使用SciPy计算FFT时出现周期性幅度误差的问题排查

计算叠加正弦信号的FFT时，所得结果与初始输入不匹配。代码逻辑为：先生成数组，叠加多个频率不同但幅度恒定的正弦信号；随后复现STFT函数的计算逻辑（STFT也得到相同错误结果）。

当前核心问题：

• 选择2^n大小的窗口（如8192）时，FFT幅度会出现周期性误差，部分误差甚至超过15%
• 选择非2^n的窗口（如10000）时，该问题不会出现
• 尝试调整窗口宽度以获取正确幅度，结果与预期完全相反

相关代码如下：

import numpy as np
from scipy.fft import fft, fftshift

# Define the sample rate and the duration of the signal
sample_rate = 10000  # in Hz, common sample rate in audio
duration = 100     # in seconds, 10 milliseconds

# Create a time array
t = np.linspace(0, duration, int(sample_rate * duration), endpoint=False)

# Frequencies of the sine waves
frequencies = np.arange(1000,4000, 100)
result = np.zeros(t.shape[0])

# Create sine waves
for freq in frequencies:
    result += np.sin(2 * np.pi * freq * t)

def compute_fft_sections(data, sample_rate, x):
    n = len(data)
    section_length = 8192
    overlap = int(section_length * (0.5))

    fft_results = []
    normalised_sections=[]
    for i in range(0, n - overlap, overlap):

        # Stop if the end of the array is reached
        if i + section_length >= n:
            break

        section = data[i:i + section_length]

        # Apply Hanning window
        window = np.hanning(len(section))
        windowed_section = section * window

        # Normalize the windowed section
        # Here we normalize by the sum of the window values
        normalized_section = windowed_section / window.sum()
        normalised_sections.append(section)
        normalised_sections.append(normalized_section)

        # fft + downsample symetric
        fft_result = fft(normalized_section)
        fft_result = fftshift(fft_result)
        fft_result = fft_result[:len(fft_result)//2]
        #fft_result = fft_result[::len(fft_result)]

        # append to results
        fft_results.append(fft_result*2)

    return fft_results, normalised_sections

# Example usage
fft_sections, normalised_sections = compute_fft_sections(result,
                                    sample_rate=0.0001, 
                                    x=120)

附上周期性误差示意图：

内容的提问来源于stack exchange，提问作者Dev0ran