# Maximize the modulus of a sum of complex exponentials on a given interval

I am trying to find $$\hat{t}=\underset{t \in [a,b]}{\text{argmax}}\,\left|\sum_{j=1}^n e^{i x_j t}\right|$$ where $t$ is a real number, $[a,b]$ is a given interval, $i$ is $\sqrt{-1}$, and $x_j$ are arbitrary real numbers.

My goal is efficient and accurate numerical computation of $\hat{t}$, so I don't necessarily need a simple formula.

What I've been doing is binning the $x_j$ values (that is, counting the number found in equally spaced intervals), taking a fast Fourier transform of the counts, and searching though the resulting vector for its maximum on the given interval. My question is: is there a better way?

In case the size of the sum turns out to matter, I'll add that in my applications, $n$ is typically around a hundred.

• Wouldn't the argmax always be at $t=0$? – Mike Earnest Sep 13 '18 at 1:36
• Perhaps there are some phase shifts missing? Like $\Big|\sum_{j=1}^n e^{itx_j + \theta_j}\Big|$? Or $t$ only ranges over some interval not containing $0$? – Mike Earnest Sep 13 '18 at 1:57
• Ah, it's actually the second: there's a specific interval. I'll edit the question to make that clear. – user54038 Sep 13 '18 at 14:41

I think this problem could be formulated as maximizing the $L_1$ norm of you given vector. This could be formulated as a nonlinear optimization problem with constraints.