Scale FFT frequency range for a bars graph

Question

I'm trying to show frequency spectrum for audio data as a bars graph using FFT.

If I have data for a sine wave at 440 Hz and with sampling rate 44100 Hz and analyze its first 1024 samples I get an array with 1024 FFT values. I know that the first half of those values (512) represents the frequency range equal to half the sampling rate (22050 Hz).

So to draw a bars graph I just take the first 15-20 values from the FFT array, calculate the frequency range for each element and plot it.

• In the case of FFT size 1024 I frequencies each ~43 Hz wide.
• But if I change FFT size to 2048 each frequency range is ~21.5 Hz. So the peak of the spectrum of the sine wave will be outside my 15-20 bars.

So I wanted to ask if it's possible to scale/normalize the X axis for a frequency range and a constant graph bars count?

The example limiting data would be:

• frequency range 0-500 Hz
• 16 bars on the graph

I guess it's some kind of bins/magnitudes grouping. But should I sum them or get max? And what could be done with magnitudes when grouped bin number isn't integer? (I hope my explanation is not too confusing, it's hard to choose the correct terminology)

Thanks.

Here is my test code in Python:
(I used a static int array imported from another file that represents a sine wave)

import scipy.io.wavfile as wavfile
import scipy
import scipy.fftpack
import numpy as np
from scipy.signal.windows import *
from matplotlib import pyplot as plt

from imp.sine_440_hex import *

def run():
  # fp = 'e:/sine_440_hz.wav'
  # FS, data = wavfile.read(fp)

  # data - array of int for a sine wave at 440Hz and amplitude 0.5
  DS = len(data)
  FS = 44100

  FN = 1024
  # FN = 2048
  MAXS = 32768

  MN = 15   # number of first frequencies in the FFT result

  ndata = np.array(data)
  ndata = ndata / MAXS

  w = np.hamming(FN)
  y = ndata[:FN] * w

  freqs = scipy.fftpack.fftfreq(FN, 1/FS)
  mags = abs(scipy.fft(y))

  dbfs = 20 * np.log10(mags * 2 / np.sum(w))
  dbs = dbfs + 120

  # -------- plot --------
  fr_res = freqs[:MN]
  db_res = dbs[:MN]

  ax1 = plt.subplot(211)
  plt.plot(fr_res, db_res)
  plt.grid(True)
  plt.xlabel('Frequency [Hz]')
  plt.ylabel('Amplitude [dB]')

  ax2 = plt.subplot(212)
  plt.bar(fr_res, db_res, fr_res[1]-10)
  plt.grid(True)
  ax2.set_ylim([80, 120])
  ax2.set_xticks(fr_res)
  ax2.set_xticklabels(fr_res.astype(int))

  plt.show()

# -------
run()

And example graphs:

For FN = 1024 For FN = 2048

Answer 1

To achieve that, use frequencies rather than bins. To convert from bin number to the frequency it represents use:

$$f = \frac{k}{N_{FFT}} \cdot Fs$$

Where:

• $k$ is a bin number whose actual frequency you want to recover
• $Fs$ is the sampling frequency
• $N_{FFT}$ is the number of points of the FFT

You can work out your $N_{FFT}$ to ensure that your scanned frequency range will be divided into 16 segments. For example, to cover the frequency range $f_l=300 f_h=3000$ Hz with $N_{bars}=16$ bars and $Fs=44100 Hz$, you would have to use:

$$N_{FFT} = \frac{Fs \cdot N_{bars}}{|f_h-f_l|}$$

Where basically, you work out the spacing between the two frequencies $f_l, f_h$ for 16 segments and then "translate" it to the $Fs$ (but the ratio has been simplified there and that's why $Fs \cdot N_{bars}$)

This works out to be:

$$N_{FFT} = \frac{44100 \cdot 16}{|3000-300|} \approx 261.3333 $$

Which will have to be rounded to $262$ points.

So, now you know that your FFT has 262 points with a spacing of $\frac{Fs}{N_{FFT}}=\frac{44100}{262}\approx 168.3206$ Hz.

Your first bin is at $\frac{300}{168.3206} \approx 1.7823$ which will have to be rounded to $2$ and 3000 Hz should be at bin $2 + 16 = 18$.

Let's see, $\frac{18}{262}\cdot 44100 \approx 3029.7709 $ Hz.

Hope this helps.