# 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 :) Commented Aug 23, 2015 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. Commented Aug 23, 2015 at 12:27

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.