# Why is the convolution of two sine waves a sinc function?

Problem/Question

After taking the convolution of two discrete sine waves of the same frequency and length the result appears to be a sinc function but I was expecting to get a sine wave since this should be the same as multiplying two impulses in the frequency domain right? There is a similar question posted here -> What is convolution of two sine waves (tones)? but the answers there suggest that the result should have been a sine wave since both have the same frequency and length. But that doesn't seem to be the case when I actually perform the operation. Can anyone help me understand why this isn't acting as a simple low pass filter allowing the 2nd sine function through unchanged?

MATLAB steps

t = 0:20
x1 = sin((2 * pi * t) / 21)
x2 = 100*sin((2 * pi * t) / 21)
y = conv(x1,x2)
plot(y)


• Hint: re-run your code with t=0:200. What do you see?
– jojeck
Commented Sep 1, 2023 at 19:56
• @jojeck Thanks. I see sine wave with some tremelo. Are you saying what I'm seeing is not really a sinc but a piece of a shifted version of a sine wave? Commented Sep 1, 2023 at 20:02
• While we know that the premise to this question is false, an interesting question is, to me, why the convolution of two identical sinc functions is another identical sinc function. It's easy to see in the frequency domain. Commented Sep 2, 2023 at 17:17