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Work in progress for a formula sheet? | Work in progress for a formula sheet? | ||
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*Fourier series of a continuous-time signal x(t) periodic with period T | *Fourier series of a continuous-time signal x(t) periodic with period T | ||
*Fourier series coefficients of a continuous-time signal x(t) periodic with period T | *Fourier series coefficients of a continuous-time signal x(t) periodic with period T | ||
| − | :<math>x(t)=\sum_{n=-\infty}^\infty a_n e^{j \frac{2\pi}{T}nt}</math> <math>a_n=\frac{1}{T} \int_{0}^T x(t) e^{-j \frac{2\pi}{T}nt}dt</math> | + | :<math>DTFS </math> <math> x(t)=\sum_{n=-\infty}^\infty a_n e^{j \frac{2\pi}{T}nt}</math> ...................... <math>a_n=\frac{1}{T} \int_{0}^T x(t) e^{-j \frac{2\pi}{T}nt}dt</math> |
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| + | :<math>CTFT</math><math>\ f(t) = \int_{-\infty}^{\infty} F(f)\ e^{j 2 \pi f t}\,df </math>.....................<math> \ F(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt</math> | ||
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| + | :<math> rep_T [x(t)] = x(t)* \sum_{k=-\infty}^{\infty}\delta(t-kT) </math>.........<math> comb_T[x(t)] = x(t) . \sum_{k=-\infty}^{\infty}\delta(t-kT) </math> | ||
| − | :<math> | + | :<math> rep_T [x(t)] \iff \frac{1}{T}comb_\frac{1}{T} [ \mathrm{X}(f)] </math>......................<math> comb_T [x(t)] \iff \frac{1}{T}rep_\frac{1}{T} [ \mathrm{X}(f)] </math> |
Revision as of 05:08, 30 September 2010
Work in progress for a formula sheet?
- Fourier series of a continuous-time signal x(t) periodic with period T
- Fourier series coefficients of a continuous-time signal x(t) periodic with period T
- $ DTFS $ $ x(t)=\sum_{n=-\infty}^\infty a_n e^{j \frac{2\pi}{T}nt} $ ...................... $ a_n=\frac{1}{T} \int_{0}^T x(t) e^{-j \frac{2\pi}{T}nt}dt $
- $ CTFT $$ \ f(t) = \int_{-\infty}^{\infty} F(f)\ e^{j 2 \pi f t}\,df $.....................$ \ F(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt $
- $ rep_T [x(t)] = x(t)* \sum_{k=-\infty}^{\infty}\delta(t-kT) $.........$ comb_T[x(t)] = x(t) . \sum_{k=-\infty}^{\infty}\delta(t-kT) $
- $ rep_T [x(t)] \iff \frac{1}{T}comb_\frac{1}{T} [ \mathrm{X}(f)] $......................$ comb_T [x(t)] \iff \frac{1}{T}rep_\frac{1}{T} [ \mathrm{X}(f)] $
