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Fourier Transform TOC
< Derivative Theorem <
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Theorem. Consider the following two Fourier Transform pairs.
Let a new function, C(t) be the convolution of F1(t) and F2(t).
The convolution theorem gives the following relationships between the convolution and multiplication operators.
Proof. Write out the Fourier transform for C(t).
Rewrite the double integral by letting T = t - t', and dT = dt.
Finally, convert the exponential into a product and separate the two integrals.
Properties of convolution.
Combinations with multiplication: Note that multiplication and convolution do not commute!
More complicated examples:
Rectangle: The convolution of two equal-sized rectangles is a triangle,
where t0 is the width of the rectangle and the Full Width at Half Maximum of the triangle. The base of the triangle is 2t0. The convolution of two different-sized rectangles is a trapezoid,
where the base of the trapezoid is (t10+t20) and the top is |t10-t20|.
Gaussian: The convolution of two Gaussian functions is a Gaussian function,
Since the standard deviation and the Full Width at Half Maximum of a gaussian function are proportional to the t0 parameters, one also has the following two relationships.
Impulse: The convolution of an impulse with any other function replicates that function. The replicate is centered at the position of the impulse. For this reason, the impulse function is called a replicator when used in convolution.
Comb: The convolution of a comb with any function having a size smaller than the comb spacing, replicates the function at the center of each comb impulse. The comb function replicates multiple copies when used in convolution.