단위 계단 응답 문서의 검증 코드다.
"""단위 계단 응답 문서: s[n] = Σ_{k<=n} h[k], h[n] = s[n] - s[n-1], s(t) = ∫h, h = s' 를 확인한다."""
import math, random
random.seed(4)
hs = {k: random.uniform(-2, 2) for k in range(0, 15)}
h = lambda n: hs.get(n, 0.0)
u = lambda n: 1 if n >= 0 else 0
s = lambda n: sum(h(k) for k in range(-5, n + 1))
for n in range(-3, 20):
conv = sum(u(k) * h(n - k) for k in range(-5, 40)) # u * h
assert abs(conv - s(n)) < 1e-12
assert abs(s(n) - s(n - 1) - h(n)) < 1e-12
# 연속 시간: h(t) = e^{-2t}u(t) -> s(t) = (1 - e^{-2t})/2 u(t), s'(t) = h(t)
st = lambda t: (1 - math.exp(-2 * t)) / 2 if t > 0 else 0.0
for t in (0.2, 0.7, 2.0):
integ = sum(math.exp(-2 * (i + 0.5) * t / 20000) for i in range(20000)) * t / 20000
assert abs(integ - st(t)) < 1e-8
d = 1e-6
assert abs((st(t + d) - st(t - d)) / (2 * d) - math.exp(-2 * t)) < 1e-6
print("ALL CHECKS PASSED")