203 lines
7.4 KiB
Python
203 lines
7.4 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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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, stable_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, stable_poles):
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basis.extend(block)
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return basis
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# ---------- 可选：离散再正交 (加权 QR) ----------
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# def trapezoid_weights(freqs: np.ndarray):
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# if len(freqs) == 1:
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# return np.ones(1)
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# df = np.diff(freqs)
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# w = np.zeros_like(freqs, dtype=float)
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# w[0] = 0.5 * df[0]
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# w[-1] = 0.5 * df[-1]
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# if len(freqs) > 2:
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# w[1:-1] = 0.5 * (df[:-1] + df[1:])
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# return w
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def weighted_qr_from_basis(basis_cols: list[np.ndarray], weights: np.ndarray | None = None):
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A = np.column_stack(basis_cols) # (Nf, M)
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if weights is None:
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sw = np.ones(A.shape[0])
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else:
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sw = np.sqrt(weights.real)
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Aw = sw[:, None] * A
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Qw, R = np.linalg.qr(Aw)
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Phi = Qw / (sw[:, None] + 1e-30)
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return Phi, R # Raw = Phi R
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def check_discrete_orthogonality(Phi: np.ndarray, w: np.ndarray):
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G = Phi.conj().T @ (w[:, None] * Phi)
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off = np.max(np.abs(G - np.eye(G.shape[0])))
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return G, off
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def verify_orthonormal(Phi: np.ndarray, w: np.ndarray, atol=1e-10, rtol=1e-8):
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"""
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返回:
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G : Gram 矩阵 (Φ^H W Φ)
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max_off : 最大非对角幅值
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diag_err : max |diag(G)-1|
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passed : 是否满足阈值
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"""
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G = Phi.conj().T @ (w[:, None] * Phi)
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I = np.eye(G.shape[0])
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diag_err = np.max(np.abs(np.diag(G) - 1.0))
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max_off = np.max(np.abs(G - I + np.diag(np.diag(G) - 1.0)))
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passed = (diag_err <= atol) and (max_off <= rtol)
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return G, max_off, diag_err, passed
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def omega_weights(freqs_hz: np.ndarray):
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"""
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基于 ω=2πf 的梯形法得到 w_ω = Δω/(2π)，使得
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(1/2π) ∫_{-∞}^{∞} → Σ w_ω,k
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"""
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f = freqs_hz
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if len(f) == 1:
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return np.ones(1)
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df = np.diff(f)
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w_f = np.zeros_like(f)
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w_f[0] = 0.5 * df[0]
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w_f[-1] = 0.5 * df[-1]
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if len(f) > 2:
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w_f[1:-1] = 0.5 * (df[:-1] + df[1:])
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# dω = 2π df, (1/2π) * dω = df ⇒ 直接 w_f 就是 w_ω
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return w_f # 已等价于 Δω/(2π)
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# ------------------ 示例 ------------------
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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("解析基函数数量 =", len(basis))
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print("前5个基函数示例 (每个前10个频点):")
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for i in range(5):
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print(f"φ_{i}:", basis[i][:10])
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w = omega_weights(freqs)
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Phi_num, R = weighted_qr_from_basis(basis, w)
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Gram, off = check_discrete_orthogonality(Phi_num, w)
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print("离散 Gram 最大非对角元素 =", off)
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print("R 形状:", R.shape)
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# 验证 Raw ≈ Phi R
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raw = np.column_stack(basis)
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err = np.max(np.abs(raw - Phi_num @ R))
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print("重构误差 ||Raw - Phi R||_∞ =", err)
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# 验证正交性
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print("离散 Gram 矩阵 (前5x5):")
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print(Gram[:5, :5])
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Gcheck, max_off, diag_err, ok = verify_orthonormal(Phi_num, w)
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print(f"Diag 误差={diag_err:.3e}, Max off={max_off:.3e}, Orthonormal={ok}")
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# 额外: 验证 R
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# raw = Φ R => R ≈ Φ^H W raw (因为 Φ^H W Φ = I)
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R_alt = Phi_num.conj().T @ (w[:,None] * raw)
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print("R 差异 ||R - R_alt||_max =", np.max(np.abs(R - R_alt)))
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