How Fourier decomposition is performed?

The Fourier decomposition explains a time series entirely as a weighted sum of sinusoidal functions and with the Fourier series,it is possible to do it.

But I have some doubts

Suppose ,for any signal we want to perform its Fourier decomposition

1. As we have to create any signal with finite number ($N$)of elementary signals .So how ( in Fourier decomposition ) to decide value of ($N$)?

2. Also, How to take into consideration the 3 important changing factors of the elementary signals i.e. phase, frequency and amplitude while performing Fourier decompositon ?

• N determines the smallest bandwidth of frequency you can resolve (you never really get at just one frequency with Fourier). Magnitude, phase, and amplitude can all be extracted using trigonometry, it is encapsulated in the complex exponential. – panthyon May 25 '15 at 15:51
• You begin with an incorrect statement "The Fourier decomposition tells us that any signal can be represented as sum of finite number of elementary signal." and so it is not to be wondered that you have doubts. How about actually opening your book and writing down what it actually says instead of just winging it? (If your book does say what you have written, I suggest you throw it away). -1 pending corrections. – Dilip Sarwate May 25 '15 at 16:02
• @Dilip is grumpy. (but maybe correct.) – robert bristow-johnson May 25 '15 at 16:11
• @robert bristow-johnson sir if possible can u explain about concept of Fourier decomposition in answer box with any example. I am totally confused now. – pandu May 25 '15 at 16:15
• well, there are textbooks and there might be online resources. what example would you propose? – robert bristow-johnson May 25 '15 at 16:30

To understand the Fourier pedagogy, first you have to deal with periodic functions. Only periodic functions, at first.

$$x(t+P) = x(t) \quad \text{for all } t$$

That has period of $P$ (and also a period of $2P$ and $3P$, but we won't worry about that for the time being).

Fourier (and Euler) postulate that this other periodic function of the same period can, within a certain limit, approach the periodic $x(t)$

$$x(t) = \sum\limits_{n=-\infty}^{+\infty} c_n e^{j 2 \pi \frac{n}{P} t}$$

and, given that postulate, the Fourier coefficients, $c_n$, come out to be

$$c_n = \frac{1}{P}\int\limits_{t_0}^{t_0+P} x(t) e^{-j 2 \pi \frac{n}{P} t} \ dt$$

Now this only decomposes periodic $x(t)$. To decompose a non-periodic $x(t)$, you have to approximate that with a periodic $x(t)$ with an extremely long period (i think in electronics, a period of a year would more than suffice) and then, in a limit, let that period go to infinity. I guess with an infinitely long period, a periodic function would no longer be periodic. This is what leads to the Fourier integral and is another big mathematical effort. But the OP should get this down first.