The Integration Property of the Fourier Transform

Back: Properties of the Fourier Transform
Fourier Transform Analsyis (Root)
Next: Fourier (Home)

On this page, we'll look at the integration property of the Fourier Transform. That is, if we have a function x(t) with Fourier Transform X(f), then what is the Fourier Transform of the function y(t) given by the integral:

integral of function for Fourier Transform
[Equation 1]

In words, equation [1] states that y at time t is equal to the integral of x() from minus infinity up to time t. Now, recall the derivative property of the Fourier Transform for a function g(t):

derivative of a function Fourier Transforms properties
[Equation 2]

Let's rewrite this Fourier property:

rewriting derivative for the fourier transform
[Equation 3]

We can substitute h(t)=dg(t)/dt [i.e. h(t) is the time derivative of g(t)] into equation [3]:

replacing the integral
[Equation 4]

Since g(t) is an arbitrary function, h(t) is as well and equation [4] gives a general result:

property of fourier transform
[Equation 5]

Equation [5] is "mostly" true. The reason it lacks completeness is that we used equation [3] to derive it - and note that the derivative of g(t) removes any constant: that is, the derivative of g(t) is equal to the derivative of the function g(t)+b, for any constant b. Because of this oversite, equation [5] is almost correct. If the total integral of g(t) is 0, then equation [5] directly follows from equation [3], and we have:

fourier transform properties of integration
[Equation 6]

If the total integral of g(t) is not zero, then there exists some constant c such that the total integral of g(t)-c is zero:

removing the dc constant term
[Equation 7]

That is, c is the "average value" of the function g(t), which is also often called the "dc term" or the "constant term". Using some math and the Fourier Transform of the impulse function, we have the general formula for the Fourier Transform of the integral of a function:

the property of fourier transform for hte integration
[Equation 8]

The Dirac-Delta impulse function in [7] is explained here.

The integration property is used and the constant in [8] is utilized on the Fourier Transform page for the unit step function, which should help clear things up if the above is not clear.

Next: Fourier Transform Pairs

Properties of the Fourier Transform

Fourier Transform Analysis (Table of Contents)

The Fourier Transform (Home)

This page on the integration property of Fourier Transforms is copyrighted. No portion can be reproduced without permission form the author. Copyright, 2010-2015, Fourier Tranform properties.