Revision as of 04:56, 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

\ F(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt

.....................

\ f(t) = \int_{-\infty}^{\infty} F(f)\ e^{j 2 \pi f t}\,df

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 *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>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>
-:<math>\ F(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt                  </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>\ f(t) = \int_{-\infty}^{\infty} F(f)\ e^{j 2 \pi f t}\,df </math>
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