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Chapter 17

Special classes of systems

Four 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 17.1
0 of 4 read2 on the essential pathabout 72 minutes

Lessons in this chapter

The z-plane with the unit circle, a pole at 0.5 and a zero at 2. The test point is on the circle at Ω = 1.00π rad/sample. The arrow from the zero is 3.000 long and the arrow from the pole 1.500, a ratio of 2. The gain, traced from 0 to 1.00π, is a flat line at 1.000.The z-plane with the unit circle, a pole at 0.5 and a zero at 2. The test point is on the circle at Ω = 1.00π rad/sample. The arrow from the zero is 3.000 long and the arrow from the pole 1.500, a ratio of 2. The gain, traced from 0 to 1.00π, is a flat line at 1.000.

Lesson 120 minYou are hereRead

All-pass systems

Put a zero at the mirror point of each pole and the gain stays at 1 for every frequency, while the group delay takes any shape you choose with the pole.

Z-plane with the unit circle and two zeros as open circles: one at −2, outside the circle (flipped from −0.5); the other at −1.25, outside the circle (flipped from −0.8). The gain curve |H| falls from 2.7 at Ω = 0 to 0.1 at π, the same for every flip. Stems of h[n] with square heads: 0.4, 1.3, 1. Faint open circles show the minimum-phase h, 1, 1.3, 0.4. Zeros inside: 0; h[0] = 0.40; phase at Ω = π: −360.0°.Z-plane with the unit circle and two zeros as open circles: one at −2, outside the circle (flipped from −0.5); the other at −1.25, outside the circle (flipped from −0.8). The gain curve |H| falls from 2.7 at Ω = 0 to 0.1 at π, the same for every flip. Stems of h[n] with square heads: 0.4, 1.3, 1. Faint open circles show the minimum-phase h, 1, 1.3, 0.4. Zeros inside: 0; h[0] = 0.40; phase at Ω = π: −360.0°.

Lesson 216 minYou are hereRead

Minimum phase

Flip a zero across the unit circle and the gain stays; see which version has the least lag, the earliest energy and a causal, stable inverse.

Three stacked stem plots against sample n, from 0 to 23. Input x[n]: the pulse 0.25, 0.75, 1, 0.75, 0.25 at n = 4 to 8, centred at n = 6. Through z⁻³: the same pulse at n = 7 to 11, centre 9.00. Through the all-pass: it dips to −0.188 at n = 5, peaks at 0.977 at n = 8, then trails a tail that halves every sample; centre 9.00. A dashed line marks each centre. Faint open circles show the input moved 3 samples later: they sit exactly on the z⁻³ stems and miss the all-pass stems.Three stacked stem plots against sample n, from 0 to 23. Input x[n]: the pulse 0.25, 0.75, 1, 0.75, 0.25 at n = 4 to 8, centred at n = 6. Through z⁻³: the same pulse at n = 7 to 11, centre 9.00. Through the all-pass: it dips to −0.188 at n = 5, peaks at 0.977 at n = 8, then trails a tail that halves every sample; centre 9.00. A dashed line marks each centre. Faint open circles show the input moved 3 samples later: they sit exactly on the z⁻³ stems and miss the all-pass stems.

Lesson 3 Essential20 minYou are hereRead

Linear-phase systems

See why a straight phase keeps a pulse's shape, show that symmetric taps give one, and read the four FIR types and their forced zeros.

The z-plane: two zeros on the unit circle at plus and minus 36 degrees (50 Hz), and two poles at the same angles. The poles are at radius r = 0.98. The gain is 0 at 50 Hz and close to 1 elsewhere; it crosses the 3 dB line at 48.4 and 51.6 Hz, a width of 3.2 Hz.The z-plane: two zeros on the unit circle at plus and minus 36 degrees (50 Hz), and two poles at the same angles. The poles are at radius r = 0.98. The gain is 0 at 50 Hz and close to 1 elsewhere; it crosses the 3 dB line at 48.4 and 51.6 Hz, a width of 3.2 Hz.

Lesson 4 Essential16 minYou are hereRead

Resonators, notches and combs

Put poles where the gain should rise and zeros where it should fall, then design a notch for mains hum, a DC blocker and a comb.

After this chapter

Where to go next.

The chapters either side, and the rest of Part VI in the library.

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