Odd Symmetry
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
• Halfwave
Shifted Triangle 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 = 4𝑡, −0.5 < 𝑡 < 0.5
𝑓 𝑡 = 𝑎0 +
𝑎𝑛 cos 𝑛𝜔𝑜 𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑜 𝑡
𝑛=1
𝑓 𝑡
-3
-2
-1
0
1
2
3
T
𝑎𝑜 = 0
𝑎𝑛 = 0
4
𝑏𝑛 =
𝑇
𝑇
0
𝑓 𝑡 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
The shifted triangle wave. Notice the left and right side of the function is a flipped mirror of
each other. This is Odd symmetry. Mathematically Odd symmetry is achieved with sine
functions.
𝑓 𝑡
-2
1
𝑎𝑜 =
𝑇
𝑇
2
−𝑇
2
0
1
T
𝑓 𝑡 𝑑𝑡
2
=
1
-1
0.5
4𝑡𝑑𝑡
−0.5
= 4𝑡 2 |0.5
−0.5
𝑎𝑜 = 0
The DC or average value of the function.
2
2
𝑎𝑛 =
𝑇
𝑇
2
𝑇
−2
2
=
1
𝑓 𝑡 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡
𝜔𝑜 =
2𝜋
= 2𝜋
𝑇
0.5
4𝑡 cos 𝑛𝜔𝑜 𝑡 𝑑𝑡
−0.5
8
=
cos 2𝜋𝑛𝑡 + 2𝜋𝑛𝑡 ∙ sin 2𝜋𝑛𝑡 |0.5
−0.5
2
(𝑛𝜔𝑜 )
=
8
cos 𝜋𝑛 + 𝜋𝑛 ∙ sin 𝜋𝑛 − cos −𝜋𝑛 + 𝜋𝑛 ∙ sin −𝜋𝑛
(2𝜋𝑛)2
8
=
𝜋𝑛 ∙ sin 𝜋𝑛 + 𝜋𝑛 ∙ sin −𝜋𝑛
(2𝜋𝑛)2
𝑎𝑛 = 0
The function has Odd symmetry,
which makes the cosine term zero.
2
𝑏𝑛 =
𝑇
𝑇
2
𝑇
−2
=
2
2
𝑓 𝑡 sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
0.5
4𝑡 ∙ sin 𝑛𝜔𝑜 𝑡 𝑑𝑡
−0.5
=
n
sin 𝒏𝜋
𝑛 ∙ cos 𝒏𝜋
1
sin 𝟏𝜋 = 0
𝟏 ∙ cos 𝟏𝜋 = −1
2
sin 𝟐𝜋 = 0
𝟐 ∙ cos 𝟐𝜋 = 2
3
sin 𝟑𝜋 = 0
𝟑 ∙ cos 𝟑𝜋 = −3
⋮
⋮
⋮
n
0
𝑛(−1)𝑛
8
0.5
sin
2𝜋𝑛𝑡
−
2𝜋𝑛𝑡
∙
cos
2𝜋𝑛𝑡
|
−0.5
(2𝜋𝑛)2
=
8
sin 𝜋𝑛 − 𝜋𝑛 ∙ cos 𝜋𝑛 − sin −𝜋𝑛 − 𝜋𝑛 ∙ cos −𝜋𝑛
(2𝜋𝑛)2
8
=
2 ∙ sin 𝜋𝑛 − 2𝜋𝑛 ∙ cos 𝜋𝑛
(2𝜋𝑛)2
𝑏𝑛 =
8
𝑛
2𝜋𝑛(−1)
(2𝜋𝑛)2
𝑏𝑛 = −
4
(−1)𝑛
𝜋𝑛
∞
𝑓 𝑡 =
𝑛=1
𝑓 𝑡 =
4
−
(−1)𝑛 𝑠𝑖𝑛 2𝜋𝑛𝑡
𝜋𝑛
4
4
4
4
4
sin 2𝜋𝑡 −
sin 4𝜋𝑡 +
sin 6𝜋𝑡 −
sin 8𝜋𝑡 +
sin 10𝜋𝑡 …
1𝜋
2𝜋
3𝜋
4𝜋
5𝜋