Introduction to the Fourier transform

Definitions & existence

The Fourier series representation of a function as discussed in this topic so far, is only applicable to periodic functions. Non-periodic functions however, can also be decomposed into Fourier components by the Fourier transform. The Fourier transform is an integral operator and, as such, it is defined with an integral. It represents a transformation as it allows mapping between domains. In particular, the Fourier transform allows the change of information from the time domain to the frequency domain. Note that the information in the time and frequency domains is the same but having this different representation can be very useful. For example, it is useful to have a signal in the time domain as the behaviour can be controlled in real life but it is also useful to have the signal in the frequency domain for the design and analysis of various control systems. The Fourier transform gives a way to transform between each domain. Note of course that the Fourier transform is not limited to functions of time and temporal frequencies; it is often used to analyse spatial frequencies as well.

Another useful integral operator is the Laplace transform which is frequently used as a powerful tool to solve ordinary differential equations. You will also see applications of the Laplace transform in your Process dynamics & Control course subject.

Definition 1.7 The Fourier transform of $f(t)$ is denoted by $\mathcal{F}(f)$ or $\hat{f}(\omega)$ and is defined by:

\mathcal{F}(f)=\hat{f}(\omega)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t} d t,

where $\omega$ denotes the frequency variable. The inverse Fourier transform is denoted by $\mathcal{F}^{-1}(\hat{f})$ and is given by:

\mathcal{F}^{-1}(\hat{f})=f(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \hat{f}(\omega) e^{i \omega t} d \omega .

A natural question that may arise is whether all functions $f(t)$ have a Fourier transform. We outline the following sufficient condition for the exis- tence of the Fourier transform: if $f(t)$ is absolutely integrable on the entire $t$ line and piecewise continuous (see Definition 1.4) on every finite interval, then the Fourier transform of $f(t)$ exists and it is given by Eq. (1.61), . The term 'absolutely integrable' implies that the integral of the absolute value of the scalar function $f(t)$ on the entire $t$ line, i.e. from $t=-\infty$ to $t=+\infty$ , is finite:

\int_{-\infty}^{\infty}|f(t)| d t<\infty

Next, let us look at an example of how the Fourier transform is applied.

Example 1.6 Find the Fourier transform of the following function,

f(t)= \begin{cases}1 & \text { if }|t|<1 \\ 0 & \text { otherwise }\end{cases}

Solution To find the Fourier transform of $f(t)$ , we use Eq. (1.61). We only integrate within $|t|<1$ since everywhere else, $f(t)=0$ , as stated by Eq. (1.64):

\begin{aligned} \mathcal{F}(f) & =\frac{1}{\sqrt{2 \pi}} \int_{-1}^{1} e^{-i \omega t} d t, \\ & =\frac{1}{\sqrt{2 \pi}}\left[\frac{e^{-i \omega t}}{-i \omega}\right]_{-1}^{1}, \\ & =-\frac{1}{i \omega \sqrt{2 \pi}}\left(e^{-i \omega}-e^{i \omega}\right) . \end{aligned}

Using Euler's formula,

e^{i \omega}=\cos \omega+i \sin \omega, e^{-i \omega}=\cos \omega-i \sin \omega

It follows that,

e^{-i \omega}-e^{i \omega}=-2 i \sin \omega .

Substituting in Eq. (1.65),

\mathcal{F}(f)=\sqrt{\frac{2}{\pi}} \frac{\sin \omega}{\omega} .

Properties of Fourier transform

Here, we outline some properties of the Fourier transform. It is noted that we can make similar claims for other integral transforms like, for example, the Laplace transform that was mentioned earlier.

Linearity

The Fourier transform is a linear operation: for any functions $f(t)$ and $g(t)$ whose Fourier transforms exist and any constants $a$ and $b$ , the Fourier transform of $a f+b g$ exists and is given by:

\mathcal{F}(a f+b g)=a \mathcal{F}(f)+b \mathcal{F}(g) .

This can be easily proven starting from the definition given by Eq. (1.61) since integration itself is a linear operation.

Convolution

Convolution is an operation on two functions, say $f(t), g(t)$ to get a third function which is also a function of $t$ . The notation we use for the convolution of two functions is:

f(t) * g(t)

and it is defined by the following convolution integral,

f(t) * g(t)=\int_{-\infty}^{\infty} f(u) g(t-u) d u .

The convolution of $f$ and $g$ is commutative, i.e.

f * g=g * f .

Note that in Eq. (1.68) the integration is done with respect to a dummy variable, $u$ . This implies that, once the integral is evaluated on the $u$ line, we are left with a function that depends on $t$ . In terms of the Fourier transform of the convolution of two functions, we have the following result:

Suppose that $f(t)$ and $g(t)$ are piecewise continuous, bounded and absolutely integrable on the $t$ -axis. Then:

\mathcal{F}(f * g)=\sqrt{2 \pi} \mathcal{F}(f) \mathcal{F}(g),

where $\mathcal{F}(f)$ and $\mathcal{F}(g)$ are the Fourier transforms of $f$ and $g$ , respectively. We can also take the inverse Fourier transform of Eq. (1.70), such that the convolution of $f$ and $g$ may also be expressed as:

\begin{aligned} f * g & =\mathcal{F}^{-1}(\sqrt{2 \pi} \mathcal{F}(f) \mathcal{F}(g)) \\ & =\int_{-\infty}^{\infty} \hat{f}(\omega) \hat{g}(\omega) e^{i \omega t} d \omega . \end{aligned}

We note that the usefulness of Eq. (1.71), is exhibited in solution techniques to partial differential equations. In particular, say we are interested in describing a system which is to be modelled as a bar of 'infinite length'. An example of where this notion applies is the spatio-temporal temperature distribution in a wire, described by solutions to the one-dimensional heat equation. We note here that in Topic B2, where we will discuss partial differential equations, we will look at solutions to the heat equation but our domain of interest will be restricted to finite lengths hence we will be making use of Fourier series expansions rather than the Fourier transform.

Plancherel's theorem

This is a generalisation of Parseval's theorem given in Section 1.4

Given a function $f(t)$ whose Fourier transform, $\hat{f}(\omega)$ exists, the following relation holds:

\int_{-\infty}^{\infty}|f(t)|^{2} d t=\int_{-\infty}^{\infty}|\hat{f}(\omega)|^{2} d \omega .

Mathematically, the theorem tells us that the integral of the squared modulus of $f$ is equal to the squared modulus of its Fourier transform, $\hat{f}$ . Now, think of a system with mechanical vibrations like, for example, a mass-spring system executing simple harmonic motion. In such a setting, in Eq. (1.62), we think of $f(t)$ as the superposition of sinusoidal oscillations of all frequencies. Then, the following integral

\int_{-\infty}^{\infty}|\hat{f}(\omega)|^{2} d \omega

may be thought of as the representation of the total energy of the system. Note that, in Eq. (1.73), using lower and upper limits of $a$ and $b$ , respectively, the term in Eq. (1.73) yields the contributions of the frequencies to the total energy of the system from $a$ to $b$ . The interpretation of Plancherel's theorem in the physical setting is that the total energy contained in a waveform $f(t)$ , summed across all of time $t$ (physical variable) is equal to the total energy of the Fourier Transform of the waveform, $\hat{f}$ , summed across all of its frequency components. Further, it implies that the Fourier transform preserves the energy of the original quantity.

Half-range series Definitions