Signal representations and system properties
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| linearity/lɪˈnɪərɪti/ | tính tuyến tính |
| time invariance | tính bất biến theo thời gian |
| causality | tính nhân quả |
| BIBO stability | ổn định BIBO |
| system/ˈsɪstəm/ | hệ thống |
Four tests, four verdicts
- Linearity: feed $ax_{1} + bx_{2}$; the output must be $ay_{1} + by_{2}$. Time invariance: shift the input by $T$; the output must shift by exactly $T$. Causality: no output before the input. BIBO stability: bounded in, bounded out — for LTI, $\int |h|\,dt < \infty$.
Is $y(t) = x(2t)$ time-invariant? Shift-then-system gives $u(2t-1) = u(t-\tfrac{1}{2})$; system-then-shift gives $u(t-1)$. Half shift vs full — time-varying ✓.
{exercise:uni-ee0611-scaling}
The system y(t) = x(2t) is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Shift-then-system gives a HALF shift; system-then-shift gives the full shift. Time scaling rescales shifts, so the system is time-varying (it is still linear).
The integrator h(t) = u(t) is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The integral of |h| diverges, so SOME bounded input (a ramp) gives an unbounded output. Boundedness of h itself is not the test.
The two that unlock everything
- Linearity + time invariance together are the ticket to convolution (6.2), Fourier (6.3) and the z-transform (6.7). The other two properties qualify the answer but never replace it.
{exercise:uni-ee0612-lti}
For an LTI system, BIBO stability is equivalent to —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The integral test covers EVERY bounded input at once. Causality is independent; one probe input proves nothing.