How can I compute the phase shift?

Question

I have a periodic function $f(t)=f(t+T)$, its period is $T>0$.

$t$ and $f(t)$ are in $\mathbb{R}$.

$f$ is unknown apart from $N$ values of $f$, namely $f(t_1)$, $f(t_2)$, $\cdots$, $f(t_N)$ and $t_i=i\frac{T}{N}$.

Then I have $g(t)=f(t-\varphi)$ and $\varphi$ is in $\mathbb{R}$.

Giving the $N$ known values of $g$, namely $g(t_1)$, $g(t_2)$, $\cdots$, $g(t_N)$ and $t_i=i\frac{T}{N}$, is it possible to compute $\varphi$?

In the following figure $\varphi=\frac{\pi}{3}$.

An approximation is to note the largest/smallest values in each curve and find the difference in argument which causes them. You should be seeing the same shape shifted horizontally, so you can do this to all special features of the graph — gt6989b, Feb 07 '20 at 17:08
@gt6989b I am afraid it will not work in the general case, see for example https://math.stackexchange.com/questions/789210/phase-reference-of-a-periodic-signal — Alessandro Jacopson, Feb 08 '20 at 17:38

score 2 · Answer 1 · answered Feb 07 '20 at 17:07

2

I randomly guessed this and confirmed on Desmos, but I don't completely understand the intuition behind it. Something to do with the idea with the functions matching up.

Either evaluate, or using a Riemann sum with your values, find the value of $a$ such that $\displaystyle \int_0^{T}f(t+a)g(t)\,dt$ is maximized. Then $a=\varphi$.

answered Feb 07 '20 at 17:07

Saketh Malyala

13,637

I am sorry, I have updated the question, $f$ is not fully known, I only know $N$ samples of $f$. – Alessandro Jacopson Feb 08 '20 at 17:43
that's alright! use the samples you have to apply a Riemann sum instead of the actual integral – Saketh Malyala Feb 08 '20 at 18:10

How can I compute the phase shift?

1 Answers1