From Amplitude or FFT to dB

Question

I've a Python code which performs FFT on a wav file and plot the amplitude vs time / amplitude vs freq graphs. I want to calculate dB from these graphs (they are long arrays). I do not want to calculate exact dBA, I just want to see a linear relationship after my calculations. I've dB meter, I will compare it. Here is my code:

#!/usr/bin/env python
# -*- coding: utf-8 -*-

from __future__ import print_function
import scipy.io.wavfile as wavfile
import scipy
import scipy.fftpack
import numpy as np
from matplotlib import pyplot as plt

fs_rate, signal = wavfile.read("output.wav")
print ("Frequency sampling", fs_rate)
l_audio = len(signal.shape)
print ("Channels", l_audio)
if l_audio == 2:
    signal = signal.sum(axis=1) / 2
N = signal.shape[0]
print ("Complete Samplings N", N)
secs = N / float(fs_rate)
print ("secs", secs)
Ts = 1.0/fs_rate # sampling interval in time
print ("Timestep between samples Ts", Ts)
t = scipy.arange(0, secs, Ts) # time vector as scipy arange field / numpy.ndarray
FFT = abs(scipy.fft(signal))
FFT_side = FFT[range(N//4)] # one side FFT range
freqs = scipy.fftpack.fftfreq(signal.size, t[1]-t[0])
fft_freqs = np.array(freqs)
freqs_side = freqs[range(N//4)] # one side frequency range
fft_freqs_side = np.array(freqs_side)

makespositive = signal[44100:]*(-1)
logal = np.log10(makespositive)

sn1 = np.mean(logal[1:44100])
sn2 = np.mean(logal[44100:88200])
sn3 = np.mean(logal[88200:132300])
sn4 = np.mean(logal[132300:176400])

print(sn1)
print(sn2)
print(sn3)
print(sn4)

abs(FFT_side)
for a in range(500):
    FFT_side[a] = 0

plt.subplot(311)
p1 = plt.plot(t[44100:], signal[44100:], "g") # plotting the signal
plt.xlabel('Time')
plt.ylabel('Amplitude')

plt.subplot(312)
p1 = plt.plot(t[44100:], logal, "r") # plotting the signal
plt.xlabel('Time')
plt.ylabel('Amplitude')

plt.subplot(313)
p3 = plt.plot(freqs_side, abs(FFT_side), "b") # plotting the positive fft spectrum
plt.xlabel('Frequency (Hz)')
plt.ylabel('Count single-sided')
plt.show()

First plot is amplitude vs time, second one is logarithm of previous graph and the last one is FFT. In sn1,sn2 part I tried to calculate dB from signal. First I took log and then calculated mean value for each second. It did not give me a clear relationship. I also tried this and did not worked.

import numpy as np
import matplotlib.pyplot as plt
import scipy.io.wavfile as wf

fs, signal = wf.read('output.wav')  # Load the file
ref = 32768  # 0 dBFS is 32678 with an int16 signal

N = 8192
win = np.hamming(N)                                                       
x = signal[0:N] * win                             # Take a slice and multiply by a window
sp = np.fft.rfft(x)                               # Calculate real FFT
s_mag = np.abs(sp) * 2 / np.sum(win)              # Scale the magnitude of FFT by window and factor of 2,
                                                  # because we are using half of FFT spectrum
s_dbfs = 20 * np.log10(s_mag / ref)               # Convert to dBFS
freq = np.arange((N / 2) + 1) / (float(N) / fs)   # Frequency axis
plt.plot(freq, s_dbfs)
plt.grid(True)

So which steps should I perform? (Sum/mean all freq amplitudes then take log or reverse, or perform it for signal etc.)

import numpy as np
import matplotlib.pyplot as plt
import scipy.io.wavfile as wf

fs, signal = wf.read('db1.wav') 

signal2 = signal[44100:]
chunk_size = 44100
num_chunk  = len(signal2) // chunk_size
sn = []
for chunk in range(0, num_chunk):
    sn.append(np.mean(signal2[chunk*chunk_size:(chunk+1)*chunk_size].astype(float)**2))

print(sn)

logsn = 20*np.log10(sn)

print(logsn)

Output:

[4.6057844427695475e+17, 5.0025315250895744e+17, 5.028593412665193e+17, 4.910948397471887e+17]
[353.26607217 353.98379668 354.02893044 353.82330741]

Answer 1

A decibel meter measures a signal's mean power. So from your time signal recording you can calculate the mean signal power with:

chunk_size = 44100
num_chunk  = len(signal) // chunk_size
sn = []
for chunk in range(0, num_chunk):
  sn.append(np.mean(signal[chunk*chunk_size:(chunk+1)*chunk_size]**2))

Then the corresponding mean signal power in decibels is simply given by:

logsn = 10*np.log10(sn)

A equivalent relationship could also be obtained for a frequency domain signal with the use of Parseval's theorem, but in your case would require unecessary FFT computations (this relationship is mostly useful when you already have to compute the FFT for other purposes).

Note however that depending on what you compare there may be some (hopefully small) discrepancies. For example the use of non-linear amplifier and speakers would affect the relationship. Similarly ambient noises would add to the measured power by the decibel meter.