2.11 Fourier representations

Fourier representations

Throughout the course we have been alluding to various Fourier representations. We first recall the appropriate transforms:

• Fourier Series (CTFS) $x\left(t\right)$ : continuous-time, finite/periodic on $[-\pi ,\pi ]$
  $$X\left[k\right]=\frac{1}{\sqrt{2\pi }}{\int }_{-\pi }^{\pi }x\left(t\right)e^{-jkt}dt$$
  $$x\left(t\right)=\frac{1}{\sqrt{2\pi }}\sum _{k=-\infty }^{\infty }X\left[k\right]e^{jkt}$$
• Discrete-Time Fourier Transform (DTFT) $x\left[n\right]$ : infinite, discrete-time
  $$X\left(e^{j\omega }\right)=\frac{1}{\sqrt{2\pi }}\sum _{n=-\infty }^{\infty }x\left[n\right]e^{-j\omega n}$$
  $$x\left[n\right]=\frac{1}{\sqrt{2\pi }}{\int }_{-\pi }^{\pi }X\left(e^{j\omega }\right)e^{j\omega n}d\omega$$
• Discrete Fourier Transform (DFT) $x\left[n\right]$ : finite, discrete-time
  $$X\left[k\right]=\frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}x\left[n\right]e^{-j\frac{2\pi }{N}kn}$$
  $$x\left[n\right]=\frac{1}{\sqrt{N}}\sum _{k=0}^{N-1}X\left[k\right]e^{j\frac{2\pi }{N}kn}$$
• Continuous-Time Fourier Transform (CTFT) $x\left(t\right)$ : infinite, continuous-time
  $$X\left(\Omega \right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }x\left(t\right)e^{-j\Omega t}dt$$
  $$x\left(t\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }X\left(\Omega \right)e^{j\Omega t}d\Omega$$

We will think of Fourier representations in two complimentary senses:

1. “Eigenbasis” representations: Each Fourier transform pair is very naturally related to an appropriate class of LTI systems. In some cases we can think of a Fourier transform asa change of basis.
2. Unitary operators: While we often use Fourier transforms to analyze certain operators, we can also think of a Fourier transform as itself being an operator.