# Can you explain about the following formulas? in Signals and System

In my textbook, any signals could be expressed like the following formula.

$x(t)=\displaystyle\frac{a_0}{2}+\displaystyle\sum_{n=1}^{\infty}[a_n\cos(2{\pi}nft)+b_n\sin(2{\pi}nft)]$

I understand that any signal could be composed of periodic signals whose frequency is $f$, $2f$, $3f$, $\cdot\cdot\cdot$.

I don't understand the part of "$\displaystyle\frac{a_0}{2}+$".

and the text book said,

$a_n=\displaystyle\frac2T\displaystyle\int_{T_1}^{T_2}x(t)\cos(2{\pi}nft)dt$

$b_n=\displaystyle\frac2T\displaystyle\int_{T_1}^{T_2}x(t)\sin(2{\pi}nft)dt$.

I also don't understand why $a_n$ and $b_n$ are like the formula above.

Thank you for kind answers, and my first language is not English. Please let me read easy English. very Thank you.

• Substitute the expressions for $a_n$ and $b_n$ into you first formula and see what you get :) – geometrikal Aug 23 '15 at 11:55
• ... and don't forget to change your summation index from $n$ to $k$ (or something else), otherwise it'll be a mess. – Matt L. Aug 23 '15 at 12:27

## 1 Answer

Your question is about the Fourier series decomposition of a function (called signal here). First, this does not apply to "any signal" in theory. In practice, it applies to most pratical "periodic" signals. Periodicity is quite important, althought Fourier development may exist for not strictly periodic functions.

In loose terms, it tells you that if a signal is periodic, then inside each period the signal can be expressed as a sum (a linear superposition) of the most simple periodic functions, sines and cosines, with frequencies taken as linear factors of the base frequency.

The $a_0/2$ term comes from the average of your signal over one period $T$. This is due to the fact that sines and cosines do not see constant signals: they are orthogonal to them, since the integral of a sine or a cosine over one period exactly sums to $0$.

The other terms $a_n$ and $b_n$ are related to orthogonality as well, and i guess that you can, with great profit, read the following Wikipedia: Fourier series page in your perferred language.