1. 7.7 Fourier Transform Theorems, Part II
We have already covered four theorems.
The following three theorems will be covered in this section
 shift theorem
 derivative theorem
 convolution theorem
The autocorrelation theorem will be covered under electronics.
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2. Shift Theorem
The Fourier transform of a shifted variable is the transform of the un-shifted variable multiplied by a complex harmonic.
Given,
then,
where t' is a constant.
Proof: Start with the integral for the forward transform.
Multiply the integrand by
e+i2pft'e-i2pft'.
Since dt = d(t-t') the previous integral can be written in the form of a forward transform.
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3. Shift Theorem Example
The Fourier transform of a single impulse at f = f 0 with amplitude, A, can be determined by the shift theorem.
First write the transform pair for the single impulse function d(f=0).
d(f=0)  C(t) = A
Now write the transform of the impulse at f 0 using the shift theorem.
d(f-f 0=0) « Aexp(-i2pf 0t) = Acos(2pf 0t)-iAsin(2pf 0t)
The temporal result can be written as a phase shifted cosine.
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4. Derivative Theorem Given, then,
The theorem can be extended to multiple levels of differentiation.
Proof:
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5. Derivative Theorem Example
What is the Fourier transform of the derivative of a cosine,
cos(2pt/t 0)?
Given,
then,
where,
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6. Convolution 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.
This is the most important and useful theorem - the convolution of two functions in the time domain has a transform given by multiplication in the frequency domain.
The theorem can eliminate the need to evaluate convolution integrals.
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7. Theorem Proof
Start by writing out the Fourier transform of the convolved function.
Substitute the convolution integral for C(t).
Let t - t' = T, and dt = dT.
Finally, separate the double integral into a product of integrals.
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8. Properties of Convolution
Commutation:
Distribution:
Association:
Combinations with multiplication: multiplication and convolution do not commute!
A more complicated example:
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9. Convolution Examples
Gaussian: The convolution of two Gaussian functions is a Gaussian function,
where the variance of the two convolved functions adds to give the variance of the result. In some cases it is more convenient to use the FWHM, G32 = G12 + G22.
Impulse: The convolution of an impulse with any other function, replicates the function and centers it at the position of the impulse. For this reason, the impulse function is called a replicator.
Comb: The convolution of a comb with any function with a size smaller than the comb spacing, replicates the function at the center of each comb impulse. Thus a comb function is a multiple copy replicator.
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