Note: PLease do no change the cheat sheet anymore as the test is now printed. -pm


Back to ECE438 Fall 2010 Prof. Boutin

Potentially Useful Formulae
$ \sum_{n=0}^{\infty} a^n = \frac{1}{1-a}, \ |a|<1 $ $ \sum_{n=0}^{N-1} a^n = \frac{1-a^N}{1-a}, \ |a|<1 $
$ W_{N}^{kn} = e^{-j\frac{2\pi}{N}kn} $
Euler's Formula $ e^{j\omega} = cos(\omega) + j sin(\omega) $
$ cos(\omega) = \frac{e^{j\omega} + e^{-j\omega}}{2} $ $ sin(\omega) = \frac{e^{j\omega} - e^{-j\omega}}{2j} $
$ \mathcal{F}[\frac{rect(t-\frac{T}{2})}{T}] \Rightarrow Tsinc(Tf)e^{-j2 \pi f \frac{T}{2}} $
DFT $ X[k] = \sum_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn} $
IDFT $ x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j\frac{2\pi}{N}kn} $
DTFT $ X(\omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\omega n} $
IDTFT $ x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega) e^{j\omega n} d\omega $
Z-Transform $ X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} $
Time Shift Property of Z-Transform $ x[n-n_0] => X(z)z^{-n_0} $
Comb/Rep $ rep_{T}(x(t)) = \sum_{k=-\infty}^{\infty} x(t-kT) $
Comb/Rep $ comb_{T}(x(t)) = \sum_{k=-\infty}^{\infty} x(kT)\delta (t-kT) $
Comb/Rep $ rep_{T}(x(t)) <=> \frac{1}{T} comb_{\frac{1}{T}}(X(f)) $
Comb/Rep $ comb_{T}(x(t)) <=> \frac{1}{T} rep_{\frac{1}{T}}(X(f)) $
Circular Convolution $ f[n]*_N g[n] = \sum_{k=0}^{N-1} f[k]g[(n-k)mod \ N] $
Short Time Fourier Transform $ X[k,m] = \sum_{n=-\infty}^{\infty} x[n]w[n-m]e^{-j\frac{2\pi}{N}kn} $
CSFT $ f(x,y) <=> F(u,v) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y)e^{-j2\pi(ux+vy)} \ dx dy $
ICSFT $ F(u,v) <=> f(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F(u,v)e^{j2\pi (ux+vy)} \ du dv $
Sinc $ sinc(\theta)= \frac{sin(\pi\theta)}{\pi\theta} $

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood