# How to synthesise an approximate 'pulse' waveform from an observed section of a similar waveform

Following on from my previous question, kindly answered by @peter-k, I have a related one...

I have, in a small interval $$a \leq x \leq b$$, a function $$q(x) := q(x;\rho,\gamma)$$ whose exact form is unknown. However, a Taylor expansion around a point $$x=\rho_0$$ shows that, to the first order, we have $$q(x;\rho_0,\gamma) \approx c(x;\rho_0,\gamma) = \frac{1}{\pi}\left[ \frac{\gamma}{(x-\rho_0)^2 + \gamma^2} \right],$$ where in general, $$c(x) := c(x;\rho,\gamma)$$ is the density function of the Cauchy probability distribution with location $$\rho$$ and scale $$\gamma$$. This is illustrated in the figure.

From prior analysis we also have that $$\arg\max_\rho q(x;\rho,\gamma) = \arg\max_\rho c(x;\rho,\gamma) = \rho_0$$ and $$\gamma > 0$$ is a (small) known constant.

My question therefore is what method(s) are available to 'fit' $$c(x)$$ to $$q(x)$$ in the interval $$[a,b]$$ so as to discover the value of $$\rho_0$$ and hence the value of $$x \in \mathbb{R}$$ at which $$q(x)$$ attains its maximum?

I have been experimenting with the discretised convolution of $$q(x)$$ with $$c(x)$$ but so far have not achieved the desired result. If this is indeed an applicable approach, any comments or guidance would be much appreciated.

I think a least square error fit should do the trick. Assuming $$c$$ is discrete at the values $$x_i$$ you can define your error as:
$$E = \sum \left[ \frac{1}{\pi} \frac{\gamma}{(x_i-\rho)^2 + \gamma^2} - c_i\right]^2$$
and you can get $$\rho_0$$ by minimizing the error, i.e. solving for
$$\frac{\partial E}{\partial \rho} = 0$$
• Thanks for this ... yes, we can discretise $c$ at points $x_i$. I actually started my approach of attacking this problem with least-squares and you're right the equation is pretty nasty! $\mathcal{O}(\rho^5)$ in the denominator (without looking back at my notes). But as you say the function is well-behaved so it is an option and I'm glad to have found your support in the approach. I'd like to leave the question open for a while to see what other ideas may be suggested, but otherwise I will persist with this one. Commented Jul 22, 2022 at 14:42