>> A non-sinusoidal **periodic function** can be expressed as an infinite sum of sinusoidal functions.

>> A **periodic function** is a function that repeats its values in regular intervals or periods.

A function is said to be periodic if >> **f (t) = f (t + nT) **(n is an integer)** **for all values of *t* in the domain. Where T is called the **fundamental period or Basic period **and T should have a **nonzero positive** value.

>> According to the Fourier theorem, any periodic function of frequency ω_{∘} can be expressed as an infinite sum of sine or cosine functions that are integral multiples of ω_{∘}.

**f (t) = a _{∘} + a_{1} cos ω_{∘}t + b_{1} sin ω_{∘}t + a_{2} cos2ω_{∘}t + b_{2 }sin2ω_{∘}t + ……….**

>> Where ω_{∘} is called the fundamental frequency in radian per second. The sinusiods **sin (nω _{∘}t) and cos (nω_{∘}t)** are called the

**n**harmonic of f(t). The constants

^{th}**a**, a

_{∘}_{n}and b

_{n }are called the fourier coefficients.

**>> ** A function that can be represented by a Fourier series must meet certain requirements, **because the infinite series may or may not be converge.**

Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

A series is **convergent** if the sequence of its partial sums (S_{1}, S_{2}, S_{3 }…. ) tends to a **limit** ; that means that the partial sums become closer and closer to a given number when the number of their terms increases. (**Any series that is not convergent is said to be divergent**)

The reciprocals of the positive integers produce a** divergent series** (harmonic series):

Alternating the signs of the reciprocals of positive integers produces a **convergent series**:

The reciprocals of factorials produce a **convergent series** :

**Dirichlet Conditions**

>> f(t) is single valued everywhere (Means f(t) is a function → A function can only have one output, f(t), for each unique input, t → **Vertical line test**).

>> f(t) has a finite number of discontinuities in any one period (Then only a_{n} and b_{n} will exist).

>> f(t) has a finite number of maxima and minima in any one period.

**f(t) = Sin (1/t)**

**y ∈ (0,1] **

A signal that does not satisfy this condition is x(t) = tan(t)

>> These conditions are not necessary conditions, they are **sufficient conditions** for a Fourier series to **exist**.

**Fourier Analysis**

>> The process of determining the Fourier coefficients is called Fourier analysis.