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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.

When a periodic function is symmetrical about the vertical axis, it is said to have ** even symmetry** or

$$\mathrm{𝑥(𝑡) = 𝑥(−𝑡)\:\:\:\:\:\: ...(1)}$$

Some examples of functions having even symmetry are shown in the figure. The even functions are always symmetrical about the vertical axis.

As we know that any periodic signal 𝑥(𝑡) can be split into even and odd components, i.e.,

$$\mathrm{𝑥(𝑡) = 𝑥_{𝑒}(𝑡) + 𝑥_{𝑜}(𝑡) … (2)}$$

If the function 𝑥(𝑡) is an even function, then,

$$\mathrm{𝑥_{𝑜}(𝑡) = 0}$$

$$\mathrm{\therefore\:𝑥(𝑡) = 𝑥_{𝑒}(𝑡)\:\:\:\:\:\: … (3)}$$

Therefore, the trigonometric Fourier series coefficients are given by,

$$\mathrm{𝑎_{0}\: =\:\frac{1}{𝑇}\int_{-\frac{𝑇}{2}}^{\frac{𝑇}{2}}𝑥(𝑡)\: dt\:=\: \frac{1}{𝑇}\:\int_{-\frac{𝑇}{2}}^{\frac{𝑇}{2}}𝑥_{e}(𝑡)\: dt}$$

$$\mathrm{⇒𝑎_{0}\: =\:\frac{1}{𝑇}(2\int_{0}^{\frac{𝑇}{2}}𝑥_{e}(𝑡) \: dt)\:=\:\frac{2}{𝑇}\int_{0}^{\frac{𝑇}{2}}𝑥(𝑡) \: dt}$$

$$\mathrm{\therefore 𝑎_{0}\: =\:\frac{2}{𝑇}\int_{0}^{\frac{𝑇}{2}}𝑥(𝑡) \: dt\:\:\:.....(4)}$$

The coefficient 𝑎_{𝑛} is given by

$$\mathrm{𝑎_{𝑛}\: =\:\frac{2}{𝑇}\int_{-\frac{𝑇}{2}}^{\frac{𝑇}{2}}𝑥(𝑡) \cos\:n\omega_{0}t\: dt\:=\:\frac{2}{𝑇}\int_{-\frac{𝑇}{2}}^{\frac{𝑇}{2}}𝑥_{e}(𝑡) \cos\:n\omega_{0}t\: dt}$$

$$\mathrm{𝑎_{𝑛}\: =\:\frac{2}{𝑇}(2\int_{0}^{\frac{𝑇}{2}}𝑥_{e}(𝑡) \cos\:n\omega_{0}t\: dt)}$$

$$\mathrm{\therefore 𝑎_{𝑛}\: =\:\frac{4}{𝑇}\int_{0}^{\frac{𝑇}{2}}𝑥(𝑡) \cos\:n\omega_{0}t\: dt\:\:\:.....(5)}$$

The coefficient 𝑏_{𝑛} is given by,

$$\mathrm{𝑏_{𝑛} =\:\frac{2}{𝑇}\int_{-\frac{𝑇}{2}}^{\frac{𝑇}{2}}𝑥_{e}(𝑡) \sin\:n\omega_{0}t\: dt}$$

Since the function (𝑥_{𝑒}(𝑡) sin 𝑛𝜔_{0}𝑡) is an odd function.

$$\mathrm{\therefore 𝑏_{𝑛}\:=\:\frac{2}{𝑇}\int_{-\frac{𝑇}{2}}^{\frac{𝑇}{2}}𝑥_{e}(𝑡) \sin\:n\omega_{0}t\: dt\:= 0\:\:\:.....(6)}$$

Hence, the Fourier series expansion of an even periodic function contains only a constant and cosine terms. Also, when the even or mirror symmetry exists in a function, then the trigonometric Fourier series coefficients for the function are given by equations (4), (5) and (6).

It is clear from the above discussion that when even symmetry exists in the function, then the trigonometric Fourier coefficient 𝑏_{𝑛} becomes zero and thus the calculation becomes simple.

- The sum of two or more even functions is always even.
- The product of two even functions is always even.
- When a constant is added to an even function, the even nature of the function still persists.

- Related Questions & Answers
- Signals and Systems – What is Odd Symmetry?
- Signals and Systems – What is Quarter Wave Symmetry?
- Signals and Systems – What is Half Wave Symmetry?
- Signals and Systems – Properties of Even and Odd Signals
- Signals and Systems: Even and Odd Signals
- Signals and Systems – What is a Linear System?
- Signals and Systems – What is Inverse Z-Transform?
- Signals and Systems – Classification of Signals
- Signals & Systems – What is Hilbert Transform?
- Signals and Systems – Time Scaling of Signals
- Signals and Systems – Fourier Transform of Periodic Signals
- Signals and Systems – Filter Characteristics of Linear Systems
- What is Convolution in Signals and Systems?
- What is Correlation in Signals and Systems?
- Signals and Systems – What is the Laplace Transform of Rectifier Function?

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