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Chapter 6

Systems described by equations

Three lessons in Part II, Systems. Read them in order, or start anywhere: a prerequisite is a link, never a gate.

Start with 6.1
0 of 3 read2 on the essential pathabout 85 minutes

Lessons in this chapter

Animation: the loop y[n] = x[n] + 0.5 × y[n−1], worked out one sample at a time from a single 1 at n = 0. 8 outputs so far: 1, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125. The delay box holds 0.015625. A smooth curve through the stem tops is labelled y[n] = 0.5 to the power n.Animation: the loop y[n] = x[n] + 0.5 × y[n−1], worked out one sample at a time from a single 1 at n = 0. 8 outputs so far: 1, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125. The delay box holds 0.015625. A smooth curve through the stem tops is labelled y[n] = 0.5 to the power n.

Lesson 1 Essential25 minYou are hereRead

Difference equations

Run a difference equation forward one sample at a time, compare a loop with no loop, and split an output into stored and pushed-in parts.

Step response y(t) of a first-order system with τ = 0.2 s, against time from 0 to 2 s. It rises from 0 toward 1. 10% point at 0.021 s. 90% point at 0.461 s. Bracket from the 10% to the 90% point: 0.44 s. Band from 0.98 to 1.02 drawn. 98% point, inside the band for good, at 0.78 s.Step response y(t) of a first-order system with τ = 0.2 s, against time from 0 to 2 s. It rises from 0 toward 1. 10% point at 0.021 s. 90% point at 0.461 s. Bracket from the 10% to the 90% point: 0.44 s. Band from 0.98 to 1.02 drawn. 98% point, inside the band for good, at 0.78 s.

Lesson 3 Essential30 minYou are hereRead

First- and second-order systems

Read a system's speed, overshoot and ringing from one or two numbers, and see why its poles come as a mirror pair.

After this chapter

Where to go next.

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

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