Multiple Definitions of the Fourier Transform
Definitions of the Fourier Transform
The definition of the Fourier transform can vary slightly between books.
When dealing with the Fourier transform, a constant multiple of $\sqrt{2 \pi} \fallingdotseq 2.5$ invariably appears, and the choice of where to apply this constant leads to different conventions.
Here, I will describe three commonly used definitions.
1. Definition Without Constants in the Fourier Transform Formula
This definition is given by:
$$ X(\omega) = \int_{-\infty}^{\infty}{x(t)e^{-i\omega t}dt} $$
In this definition, no constants appear in the Fourier transform formula.
Instead, constants are applied elsewhere (such as in the inverse Fourier transform).
If the input is a time-domain signal, the output is a signal in the angular frequency domain.
I generally use this convention because it is easy to remember.
Definition Using (Standard) Frequency Domain Instead of Angular Frequency
This definition is given by:
$$ X(f) = \int_{-\infty}^{\infty}{x(t)e^{-2 \pi i f t}dt} $$
In this definition, constants are included in the integrand.
If the input is a time-domain signal, the output is a signal in the (standard) frequency domain.
This convention is sometimes used in fields like communications engineering.
In practice, when referring to the frequency of radio waves or sound, terms like “Hz” are used, rather than angular frequency in “rad/s,” making this representation seem more natural.
3. Definition Using Angular Frequency with Constants Outside the Integral
This definition is given by:
$$ X(\omega) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}{x(t)e^{-i\omega t}dt} $$
In this definition, a constant is included outside the integral.
If the input is a time-domain signal, the output is a signal in the angular frequency domain.
This definition is essentially the same as Definition 1, but scaled by a constant, making it symmetrical with the inverse Fourier transform.
This convention is often used in mathematical texts.
Parseval’s Theorem
An important theorem related to the Fourier transform is Parseval’s theorem.
Parseval’s theorem in Definitions 1, 2, and 3 is expressed as follows:
Let the Fourier transforms of functions $x(t)$, $y(t)$ be $X_1(\omega), Y_1(\omega)$ for Definition 1, $X_2(f), Y_2(f)$ for Definition 2, and $X_3(\omega), Y_3(\omega)$ for Definition 3. Then:
$$ \int_{-\infty}^{\infty}X_1(\omega)Y_1^*(\omega)d\omega = 2 \pi \int_{-\infty}^{\infty}x(t)y^*(t)dt $$
$$ \int_{-\infty}^{\infty}X_2(f)Y_2^*(f)df = \int_{-\infty}^{\infty}x(t)y^*(t)dt $$
$$ \int_{-\infty}^{\infty}X_3(\omega)Y_3^*(\omega)d\omega = \int_{-\infty}^{\infty}x(t)y^*(t)dt $$
Only Definition 1 includes a constant.
From this, it can be seen that in Definitions 2 and 3, the Fourier transform is unitary, but in Definition 1, the Fourier transform is not a unitary transformation.
Since unitary transformations are useful for preserving length in two vector spaces, it is preferable to use Definitions 2 or 3 if one wishes to utilize this property in their analysis.