Lecture 5: Signal Energy and Parseval's Formula
If we assume for definiteness a unit load (a 1
resistor),
then if
is the voltage across the resistor or the current
through the resistor, then the energy dissipated is
If this quantity is finite, then is said to be an energy signal . Not every signal that we consider analytically is an energy signal: for example, is not an energy signal.
Substituting in for the
in terms of the inverse F.T.,
So we can write
For a real , the F.T. satisfies (they are conjugates), so that
This is Parseval's theorem for Fourier Transforms. It can be written using inner product notation as
It can also be generalized for products of different functions as
We can think of an increment of energy lying in an increment of bandwidth:
To get the total energy, add up all of the pieces. The function is therefore referred to as the energy spectral density of the function: it tells where the energy is distributed in frequency.
The energy which is passed by a bandpass filter with cutoff
frequencies
and
is
Some observations:

We can use the Parseval's theorem to integrate some messy
functions by simply applying the theorem. For example, we can
compute
by writing
The previous example also demonstrated how we can sometimes do easier integrals in one domain than the other.  We can also use the idea of the last example to introduce the idea of an essential bandwidth : include enough bandwidth that most of the energy of the signal can get through. Often the 95% or 99% level is sufficient.