Fourier Transforms, Theorems, Convolution

Convolution Theorem

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Theorem. Consider the following two Fourier Transform pairs.

F 1 ( t ) Φ 1 ( f )and F 2 ( t ) Φ 2 ( f )

Let a new function, C(t) be the convolution of F1(t) and F2(t).

F 1 F 2 C( t )= F 1 ( t ) F 2 ( t t )d t

The convolution theorem gives the following relationships between the convolution and multiplication operators.

F 1 ( t ) F 2 ( t ) Φ 1 ( f ) Φ 2 ( f ) F 1 ( t ) F 2 ( t ) Φ 1 ( f ) Φ 2 ( f )

Proof. Write out the Fourier transform for C(t).

Φ C ( f )= C( t ) e i2πft dt Φ C ( f )= F 1 ( t ) F 2 ( t t )d t e i2πft dt

Rewrite the double integral by letting T = t - t', and dT = dt.

Φ C ( f )= F 1 ( t ) F 2 ( T ) e i2πf( t +T ) d t dT

Finally, convert the exponential into a product and separate the two integrals.

Φ C ( f )= F 1 ( t ) e i2πf t d t F 2 ( T ) e i2πfT dT Φ C ( f )= Φ 1 ( f ) Φ 2 ( f )

Properties of convolution.

Commutation:

F 1 ( t ) F 2 ( t )= F 2 ( t ) F 1 ( t )

Distribution:

F 1 ( t )[ F 2 ( t )+ F 3 ( t ) ]= F 1 ( t ) F 2 ( t )+ F 1 ( t ) F 3 ( t )

Association:

F 1 ( t )[ F 2 ( t ) F 3 ( t ) ]=[ F 1 ( t ) F 2 ( t ) ] F 3 ( t )

Combinations with multiplication: Note that multiplication and convolution do not commute!

[ F 1 ( t ) F 2 ( t ) ] F 3 ( t )[ Φ 1 ( f ) Φ 2 ( f ) ] Φ 3 ( f ) [ F 1 ( t ) F 2 ( t ) ][ F 3 ( t ) F 4 ( t ) ][ Φ 1 ( f ) Φ 2 ( f ) ][ Φ 3 ( f ) Φ 4 ( f ) ]

More complicated examples:

[ F 1 ( t )+ F 2 ( t ) ][ F 3 ( t )+ F 4 ( t ) ][ Φ 1 ( f )+ Φ 2 ( f ) ][ Φ 3 ( f )+ Φ 4 ( f ) ] [ F 1 ( t ) F 2 ( t )+ F 3 ( t ) F 4 ( t ) ] F 5 ( t )[ F 1 ( t ) F 2 ( t )+ F 3 ( t ) F 4 ( t ) ] F 5 ( t )

Common convolutions.

Rectangle: The convolution of two equal-sized rectangles is a triangle,

rect( t 0 )rect( t 0 )=Λ( t 0 )

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,

rect( t 1 0 )rect( t 2 0 )=trap( t 1 0 + t 2 0 )

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,

gauss( t 1 0 )gauss( t 2 0 )=gauss( t 3 0 )

where,

( t 3 0 ) 2 = ( t 1 0 ) 2 + ( t 2 0 ) 2

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.

σ 3 2 = σ 1 2 + σ 2 2 Γ 3 2 = Γ 1 2 + Γ 2 2

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.

Fourier Transform Theorems

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Copyright 2004 by F. E. Lytle