Z-Transform Properties and Convolution

Z-Transform Properties

Property	Time Domain	Z-Domain
Linearity	$a x_1[n] + b x_2[n]$	$a X_1(z) + b X_2(z)$
Time Shift	$x[n-k]$	$z^{-k} X(z)$
Scaling in z	$a^n x[n]$	$X(z/a)$
Time Reversal	$x[-n]$	$X(1/z)$
Differentiation	$n x[n]$	$-z \frac{dX(z)}{dz}$
Convolution	$x_1[n] * x_2[n]$	$X_1(z) X_2(z)$
Initial Value	$x[0]$	$\lim_{z\to\infty} X(z)$ (causal)
Final Value	$\lim_{n\to\infty} x[n]$	$\lim_{z\to 1} (1-z^{-1})X(z)$

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The Convolution Property: Why Z-Transform Matters

\[y[n] = x[n] * h[n] \quad \overset{\mathcal{Z}}{\longleftrightarrow} \quad Y(z) = X(z) H(z)\]

Convolution becomes multiplication. This is the entire foundation of digital filter analysis.

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