Fourier series
Periodic function:
• is a function that repeats its values at regular intervals
𝒇 𝒕+𝑻 =𝒇 𝒕
after every interval of T the function f(t) repeats all its values
Ex: 𝑓 𝑡 + 2𝜋 = 𝑓 𝑡
after every interval of 2𝜋t he function f(t) repeats all its values
Types of periodic function:
1. Even function:
even functions have symmetry about the y-axis.
2. Odd function:
odd functions have symmetry around the origin.
3. Neither odd nor even:
No symmetry.
Ex:
𝑡2
0, 𝜋
𝑓 𝑡+𝜋 =𝑓 𝑡
Sol:
-3𝜋
-2𝜋
-𝜋
𝜋
2𝜋
3𝜋
Fourier series:
• A series of infinite sine or cosine waves added together to make another
function.
• Any periodic function can make using sin and cos wave.
𝑓 𝑡 + 𝑇 = 𝑓(𝑡)
∞
∴ 𝑓 𝑡 = 𝑎0 + 𝑎 𝑛
𝑛=1
𝑛𝑡𝜋
cos
+ 𝑏𝑛
𝐿
• We need to calculate 𝑎0 , 𝑎𝑛 , 𝑏𝑛 .
∞
𝑛=1
𝑛𝑡𝜋
sin
𝐿
𝑇
𝐿=
2
1
𝑎0 =
2𝐿
1
𝑎𝑛 =
𝐿
1
𝑏𝑛 =
𝐿
𝐿
𝑓 𝑡 𝑑𝑡
−𝐿
𝐿
𝑛𝑡𝜋
𝑓 𝑡 cos
𝑑𝑡
𝐿
−𝐿
𝐿
𝑛𝑡𝜋
𝑓 𝑡 𝑠𝑖𝑛
𝑑𝑡
𝐿
−𝐿
• In odd function (f(t) is odd):
integratio of odd function in −L, L interval is zero so:
𝑎0 , 𝑎𝑛 𝑖𝑠 𝑧𝑒𝑟𝑜
and we need to calculate 𝑏𝑛 𝑜𝑛𝑙𝑦
• In even function (f(t) is even):
integratio of odd function in −L, L interval is zero so:
𝑏𝑛 𝑖𝑠 𝑧𝑒𝑟𝑜
and we need to calculate 𝑎0 , 𝑎𝑛 𝑜𝑛𝑙𝑦
• Note: integration in interval (-L, L) convert to integration in (0,L) multiple by 2
−3
EX1: 𝑓 𝑡 =
3
Sol:
−𝜋 <𝑡 <0
0<𝑡<𝜋
𝑇 = 2𝜋
function have symmetry around the origin
∴ function is odd
𝑎0 , 𝑎𝑛 = 0
𝑏𝑛 =
1 𝐿
𝑓
𝐿 −𝐿
𝑡 𝑠𝑖𝑛
𝑛𝑡𝜋
𝐿
3
𝑑𝑡
3
𝑜𝑑𝑑 ∗ 𝑜𝑑𝑑 = 𝑒𝑣𝑒𝑛
𝐿=
2𝜋
2
=𝜋
∴ 𝑏𝑛 =
2 𝜋
𝑛𝑡𝜋
3𝑠𝑖𝑛
𝜋 0
𝜋
𝑑𝑡 =
6 −cos(𝑛𝑡) 𝜋
0
𝜋
𝑛
−6
𝑓 𝑡 =
cos 𝑛𝜋 − 1
𝜋𝑛
=
−6
𝜋𝑛
cos 𝑛𝜋 − 1
∞
sin 𝑛𝑡
𝑛=1