Discrete-time systems
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| DC gain | DC gain |
| iteration/ˌɪtəˈreɪʃn/ | Repeating a sequence of instructions or calculations. |
Iterate, don't guess
- $y[n] = a\,y[n-1] + x[n]$ on a step gives geometric partial sums: $y[n] = \frac{1-a^{n+1}}{1-a} \to \frac{1}{1-a}$ for $|a| < 1$ — the DC gain.
![Iteration of y[n] = 0.5·y[n−1] + x[n] onto the limit 2](/handout-media/university_electrical_engineering/assets/06-iteration.png?v=1793069544)
$a = 0.5$: $1, 1.5, 1.75, 1.875 \to 2$ ✓; the gap halves per step, $0.5^{11} \approx 0.05\%$ after 10 steps.
{exercise:uni-ee0661-iterate}
y[n] = 0.5·y[n−1] + x[n], y[−1] = 0, unit step: y[2] equals —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
y[0] = 1, y[1] = 1.5, y[2] = 0.75 + 1 = 1.75 — the geometric partial sum closing on the DC gain 2.
The DC gain of y[n] = a·y[n−1] + x[n] with |a| < 1 is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The geometric series sums to 1/(1−a) — the discrete echo of the continuous integral condition.
Roots can ring
- $a = -0.8$ oscillates toward its limit (sign flips every sample) yet stays stable — $|a| < 1$ is the whole condition; a negative or complex root is the discrete fingerprint of ringing.
{exercise:uni-ee0662-osc}
y[n] = −0.8·y[n−1] + x[n] on a unit step —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
a magnitude 0.8 below 1 is the only stability condition; the negative sign makes the approach oscillate (1, 0.2, 0.84, 0.328, …) toward 1/1.8 = 0.556.