Impulse Response

Define the impulse response h: Integers ® Reals by

" n Î Integers,

h(n) = d, if n = 0

h(n) = c^T A^{n -1} b , if n > 0

Then the output can be written,

y(n) = c^T Aⁿ s₀ + S _{(m = 0 to n)} h(n- m) x(m)

Notice that if s₀ = 0 and the input is the Kronecker delta function, or impulse:

x(m) = 1; if m = 0,

x(m) = 0; otherwise,

then " n Î Integers,

y(n) = h(n).

So h is called the (zero state) impulse response of the system.