# Implementing a sine wave with linearly changing frequency

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Paul Mattke on 20 May 2015
Commented: Talmon Alexandri on 22 Dec 2022
Hi everyone
I was trying to implement a sine wave that changes its frequency linearly between f_start and f_end in a specific time t. So let's say:
f_start = 50 Hz
f_end = 100 Hz
t = 10 s
That means, i want the frequency of my sine to change linearly from 50 to 100 Hz in 10 seconds. In my mind, that seemed to be easy, so I started with the following approach:
clear;
t = 0:0.00001:10;
f_in_start = 50;
f_in_end = 100;
f_in = linspace(f_in_start, f_in_end, length(t));
y = sin(2*pi * f_in .* t);
plot(t,y)
f_out = 1./(2*diff(find_y_zeros(t,y)));
f_out_start = f_out(1);
f_out_end = f_out(end);
find_y_zeros is a function that I wrote to calculate the times where my y goes 0 (via interpolation). I tested this function seperately and everything seemed to work fine.
However, when I calculate the differences between the zeros, multiply that by 2 (to get the period length) and divide 1 by this, the frequencies at the beginning and the end of the oscillation are 50 and 150 Hz which was definitely not what I wanted above.
Then, by 'try-and-error', I found out that I could get what I wanted by changing line 4 of my code above to this:
f_in_end = f_in_start + 50/2;
Generally speaking, it seems that I get what I want when I divide the difference between my aimed f_start and f_end by 2. So, in this case, to get from 50 to 100 Hz, the difference is 50, and that is divided by 2, so I get f_in_start = 50 and f_in_end = 75.
From the distances of the zeros I can see that now, the frequency of my sine goes from 50 to 100 Hz and that seems to work on any other example (other frequencies and other times) too.
So... it finally works. But I want to know why. Could anybody tell my why the first naive approach did not work? I would be grateful for any hint on this.

Alfonso Nieto-Castanon on 22 May 2015
Edited: Alfonso Nieto-Castanon on 22 May 2015
the formula:
sin( 2*pi*f(t)*t )
does not result in the desired sine wave with time-varying frequency. Rather the appropriate formula would use, instead of f(t)*t, the integral between 0 and t of f(t):
sin( 2*pi* f(t)dt )
Only when f(t) is a constant f value, its integral is f*t, and the sine wave is the familiar sin(2*pi*f*t).
The rational for this is that the sinusoid frequency is the rate-of-change / derivative of the sinusoid phase. If you are given a desired frequency profile f(t) you need to integrate that to compute the desired sinusoid phase (up to a constant additive term), from which you can then compute your desired waveform.
One possible way to produce your desired signal would be:
Fs = 100000;
t = 0:1/Fs:10;
f_in_start = 50;
f_in_end = 100;
f_in = linspace(f_in_start, f_in_end, length(t));
phase_in = cumsum(f_in/Fs);
y = sin(2*pi*phase_in);
plot(t,y)
##### 2 CommentsShow 1 older commentHide 1 older comment
What is making the phase in here
Fs = 100000;
t = 0:1/Fs:10;
f_in_start = 50;
f_in_end = 100;
f_in = linspace(f_in_start, f_in_end, length(t));
phase_in = cumsum(f_in/Fs);
y = sin(2*pi*phase_in);
plot(t,y)

Alfonso Nieto-Castanon on 22 May 2015
Edited: Alfonso Nieto-Castanon on 22 May 2015
someone deleted his own answer including my own comments to it so, just in case these are interesting to anybody else, I am repeating a hopefully illustrative example below:
Using sin(2*pi*f(t)*t) instead of sin(2*pi*∫f(t)dt) is not really an uncommon mistake at all when building frequency-modulated signals. This is aggravated by the fact that, when f(t) is a polynomial function, the former leads to an "almost correct-looking" solution. Trying instead a non-polynomial function (e.g. f(t)=2+sin(t), a frequency modulated tone oscillating between 1Hz and 3Hz) one is able to see the differences between the two methods quite more clearly.
In the example below, we are trying to build a frequency modulated signal with frequency oscillating between 1Hz and 3Hz. The top graph shows the signal built using sin(2*pi*f(t)*t) which does not show the expected 1Hz to 3Hz frequency oscillation at all (things get even worse it you continue plotting beyond t=10s, as the signal frequency starts making wilder and wilder oscillations, then it starts aliasing, etc.), the middle graph shows the integral approximation using the cumsum trick sin(2*pi*cumsum(f(t))/Fs), and the bottom graph shows the exact-integral solution sin(2*pi*∫f(t)dt))
t = 0:1/1000:10;
f = 2+sin(t);
y_wrong = sin(2*pi*f.*t);
y_approx = sin(2*pi*cumsum(f)/1000);
y_exact = sin(2*pi*(2*t-cos(t)));
subplot(311); plot(t,y_wrong,'.-')
subplot(312); plot(t,y_approx,'.-');
subplot(313); plot(t,y_exact,'.-'); Talmon Alexandri on 22 Dec 2022
Thanks a lot for the nice explanation and good example. Helped me so much!!!!!

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