Digital signal and image processing
Before / after
Before the z-transform, filters were painful
Before the z-transform, analysing a filter meant solving difference equations
Quick check · tap an answer to continue
Lesson notes
Key points
- Before it, you stepped through difference equations sample by sample, and stability was guesswork from long output sequences.
- The z-transform turns convolution into multiplication, so a system becomes H(z), and it is stable if every pole lies inside the unit circle.
Takeaway: Z-transform: convolution becomes multiplication; poles reveal stability.
Quick check: Where must all poles of a stable causal discrete system lie?
- On the imaginary axis
- Inside the unit circle
- Outside the unit circle
Poles with magnitude below one give decaying, stable responses.
Difference equationsZ-transform
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