1. 7.4 Fourier Transform Theorems, Part I
Right now there are four theorems that need to be memorized. Four more to come!
scaling theorem (multiplying a function by a constant)
addition theorem (adding two functions)
similarity theorem (multiplying the variable by a constant)
power theorem (conservation of power between the temporal and frequency domains)
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2. Scaling Theorem
A function whose amplitude is scaled by a constant has a Fourier transform scaled by the same constant.
Given,
then,
Proof:
Example: What is the Fourier transform of a 1.38 V cosine having a period of 1 ms?
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3. Addition Theorem
The Fourier transform of a sum of functions is the sum of the individual Fourier transforms.
Given,
then,
Proof:
Example: What is the Fourier transform of cos(2pt)cos(2p2t)?
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4. Similarity Theorem
The Fourier transform of a stretched temporal variable, at, is a compressed frequency variable, f/a. Additionally, the spectrum amplitude is scaled by 1/|a|. Note: this theorem has already been built into the basis set definitions.
Given,
Then,
Proof: Start with the integral definition of the forward transform using the variables T and F.
Create a new dummy variable of integration, T = at, with dT = a dt.
Finally, create a new frequency variable, f = aF to eliminate the "stretching constant", a, in the exponent.
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5. Power Theorem
Since electrical signals can be represented by a Fourier transform pair, and since the power contained in the signal has to be independent of the domain (time or frequency), it makes sense that a Fourier transform conserves power.
Given,
Then,
or, alternatively,
The theorem can be extended to two functions.
In this expression the complex conjugate can be put on the other functions, e.g. F1*(t)F2(t).
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6. Power Theorem Example
Show that the power theorem is obeyed for the following Fourier transform pair.
The rectangle area can be computed remembering that
rect2(t 0) = rect(t 0).
The area under the sinc2 function requires using a tabulated integral, where a = p/f 0.
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