# Equation for a sinusoidal wave with changing frequency

I would like to find a sinusoidal wave whose period or frequency change to half (or double) with every step, someting like this But I cant find the precise coefficients for the period to decrease (or increase) twice. Can someone help me?

I want to use it to map the different pitch (or notes) of voice, to certain values.

In our speech, and singing, the range of the first octave is half of the next octave, so I need a function whose period is twice as big each time to cover all the octaves. it need not be a sinusoidal it can be any periodical function.

• $\sin \left( 2 \pi \; 2^x \right)$ or $\sin \left( 2 \pi \; 2^{-x} \right)$ – Will Jagy Jun 9 '16 at 18:52
• Let $p_k$ be the distance between two successive zero crossings. If $p_{k+1} = {1 \over 2} p_k$, then the zero crossings must accumulate at some finite point. – copper.hat Jun 9 '16 at 19:10
• @WillJagy I don't see the period duplicating :( – lesolorzanov Jun 9 '16 at 21:13
• @copper.hat , I'm not sure about that, we would just divide by half and keep diving to infinity they would be very very close but never accumulate, I think. – lesolorzanov Jun 9 '16 at 21:14
• The zero crossings would be at $p_0+p_1+p_2+...$ which is bounded above... – copper.hat Jun 9 '16 at 21:23

## 1 Answer

The "correct" way of specifying a signal with time-varying frequency $f(t)$ is

$s(t) = \sin(2 \pi \int_0^t f(t) dt)$

In other words the phase is the integral of the instantaneous frequency $f(t)$. This is best checked when you consider the case when $f(t)$ is a step function, or linearly increasing.

If you now want the frequency to increase from a certain value at $t_0$ to twice the value at some time $t_1$, you need to choose an appropriate function $f(t)$ which does

$f(t_1) = 2 f(t_0)$

The precise form of $f$ can be chosen freely to meet your demand. If you want exponential behavior, i.e. doubling every $T_0$ with frequency of $f_0$ at time = 0 the following would be a possible choice

$f(t) = f_0 2^{t/T_0}$

• Maybe I am confused, I don't want to double every To,by an f0. I mean every period being twice as long as the one before. If I leave the integral of 2^t, then I have sin(2^(x)*2*pi/ln(2)) which doesn't work either – lesolorzanov Jun 9 '16 at 21:25