The Integration Property of the Fourier Transform

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):

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 [3]:

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

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:

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 [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.

Properties of the Fourier Transform

Fourier Transform Analysis (Table of Contents)

The Fourier Transform (Home)

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 [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):

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 [3]:

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

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:

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 [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.

Properties of the Fourier Transform

Fourier Transform Analysis (Table of Contents)

The Fourier Transform (Home)