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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> | + | :<math>CTFS </math> <math> 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</math> |
| + | :<math>CTFT</math><math>\ x(t) = \int_{-\infty}^{\infty} \chi(f)\ e^{j 2 \pi f t}\,df \;\;\;\;\;\;\;\;\;\;\;\;\;\ \chi(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) \;\;\;\;\;\;\;\;\;comb_T[x(t)] = x(t) . \sum_{k=-\infty}^{\infty}\delta(t-kT) </math> | ||
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| + | :<math> 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)] </math> | ||
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| + | <math>\displaystyle\delta(\alpha f)= \frac{1}{\alpha}\delta(f)\;\;\;\;\;\;for\;\;\alpha>0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;sinc(\theta)= \frac{sin(\pi\theta)}{\pi\theta} </math> | ||
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| + | <math> \displaystyle e^{j\pi}=-1 \;\;\;\;\;\;\; \cos(t\theta) = \frac{(e^{j\theta}+e^{-j\theta})}{2}\;\;\;\;\;\;\;\;\;\;\;\; sin(t\theta) = \frac{(e^{j\theta}-e^{-j\theta})}{2j}</math> | ||
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| + | <math> \mathcal{F}(\frac{rect( (t-\frac{T}{2})}{T})) \Rightarrow Tsinc(Tf)(e^{-j2 \pi f \frac{T}{2} }) </math> | ||
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[[ 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]] | ||
Latest revision as of 05:50, 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
- $ CTFS $ $ 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 $$ \ x(t) = \int_{-\infty}^{\infty} \chi(f)\ e^{j 2 \pi f t}\,df \;\;\;\;\;\;\;\;\;\;\;\;\;\ \chi(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)] $
$ \displaystyle\delta(\alpha f)= \frac{1}{\alpha}\delta(f)\;\;\;\;\;\;for\;\;\alpha>0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;sinc(\theta)= \frac{sin(\pi\theta)}{\pi\theta} $
$ \displaystyle e^{j\pi}=-1 \;\;\;\;\;\;\; \cos(t\theta) = \frac{(e^{j\theta}+e^{-j\theta})}{2}\;\;\;\;\;\;\;\;\;\;\;\; sin(t\theta) = \frac{(e^{j\theta}-e^{-j\theta})}{2j} $
$ \mathcal{F}(\frac{rect( (t-\frac{T}{2})}{T})) \Rightarrow Tsinc(Tf)(e^{-j2 \pi f \frac{T}{2} }) $
Back to 2010 Fall ECE 438 Boutin/ECE438Mid1FormulaSheet Work
