Continous phase transition between two channels of sine wave

Question

I have:

I want to make smooth transition of phase over time for 2nd channel. I achieved it, but it was discrete transition, here is mine:

0-2 sec: 0 degree
2-4 sec: 90 degree
4-6 sec: 180 degree
6-8 sec: 270 degree

Instead of discrete transition, I want to make continous transition:

From 0 sec until 10 sec, the phase is changing over time from 0 degree to 360 degree.

How do I achieve this?

Here is the picture of mine, which is discrete transition:

This answer should help you as well: dsp.stackexchange.com/a/83014/21048 — Dan Boschen, Commented Oct 22 at 19:32

Hilmar · Accepted Answer · 2024-10-22 16:04:31Z

From 0 sec until 10 sec, the phase is changing over time from 0 degree to 360 degree.

A linear phase shift is the same thing as a change in frequency.

I think what you want is something like

$$x(t) = A\cos(\omega_0 t + 2\pi\frac{t}{10s}) $$

That is a phase offset that linearly increases by $2\pi$ in a 10 second period. However this simply another sine wave at a slightly higher frequency

$$x(t) = A\cos(\omega_0 t + 2\pi\frac{t}{10s}) = A\cos((\omega_0 + 2\pi\frac{1}{10s})t) $$

So this just a sine wave at a frequency of $\omega_1 = (\omega_0 + 2\pi\frac{1}{10s})$.

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