Even 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.
This time we are going to focus on the advantages to the symmetry
in functions, which can be used to make our calculation of
coefficients easier.
∞
∞
𝑐𝑛 𝑒 −𝑗𝑛𝜔𝑜 𝑡
𝑓 𝑡 =
𝑛=−∞
or
𝑓 𝑡 = 𝑎0 +
𝑎𝑛 cos 𝑛𝜔𝑜 𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑜 𝑡
𝑛=1
There are two types of symmetry we will examine:
• Even
• Odd
Shifted Square Wave
For the shifted square wave solve for the Fourier Coefficients and plot
the resulting sum. The period (T) of the sequence is 2 sec
𝑓 𝑡 =
𝐴,
0,
∞
−0.5 < 𝑡 < 0.5
0.5 ≤ 𝑡 < 1.5
𝑓 𝑡 = 𝑎0 +
𝑎𝑛 cos 𝑛𝜔𝑜 𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑜 𝑡
𝑛=1
𝑓 𝑡
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
3
2
𝑏𝑛 =
𝑇
𝑇
0
𝑓 𝑡 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
The shifted square wave. Notice the left and right side of the function is a mirror of each
other. This is Even symmetry. Mathematically even symmetry is achieved with cosine
functions.
𝑓 𝑡
−
1
𝑎𝑜 =
𝑇
𝑇
2
2
𝑓 𝑡 𝑑𝑡 =
−𝑇
𝑇
2
2
=
2
𝑇
2
-2
-1
𝑻
𝟐
𝑻
𝟐
0
1
2
𝑓 𝑡 𝑑𝑡
0
0.5
1𝑑𝑡
0
= 𝑡|0.5
0
1
𝑎𝑜 =
2
The DC or average value of the function.
3
n is an integer (1,2,3…)
4
𝑎𝑛 =
𝑇
𝑇
2
0
4
=
2
For the sin
𝑓 𝑡 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡
0.5
0
4
1 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡 +
2
1𝜋
=1
2
2𝜋
sin 2 = 0
3𝜋
sin 2 = −1
4𝜋
sin 2 = 0
1
0 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡
0.5
…
sin
sin
2
𝑛𝜔𝑜
sin
− sin 𝑛𝜔𝑜 0
𝑛𝜔𝑜
2
2
:
sin
2
=
sin 𝑛𝜔𝑜 𝑡 |0.5
0
𝑛𝜔𝑜
=
𝑛𝜋
2
𝑛𝑜𝑑𝑑 𝜋
𝑎𝑛 =
sin
𝑛𝑜𝑑𝑑 𝜋
2
2𝜋
𝜔𝑜 =
=𝜋
𝑇
𝑛𝑒𝑣𝑒𝑛 𝜋
=0
2
𝑛𝑜𝑑𝑑 𝜋
= ±1
2
𝑏𝑛 =
2
𝑇
4
=
2
𝑇
2
𝑇
−2
𝑓 𝑡 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
−0.5
−1
0
0 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡 +
0.5
1 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡 +
−0.5
0
=2
0.5
1 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡 +
−0.5
=−
0
1 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡 +
0
1 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
2
cos 𝑛𝜋𝑡 |0−0.5 + cos 𝑛𝜋𝑡 |0.5
0
𝑛𝜔𝑜
𝑏𝑛 = −
2
𝑛𝜋
𝑛𝜋
1 − cos −
+ cos
−1
𝑛𝜔𝑜
2
2
𝑏𝑛 = 0
1
0 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
0.5
1
𝑓 𝑡 = +
2
𝑓 𝑡 =
∞
𝑛𝑜𝑑𝑑 =1
2
𝑛𝜋
sin
c𝑜𝑠 𝑛𝜋𝑡
𝑛𝜋
2
1
2
2
2
2
2
+
cos 1𝜋𝑡 −
cos 3𝜋𝑡 +
cos 5𝜋𝑡 −
cos 7𝜋𝑡 +
cos 9𝜋𝑡 …
2 1𝜋
3𝜋
5𝜋
7𝜋
9𝜋