Revision as of 05:00, 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(t) = \int_{-\infty}^{\infty} F(f)\ e^{j 2 \pi f t}\,df

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

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

$comb_T [x(t)] \iff \frac{1}{T}rep_\frac{1}{T} [ \mathrm{X}(f)]$

$rep_T [x(t)] \iff \frac{1}{T}comb_\frac{1}{T} [ \mathrm{X}(f)]$

@@ Line 7: / Line 7: @@
-:<math>CTFT</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(t) = \int_{-\infty}^{\infty} F(f)\ e^{j 2 \pi f t}\,df </math>.....................
+<math>CTFT</math><math> \ F(f) = \int_{-\infty}^{\infty} x(t)\ e^{- j 2 \pi f t}\,dt</math>
+<math> comb_T [x(t)] \iff \frac{1}{T}rep_\frac{1}{T} [ \mathrm{X}(f)] </math>
+<math> rep_T [x(t)] \iff \frac{1}{T}comb_\frac{1}{T} [ \mathrm{X}(f)] </math>
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