(New page: Saving location for potential formula sheet. <math>W_{N}^{kn} = e^{-j\frac{2\pi}{N}kn}</math>) |
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| + | ! ! style="background: rgb(228, 188, 126) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Potentially Useful Formulae | ||
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| + | | align="right" style="padding-right: 1em;" | | ||
| + | | <math>\sum_{n=-\infty}^{\infty} a^n = \frac{1}{1-a}, \ |a|<1</math> | ||
| + | | <math>\sum_{n=0}^{N-1} a^n = \frac{1-a^N}{1-a}, \ |a|<1</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | <math>W_{N}^{kn} = e^{-j\frac{2\pi}{N}kn}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Euler's Formula | ||
| + | | <math>e^{j\omega} = cos(\omega) + j sin(\omega)</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | <math>cos(\omega) = \frac{e^{j\omega} + e^{-j\omega}}{2}</math> | ||
| + | | <math>sin(\omega) = \frac{e^{j\omega} - e^{-j\omega}}{2j}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | DFT | ||
| + | | <math>X[k] = \sum_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | IDFT | ||
| + | | <math>x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j\frac{2\pi}{N}kn}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | DTFT | ||
| + | | <math>X(\omega) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\omega n}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | IDTFT | ||
| + | | <math>x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega) e^{j\omega n} d\omega</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Z-Transform | ||
| + | | <math>X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Time Shift Property of Z-Transform | ||
| + | | <math>x[n-n_0] => X(z)z^{-n_0}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Comb/Rep | ||
| + | | <math>rep_{T}(x(t)) = \sum_{k=-\infty}^{\infty} x(t-kT)</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Comb/Rep | ||
| + | | <math>comb_{T}(x(t)) = \sum_{k=-\infty}^{\infty} x(kT)\delta (t-kT)</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Comb/Rep | ||
| + | | <math>rep_{T}(x(t)) <=> \frac{1}{T} comb_{\frac{1}{T}}(X(f))</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Comb/Rep | ||
| + | | <math>comb_{T}(x(t)) <=> \frac{1}{T} rep_{\frac{1}{T}}(X(f))</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Circular Convolution | ||
| + | | <math>f[n]*_N g[n] = \sum_{k=0}^{N-1} f[k]g[(n-k)mod \ N]</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Short Time Fourier Transform | ||
| + | | <math>X[k,m] = \sum_{n=-\infty}^{\infty} x[n]w[n-m]e^{-j\frac{2\pi}{N}kn}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | CSFT | ||
| + | | <math>f(x,y) <=> F(u,v) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y)e^{-j2\pi(ux+vy)} \ dx dy</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | ICSFT | ||
| + | | <math>F(u,v) <=> f(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F(u,v)e^{j2\pi (ux+vy)} \ du dv</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Acoustic Pressure (outside last tube) | ||
| + | | <math>b(t) = \frac{\rho c}{A_k}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Total acoustic Pressure (inside first tube) | ||
| + | | <math>u(t) = (r+l)\frac{\rho c}{A_k}</math> | ||
| + | |- | ||
| − | + | ||
| + | |||
| + | ---- | ||
| + | [[2010 Fall ECE 438 Boutin|Back to ECE438 Fall 2010 Prof. Boutin]] | ||
Revision as of 08:22, 1 December 2010
| 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} $ | |
