# Analysis of Exponential Fourier Series

## Fourier Series at a Glance

A continuous time signal x(t) is said to be periodic if there is a positive non-zero value of T for which As we know any periodic signal can be classified into harmonically related sinusoids or complex exponential, provided it satisfies the Dirichlet’s Conditions. This decomposed representation is called**FOURIER SERIES**. Two type of Fourier Series representation are there. Both are equivalent to each other.

**Exponential Fourier Series****Trigonometric Fourier Series**

## Exponential Fourier Series

A periodic signal is analyzed in terms of**Exponential Fourier Series**in the following three stages:

**Representation of Periodic Signal**.**Amplitude and Phase Spectra of a Periodic Signal**.**Power Content of a Periodic Signal**.

### Representation of Periodic Signal

A periodic signal in Fourier Series may be represented in two different time domains:**Continuous Time Domain**.**Discrete Time Domain**.

#### Continuous Time Domain

The complex**Exponential Fourier Series**representation of a periodic signal x(t) with fundamental period T

_{o}is given by Where, C is known as the

**Complex Fourier Coefficient**and is given by, Where ∫

_{0}

^{T0}, denotes the integral over any one period and, 0 to T

_{0}or –T

_{0}/2 to T

_{0}/2 are the limits commonly used for the integration. The equation (3) can be derived be multiplying both sides of equation (2) by e

^{(-jlω0t)}and integrate over a time period both sides. On interchanging the order of summation and integration on R.H.S., we get When, k≠l, the right hand side of (5) evaluated at the lower and upper limit yields zero. On the other hand, if k=l, we have Consequently equation (4) reduces to which indicates average value of x(t) over a period. When x (t) is real, C

_{-k}= C

_{k}

^{*}Where * indicates conjugate

#### Discrete Time Domain

Fourier representation in discrete is very much similar to Fourier representation of periodic signal of continuous time domain. The discrete Fourier series representation of a periodic sequence x[n] with fundamental period N_{o}is given by Where C

_{k}, are the Fourier coefficients and are given by This can be derived in the same way as we derived it in continuous time domain.

### Amplitude and Phase Spectra of a Periodic Signal

We can express Complex Fourier Coefficient, C_{k}as A plot of |C

_{k}| versus the angular frequency w is called the amplitude spectrum of the periodic signal x(t), and a plot of Ф

_{k}, versus w is called the phase spectrum of x(t). Since the index k assumes only integers, the amplitude and phase spectra are not continuous curves but appear only at the discrete frequencies kω

_{0}, they are therefore referred to as discrete frequency spectra or line spectra. For a real periodic signal x ( t ) we have C

_{-k}= C

_{k}

^{ *}. Thus, Hence, the amplitude spectrum is an even function of ω, and the phase spectrum is an odd function of

*0*for a real periodic signal.

### Power Content of a Periodic Signal

Average**Power Content of a Periodic Signal**is given by If x (t) is represented by the complex exponential Fourier Series, then This equation is known as Parseval’s identity or Parseval’s Theorem.