0
6.5kviews
Determine the trigonometric form of Fourier series of the waveform shown below.

enter image description here

Subject : Signals & Systems

Topic : Fourier Series.

Difficulty: Medium

1 Answer
0
136views

From the figure,

Period of the signal x(t) = T x(t)= {A,0<x<t 2=""\[2ex]=""a,=""amp;=""t="" 2<x<t=""<="" p="">

The trigonometric Fourier series equation is given by

x(t)=a(0)+k=1a(k)cos(kωOt) +k=1b(k)sin(kωOt) 

wherea(0)=1T\ltT>x(t)dt

a(k)=2T\ltT>x(t)cos(kωOt)dt

b(k)=2T\ltT>x(t)sin(kωOt)dt

Step 1: To calculate a(0) a(0)=1T\ltT>x(t)dt

 a(0)=1T[T/20Adt+TT/2Adt]

 a(0)=AT[T/20dt+TT/2dt]

=AT[T2T+T2]

∴a(0) = 0 ----------- 1

Step 2: To calculate a(k)

a(k)=2T\ltT>x(t)cos(kωOt)dt

a(k)=2TT0x(t)cos(kωOt)dt

=2TT0x(t)cos(kωOt)dtTT/2Acos(kωOt)dt

Here ω_O= 2π/T

∴a(k) = 2T\int_0^{T/2} x(t) cos⁡(k 2π/T t) \,dt - \int_{T/2}^T A cos⁡(k 2π/T t) \, dt

= 2T\int_0^{T/2} A cos⁡(k 2π/T t) \,dt - \int_{T/2}^T A cos⁡(k 2π/T t) \, dt

= 2T\int_0^{T/2} cos⁡(k 2π/T t) \,dt - \int_{T/2}^T cos⁡(k 2π/T t) \, dt

enter image description here

enter image description here

enter image description here

Please log in to add an answer.