Revision as of 06:11, 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 $

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