1 2 t _ (ant 
X(t) = B ao + » an cos( =) + b, sin( =) (2.29) 
where 
I 
1 | Xo) Jtnt a 
n= cos — 
ser T 
ie 
: (2.30) 
2 
t 
pone [ xo fin ae 
T T 
ae Ta 
2 
The a, 
and 6, functions are called Euler—Fourier coefficients. Equation 2.30 is 
t t 
based on the orthogonality of cos, sin. 
For the existence of a,, b,, and for the convergence of the series, the conditions 
I I 
2 2 
IX(t)ldt < «, | IX(t)P?dt < © (2.31) 
RIN 
are sufficient. 
1 
1 
Replacing a,b, by Co =o, Cp = E (a? + Ale gives 
the total energy [-T/2 to T/2] as 
I 
Sl 1 
| X(t = ST 7% +5 (a + bi =T c3 (2.32) 
n=1 
ae ea 
2 
and the total power [- 7/2 to T/2] as 
13 
