... below time zero is not a realistic situation.
I don't agree; n=0 can be an arbitrary time reference. It is as realistic as to consider the time before 0:00 of today.
A pole zero filter design gave the following difference equation:
y
= x
+ b.y(n - 1)
Describe the output when using an input signal given as:
x
= 0 for n <0
1 for n >= 0
(I removed the "1" character at the end of the difference equation considering it's a typo.)
It should be specified an initial condition in order that the problem b completely specified.
Let's consider that, for example y(-2)=0, and let's see what happens for n>=-1
For any n:
y
= x
+ b.y(n - 1)
Knowing that y(-2)=0 and that x
=0 for n<0, now we replace n for the different values starting at n=-1:
y(-1) = x(-1) + b.y(-1 - 1) = x(-1) + b.y(-2) = 0 + b.0 = 0
y(0) = x(0) + b.y(-1) = 1 + b.0 = 1
y(1) = x(1) + b.y(0) = 1 + b.1 = 1 + b
y(2) = x(2) + b.y(1) = 1 + b.(1+b) = 1 + b + b^2
... an so on ...
The next step is to realize that methods of signal processing allow to get the same result for this problem (an for others, much more complicated) in a simple and clever way.
Regards
Z