Intro to Fourier transforms: how to calculate them
Intro to Fourier transforms: how to calculate them

The Integration Property of the Fourier Transform

In words, equation  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):
Let’s rewrite this Fourier property:
We can substitute h(t)=dg(t)/dt [i.e. h(t) is the time derivative of g(t)] into equation :
Since g(t) is an arbitrary function, h(t) is as well and equation  gives a general result:
Equation  is “mostly” true. The reason it lacks completeness is that we used equation  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  is almost
correct. If the total integral of g(t) is 0, then equation  directly follows from equation , and we have:
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:
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 Dirac-Delta impulse function in  is explained here.
The integration property is used and the constant in  is utilized on
the Fourier Transform page for the unit step function, which should help clear
things up if the above is not clear.
Properties of the Fourier Transform
The Fourier Transform (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: [Equation 1] [Equation 2] [Equation 3] [Equation 4] [Equation 5] [Equation 6] [Equation 7] [Equation 8]

Next: Fourier Transform Pairs

In words, equation  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):

Let’s rewrite this Fourier property:

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

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

Equation  is “mostly” true. The reason it lacks completeness is that we used equation  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  is almost correct. If the total integral of g(t) is 0, then equation  directly follows from equation , and we have:

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:

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 Dirac-Delta impulse function in  is explained here.

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

Properties of the Fourier Transform