Z-Transform: Convergence, Inversion, Frequency Analysis, and Systems

All flashcards

Region of convergence (ROC)

The set of complex values of $z$ for which the Z-transform sum converges absolutely.

Transfer function

For zero-initial-condition analysis, the ratio $H(z)=Y(z)/X(z)$ relating a system's output and input transforms.

Pole and zero

A zero is a root of a transfer-function numerator; a pole is a root of its denominator.

Inverse Z-transform

The operation that recovers a time sequence from its transform, using the ROC to select the appropriate sequence.

Bilinear transform

The Tustin mapping $s=\frac{2}{T}\frac{z-1}{z+1}$ used to convert continuous-time filter models to discrete-time models.

Quiz questions

1. For $X(z)=1/(1-0.5z^{-1})$, which ROC corresponds to the causal sequence?Answer: $|z|>0.5$
2. For a discrete-time system to be both causal and stable, where must its poles lie?Answer: Inside the unit circle
3. When does evaluating a bilateral Z-transform on $z=e^{i\omega}$ yield the DTFT?Answer: When the unit circle lies in the ROC
4. What does the bilinear transform map from the $s$-plane to the $z$-plane?Answer: The imaginary axis to the unit circle
5. What distinguishes the discrete Fourier transform (DFT) from the DTFT?Answer: The DFT uses a finite set of frequency samples