| Potentially Useful Formulae | ||
|---|---|---|
| $ \sum_{n=-\infty}^{\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} $ | |
| 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 $ | |
| Acoustic Pressure (outside last tube) | $ b(t) = \frac{\rho c}{A_k} $ | |
| Total acoustic Pressure (inside first tube) | $ u(t) = (r+l)\frac{\rho c}{A_k} $ | |
