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*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>DTFS </math> <math> x(t)=\sum_{n=-\infty}^\infty a_n e^{j \frac{2\pi}{T}nt}</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> |
| − | :<math>CTFT</math><math>\ f(t) = \int_{-\infty}^{\infty} F(f)\ e^{j 2 \pi f t}\,df </math> | + | :<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> |
| − | :<math> rep_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> |
<math>\displaystyle\delta(\alpha | <math>\displaystyle\delta(\alpha | ||
| − | f)=\frac{1}{\alpha}\delta(f) | + | f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;for\;\;\alpha>0</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 05:10, 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)] <\;\;\;\;\;\;\;\;\;\/math><math> comb_T [x(t)] \iff \frac{1}{T}rep_\frac{1}{T} [ \mathrm{X}(f)] $
$ \displaystyle\delta(\alpha f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;for\;\;\alpha>0 $
Back to 2010 Fall ECE 438 Boutin/ECE438Mid1FormulaSheet Work
