Excerpts from SPM-help#: 9-Jun-100 Convolution with delta func.. by John
[log in to unmask]
> My question is, if u_i(t) is an event-related sequence then presumably it
> is 0 everywhere but a finite number of points (i.e. a delta or "stick"
> function), in which case why doesn't the integral
>
> int_0^T ita_i(tau).u_i(t - tau) dtau
>
> always equal zero?
The dirac delta function has a special definition; it is the limit of an
ever-narrowing box car of ever-increasing height.
Precisely, a delta function at point A is the limit of the following
function as e->0
1/e I_{A-e/2,A+e/2}(x)
where I_{,}() is the indicator function: I_{a,b}(x) is a function of x
that is 1 between a and b and 0 otherwise.
Note that for any e>0 this function integrates to 1. Hence when you
integrate a dirac delta function you get 1, or 1 times whatever your
multiplying it against.
Any signal processing text should have more details if you're interested.
-Tom
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|