VE216 Lecture 18

DT Fourier Representations

FFT

This is actually 64 multiplications.

So we can generate the FFT calculation method.

Even numbered	Odd Numbered

Total 32 multiplications.

$x[n] \to$ aperiodic DT signal.

Then we have the periodic extension: $\begin{align}x_N[n] = \sum^ \infty_{k=-\infty}x[n+kN]\end{align}$

So $x[n] = \lim_{N\to\infty} x_N[n]$

Analysis Equation: $\begin{align} X(e^{j\Omega}) = \sum^\infty_{n=-\infty} x[n] e^{-j\Omega n} = H(z)_{z=e^{j\Omega}} \end{align}$ with $H(z) = \sum x[n]z^{-n}$
Synthesis Equation: $\begin{align} x[n] = \frac 1 {2\pi} \int_{2\pi}X(e^{j\Omega})e^{j\Omega n} d\Omega \end{align}$