booklet format - inaf iasf bologna

Temporal Data Analysis A.A. 2011/2012
Here, we tacitly assumed a > 0. For a < 0 we would get a minus sign in the prefactor; however,
we would also have to interchange the integration limits and thus get together the factor 1/|a|.
This means: stretching (compressing) the time-axis results in the compression (stretching) of the
frequency-axis. For the special case a = −1 we get:
f (t) →F (ω);
f (−t) →F (−ω);
(2.41)
Therefore, turning around the time axis (“looking into the past”) results in turning around the
frequency axis.
2.2.3 Convolution, Parseval Theorem
2.2.3.1 Convolution
The convolution of a function f (t) with another function g (t) is defined as:
Definition 2.5 (Convolution).
h(t) =
∫ +∞
−∞
f (ξ) g (t − ξ)dξ ≡ f (t) ⊗ g (t) (2.42)
Please note that there is a minus sign in the argument of g (t). The convolution is commutative,
distributive, and associative. This means
commutative:
distributive:
associative:
f (t) ⊗ g (t) = g (t) ⊗ f (t)
f (t) ⊗ (g (t) + h(t)) = f (t) ⊗ g (t) + f (t) ⊗ h(t)
f (t) ⊗ (g (t) ⊗ h(t)) = (f (t) ⊗ g (t)) ⊗ h(t)
As an example of convolution, let us take a pulse that looks like an unilateral exponential
function
{ e
−t/τ
for t ≥ 0
f (t) =
0 else
(2.43)
Any device that delivers pulses as a function of time, has a finite rise-time/decay-time, which
for simplicity’s sake we’ll assume to be a Gaussian
g (t) = 1 (−
σ 2π exp 1 t 2 )
(2.44)
2 σ 2
That is how our device would represent a δfunction – we can’t get sharper than that. The
function g (t), therefore, is the device’s resolution function, which we’ll have to use for the
convolution of all signals we want to record. An example would be the bandwidth of an
oscilloscope. We then need:
26 M.Orlandini