如何根据音频进行直方图均衡？

Question

I have already tried histogram equalization based on image and it works just fine.

But now I want to implement the same approach using audio frequency instead of image gray scale. Which means I would like to make the spectrum flatter. The sampling rate I use is 44.1kHz and want to make the frequency evenly spread to range 0-22050Hz, but the peak is still the highest.

Here is the spectrum:

And this is what I have tried:

I think the original histogram I plot is already wrong, I can't count the number of occurrences per frequency, or maybe I shouldn't do this at all. Somebody told me I need to use fft() but I have no idea how to do it.

Any help would be appreciated! Thanks

Here is the code for how I plot the spectrum:

import librosa
import numpy as np
import matplotlib.pyplot as plt
import math

file = 'example.wav'
y, sr = librosa.load(file, sr=None)

n_fft = 2048
S = librosa.stft(y, n_fft=n_fft, hop_length=n_fft//2)

S = abs(S)
D_AVG = np.mean(S, axis=1)

plt.figure(figsize=(25, 12))

plt.bar(np.arange(D_AVG.shape[0]), D_AVG)
x_ticks_positions = [n for n in range(0, n_fft // 2, n_fft // 16)]
x_ticks_labels = [str(sr / 2048 * n) + 'Hz' for n in x_ticks_positions]
plt.xticks(x_ticks_positions, x_ticks_labels)
plt.xlabel('Frequency')
plt.ylabel('dB')
plt.savefig('spectrum.png')

Answer 1

"Equalization" in the sense of making a flat frequency spectrum is usually done by a whitening transformation . This post on dsp.stackexchange might also be helpful. As Mark mentioned, this spectral equalization is different from histogram equalization in image processing.

Equalizing/whitening the spectrum of a signal:

1. Estimate the PSD. Given an array of samples x with sample rate fs , you can compute a robust estimate of the power spectral density (PSD) with scipy.signal.welch :

    f, psd = scipy.signal.welch(x, fs=fs)

   This function performs Welch's method . Basically, it divides up the signal into several segments, does FFT on each one, and averages the power spectra to get a good estimate of how much power x has at each frequency on average. The point of all this is it gets a more reliable frequency characterization than just taking one FFT of x as a whole.

2. Compute equalizer gain. Use eq_gain = 1 / (1e-6 + psd)**0.5 , or something similar, to determine the gain of the equalizer. The 1e-6 denominator offset is to avoid division by zero. It often happens that the PSD extremely small for some frequencies because, say, x went through an anti-aliasing filter that made some high frequency powers nearly zero.

3. Apply the equalizer gain. Finally, eq_gain needs to be applied to the signal x to equalize it. There are many ways this could be done, but one way is to use scipy.signal.firwin2 to turn the gains into an FIR filter,

    eq_filter = scipy.signal.firwin2(99, f, eq_gain, fs=fs)

   and use a convolution or scipy.signal.lfilter to apply the filter to x . You can then use scipy.signal.welch again to check that the PSD is flatter than before.