Fourier Series
ECE 3100
John Stahl
Western Michigan University
Any periodic function of frequency wo can be represented as an
infinite sum of sine or cosine terms which are integer multiples of
wo.
𝑓 𝑡 = 𝑓 𝑡 + 𝑛𝑇
∞
∞
𝑐𝑛 𝑒 −𝑗𝑛𝜔𝑜 𝑡
𝑓 𝑡 =
𝑛=−∞
or
𝑓 𝑡 = 𝑎0 +
𝑎𝑛 cos 𝑛𝜔𝑜 𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑜 𝑡
𝑛=1
In the equation ao is the DC or Average Value of the function and the
coefficients an and bn represent the contributions of the sine and
cosine terms.
Terms and Conditions
Dirchlet Conditions
1. 𝑓 𝑡 is single valued everywhere
2. 𝑓 𝑡 has a finite number of finite number of
discontinuities in any one period.
3. 𝑓 𝑡 has a finite number of finite number of
maxima or minima in any one period.
4. The intergral
𝑡𝑜 +𝑇
𝑡𝑜
𝑓 𝑡 𝑑𝑡 < ∞ for any t0.
Fourier Coefficients
The DC or average of a function is the coefficient 𝑎𝑜 .
1
𝑎𝑜 =
𝑇
𝑇
𝑓 𝑡 𝑑𝑡
0
The coefficients 𝑎𝑛 and 𝑏𝑛 are found with the following integrations.
2
𝑎𝑛 =
𝑇
𝑇
0
𝑓 𝑡 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡
2
𝑏𝑛 =
𝑇
𝑇
0
𝑓 𝑡 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
Square Wave
For the square wave solve for the Fourier Coefficients and plot the
resulting sum. The period (T) of the sequence is 2 sec
𝑓 𝑡 =
1,
0,
∞
0<𝑡<1
1≤𝑡<2
𝑓 𝑡 = 𝑎0 +
𝑎𝑛 cos 𝑛𝜔𝑜 𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑜 𝑡
𝑛=1
𝑓 𝑡
-2
1
𝑎𝑜 =
𝑇
𝑇
𝑓 𝑡 𝑑𝑡
0
-1
2
𝑎𝑛 =
𝑇
0
1
𝑇
0
𝑓 𝑡 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡
2
2
𝑏𝑛 =
𝑇
3
𝑇
0
𝑓 𝑡 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
𝑓 𝑡 =
1
𝑎𝑜 =
𝑇
1,
0,
0<𝑡<1
1≤𝑡<2
𝑇
𝑓 𝑡 𝑑𝑡
0
1
=
2
1
0
1
1𝑑𝑡 +
2
=
2
0𝑑𝑡
1
1 1
𝑡|
2 0
𝑎𝑜 =
1
2
The DC or average value of the function.
2
𝑎𝑛 =
𝑇
𝑇
0
2
=
2
𝑓 𝑡 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡
1
0
2
1 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡 +
2
2
1
0 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡
1
=
sin 𝑛𝜔𝑜 𝑡 |10
𝑛𝜔𝑜
=
1
sin 𝑛𝜔𝑜 1 − sin 𝑛𝜔𝑜 0
𝑛𝜔𝑜
1
𝑎𝑛 =
sin 𝑛𝜋
𝑛𝜋
𝑎𝑛 = 0
2𝜋
𝜔𝑜 =
=𝜋
𝑇
n is an integer (1,2,3…)
For the sin(np):
sin(1p)=0
sin(2p)=0
sin(3p)=0
…
sin(np)=0
2
𝑏𝑛 =
𝑇
2
=
2
𝑇
𝑓 𝑡 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
0
1
0
2
1 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡 +
2
2
1
0 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
1
=
0
sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
=−
1
cos 𝑛𝜔𝑜 𝑡 |10
𝑛𝜔𝑜
𝑏𝑛 = −
1
cos 𝑛𝜋 − cos 0
𝑛𝜔𝑜
1
𝑏𝑛 =
1 − cos 𝑛𝜋
𝑛𝜋
2
𝑏𝑛 = 𝑛𝜋 ,
0,
n is an integer (1,2,3…)
For the 1 − cos 𝑛𝜋 :
1 − cos 1𝜋 = 2
1 − cos 2𝜋 = 0
1 − cos 3𝜋 = 2
1 − cos 4𝜋 = 0
…
1 − cos 𝑛𝑜𝑑𝑑 𝜋 = 2
1 − cos 𝑛𝑒𝑣𝑒𝑛 𝜋 = 0
𝑛 = 𝑜𝑑𝑑
𝑛 = 𝑒𝑣𝑒𝑛
1
𝑓 𝑡 = +
2
∞
𝑛𝑜𝑑𝑑 =1
2
sin 𝑛𝜋𝑡
𝑛𝜋
1
2
2
2
2
2
𝑓 𝑡 = +
sin 1𝜋𝑡 +
sin 3𝜋𝑡 +
sin 5𝜋𝑡 +
sin 7𝜋𝑡 +
sin 9𝜋𝑡 …
2 1𝜋
3𝜋
5𝜋
7𝜋
9𝜋