written 6.3 years ago by | • modified 6.3 years ago |
Subject : Signals & Systems
Topic : Fourier Series.
Difficulty: Medium
written 6.3 years ago by | • modified 6.3 years ago |
Subject : Signals & Systems
Topic : Fourier Series.
Difficulty: Medium
written 6.3 years ago by | • modified 6.3 years ago |
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/2−Adt]
a(0)=AT[∫T/20dt+∫TT/2dt]
=AT[T2−T+T2]
∴a(0) = 0 ----------- 1
Step 2: To calculate a(k)
a(k)=2T∫\ltT>x(t)cos(kωOt)dt
a(k)=2T∫T0x(t)cos(kωOt)dt
=2T∫T0x(t)cos(kωOt)dt−∫TT/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