2/5/08

with diff equations, its convenient to use what is called operational notation
E is used to rep an advancement of one sample (Fig1)
therefore, we can rep an arbitrary diff equation in this way
ex (Fig2)

a LTI system's response is made up of two components: the zero-input response and the zero-state response
the zero-input response is the response due only to initial conditions, and is the solution of (Fig3)

an exponention f() turns out to be the only f() with this property -> zero-input response takes form of (Fig4)
advancements take the same form
substitution of our solution into the general equation gives (equation with an exponential in place of the f())
there are K roots to the equation where K is the highest power of E
the zero-input response could be any linear combination of these solutions

a discrete time system is described by the fillowing diff equation (Fig5)
the solution is made up of the zero-input response and the zero-state response
zero-input response described by (Fig6)
the zero-input solution now becomes (Fig7)
make use of init conds to solve for coefficients (Fig8)
finally, solution for zero-input is (Fig9)

repeted roots add factors of n into the solution with each repeated exponential
another example (Fig10)
another example (Fig11)