Determine the Fourier Series co-efficient for the following continuous time periodic signals.Show the details of your calculations and simplify your answers.

$a_{k} = 1/T \int_{T} x(t) e^{-jkw_{o}t} dt$

     $= 1/T \int_{-T_{1}} ^ {T_{1}} 1*e^{-jk2\frac{\pi}{T} t} dt$  (x(t)=1)

$= 1/T \int_{-T_{1}} ^ {T_{1}} e^{-jk2\frac{\pi}{T} t} dt$

     $= 1/T [\frac{e^{-jk2\frac{\pi}{T} t}}{-jk2\frac{\pi}{T}}]_{-T_{1}} ^ {T_{1}}$

     $= \frac{-1}{jk2\pi} (e^{-jk2\frac{\pi}{T} T_{1}} - e^{jk2\frac{\pi}{T} T_{1}})$

     $= \frac{1}{k\pi} (Sin(\frac{2k\pi}{T} T_{1})$


Now For K = 0 Condition

$a_{k} = 1/T \int_{T} x(t) e^{-jkw_{o}t} dt$

     PUT THE VALUE  K=0 IN ABOVE EQUATION

     $= 1/T \int_{-T_{1}} ^ {T_{1}} 1 dt$

$= \frac{T_{1} + T_{1}}{T}$

     $= \frac{2T_{1}}{T}$


## Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett