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Week 7

• Linearity
• Matrices
• Vector LTI state
• Evaluation
• SISO
• Impulse response
• Convolution
• Moving average
• Echo
• Echo Response

Impulse Response

Define the impulse response h: Integers → Reals by

∀ n ∈ 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 + ∑ _{(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 ∈ Integers,

y(n) = h(n).

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