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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> 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> rep_T [x(t)] \iff \frac{1}{T}comb_\frac{1}{T} [ \mathrm{X}(f)] </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> |
| − | + | ||
[[ 2010 Fall ECE 438 Boutin/ECE438Mid1FormulaSheet Work|Back to 2010 Fall ECE 438 Boutin/ECE438Mid1FormulaSheet Work]] | [[ 2010 Fall ECE 438 Boutin/ECE438Mid1FormulaSheet Work|Back to 2010 Fall ECE 438 Boutin/ECE438Mid1FormulaSheet Work]] | ||
Revision as of 06:08, 30 September 2010
2010_Fall_ECE_438_Boutin/ECE438Mid1FormulaSheet_Work_wrk
- 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)] $
Back to 2010 Fall ECE 438 Boutin/ECE438Mid1FormulaSheet Work
