Continuous-Time Fourier Transforms

Representation Theory

An important concept from signal processing is that we can often uniquely and completely represent a signal as a weighted sum (or weighted integral) of others signals. Mathematiclly, we can write this as

$x(t) = \sum_{m = \mathcal{M}} c_{m} \phi_{m}(t) \; .$

The notation $m = \mathcal{M}$ denotes that we are summing over some signal collection (which could be finite or infinite in size). As an integral equation, this would be represented by

$x(t) = \int_{\mathcal{M}} c(w) \phi(t,w) \; dw \; .$

If a signal can be uniquely represented by a weighted collection of $\phi_{m}(t)$ or $\phi(t,w)$ functions, then that collection of signals is known as a basis.

Frequency Representations

In the rest of this course, we will learn about four closely related frequency representations and their bases — (1) the frequency (Fourier) representation for continuous power signals, (2) the frequency (Fourier) representation for continuous energy signals, (3) the frequency (Fourier) representation for discrete energy signals, (4) the frequency (Fourier) representation for discrete power signals.

Signal consisting of three basis components.

First component of Fourier series. — First component.

Second component of Fourier series. — Second component.

Third component of Fourier series. — Third component.

The Fourier Series

Continuous-Time Power Signals

For the Fourier series, we can express any continuous-time, power (periodic) signal as a summation of weighted complex exponentials (or equivalently, of weighted sines and cosines).

Fourier Series

We can express the Fourier Series in a compact manner with complex exponentials. In this case, continuous-time Fourier Series synthesis is defined by

$x(t) = \sum_{k=-\infty}^{\infty} c_k e^{j k \Omega_0 t }$

Continuous-time Fourier Series analysis is defined by

$c_k = \frac{1}{T_0} \int_{T_0} x(t) e^{-j k \Omega_0 t} dt$

where $\Omega_0 = 2 \pi / T_0$ and $T_0$ is the fundamental period of the wave.

Existence and convergence

The Fourier Series of a continuous-time, periodic signal $x(t)$ is guaranteed to exist if

$\frac{1}{T_0} \int_{T_0} |x(t)| dt < \infty$

This is known as the weak Dirchlet condition. Note that if the signal $x(t)$ is a power signal, it will satisfy this property.

The Fourier Series converges to $x(t)$ (i.e. it reconstructs $x(t)$ perfectly) when three conditions are satisfied:

$\frac{1}{T_0} \int_{T_0} |x(t)| dt < \infty$

$\text{Finite number of maxima of minima over one period }T$

$\text{Finite number of discontinuities over one period }T$

These are known as the strong Dirchlet conditions.

The Fourier Transform

Continuous-Time Energy Signals

For the Fourier Transform, we can express any continuous-time, energy (aperiodic) signal as a continuous summation (i.e., an integral) of weighted complex exponentials (or equivalently, of weighted sines and cosines).

Continuous-time Fourier Transform

Continuous-time Fourier transform analysis (i.e., the Fourier transform), is

$X(\Omega) = \int_{-\infty}^{\infty} x(t) e^{-j \Omega t } dt$

Continuous-time Fourier transform synthesis (i.e., the inverse Fourier transform) is

$x(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X(\Omega) e^{j \Omega t } d \Omega$

Note that the constant (i.e., the normalization) in front of the Fourier transform and inverse Fourier transform may sometimes differ depending on your field of study.

Fourier Transform Derivation

Consider Fourier series analysis:

$c_k = \frac{1}{T_0} \int_{T_0} x(t) e^{-j k \Omega_0 t} dt$

An aperiodic signal is a periodic signal with a fundamental period that approaches $\infty$ . In this condition, we get

$c_k = \lim_{T_0 \rightarrow \infty} \frac{1}{T_0} \int_{T_0} x(t) e^{-j k \Omega_0 t} dt \; .$

As $T_0 \rightarrow \infty$ , then $\Omega_0 = 2\pi/T_0 \rightarrow 0$ . Since $k$ is a variable that spans from $0$ to $\infty$ , then we can say product $k \Omega_0 \rightarrow \omega$ becomes a continuous variable. Therefore, our equation becomes

$X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j \omega t} dt \; .$

This is the Fourier Transform analysis equation.

Existence

The Fourier Transform of a continuous-time, aperiodic signal $x(t)$ is guaranteed to exist if

$\int_{-\infty}^{\infty} |x(t)| dt < \infty \; .$

Note that if the signal $x(t)$ is a energy signal, it will satisfy this property.

Caveat: Some periodic signals (e.g., sines and cosines) can have a Fourier transform through the Fourier Transform’s connection to the Fourier Series and through the use of the Dirac delta function, even though periodic signals fail to satify the above property.