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Commit d52173a8 authored by Eric Kooistra's avatar Eric Kooistra
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Add z-transform properties.

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......@@ -175,13 +175,20 @@ b) z-transform [LYONS 6.3, MATLAB]
- Causal systems are 0 for n < 0, so sum from n = 0 yields one sided z-transform
- For r = 1 the z-transform yields the DTFT, so DTFT evaluates the
z-transform on the unit circle.
- Properties [PROAKIS 3.3]:
. shift property: x[n - k] <--> X(z) z^-k
. time reversal: x[-n] <--> X(z^-1]
. convolution: x[n] * y[n] <--> X(z) Y(z)
. conjugation: x*[n] <--> X*(z*)
. real part: Re{x[n]} <--> 1/2 [X(z) + X*(z*)]
. imag part: Im{x[n]} <--> 1/2 [X(z) - X*(z*)]
- Examples:
. wire: H(z) = 1
. delay: H(z) = z^-1, so x[n - 1] <--> X(z) z^-1 (shift property)
. integrator: d[n] pulse at n = 0 --> u[n] step,
Y(z) = 1 + z^-1 + z^-2 + ..., multiply by (z - 1) / (z - 1)
= z / (z - 1) = 1 / (1 - z^-1)
. difference equation
. difference equation <--> rational z-transform
y[n] + 0.5 y[n-1] = u[n] --> Y(z) + 0.5 z^-1 Y(z) = U(z)
H(z) = Y(z) / U(Z) = 1 / (1 + 0.5 z^-1)
......
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