trade between supressing sidelobes, and maintaining the resolution of the mainlobe. A window with lower sidelobes has a wider mainlobe.

But another characteristic that is desirable in some situations is to control where the first (or second or third) null of the window falls in the spectral domain.

Are there window functions that allow you to control the first or second null location, while also preserving other desirable characteristics such as sidelobe suppression?

Edit

I've heard that one way to do this is to modify the original Taylor window formulation by moving some of the $\bar n$ zeros. Can anyone confirm this and describe the procedure?

Answer 1

This is an interesting question. Other than thinking about the sinc function (and higher powers of it), which have nulls that are equally spaced from DC, and that spacing can be controlled by the width of the rect function that you convolve your window kernel with, I dunno exactly how you'ld do it.

Or you could convolve your window kernel with the FIR of a notch filter placing the null exactly where you want it.

Consider:

$$\begin{align} W_n(z) &= \frac{(z-e^{j \omega_n})(z-e^{-j \omega_n})}{z^2} \\ \\ &= \frac{z^2 - 2\cos(\omega_n)z + 1}{z^2} \\ \\ &= 1 - 2\cos(\omega_n)z^{-1} + z^{-2} \\ \\ &= w_n[0] + w_n[1]z^{-1} + w_n[2] z^{-2} \\ \end{align} $$

Two poles at the origin and two zeros notching frequency $\omega_n$, your $n$th null. It's a simple 3-tap FIR filter.

$$\begin{align} w_n[0]&=1 \\ w_n[1]&=-2\cos(\omega_n) \\ w_n[2]&=1 \\ \end{align}$$

Each convolving iteration (with a single-frequency notch FIR filter) makes your window function longer by two samples. Now choose your $N$ desired null frequencies and convolve your window kernel with these $N$ 3-tap FIRs:

$$ W(z) = \prod\limits_{n=1}^{N} W_n(z)$$

Now, if you started out with a gaussian kernel, the result would be pretty smooth and have nulls pretty much at your $N$ locations. (Truncating the gaussian function to a finite length won't change the main lobe much but will result in noise in the stopband of the window frequency response. So there may be some nulls at high frequencies that are not your $N$ specified nulls.)

I dunno, that's how I might begin looking at it.