Supposing a piston sequence of duration T. Let us define the standard
deviation of piston during that sequence:
where 
is the average value of piston:
Let us consider the T-periodic function:
where 
and 
are, respectively, the window
function of width T and the Dirac comb with spacing T. Then, 
equals
on the interval 
and the
moving average: 
is a constant equal to 
by definition of 
. Besides, the
moving average acts as a low pass filter with transfer function
on 
. It is thus straightforward to
derive the spectrum of the periodic function 
which
matches 
on 
:
with:
has discrete values with spacing 
and
. Hence, for 
:
and the spectrum is zero except for harmonics:
Computing the standard deviation for a piston sequence of duration T thus amounts
to computing the standard deviation of the periodic function 
which is
the square root of the infinite sum of the squared harmonics. Eventually:
