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

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