Signals & Systems – Complex Exponential Fourier Series

Signals and SystemsElectronics & ElectricalDigital Electronics

Exponential Fourier Series

Periodic signals are represented over a certain interval of time in terms of the linear combination of orthogonal functions. If these orthogonal functions are the exponential functions, then the Fourier series representation of the function is called the exponential Fourier series.

The exponential Fourier series is the most widely used form of the Fourier series. In this representation, the periodic function x(t) is expressed as a weighted sum of the complex exponential functions. The complex exponential Fourier series is the convenient and compact form of the Fourier series, hence, its findsextensive application in communication theory.

Explanation

Let a set of complex exponential functions as,

$$\mathrm{\left \{e^{jn\omega_{0}t},n=0,\pm 1,\pm 2,\pm 3,....\right \}}$$

This set of exponential functions forms a closed orthogonal set over a time interval [𝑡0, (𝑡0 + 𝑇)] for any value of 𝑡0. Therefore, it can be used as a Fourierseries. Here, the parameter T is the period of the function and is given by,

$$\mathrm{T=\frac{2\pi}{\omega_{0}}}$$

The cosine Fourier series of a periodic function is defined as,

$$\mathrm{x(t)=A_0+\sum_{n=1}^{\infty}A_n\cos(n\omega_{0}t+\theta_n)\:\:\:\:....(1)}$$

Now, by using the Euler’s rule, we can write,

$$\mathrm{A_n\cos(n\omega_{0}t+\theta_n)=A_n[\frac{e^{j(n\omega_{0}t+\theta_n)}+e^{-j(n\omega_{0}t+\theta_n)}}{2}]\:\:\:\:.....(2)}$$

From the equations (1) & (2), we get,

$$\mathrm{x(t)=A_0+\sum_{n=1}^{\infty}A_n[\frac{e^{j(n\omega_0t+\theta_n)}+e^{-j(n\omega_0t+\theta_n})}{2}] }$$

$$\mathrm{\Rightarrow x(t)= A_0+\sum_{n=1}^{\infty}\frac{A_n}{2}[e^{jn \omega_0t}e^{j\theta_n}+e^{-jn \omega_0t}e^{-j\theta_n}]}$$

$$\mathrm{\Rightarrow x(t)=A_0+\sum_{n=1}^{\infty}[\frac{A_n}{2}e^{jn \omega_0t}e^{j\theta_n}]+\sum_{n=1}^{\infty}[\frac{A_n}{2}e^{-jn \omega_0t}e^{-j\theta_n}]}$$

$$\mathrm{\Rightarrow x(t)=A_0+\sum_{n=1}^{\infty}[\frac{A_n}{2}e^{j\theta_n}]e^{jn\omega_0t} + \sum_{n=1}^{\infty}[\frac{A_n}{2}e^{-j\theta_n}]e^{-jn\omega_0t}\:\:\:\:\:......(3)}$$

Now, by replacing 𝑛 = (−𝑚) in the second summation term of the equation (3), we get,

$$\mathrm{\Rightarrow x(t)=A_0+\sum_{n=1}^{\infty}[\frac{A_n}{2}e^{j\theta_n}]e^{jn\omega_0t} + \sum_{m=-1}^{-\infty}[\frac{A_m}{2}e^{j\theta_m}]e^{jm\omega_0t}\:\:\:\:........(4)}$$

On comparing equations (3) and (4), we have

$$\mathrm{A_n=A_m\:and\:(-\theta_n)=\theta_m\:for\:n>0\: \&\:k<0}$$

Now, let us define the exponential Fourier coefficients as,

$$\mathrm{C_0=A_0\:and\:C_n=\frac{A_n}{2}e^{j\theta_n}\:for\:n> 0}$$

Hence, the equation (4) can be written as,

$$\mathrm{\Rightarrow x(t)=A_0+\sum_{n=1}^{\infty}[\frac{A_n}{2}e^{j\theta_n}]e^{jn\omega_0t} + \sum_{n=-1}^{-\infty}[\frac{A_n}{2}e^{j\theta_n}]e^{jn\omega_0t}}$$

$$\mathrm{\Rightarrow x(t)=\sum_{n=-\infty}^{\infty}C_ne^{jn\omega_{0}t}\:\:\:\:.....(5)}$$

The expression in equation (5) is known as the exponential form of Fourier series. It is also called as the synthesis equation.