0.2 Fourier integral

Fourier integral

Continuous-time signals $x\left(t\right)$ which are not periodic can still be understood as superpositions of pure oscillations $e^{j2\pi ft}$ where now all frequencies are present in the signal. The coefficients $X\left(f\right)$ of the oscillations can be computed as follows:

The representation as a superposition takes then the following form:

We call $X\left(f\right)$ the Fourier transform of $x$ and write also $\mathcal{F}\left\{x\right\}\left(f\right)$ instead of $X\left(f\right)$ to indicate clearly which signal has been transformed. The “Fourier spectrum”, or simply the spectrum , or also the “power spectrum” of the signal is the squared amplitude ${\left|X\left(f\right)\right|}^2$ . This is the function usually plotted, while the phase of $X$ is not shown. Nevertheless, the plots are usually —and erroneously— labeledwith $X$ instead of ${\left|X\right|}^2$ (see [link] ).

A signal is called bandlimited if its Fourier transform $X\left(f\right)$ is zero for high frequencies, i.e. for large $\left|f\right|$ . Similarly we say that a signal is time-limited if it is zero for large times, i.e., for large $\left|t\right|$ . By Heisenberg's principle a bandlimited signal can not be time-limited.Since bandlimited signals are of great importance, there is a need to study signals which are not time-limited and, thus, the Fourier integral.

Properties

• Linearity:
  $$\mathcal{F}\left\{a,x,+,y\right\}\left(f\right)=aX\left(f\right)+Y\left(f\right)$$
• Convolution
  $$\mathcal{F}\left\{x,*,y\right\}\left(f\right)=X\left(f\right)·Y\left(f\right)\mathcal{F}\left\{x,·,y\right\}\left(f\right)=X\left(f\right)*Y\left(f\right)$$
• Change of time scale
  $$\mathcal{F}\left\{\frac{1}{a},x,\left(\frac{t}{a}\right)\right\}\left(f\right)=X\left(af\right)\mathcal{F}\left\{x,\left(,a,t,\right)\right\}\left(f\right)=\frac{1}{a}X\left(\frac{f}{a}\right)$$
• Translation in time and frequency
  $$\mathcal{F}\left\{x,\left(t,-,b\right)\right\}\left(f\right)=X\left(f\right)e^{-j2\pi bf}\mathcal{F}\left\{x,\left(t\right),e^{j2\pi bt}\right\}\left(f\right)=X(f-b)$$
• Symmetries and Fourier pairs The symmetry of [link] and [link] leads one to consider $x\left(t\right)$ and $X\left(f\right)$ as a Fourier pair. Indeed, the Fourier transform of $X$ is almost $x$ : $\mathcal{F}\left\{X\right\}\left(f\right)=x(-f)$ . Clearly, the symmetry is not perfect since $X\left(f\right)$ is in general complex, while $x$ is real. However: If $x\left(t\right)$ is symmetric, i.e. $x(-t)=x\left(t\right)$ then $X\left(f\right)$ is real-valued, and vice versa!

In summary: Symmetric real signals have symmetric real Fourier transforms and vice versa. As we will see below, they also possess the same energy.