174 lines
6.6 KiB
Python
174 lines
6.6 KiB
Python
import numpy as np
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# ------------------------------------------------------------
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# 按论文公式 (9)(10)(11) 生成 Muntz–Laguerre 正交有理基 (解析形式)：
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#
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# 给定稳定极点集合 {p_k} (Re(p_k)<0)。论文记法中使用 -a_k，其中 Re(a_k)>0。
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# 对应关系：p_k = -a_k ⇒ a_k = -p_k, Re(a_k)= -Re(p_k) >0
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#
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# 连续内积意义下（沿 jω 轴积分）这些 φ_k 解析正交。离散频率采样后数值上
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# 可能偏离，可再用加权 QR 做数值再正交（可选）。
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#
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# 公式在“稳定极点 p 表达”下的改写：
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# (原) 实极点: φ_p(s) = sqrt(2 Re(a_p)) / (s + a_p) * Π (s - a_i^*)/(s + a_i)
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# 变换 a_p = -p ⇒ Re(a_p)= -Re(p) = σ >0 且 (s + a_p) = (s - p)
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# 且乘积 (s - a_i^*)/(s + a_i) = (s + p_i^*)/(s - p_i)
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# ⇒ φ_p(s) = sqrt(-2 Re(p)) / (s - p) * Π_{i<p} (s + p_i^*)/(s - p_i)
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#
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# 复对 (p, p*)，取 imag(p)>0 的 p 作为首：
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# (原) φ_p = sqrt(2 Re(a_p)) (s - |a_p|)/[(s + a_p)(s + a_p^*)] * Π(...)
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# (原) φ_{p+1}= sqrt(2 Re(a_p)) (s + |a_p|)/[(s + a_p)(s + a_p^*)] * Π(...)
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# 代入 a_p=-p：
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# Re(a_p)= -Re(p)=σ>0, (s + a_p) = (s - p), (s + a_p^*)=(s - p^*)
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# |a_p| = |p|
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# 乘积同上 ⇒ Π_{i<p} (s + p_i^*)/(s - p_i)
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#
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# ⇒ φ_p(s) = sqrt(-2 Re(p)) (s - |p|)/[(s - p)(s - p^*)] * Π_{i<p} (s + p_i^*)/(s - p_i)
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# φ_{p^*}(s)= sqrt(-2 Re(p)) (s + |p|)/[(s - p)(s - p^*)] * Π_{i<p} (s + p_i^*)/(s - p_i)
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#
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# product 递推维护：prod_k = Π_{i≤k} (s + p_i^*)/(s - p_i)
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# 对复对要顺序乘两次（p 与 p*）。
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# ------------------------------------------------------------
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def generate_laguerre_basis(poles:list|np.ndarray, s: np.ndarray):
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basis = np.zeros((len(poles)+1,len(s)),dtype=complex)
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product = np.ones(len(s),dtype=complex)
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basis[0] = np.ones(len(s),dtype=complex) # φ_0 = 1
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i = 0
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while i < len(poles):
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if np.real(poles[i]) >= 0:
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raise ValueError(f"极点必须在左半平面: {poles[i]}")
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# 复对首 (正虚部)
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if np.iscomplex(poles[i]) and np.imag(poles[i]) > 0:
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if i + 1 >= len(poles):
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raise ValueError("复极点缺少共轭")
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pc = poles[i + 1]
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if not np.isclose(pc, np.conj(poles[i])):
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raise ValueError("复极点未按 (p, p*) 顺序排列 (正虚部在前)")
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sigma = -np.real(poles[i]) # >0
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scale = np.sqrt(2 * sigma)
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r = np.abs(poles[i])
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denom = (s - poles[i]) * (s - pc)
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# 两个基函数
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phi_p = scale * (s - r) / denom * product
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phi_pc = scale * (s + r) / denom * product
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# product 先乘 (s + p^*)/(s - p)，再乘 (s + p)/(s - p^*)
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product = product * (s + pc) / (s - poles[i])
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product = product * (s + poles[i]) / (s - pc)
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basis[i + 1] = phi_p
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basis[i + 2] = phi_pc
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i += 2
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continue
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# 复对次 (负虚部) —— 应该被首元素处理，出现表示顺序错误
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if np.iscomplex(poles[i]) and np.imag(poles[i]) < 0:
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raise ValueError("检测到负虚部复极点但其共轭尚未处理，请将正虚部成员放在前面。")
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# 实极点
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sigma = -np.real(poles[i])
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if sigma <= 0:
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raise ValueError("实极点实部应为负 (稳定)。")
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scale = np.sqrt(2 * sigma)
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phi = scale / (s - poles[i]) * product
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# 更新乘积
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product = product * (s + poles[i]) / (s - poles[i])
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i += 1
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basis[i + 1] = phi
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return basis
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# class MuntzLaguerreIterator:
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# def __init__(self, s: np.ndarray, stable_poles: list | np.ndarray):
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# """
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# s: 复频率数组 (Nf,), s = j 2π f
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# stable_poles: 稳定极点列表 (Re<0). 复共轭对要求正虚部在前 (p, p*).
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# """
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# self.s = np.asarray(s, dtype=complex)
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# self.poles = list(stable_poles)
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# self.N = len(self.poles)
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# self.k = 0
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# # 初始化乘积 Π_{i<p} (s + p_i^*)/(s - p_i)
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# self.product = np.ones_like(self.s, dtype=complex)
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# def __iter__(self):
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# return self
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# def __next__(self):
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# if self.k >= self.N:
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# raise StopIteration
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# p = self.poles[self.k]
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# if np.real(p) >= 0:
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# raise ValueError(f"极点必须在左半平面: {p}")
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# # 复对首 (正虚部)
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# if np.iscomplex(p) and np.imag(p) > 0:
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# if self.k + 1 >= self.N:
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# raise ValueError("复极点缺少共轭")
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# pc = self.poles[self.k + 1]
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# if not np.isclose(pc, np.conj(p)):
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# raise ValueError("复极点未按 (p, p*) 顺序排列 (正虚部在前)")
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# sigma = -np.real(p) # >0
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# scale = np.sqrt(2 * sigma)
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# r = np.abs(p)
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# denom = (self.s - p) * (self.s - pc)
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# # 两个基函数
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# phi_p = scale * (self.s - r) / denom * self.product
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# phi_pc = scale * (self.s + r) / denom * self.product
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# # product 先乘 (s + p^*)/(s - p)，再乘 (s + p)/(s - p^*)
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# self.product = self.product * (self.s + pc) / (self.s - p)
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# self.product = self.product * (self.s + p) / (self.s - pc)
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# self.k += 2
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# return [phi_p, phi_pc]
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# # 复对次 (负虚部) —— 应该被首元素处理，出现表示顺序错误
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# if np.iscomplex(p) and np.imag(p) < 0:
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# raise ValueError("检测到负虚部复极点但其共轭尚未处理，请将正虚部成员放在前面。")
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# # 实极点
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# sigma = -np.real(p)
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# if sigma <= 0:
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# raise ValueError("实极点实部应为负 (稳定)。")
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# scale = np.sqrt(2 * sigma)
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# phi = scale / (self.s - p) * self.product
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# # 更新乘积
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# self.product = self.product * (self.s + p) / (self.s - p)
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# self.k += 1
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# return [phi]
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# def generate_muntz_laguerre_basis(s: np.ndarray, init_poles: list | np.ndarray):
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# """
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# 生成完整基函数列表: [φ_0=1, φ_1, φ_2, ...]
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# """
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# basis = [np.ones_like(s, dtype=complex)]
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# for block in MuntzLaguerreIterator(s, init_poles):
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# basis.extend(block)
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# return basis
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# if __name__ == "__main__":
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# # 示例稳定极点 (复对正虚部在前)
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# stable_poles = [
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# -0.8e9,
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# -1.0e9 + 2.5e9j,
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# -1.0e9 - 2.5e9j,
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# -2.2e9
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# ]
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# freqs = np.linspace(1e8, 8e9, 400)
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# s = 1j * 2 * np.pi * freqs
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# basis = generate_muntz_laguerre_basis(s, stable_poles)
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# print(f"生成 {len(basis)} 个基函数,{basis}") |