Relation between Trigonometric & Exponential Fourier Series

Trigonometric Fourier Series

A periodic function can be represented over a certain interval of time in terms of the linear combination of orthogonal functions. If these orthogonal functions are the trigonometric functions, then it is known as trigonometric Fourier series.

Mathematically, the standard trigonometric Fourier series expansion of a periodic signal is,

$$\mathrm{x(t)=a_{0}+ \sum_{n=1}^{\infty}a_{n}\:cos\:\omega_{0}nt+b_{n}\:sin\:\omega_{0}nt\:\:… (1)}$$

Exponential Fourier Series

A periodic function can be 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 it is known as exponential Fourier series.

Mathematically, the standard exponential Fourier series expansion for a periodic function is given by,

$$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}e^{jn\omega_{0} t}\:\:… (2)}$$

Obtain Trigonometric Fourier Series from Exponential Fourier Series

The exponential Fourier series of a periodic function $x(t)$ is given by,

$$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}e^{jn\omega_{0} t}}$$

$$\mathrm{\Rightarrow\:x(t)=C_{0}+\sum_{n=−\infty}^{-1}C_{n}e^{jn\omega_{0} t}+\sum_{n=1}^{\infty}C_{n}e^{jn\omega_{0} t}}$$

$$\mathrm{\Rightarrow\:x(t)=C_{0}+\sum_{n=1}^{\infty}(C_{-n}e^{-jn\omega_{0} t}+C_{n}e^{jn\omega_{0} t})}$$

$$\mathrm{\Rightarrow\:x(t)=C_{0}+\sum_{n=1}^{\infty}[C_{-n}(cos\:n\omega_{0}t-j\:sin\:n\omega_{0}t)+C_{n}(cos\:n\omega_{0}t + j\:sin\:n\omega_{0}t)]}$$

$$\mathrm{\Rightarrow\:x(t)=C_{0}+\sum_{n=1}^{\infty}[(C_{n}+C_{-n})cos\:n\omega_{0}t+j(C_{n}-C_{-n})sin\:n\omega_{0}t]\:\:… (3)}$$

Now, on comparing equation (3) with the standard trigonometric Fourier series given in the eq. (1), we obtain the coefficients of trigonometric Fourier series as follows −

$$\mathrm{a_{0}=C_{0}}$$

$$\mathrm{a_{n}=C_{n}+C_{-n}}$$

$$\mathrm{b_{n}=j(C_{n}+C_{-n})}$$

By evaluating these trigonometric coefficients, we can write the trigonometric Fourier series expansion of the periodic function.

Obtain Exponential Fourier Series from Trigonometric Fourier Series

The exponential Fourier series can be obtained from the trigonometric Fourier series as follows −

The trigonometric Fourier series expansion of a periodic function is given by,

$$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\:\omega_{0}nt+b_{n}\:sin\:\omega_{0}nt}$$

Where, the trigonometric Fourier coefficients are given by,

$$\mathrm{a_{0}=\frac{1}{T}\int_{0}^{T}x(t)dt\:\:… (4)}$$

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

$$\mathrm{b_{n}=\frac{2}{T}\int_{0}^{T}x(t)sin\:n\omega_{0}t\:\:dt\:\:\:… (6)}$$

From the exponential Fourier series, the exponential Fourier coefficient $C_{n}$ is given by,

$$\mathrm{C_{n}=\frac{1}{T}\int_{0}^{T}x(t) e^{-jn\omega_{0} t}dt}$$

By using Euler’s formula, we obtain,

$$\mathrm{C_{n}=\frac{1}{T}\int_{0}^{T} x(t)(cos\:n\omega_{0} t - j\:sin\:n\omega_{0} t)dt}$$

$$\mathrm{\Rightarrow\:C_{n}=\frac{1}{T}\left ( \frac{2}{T}\int_{0}^{T} x(t)\:cos\:n\omega_{0} t\: dt-j\frac{2}{T}\int_{0}^{T} x(t)\: sin \:n\omega_{0} t dt \right )… (7)}$$

On comparing equation (7) with (5) & (6), we get,

$$\mathrm{C_{n}=\frac{1}{2}[a_{n}-jb_{n}]\:… (8)}$$

Similarly, the exponential Fourier coefficient $C_{-n}$ is,

$$\mathrm{C_{-n}=\frac{1}{T}\int_{0}^{T}x(t) e^{jn\omega_{0} t}dt}$$

By using Euler’s formula, we obtain,

$$\mathrm{C_{-n}=\frac{1}{T}\int_{0}^{T}x(t)(cos\:n\omega_{0}t+j\:sin\:n\omega_{0}t)\:dt}$$

$$\mathrm{\Rightarrow\:C_{-n}= \frac{1}{2}\left(\frac{1}{T}\int_{0}^{T}x(t)cos\:n\omega_{0}t\:dt + j\frac{2}{T}\int_{0}^{T} x(t)\:sin\:n\omega_{0} t dt \right)\:\:… (9)}$$

On comparing equation (9) with (5) & (6), we get,

$$\mathrm{C_{-n}=\frac{1}{2}[a_{n}+jb_{n}]\:… (10)}$$

And, the exponential Fourier coefficient $C_{0}$ is,

$$\mathrm{C_{0}=\frac{1}{T}\int_{0}^{T}x(t)\:dt=a_{0}… (11)}$$

Using equations (8), (10) & (11), we can obtain the values of exponential Fourier coefficient from trigonometric Fourier coefficients and then the exponential Fourier series from the trigonometric Fourier series.