## 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**

Both representations give the same result. Depending upon the type of signal, we choose any of the representation according to our convenience.

## 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,

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.