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I am interested in the convolution of a triangle function of width $2d$ with a cosine function (it has a useful analogy with a physics problem). I think I should be able to break the problem down using the following:

  • triangle function of width $2d$ is two convolved box-cars of width $d$

  • a box-car is two Heaviside functions (one positive, one negative... I think)

  • derivative of Heaviside function $H$: $\partial H = \delta$

  • for any convolution $\partial f * g = f*\partial g$

  • $\partial (f*g) = \partial f *\partial g$ (I think)

  • $\delta*-\delta = 0$ (I think)

The general idea is to convert a triangle function into an equivalent set of convolved Heaviside functions with some offsets, then take the derivative and convolve a bunch of Dirac delta functions with the sinusoid. Therefore, for triangle function $T$, box-cars $B$:

$T(x)*\cos = T(x)*\partial(\sin(x))\\ = (B(x-d/2)*B(x+d/2))*\partial(\sin)\\ = ((H(x-d)-H(x))*(H(x)-H(x+d)))*\partial(\sin)\\ = \partial\big[(H(x-d)-H(x))*(H(x)-H(x+d))\big]*\sin\\ = \big[(\delta(x-d)-\delta(x))*(\delta(x)-\delta(x+d))\big]*\sin\\ = 0*\sin(x)$

Clearly, that is hilariously wrong. This convolution does not generally self-cancel. I'm sure there is a nice analytical expression here, but I am not sure how to proceed.

PS. For future readers, the flaw is using $\partial (f*g) = \partial f *\partial g$. Differentiation is not distributive across convolution. Error thus propagates from line 4.

Rodrigo de Azevedo
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Mark_Anderson
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2 Answers2

1

Here is a valid derivation. Matches the numerical solution in MATLAB, although off by a constant scaling factor.

Taking a triangle from $0$ to $2d$,

$$ \begin{aligned} T(x)*\cos(x) &= T(x)*\partial(\sin(x)) \\ &= \partial(T)*\sin(x) \\ &= (B(d/2)-B(3d/2))*\sin(x) \\ &= (B(d/2)-B(3d/2))*\partial(-\cos(x)) \\ &= \partial \big[ B(d/2)-B(3d/2) \big]*-\cos(x) \\ &= \big[ \delta(x) - 2\delta(x-d) + \delta(x-2d) \big]*-\cos(x) \\ &= -\cos(x) + 2 \cos(x - d) - \cos(x - 2d) \end{aligned} $$

Generalising to an input $A k\cos(kx)$ [weird I know, but that's my actual physics equation] I get the result

$$ T(x)*\cos = \frac{A}{k}\bigg(-\cos(x) + 2 \cos(x - d) - \cos(x - 2d)\bigg) $$

Rodrigo de Azevedo
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Mark_Anderson
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0

There are a couple of mistakes in your derivation. First, if $$B(x-d/2)=H(x)-H(x-d),$$ then $$B(x+d/2)=H(x+d)-H(x).$$ Second, $\partial(f*g)=\partial(f)*g$. On the other hand, in the second last step, you used $\partial(f*g) = \partial(f)*\partial(g)$. Correcting them might help.

Math Lover
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  • Heaviside conversion had a typo. thanks! Those lines now agree with my final answer. However, I did not state $\partial(fg)=\partial(f)g$, I stated $\partial(fg)=\partial f\partial g$, so I think that bit is okay, unless the derivative is not actually distributive. – Mark_Anderson Aug 04 '17 at 18:22
  • @Mark_Anderson Check https://math.stackexchange.com/questions/177239/derivative-of-convolution for the derivative of convolution. – Math Lover Aug 04 '17 at 18:25
  • Cool, so the derivative is not distributive across convolution. Thanks for the correction. – Mark_Anderson Aug 04 '17 at 19:27