频域分片采样，ifft重构后，子带交叠部分是否会有失真？

云深无际

发布于 2026-01-07 13:38:30

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稿子一大堆，写的多了又怕大家说我串稀式更新，嘤嘤嘤！

之前不是写频域拼接吗？有人问：频域分片采样，ifft重构后，子带交叠部分是否会有失真？非常好的问题，这其实是频谱拼接式 ADC / 频域分片采样 (Frequency-Sliced Sampling, FSS) 的核心关键之一。（我不确定有没有什么实际的意义）

基本原理回顾

假设原信号的频谱在内，我们将其划分为多个频率子带：

其中第个子带的带宽为，中心频率为。

每个子带经过带通滤波后，被分别下变频到基带：

再各自以较低采样率采样；最后在数字域中，通过上变频 + 拼接 + IFFT（或 IDFT 合成）重构回去。

频域拼接（IFFT 重构）的关键条件

在理想情况下，如果：各个子带的 带宽完全等分；各子带间 频率分界点无重叠也无空隙；每个子带采样前的 模拟带通滤波器 边缘响应理想（即矩形带通）；数字重构时 子带的幅度和相位都完美对齐（含 group delay）；

则在 IFFT 拼接后：

不会有任何失真。

实际中出现子带交叠失真的三种典型原因

（1）滤波器过渡带引起的幅度交叠

如果每个子带的带通滤波器不是理想矩形（实际上不可能），其过渡带会造成频谱交叠，如下图所示：

  X0(f)    X1(f)
  ┌────┐   ┌────┐
     └───┬──┘
         ↑ 过渡带交叠区

在重构时，如果两个子带的重叠部分都被保留，则叠加后该频段能量会双计 → 幅度失真（Amplitude Ripple），解决办法是在交叠区加 cosine taper / overlap-add 窗函数（类似 STFT 拼接）。

（2）相位未对齐（group delay mismatch）

每个子带可能来自不同通道（不同滤波器、不同ADC路径），其相位响应或群延迟略有不同：

拼接时，频域相位突变 → 时域表现为波形边界振铃、过冲，类似“stitching glitch”或“Gibbs效应”；需要校准每个子带的 group delay；在数字域中补偿相位；或使用平滑的 crossfade 方式拼接。

（3）FFT/IFFT 网格不匹配或带宽边界对齐误差

如果分片不是在 FFT 网格上整齐切割（例如频率分辨率 Δf 不对齐），则 IFFT 时每个子带对应的 bin 不连续，出现频谱泄漏 → 时域失真；需要统一采样点数与 FFT 长度；子带频率边界必须是 FFT bin 的整数倍；或使用小重叠（50% overlap）+ 窗函数拼接法。

用数学公式看“交叠失真”

设两个相邻子带为：

其中与的过渡带有重叠区。

拼接结果为：

如果在交叠区，就会出现失真：若 >1 → 幅度过高；若 <1 → 幅度缺损；若相位不一致 → 相位失真。

因此理想拼接需满足：

这正是“Perfect Reconstruction Filter Bank（完美重构滤波器组）”的基本条件。

总结一下

要恢复的部分	理想情况	现实失真来源	解决方案
幅度响应	子带边界无重叠	过渡带重叠导致幅度双计	overlap-add 窗平滑
相位响应	各通道群延一致	相位不对齐造成振铃	group delay 校准
频率分辨率	FFT bin 完全对齐	不对齐导致泄漏	统一采样点/FFT 长度
滤波器设计	PRFB 满足 H₁+H₂=1	滤波器设计不完美	采用Cosine-Modulated FB

仿真

读者肯定说，教练，我看不懂！

“两个频域子带有 10% 过渡带交叠时，IFFT 重构的时域波形与理想波形对比（含幅度失真与相位畸变）”

绘制出：频域拼接前后幅度响应；IFFT 后时域波形差异；幅度误差随交叠比例变化曲线。

STDOUT/STDERR
=== Reconstruction RMS Error (time-domain, normalized) ===
Case A (Bad rectangular + overlap double-count): 0.099867
Case B (Good complementary raised-cosine):      0.000000
Case C (Good mag, but subband-2 delay τ=3.0):  0.490463

Case A: Bad masks (rectangular + overlap double-count)

子带有重叠但直接相加（双计）→ 失真

图“Case A”里 H1+H2 在交叠区等于 2，导致频域能量双计；时域重构 RMS 误差 ≈ 0.0999；误差谱主要集中在分界处（图“Error spectra around crossover”）。

在交叠区做互补 crossfade（raised-cosine，保证 H1+H2=1）→ 无幅度失真

图“Case B”里 H1+H2 ≡ 1；时域重构 RMS 误差 = 0（数值精度内）；这就是完美重构滤波器组（PRFB）的核心条件：交叠区满足。

群延迟不一致（相位错配）→ 即使幅度互补也会失真

给子带2加了 τ=3 个采样的额外延迟；有效合成增益在分界附近出现凹陷（图“Case C”）；时域 RMS 误差升至 0.4905，局部波形出现振铃（最后一张图）。

扩展到 K>2 子带（均匀或非均匀滤波器组）

仿真完成过渡带宽度 vs 误差、延迟偏差 vs 误差画参数扫描热力图。

=== Single-case RMS (vs x_ref) ===
Ideal: 6.274424e-17
Delay mismatch τ=2 on band2: 0.453603
Gain mismatch +0.5 dB on band2: 0.014614

=== RMSE vs overlap for τ in {0,1,2,3,5} (K=4) ===
tau=0.0: 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000
tau=1.0: 0.4081, 0.4071, 0.4045, 0.4019, 0.3993, 0.3967, 0.3941, 0.3914, 0.3888
tau=2.0: 0.4600, 0.4573, 0.4547, 0.4519, 0.4492, 0.4463, 0.4435, 0.4406, 0.4376
tau=3.0: 0.1978, 0.1941, 0.1918, 0.1895, 0.1873, 0.1852, 0.1833, 0.1814, 0.1797
tau=5.0: 0.4305, 0.4293, 0.4279, 0.4264, 0.4247, 0.4229, 0.4210, 0.4189, 0.4167

=== RMSE vs gain mismatch (dB) for overlap in {0,0.1,0.2,0.3,0.4} (τ=0) ===
overlap=0.0: 0.0272, 0.0207, 0.0140, 0.0071, 0.0000, 0.0073, 0.0148, 0.0225, 0.0305
overlap=0.1: 0.0269, 0.0205, 0.0138, 0.0070, 0.0000, 0.0072, 0.0147, 0.0223, 0.0302
overlap=0.2: 0.0265, 0.0202, 0.0137, 0.0069, 0.0000, 0.0071, 0.0145, 0.0220, 0.0298
overlap=0.3: 0.0262, 0.0199, 0.0135, 0.0068, 0.0000, 0.0070, 0.0143, 0.0217, 0.0294
overlap=0.4: 0.0258, 0.0197, 0.0133, 0.0067, 0.0000, 0.0069, 0.0141, 0.0214, 0.0290

只要交叠区做“互补窗”使，则在幅度层面可以做到完美重构。

把 4 子带的互补 raised-cosine 交叉淡入淡出实现为：每个边界两侧各一半过渡，交叠内与互补，整带求和恒等于 1；这就是 K 通道完美重构滤波器组（PRFB）的“Partition of Unity”。

相位/群延迟不一致会引入显著失真（即使幅度互补）。

=== K=4 single-case RMS errors ===
Ideal (mag+phase perfect): 5.196418e-01
Delay mismatch (band2 τ=2.0): 0.836231
Gain mismatch (band2 +0.5 dB):  0.512645
=== Sweep summary (delay mismatch) ===
Min RMSE=5.008023e-01 at τ=0.000 samples, overlap=0.000 of subband
=== Sweep summary (gain mismatch) ===

仿真（K=4，子带2额外延迟）的单例结果：幅度与相位都理想：RMS 误差 ≈ 6.5e−17（数值零）。

仅子带2 延迟个采样：RMS ≈ 0.4536（很大）。

仅子带2 增益 +0.5 dB：RMS ≈ 0.0146（比相位错配小很多）。

对齐的时候先做相位/群延迟标定，再谈幅度微调相位错一点点，代价远大于轻微增益误差。

后记（

“频域分片采样，IFFT 重构后，子带交叠部分是否会有失真？”

不会——前提是：交叠区采用互补窗，满足；子带之间相位/群延迟已对齐（或在数字域补偿）；分界点与 FFT 网格尽可能对齐。

会——如果：

交叠直接相加导致双计（幅度起伏）；相位/群延迟不一致（会出现明显失真/振铃）；网格不对齐且无交叠窗，导致泄漏与拼接撕裂。

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(2)

# ---------- Test signal ----------
N = 65536
fs = 1.0
t = np.arange(N) / fs

# Multisine + noise (real)
num_tones = 160
freqs = np.linspace(0.005, 0.49, num_tones)
phases = 2*np.pi*np.random.rand(num_tones)
x = np.zeros(N)
for f, phi in zip(freqs, phases):
    x += np.cos(2*np.pi*f*t + phi)
x += np.random.randn(N)*0.15
x = x / np.std(x)

X = np.fft.fftshift(np.fft.fft(x))
freq = np.fft.fftshift(np.fft.fftfreq(N, d=1/fs))

# ---------- Helper: build K complementary masks with raised-cosine overlaps ----------
def build_K_masks(freq, K=4, f_low=0.0, f_high=0.49, overlap_frac=0.1):
    Hs = [np.zeros_like(freq, dtype=float) for _ in range(K)]
    B = (f_high - f_low) / K
    ov = overlap_frac * B
    for k in range(K):
        fk_left = f_low + k*B
        fk_right = fk_left + B
        a = fk_left + ov/2.0
        b = fk_right - ov/2.0
        if a < b:
            idx_core = (freq >= a) & (freq <= b)
            Hs[k][idx_core] = 1.0
        if k > 0 and ov > 0:
            aL = fk_left - ov/2.0
            bL = fk_left + ov/2.0
            idx = (freq >= aL) & (freq <= bL)
            w = 0.5*(1 - np.cos(np.pi*(freq[idx]-aL)/(bL-aL)))  # 0->1
            Hs[k][idx] += w
        if k < K-1 and ov > 0:
            aR = fk_right - ov/2.0
            bR = fk_right + ov/2.0
            idx = (freq >= aR) & (freq <= bR)
            w = 0.5*(1 + np.cos(np.pi*(freq[idx]-aR)/(bR-aR)))  # 1->0
            Hs[k][idx] += w
    return Hs

def reconstruct(X, freq, Hs, tau_list=None, gain_list=None):
    K = len(Hs)
    Y = np.zeros_like(X, dtype=complex)
    for k in range(K):
        Hk = Hs[k]
        phase = 1.0
        if tau_list is not None:
            tau = tau_list[k]
            phase = np.exp(-1j*2*np.pi*freq*tau)
        gain = 1.0 if gain_list is None else gain_list[k]
        Y += (gain * Hk) * phase * X
    y = np.fft.ifft(np.fft.ifftshift(Y)).real
    return y, Y

def rms(a):
    return np.sqrt(np.mean(a**2))

# Target band for perfect reconstruction
K = 4
overlap_frac = 0.12
f_low = 0.0
f_high = 0.49
Hs = build_K_masks(freq, K=K, f_low=f_low, f_high=f_high, overlap_frac=overlap_frac)

# Reference: bandlimited version of x (what the system intends to pass)
Hsum = np.zeros_like(freq)
for Hk in Hs:
    Hsum += Hk
Y_ref = Hsum * X
x_ref = np.fft.ifft(np.fft.ifftshift(Y_ref)).real

# Perfect case
y_ideal, Y_ideal = reconstruct(X, freq, Hs)
e_ideal = y_ideal - x_ref

# Delay mismatch on band2
tau = 2.0
tau_list = [0.0, 0.0, tau, 0.0]
y_delay, Y_delay = reconstruct(X, freq, Hs, tau_list=tau_list)
e_delay = y_delay - x_ref

# Gain mismatch on band2 (+0.5 dB)
g_lin = 10**(0.5/20)
gain_list = [1.0, 1.0, g_lin, 1.0]
y_gain, Y_gain = reconstruct(X, freq, Hs, gain_list=gain_list)
e_gain = y_gain - x_ref

print("=== K=4 single-case RMS errors (vs x_ref in [0, 0.49]) ===")
print(f"Ideal (mag+phase perfect): {rms(e_ideal):.6e}")
print(f"Delay mismatch (band2 τ={tau}): {rms(e_delay):.6f}")
print(f"Gain mismatch (band2 +0.5 dB):  {rms(e_gain):.6f}")

# Plot masks
plt.figure(figsize=(8,4.5))
for k, Hk in enumerate(Hs):
    plt.plot(freq, Hk, label=f"H{k}")
plt.plot(freq, Hsum, label="Sum", linewidth=2)
plt.xlim(0.0, 0.5)
plt.ylim(-0.05, 1.2)
plt.title("K=4 complementary masks and their sum")
plt.xlabel("Normalized frequency (cycles/sample)")
plt.ylabel("Gain")
plt.legend()
plt.tight_layout()
plt.show()

# Heatmap sweep: RMSE vs overlap and delay (vs x_ref)
ov_list = np.linspace(0.0, 0.4, 21)
tau_list_grid = np.linspace(0.0, 5.0, 26)
RMSE = np.zeros((len(tau_list_grid), len(ov_list)))

for i, tau_val in enumerate(tau_list_grid):
    for j, ov_frac in enumerate(ov_list):
        Hs_tmp = build_K_masks(freq, K=K, f_low=f_low, f_high=f_high, overlap_frac=ov_frac)
        Hsum_tmp = np.zeros_like(freq)
        for Hk in Hs_tmp: Hsum_tmp += Hk
        x_ref_tmp = np.fft.ifft(np.fft.ifftshift(Hsum_tmp * X)).real
        tl = [0.0, 0.0, tau_val, 0.0]
        y_tmp, _ = reconstruct(X, freq, Hs_tmp, tau_list=tl)
        RMSE[i,j] = rms(y_tmp - x_ref_tmp)

plt.figure(figsize=(8,4.5))
plt.imshow(RMSE, aspect='auto', origin='lower',
           extent=[ov_list[0], ov_list[-1], tau_list_grid[0], tau_list_grid[-1]])
plt.colorbar(label="RMS error")
plt.xlabel("Overlap fraction of subband width")
plt.ylabel("Delay mismatch τ (samples)")
plt.title("RMS error vs overlap and delay mismatch (K=4)")
plt.tight_layout()
plt.show()

# Slices: RMSE vs overlap for selected τ
plt.figure(figsize=(8,4.5))
for tau_val in [0.0, 1.0, 2.0, 3.0, 5.0]:
    idx = np.argmin(np.abs(tau_list_grid - tau_val))
    plt.plot(ov_list, RMSE[idx,:], label=f"τ={tau_val}")
plt.xlabel("Overlap fraction of subband width")
plt.ylabel("RMS error")
plt.title("RMSE vs overlap for different delay mismatches")
plt.legend()
plt.tight_layout()
plt.show()

# Heatmap: RMSE vs overlap and gain mismatch (no delay)
g_db_list = np.linspace(-1.0, 1.0, 21)
RMSEg = np.zeros((len(g_db_list), len(ov_list)))
for i, gdb in enumerate(g_db_list):
    glin = 10**(gdb/20)
    for j, ov_frac in enumerate(ov_list):
        Hs_tmp = build_K_masks(freq, K=K, f_low=f_low, f_high=f_high, overlap_frac=ov_frac)
        Hsum_tmp = np.zeros_like(freq)
        for Hk in Hs_tmp: Hsum_tmp += Hk
        x_ref_tmp = np.fft.ifft(np.fft.ifftshift(Hsum_tmp * X)).real
        y_tmp, _ = reconstruct(X, freq, Hs_tmp, gain_list=[1.0, 1.0, glin, 1.0])
        RMSEg[i,j] = rms(y_tmp - x_ref_tmp)

plt.figure(figsize=(8,4.5))
plt.imshow(RMSEg, aspect='auto', origin='lower',
           extent=[ov_list[0], ov_list[-1], g_db_list[0], g_db_list[-1]])
plt.colorbar(label="RMS error")
plt.xlabel("Overlap fraction of subband width")
plt.ylabel("Gain mismatch on band2 (dB)")
plt.title("RMS error vs overlap and gain mismatch (τ=0, K=4)")
plt.tight_layout()
plt.show()

# Slices: RMSE vs gain mismatch for selected overlaps
plt.figure(figsize=(8,4.5))
for ov_frac in [0.0, 0.1, 0.2, 0.3, 0.4]:
    j = np.argmin(np.abs(ov_list - ov_frac))
    plt.plot(g_db_list, RMSEg[:,j], label=f"overlap={ov_frac}")
plt.xlabel("Gain mismatch on band2 (dB)")
plt.ylabel("RMS error")
plt.title("RMSE vs gain mismatch for different overlaps (τ=0)")
plt.legend()
plt.tight_layout()
plt.show()

# Summary
i_min, j_min = np.unravel_index(np.argmin(RMSE), RMSE.shape)
print("=== Sweep summary (delay mismatch) ===")
print(f"Min RMSE={RMSE[i_min,j_min]:.6e} at τ={tau_list_grid[i_min]:.3f} samples, overlap={ov_list[j_min]:.3f}")

i2_min, j2_min = np.unravel_index(np.argmin(RMSEg), RMSEg.shape)
print("=== Sweep summary (gain mismatch) ===")
print(f"Min RMSE={RMSEg[i2_min,j2_min]:.6e} at gain_err={g_db_list[i2_min]:.3f} dB, overlap={ov_list[j2_min]:.3f}"

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原始发表：2025-11-02，如有侵权请联系 cloudcommunity@tencent.com 删除

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