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Signals and Systems – What is Odd Symmetry?
Importance of Wave symmetry
If a periodic signal $x(t)$ has some type of symmetry, then some of the trigonometric Fourier series coefficients may become zero and hence the calculation of the coefficients becomes simple.
Odd or Rotation Symmetry
When a periodic function $x(t)$ is antisymmetric about the vertical axis, then the function is said to have the odd symmetry or rotation symmetry.
Mathematically, a function $x(t)$ is said to have odd symmetry, if
$$\mathrm{x(t)=-x(-t)… (1)}$$
Some functions having odd symmetry are shown in the figure. It is clear that the odd symmetric functions are always antisymmetrical about the vertical axis.
Explanation
As we know that any periodic signal $x(t)$ can be split into even and odd components, i.e.,
$$\mathrm{x(t)=x_{e}(t)+x_{0}(t)… (2)}$$
If the function $x(t)$ is an odd function, then,
$$\mathrm{x_{e}(t)=0}$$
$$\mathrm{\therefore\:x(t)=x_{0}(t)… (3)}$$
The trigonometric Fourier coefficients of the function can be evaluated as follows −
The coefficient $a_{0}$ is given by,
$$\mathrm{a_{0}=\frac{1}{T} \int_{−T/2}^{T/2}x(t)\:dt=\frac{1}{T}\int_{−T/2}^{T/2}x_{0}(t)\:dt}$$
For an odd function, the area under the curve over one period is zero, i.e.,
$$\mathrm{\int_{−T/2}^{T/2}x_{0}(t)\:dt=0}$$
$$\mathrm{\therefore\:a_{0}=0… (4)}$$
The coefficient $a_{n}$ is given by,
$$\mathrm{a_{n}=\frac{2}{T} \int_{−T/2}^{T/2}x(t)cos\:n\omega_{0} t\:dt}$$
$$\mathrm{\Rightarrow\:a_{n}=\frac{2}{T} \int_{−T/2}^{T/2}x_{0}(t)cos\:n\omega_{0} t\:dt}$$
Since the function $(x_{0}(t)\:cos\:n\omega_{0𝑡}t)$ is an odd function, hence its integral over one complete is zero.
$$\mathrm{\therefore\:a_{n}=0… (5)}$$
And the coefficient $b_{n}$ is given by,
$$\mathrm{b_{n}=\frac{2}{T}\int_{−T/2}^{T/2}x(t)sin\:n\omega_{0}t\:dt}$$
$$\mathrm{\Rightarrow\:b_{n}=\frac{2}{T}\int_{−T/2}^{T/2}x_{0}(t)sin\:n\omega_{0}t\:dt=\frac{2}{T}\left ( \int_{0}^{T/2}x_{0}(t)sin\:n\omega_{0}t\:dt \right )}$$
$$\mathrm{\therefore\:b_{n}=\frac{4}{T}\int_{0}^{T/2}x(t)sin\:n\omega_{0}t\:dt… (6)}$$
Therefore, the Fourier series expansion of an odd periodic function contains only sine terms. When odd or rotation symmetry exists in a function, then the trigonometric Fourier series coefficients for the functions are given by the equations (4), (5) and (6).
Properties of Odd Functions
The sum of two or more odd functions is always an odd function.
The product of two odd functions is an even function.
When a constant is added to an odd function, then the odd nature of the function is removed.
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