Odd Symmetry

A function $f(t)$ is said to be odd if its plot is anti-symmetrical about the vertical axis:
$$f(-t)=-f(t) \tag{1}$$
Examples of odd functions are $t, t^{3}$, and $\sin t$. Figure $1$ shows more examples of periodic odd functions.
Fig. 1: Typical examples of odd periodic functions.
All these examples satisfy Eq. (1). An odd function $f_{o}(t)$ has this major characteristic:
$$\int_{-T / 2}^{T / 2} f_{o}(t) d t=0 \tag{2}$$
because integration from $-T / 2$ to 0 is the negative of that from 0 to $T / 2$. With this property, the Fourier coefficients for an odd function become
$$\begin{array}{c}a_{0}=0, \quad a_{n}=0 \\b_{n}=\frac{4}{T} \int_{0}^{T / 2} f(t) \sin n \omega_{0} t d t\end{array} \tag{3}$$
Previous Page Equations
$$\begin{array}{l}a_{0}=\frac{2}{T} \int_{0}^{T / 2} f(t) d t \\a_{n}=\frac{4}{T} \int_{0}^{T / 2} f(t) \cos n \omega_{0} t d t \\b_{n}=0\end{array} \tag{A}$$ \begin{aligned}a_{0} &=\frac{1}{T}\left[\int_{T / 2}^{0} f(x)(-d x)+\int_{0}^{T / 2} f(t) d t\right] \\&=\frac{1}{T}\left[\int_{0}^{T / 2} f(x) d x+\int_{0}^{T / 2} f(t) d t\right]\end{aligned} \tag{B} \begin{aligned}a_{n} &=\frac{2}{T}\left[\int_{T / 2}^{0} f(-x) \cos \left(-n \omega_{0} x\right)(-d x)+\int_{0}^{T / 2} f(t) \cos n \omega_{0} t d t\right] \\&=\frac{2}{T}\left[\int_{T / 2}^{0} f(x) \cos \left(n \omega_{0} x\right)(-d x)+\int_{0}^{T / 2} f(t) \cos n \omega_{0} t d t\right] \\&=\frac{2}{T}\left[\int_{0}^{T / 2} f(x) \cos \left(n \omega_{0} x\right) d x+\int_{0}^{T / 2} f(t) \cos n \omega_{0} t d t\right]\end{aligned} \tag{C} $$b_{n}=\frac{2}{T}\left[\int_{-T / 2}^{0} f(t) \sin n \omega_{0} t d t+\int_{0}^{T / 2} f(t) \sin n \omega_{0} t d t\right] \tag{D}$$ $$f(t)=f(-t) \tag{E}$$
which give us a Fourier sine series. Again, this makes sense because the sine function is itself an odd function. Also, note that there is no dc term for the Fourier series expansion of an odd function. The quantitative proof of Eq. (3) follows the same procedure taken to prove Eq. (A) except that $f(t)$ is now odd, so that $f(t)=$ $-f(t)$. With this fundamental but simple difference, it is easy to see that $a_{0}=0$ in Eq. (B), $a_{n}=0$ in Eq. (C), and $b_{n}$ in Eq. (D) becomes
\begin{aligned}b_{n} &=\frac{2}{T}\left[\int_{T / 2}^{0} f(-x) \sin \left(-n \omega_{0} x\right)(-d x)+\int_{0}^{T / 2} f(t) \sin n \omega_{0} t d t\right] \\ &=\frac{2}{T}\left[-\int_{T / 2}^{0} f(x) \sin n \omega_{0} x d x+\int_{0}^{T / 2} f(t) \sin n \omega_{0} t d t\right] \\ &=\frac{2}{T}\left[\int_{0}^{T / 2} f(x) \sin \left(n \omega_{0} x\right) d x+\int_{0}^{T / 2} f(t) \sin n \omega_{0} t d t\right] \\ b_{n}&=\frac{4}{T} \int_{0}^{T / 2} f(t) \sin n \omega_{0} t d t\end{aligned}
as expected. It is interesting to note that any periodic function $f(t)$ with neither even nor odd symmetry may be decomposed into even and odd parts. Using the properties of even and odd functions from Eqs. (E) and (1), we can write
$$f(t)=\underbrace{\frac{1}{2}[f(t)+f(-t)]}_{\text {even }}+\underbrace{\frac{1}{2}[f(t)-f(-t)]}_{\text {odd }}=f_{e}(t)+f_{o}(t)$$
Notice that $f_{e}(t)=\frac{1}{2}[f(t)+f(-t)]$ satisfies the property of an even function in Eq. (E), while $f_{o}(t)=\frac{1}{2}[f(t)-f(-t)]$ satisfies the property of an odd function in Eq. $(1)$. The fact that $f_{e}(t)$ contains only the dc term and the cosine terms, while $f_{o}(t)$ has only the sine terms, can be exploited in grouping the Fourier series expansion of $f(t)$ as
$$f(t)=\underbrace{a_{0}+\sum_{n=1}^{\infty} a_{n} \cos n \omega_{0} t}_{\text {even }}+\underbrace{\sum_{n=1}^{\infty} b_{n} \sin n \omega_{0} t}_{\text {odd }}=f_{e}(t)+f_{o}(t) \tag{4}$$
It follows readily from Eq. (4) that when $f(t)$ is even, $b_{n}=0$, and when $f(t)$ is odd, $a_{0}=0=a_{n}$. Also, note the following properties of odd and even functions:
• The product of two even functions is also an even function.
• The product of two odd functions is an even function.
• The product of an even function and an odd function is an odd function.
• The sum (or difference) of two even functions is also an even function.
• The sum (or difference) of two odd functions is an odd function.
• The sum (or difference) of an even function and an odd function is neither even nor odd.
Each of these properties can be proved using Eqs. (E) and (1).