- #1

physiks

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## Homework Statement

According to my notes, if we have a sinusoidal aperture/transmission function of the form a(x)=1+sin(wx) and a 'top-hat' aperture function given by b(x)=1, -0.5d≤x≤0.5d, b(x)=0 otherwise, then their convolution should give a finite sinusoidal aperture function, i.e sinusoidal but only over a certain range - outside of this it is zero.

## Homework Equations

Convolution of a(x) and b(x) is ∫

_{-∞}

^{∞}a(y)b(x-y)dy

## The Attempt at a Solution

We need ∫

_{-∞}

^{∞}[1+sin(wy)]b(x-y)dy, and my thinking is that the integrand is zero everywhere apart from between x-0.5d and x+0.5d, where it is simply [1+sin(wy)]. Then the convolution integral reduces to ∫

_{x-0.5d}

^{x+0.5d}[1+sin(wy)]dy giving a-{cos[w(x+d/2)]+cos[w(x-d/2)]}/w which is not the finite sinusoidal aperture function I've been told I should get. Is this right?

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