![Stems panel: x[n] r⁻ⁿ for samples n = 0 to 15, with test radius r = 2.50: each stem is 0.500 times the last. Strip panel: test radius r from 1 to 3. The part from 1 to 1.25 is hatched, "no value", and the rest is plain, "adds up"; a line at 1.25 is labelled "growth of x: 1.25". The marker is at r = 2.50. Readouts: test radius r 2.50, sum of x[n] r⁻ⁿ 2.00.](/thumbs/z-transform-dark.webp)
![Stems panel: x[n] r⁻ⁿ for samples n = 0 to 15, with test radius r = 2.50: each stem is 0.500 times the last. Strip panel: test radius r from 1 to 3. The part from 1 to 1.25 is hatched, "no value", and the rest is plain, "adds up"; a line at 1.25 is labelled "growth of x: 1.25". The marker is at r = 2.50. Readouts: test radius r 2.50, sum of x[n] r⁻ⁿ 2.00.](/thumbs/z-transform-light.webp)
Lesson 1★ Essential30 minYou are hereRead
The z-transform
Divide a sampled signal by r to the n, find which radii make the sum settle, and read the DTFT off the unit circle of the resulting plane.
Chapter 16
The z-transform
Five lessons in Part VI, The z-transform and system analysis. Read them in order, or start anywhere: a prerequisite is a link, never a gate.
Start with 16.10 of 5 read4 on the essential pathabout 122 minutes
Lessons in this chapter
![Z-plane with the unit circle and two poles marked by crosses, at 0.5 and −0.25 on the real axis. Term (2/3)·0.5ⁿ as stems with dot heads, all 11 samples, starting at 0.667 and halving every sample. Term (1/3)·(−0.25)ⁿ as stems with square heads, all 11 samples, starting at 0.333 and flipping sign every sample. Their sum x[n] as diamonds: 1, 0.25, 0.1875, 0.0781, then smaller. Weight of pole 0.5: 0.667; weight of pole −0.25: 0.333.](/thumbs/inverse-z-transform-dark.webp)
![Z-plane with the unit circle and two poles marked by crosses, at 0.5 and −0.25 on the real axis. Term (2/3)·0.5ⁿ as stems with dot heads, all 11 samples, starting at 0.667 and halving every sample. Term (1/3)·(−0.25)ⁿ as stems with square heads, all 11 samples, starting at 0.333 and flipping sign every sample. Their sum x[n] as diamonds: 1, 0.25, 0.1875, 0.0781, then smaller. Weight of pole 0.5: 0.667; weight of pole −0.25: 0.333.](/thumbs/inverse-z-transform-light.webp)
![Plane with the unit circle, the pole 0.60∠90.0° and its mirror 0.60∠−90.0°, and two zeros at 0. The impulse response h[n], n = 0 to 24, turns 90.0° and shrinks to 0.60 of its size each sample. It rings once every 4.0 samples and its envelope rⁿ falls to 2 % by n = 7.7.](/thumbs/poles-and-zeros-dark.webp)
![Plane with the unit circle, the pole 0.60∠90.0° and its mirror 0.60∠−90.0°, and two zeros at 0. The impulse response h[n], n = 0 to 24, turns 90.0° and shrinks to 0.60 of its size each sample. It rings once every 4.0 samples and its envelope rⁿ falls to 2 % by n = 7.7.](/thumbs/poles-and-zeros-light.webp)
![The z-plane with the pole pair 1.05 e^(±jπ/4), at ±45°, outside the unit circle. Stems of h[n] for sample n from 0 to 40 with the dashed envelope: h grows. Pole radius r 1.05, sum of |h[n]| no limit.](/thumbs/z-stability-dark.webp)
![The z-plane with the pole pair 1.05 e^(±jπ/4), at ±45°, outside the unit circle. Stems of h[n] for sample n from 0 to 40 with the dashed envelope: h grows. Pole radius r 1.05, sum of |h[n]| no limit.](/thumbs/z-stability-light.webp)
![Stems of y[n] against sample n from −2 to 14 for y[n] = y[n−1] − 0.5y[n−2] + x[n] with a step input. Stored values y[−1] = 2 and y[−2] = 2, drawn as open squares at n = −1 and n = −2. Each output stem stacks the zero-input part, with an open-circle head, and the zero-state part, a bare stem, with the total as a diamond. The total is 2, 2, 2, 2, 2, 2 for n = 0 to 5. A dashed line marks the final value 2.](/thumbs/unilateral-z-dark.webp)
![Stems of y[n] against sample n from −2 to 14 for y[n] = y[n−1] − 0.5y[n−2] + x[n] with a step input. Stored values y[−1] = 2 and y[−2] = 2, drawn as open squares at n = −1 and n = −2. Each output stem stacks the zero-input part, with an open-circle head, and the zero-state part, a bare stem, with the total as a diamond. The total is 2, 2, 2, 2, 2, 2 for n = 0 to 5. A dashed line marks the final value 2.](/thumbs/unilateral-z-light.webp)
After this chapter
Where to go next.
The chapters either side, and the rest of Part VI in the library.