Convolution
[1] We learn about a system by studying it responses to various input
signals.
I claim that the weight function
w(t)  the solution
to
p(D)x = delta(t)
with rest initial conditions  contains complete
data about the LTI operator
p(D)
(and so about the system it
represents).
In fact there is a formula which gives the system response (with rest
initial conditions)
to any input signal
q(t)
as a kind of "product"
of
w(t)
with
q(t) .
Suppose phosphates from a farm run off fields into a lake, at a rate
q(t)
which varies with the seasons. For definiteness let's say
q(t)
=
1 + cos(bt)
Once in the lake, the phosphate decays: it's carried out of the stream
at a rate proportional to the amount in the lake:
x' + ax
=
q(t)
,
x(0) = 0
The weight function for this system is
w(t)
=
e^{at}
for
t > 0
=
0
for
t < 0 .
This tells you how much of each pound is left after t units of time have
elapsed. If c pounds go in at time tau, then
w(ttau) c
is the amount left at time
t > tau.
Fix a time
t .
We'll think about how
x(t)
gets built up from the
contributions made during the time between
0
and
t .
We'll need
another
letter to denote that changing time;
tau.
We'll replace the continuous input represented by
q(t)
by a discrete
input.
Divide time into very small intervals,
Delta tau
(1 second maybe) .
During the
Delta tau
time interval around time
tau ,
the quantity of phosphate entering the lake is
q(tau) Delta tau
How much of that drop remains at time
t?
Well, the weight function tells you!
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 Winter '08
 Staff
 Digital Signal Processing, LTI system theory, Impulse response, Tau, rest initial conditions

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