# Fourier Series representation

## Definition
The formal definition of Fourier series representation goes as follows:

> The Fourier series representation of a periodic signal is the linear combination of harmonically-related complex exponentials.

Let's look in detail at each of the terms in this definition.

### Complex Exponentials

A complex exponential is of the form : $e^{st}$
where $s$ is a complex number; $s = \sigma + j\omega$.
When $s$ is purely imaginary, i.e. $\sigma =0$, then the exponential is *periodic*.
Since we're dealing with periodic signals, we'll consider the complex exponential of the form: $e^{j\omega t}$.

#### Why use complex exponential?

#### Usefullness

### Harmonically - Related

### Linear Combination
Since we're dealing with *Linear* Time-invariant systems, we can represent a periodic signal as the ***linear combination*** of harmonically-related exponentials.

So, finally a signal can be represnted as:
$$x(t) = \sum_{k=-\infty}^{+\infty} x_{k}e^{jk\omega_{o}t} = \sum_{k=-\infty}^{+\infty} x_{k}e^{jk\frac{2\pi}{T}t}$$

where the summation symbol represents the sum in the linear combination and $x_{k}$'s represent the weights with which each of the term in the linear combination is multiplied.

The above represntation is also called the: **Synthesis Equation**.
And, the weights $x_{k}$ are called: **Fourier series Coefficients** or Spectral coefficients.

Note, that in the equation above, the linear combination of the complex exponentials actually represents an **infinite series**.
This infinite series convereges to the actual signal $x(t)$.

| | |
|:------------------------------------------------------------:| ------------------ |
| $$x(t) = \sum_{k=-\infty}^{+\infty} x_{k}e^{jk\omega_{o}t}$$ | Synthesis Equation |
| $$x_{k} = \frac{1}{T}\int_{0}^{T}x(t) e^{-jk\omega_{o}t}dt$$ | Analysis Equation |

## What class of periodic signals can be represented using F.S.?
Now, that we have defined Fourier series, a very important question to ask at this junction is:

What class of periodic signals can be represented by the Fourier Series?
In other words, for what class of periodic signals does the Fourier series work?

What we commonly say as an answer to this question is that, a _very large class_ of periodic signals can be represented using F.S.
That is, F.S. works for _almost all_ periodic signals of practical utility.

But if it is really the case, then why do we even bother to ask this question?

Indeed for most practical scenarios we can choose to ignore this question. However, there does exist a scenario which we'll be discussing now, for which this question becomes important.

Suppose we've a signal $x(t)$. We'll consider two cases, one when $x(t)$ is continuous, and another when $x(t)$ is discontinuous.

Does, Fourier series work when $x(t)$ is continuous? Of course it does. In fact all periodic signals that are continuous can be represented by Fourier series and that their F.S. representation converges to the original signal $x(t)$. We can actually give a very simple reasoning for this.

Consider the F.S. representation of a continuous periodic signal $x(t)$ :

$$x(t) = \sum_{k=-\infty}^{+\infty} x_{k}e^{jk\omega_{o}t}$$

Since exponentials are continuous functions, in this representation, each of the individual components $e^{jk\omega_{o}t}$ are continuous. We also know that, the sum of continuous functions is also continuous. With this reasoning, we can very well say that $x(t)$ will be continuous.

So everything is fine for continous functions. But what if $x(t)$ is discontinous?
Does F.S. work for discontinous signals like a square wave? The answer is simple: yes. But if we ask how, we see that with the level of understanding we've attained till now about F.S. we won't be able to reason for the F.S. representaion of discontinous signals. After all, how will you reason that the sum of continous functions can ever be discontinous?!

In fact, to give a bit of historical perspective: when Fourier first introduced the F.S. representation he firmly believed that F.S. _can_ be used to represent discontinous functions. However, several other influential mathematicians of the time, like Lagrange were very skeptical of this idea, for the simple reason that we just discussed. They simply couldn't digest how a series of continous functions can be used to represent a discontinous function like a square wave.

However it turns out that discontinous signals like a square wave can be represnted using Fourier series and that now we need to develop a reasoning for it. It is for the scanario of these discontinous signals that the question originally asked at the beginning was important.

In this regard we would be happy, if we get a concrete **set of conditions** that a periodic signal can satisfy to _guarantee_ that it can be represented using Fourier series.

There do exist such a set of $3$ conditions known as: **Dirichlet conditions**.

### Dirichlet Conditions
1. The periodic signal $x(t)$ must be **absoulutely integrable** :
$$\int_{0}^{T}|x(t)| dt < \infty$$

2. $x(t)$ is of **Bounded Variation** : $x(t)$ is allowed to have atmost only a **finite number of maxima and minima** over a period 0 to T.

3. $x(t)$ is allowed to have atmost only a **finite number of finite discontinuities**.
Finite discontinuities: Discontinuity with finite height.

For a signal $x(t)$ that satisfies the $3$ Dirichlet conditions, we can conclude that:
1. If $x(t)$ is **continous** : then the F.S. represntation converges and equals to $x(t)$ at all points $t$ where $x(t)$ is continous.

2. If $x(t)$ has **finite discontinuties** : then at points of those discontinuities, the F.S. representation converges to : the average value of the signal on either side of the discontinuity, i.e. :
$$\frac{x(t+\epsilon)+x(t-\epsilon)}{2}$$

So, this now answers our initial question: what class of periodic signals can be represented using F.S? All those peiodic signals that satisfy the above set of conditions can be represented using F.S.

## Properties of Fourier Series representaion for Continuous-time signals

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