# Optimally approximating the sign function by functions with compactly supported Fourier transform

I'm re-posting a question of mine from math.stackexchange in hopes that folks here might have the right kind of expertise.

I'm looking for a systematic way to approximate the sign function

$$\operatorname{sgn}(x) \triangleq \begin{cases}1&\text{ if }x > 0\\ 0&\text{ if }x=0\\ -1&\text{ if }x<0\\\end{cases}$$

by functions $$f$$ whose (distributional) Fourier transforms $$\hat{f}$$ are compactly supported.

More precisely, let us fix some $$\epsilon$$ small. Then I am interested in odd, continuous $$f$$ such that:

• $$\text{supp}{\hat{f}}\subseteq[-\pi,\pi];$$
• there exists some $$a_f>0$$ such that $$|f^2(x)-1|<\epsilon$$ whenever $$|x|\geq a_f$$.

Goal: I am looking for a "best" $$f$$ in the sense $$a_f$$ is as small as possible.

I figured that one way to do this would to be to look at convolutions $$f=a*sgn$$ where $$a$$ is a function that satisfies $$\int_\mathbb{R} a(s)\,ds=1$$ and $$\text{supp }{\hat{a}}\subseteq[-\pi,\pi]$$. For example, we could take

1. $$a_1(s)=\dfrac{\sin(\pi s)}{\pi s}$$; or
2. $$a_2(s)=\frac{1}{2}\left(\dfrac{\sin(\frac{\pi}{2}s)}{\frac{\pi}{2}s}\right)^2,$$

where the Fourier transforms of $$a_1$$ and $$a_2$$ are a rectangular and a triangle pulse respectively.

Doing some numerical experiments, I find that for the choice $$\epsilon=0.05$$, $$a_2$$ is "better" than $$a_1$$ by the above criterion.

Question: Is there a more systematic way to find better (or a best) $$f$$? References that deal with this type of problem would also be appreciated.

• Here's one sign(x) approximation desmos.com/calculator/0k9gqmbqxp ... but dunno if it suites for you.... Nov 2 '20 at 17:05
• That's all well and good, but while the Fourier transform of your $a_1$ is zero for $\left | \omega \right| > \pi$, the Fourier transform of your $a_2$ doesn't go to zero until $\left | \omega \right | = 2 \pi$. Nov 2 '20 at 23:06
• Certainly, to minimize $\int_{-\infty}^{\infty} \left ( f(x) - \mathrm{sgn}(x) \right)^2 dt$ under your frequency-domain criteria, convolution with $a_1$ is as good as you can get. Just review Parseval's theorem and do some ciphering. Nov 2 '20 at 23:09
• you might wanna look at sigmoid functions. an example of that is what @JuhaP was pointing you to. you can take one of them and then scale the argument to such a degree to approximate the signum function. Nov 3 '20 at 4:49
• The sigmoid function is not band-limited. Nov 3 '20 at 12:41

This problem is in a way the dual of the problem of approximating a Hilbert transformer by a filter with finite memory. The frequency response of an ideal Hilbert transformer is

$$H(\omega)=-j\,\textrm{sgn}(\omega)\tag{1}$$

and the corresponding ideal impulse response is

$$h(t)=\frac{1}{\pi t}\tag{2}$$

We can approximate $$(1)$$ and $$(2)$$ by multiplying the ideal impulse response by a window of finite length:

$$\tilde{h}(t)=h(t)w(t)\tag{3}$$

where the window function $$w(t)$$ is a real-valued even and smooth function with finite support.

In your case you could use a window function $$W(\omega)$$ in the frequency domain to approximate a sign function:

$$f(t)=\textrm{p.v.}\frac{1}{j\pi}\int_{-\pi}^{\pi}\frac{W(\omega)}{\omega}e^{j\omega t}d\omega=\textrm{p.v.}\frac{1}{\pi}\int_{-\pi}^{\pi}\frac{W(\omega)}{\omega}\sin(\omega t)d\omega\tag{4}$$

Note that this is just a generalization of what you already did. You used a rectangle and a triangle for $$W(\omega)$$. However, there are many more "nicer" window functions that are smooth, and that may result in better approximations of the sign function.