define
i.e. energy in a period add up energy in spectral components
Just spread period out:
Look! no and integrals have f not . This is the FT pair - f(t) must satisfy Dirichlet
(There are others comparable definition) then
Likewise,
Note that if x(t) has no DC component then this disappears
(will be useful for probability functions)
from differentiation theorem
(might want to memorize them)
(from integration theorem)
Qualitative
energy concentrated at harmonics
So
x'(t)=x(t+T)
i.e. Fourier series coeff.
as we used before
If
is an EIGENFUNCTION of LTI systems
that's why we decompose signals onto
energy signal
power signal
or OTHER signals (don't really care
for this designation, but it's analytically useful sometimes)
since
rough units: F[joules(t)] joules.sec=joules/(1/sec)=joules/Hz
define
as with energy signals
so
Periodic signals are power signals x(t) periodic,