This post gives an extremely terse and brief overview of Fourier analysis (Fourier series and transform). Full derivations are given but other supplementary material should be used in addition to this post.
Fourier Series
Fourier’s theorem
Statement
A function is periodic with period if
(1)
need only be given on the half-open interval for it to be specified everywhere.
Fourier’s theorem states that any such which is well behaved can be written as one of the three following forms:
(2)
The different sets of coefficients , , can be converted between easily via trigonometric relations.
Formulae for the coefficients
We are going to make use of the integrals
(3)
If we integrate over its range, , we see that since the average value of and is zero. The integral reduces to , so we find
(4)
Note: we can integrate over at any point in the function’s range, so this is perfectly equivalent to
(5)
In other words, is the average value of .
For the coefficients, multiply by and integrate over the range:
(6)
finally, relabelling to gives
(7)
Exactly the same argument holds for the , so similarly
(8)
Examples
Square wave example
Given the function
(9)
We see that since it is odd, must be zero, and we find the to be .
So the Fourier series representation of is
(10)
Triangle wave example
Given the function
(11)
We see that since it is even, must be zero, and we find the to be
(12)
So the Fourier series representation of is
(13)
Extension to other periods
Suppose , that is is periodic with some arbitrary period. Then we would write
(14)
And the formulae for the coefficients would become
(15)
This can be simplified by rewriting in terms of angular frequency , giving
(16)
Fourier Transform
Derivation from Fourier Series
Earlier, we noted that the Fourier series can be written
(17)
Using the fact that and integrating over the series representation, we extract the coefficients
(18)
For a period of , the series becomes
(19)
Now we define , that is, a function that is only defined at the points . The series now looks like
(20)
Further, if we define we get
(21)
Rewriting as and taking finally gives us
(22)
Examples
Gaussian function
Suppose we have . The Fourier transform of this is
(23)
Shifting this by the substitution gives
(24)
Another Gaussian, but stretched.
Square pulse
Given the function
(25)
its Fourier transform is
(26)
Convolutions
Definition
A convolution is an integral transform that takes two functions and composes them, forming a new function. The definition is
(27)
Fourier Transform of a Convolution
The Fourier transform of the convolution of two functions is times the product of the Fourier transforms.
This post gives an extremely terse and brief overview of Fourier analysis (Fourier series and transform). Full derivations are given but other supplementary material should be used in addition to this post.
Fourier Series
Fourier’s theorem
Statement
A function is periodic with period if
(1)
need only be given on the half-open interval for it to be specified everywhere.
Fourier’s theorem states that any such which is well behaved can be written as one of the three following forms:
(2)
The different sets of coefficients , , can be converted between easily via trigonometric relations.
Formulae for the coefficients
We are going to make use of the integrals
(3)
If we integrate over its range, , we see that since the average value of and is zero. The integral reduces to , so we find
(4)
Note: we can integrate over at any point in the function’s range, so this is perfectly equivalent to
(5)
In other words, is the average value of .
For the coefficients, multiply by and integrate over the range:
(6)
finally, relabelling to gives
(7)
Exactly the same argument holds for the , so similarly
(8)
Examples
Square wave example
Given the function
(9)
We see that since it is odd, must be zero, and we find the to be .
So the Fourier series representation of is
(10)
Triangle wave example
Given the function
(11)
We see that since it is even, must be zero, and we find the to be
(12)
So the Fourier series representation of is
(13)
Extension to other periods
Suppose , that is is periodic with some arbitrary period. Then we would write
(14)
And the formulae for the coefficients would become
(15)
This can be simplified by rewriting in terms of angular frequency , giving
(16)
Fourier Transform
Derivation from Fourier Series
Earlier, we noted that the Fourier series can be written
(17)
Using the fact that and integrating over the series representation, we extract the coefficients
(18)
For a period of , the series becomes
(19)
Now we define , that is, a function that is only defined at the points . The series now looks like
(20)
Further, if we define we get
(21)
Rewriting as and taking finally gives us
(22)
Examples
Gaussian function
Suppose we have . The Fourier transform of this is
(23)
Shifting this by the substitution gives
(24)
Another Gaussian, but stretched.
Square pulse
Given the function
(25)
its Fourier transform is
(26)
Convolutions
Definition
A convolution is an integral transform that takes two functions and composes them, forming a new function. The definition is
(27)
Fourier Transform of a Convolution
The Fourier transform of the convolution of two functions is times the product of the Fourier transforms.
This post gives an extremely terse and brief overview of Fourier analysis (Fourier series and transform). Full derivations are given but other supplementary material should be used in addition to this post.
Fourier Series
Fourier’s theorem
Statement
A function is periodic with period if
(1)
need only be given on the half-open interval for it to be specified everywhere.
Fourier’s theorem states that any such which is well behaved can be written as one of the three following forms:
(2)
The different sets of coefficients , , can be converted between easily via trigonometric relations.
Formulae for the coefficients
We are going to make use of the integrals
(3)
If we integrate over its range, , we see that since the average value of and is zero. The integral reduces to , so we find
(4)
Note: we can integrate over at any point in the function’s range, so this is perfectly equivalent to
(5)
In other words, is the average value of .
For the coefficients, multiply by and integrate over the range:
(6)
finally, relabelling to gives
(7)
Exactly the same argument holds for the , so similarly
(8)
Examples
Square wave example
Given the function
(9)
We see that since it is odd, must be zero, and we find the to be .
So the Fourier series representation of is
(10)
Triangle wave example
Given the function
(11)
We see that since it is even, must be zero, and we find the to be
(12)
So the Fourier series representation of is
(13)
Extension to other periods
Suppose , that is is periodic with some arbitrary period. Then we would write
(14)
And the formulae for the coefficients would become
(15)
This can be simplified by rewriting in terms of angular frequency , giving
(16)
Fourier Transform
Derivation from Fourier Series
Earlier, we noted that the Fourier series can be written
(17)
Using the fact that and integrating over the series representation, we extract the coefficients
(18)
For a period of , the series becomes
(19)
Now we define , that is, a function that is only defined at the points . The series now looks like
(20)
Further, if we define we get
(21)
Rewriting as and taking finally gives us
(22)
Examples
Gaussian function
Suppose we have . The Fourier transform of this is
(23)
Shifting this by the substitution gives
(24)
Another Gaussian, but stretched.
Square pulse
Given the function
(25)
its Fourier transform is
(26)
Convolutions
Definition
A convolution is an integral transform that takes two functions and composes them, forming a new function. The definition is
(27)
Fourier Transform of a Convolution
The Fourier transform of the convolution of two functions is times the product of the Fourier transforms.
