Ok, I was tired of using Audacity filters, so it is time to build our own low-pass filter.

AFAIK there are three main ways to design a digital filter, like a lowpass filter that we need in AM modulation. The most common is the FIR filter. Another way is the IIR filter. And the third way is using a Discrete Fourier Transform (DFT).

In this post, I will use the easiest way, that happens to be DFT. Other posts demonstrate FIR and IIR.

DFT is the discrete, or "digital" flavor of the Fourier Transform. It is a mathematical concept that can be implemented in many ways. It is often implemented by a fast algorithm called FFT (Fast Fourier Transform), and most mathematical libraries include FFT. Python-Numeric is no exception.

The Fourier transform converts a time-domain signal, like an audio wave, to a frequency-domain representation. For example, a spectrum analyzer (see figure below) is one application of the Fourier transform; and converserly the easiest method of implementing a spectrum analyzer is using DFT.

DFT is not just a one-way signal processing feature. Given a frequency domain representation, we can calculate the IDFT (Inverse DFT), to get back the time-domain signal. So, we can get some audio, transform into DFT, manipulate the frequencies (e.g. increase bass, decrease midrange, etc.) and then convert back to time-domain. It is dead easy to implement an equalizer using DFT!

In our case, we just want to build a lowpass filter, so we just have to wipe the frequencies we don't want, that is easy to do in the frequency domain. Note that a sharp removal of some range of frequencies is not a good filter, because it causes distortion. But it works, and it is ok for a simple demonstration.

Executing DFT and IDFT just to filter a signal seems wasteful. Indeed a FIR filter needs less processing power, and it is normally the #1 choice. But a DFT filter is easier to understand, and knowing the DFT filter first is important to understand the FIR filter.

For this article, I will use a female singer's voice that I really like: Núbia Lafayette. Another reason I used this particular sample is that I own the CD, so nobody can complain about piracy etc. Let's hear the original audio:

Now, the filtered version (1000-4000Hz bandpass filter), that sounds like a pocket radio:

And the code that does the trick:

#!/usr/bin/env python LOWPASS = 4000 # Hz HIGHPASS = 1000 # Hz SAMPLE_RATE = 44100 # Hz import wave, struct, math from numpy import fft FFT_LENGTH = 2048 # The higher the better, but this seems enough OVERLAP = 512 FFT_SAMPLE = FFT_LENGTH - OVERLAP # Nyquist rate = bandwidth available for a given sample rate NYQUIST_RATE = SAMPLE_RATE / 2.0 # Convert frequencies from Hz to our digital sampling units LOWPASS /= (NYQUIST_RATE / (FFT_LENGTH / 2.0)) HIGHPASS /= (NYQUIST_RATE / (FFT_LENGTH / 2.0)) zeros = [ 0 for x in range(0, OVERLAP) ] # Builds filter mask. Note that this filter is BAD, a # good filter must have a smooth curve, respecting a # dB/octave attenuation ramp! mask = [] for f in range(0, FFT_LENGTH / 2 + 1): rampdown = 1.0 if f > LOWPASS: rampdown = 0.0 elif f < HIGHPASS: rampdown = 0.0 mask.append(rampdown) def bound(sample): # takes care of "burnt" samples, due some mistake in filter design if sample > 1.0: print "!", sample = 1.0 elif sample < -1.0: print "!", sample = -1.0 return sample original = wave.open("NubiaCantaDalva.wav", "r") filtered = wave.open("NubiaFiltered.wav", "w") filtered.setnchannels(1) filtered.setsampwidth(2) filtered.setframerate(SAMPLE_RATE) n = original.getnframes() original = struct.unpack('%dh' % n, original.readframes(n)) # scale from 16-bit signed WAV to float original = [s / 32768.0 for s in original] saved_td = zeros for pos in range(0, len(original), FFT_SAMPLE): time_sample = original[pos : pos + FFT_LENGTH] # convert frame to frequency domain representation frequency_domain = fft.fft(time_sample, FFT_LENGTH) l = len(frequency_domain) # mask positive frequencies (f[0] is DC component) for f in range(0, l/2+1): frequency_domain[f] *= mask[f] # mask negative frequencies # http://stackoverflow.com/questions/933088/clipping-fft-matrix for f in range(l-1, l/2, -1): cf = l - f frequency_domain[f] *= mask[cf] # convert frame back to time domain time_domain = fft.ifft(frequency_domain) # modified overlap-add logic: previously saved samples # prevail in the beginning, and they are ramped down # in favor of this frame's samples, to avoid 'clicks' # in either end of frame. for i in range(0, OVERLAP): time_domain[i] *= (i + 0.0) / OVERLAP time_domain[i] += saved_td[i] * (1.0 - (i + 0.00) / OVERLAP) # reserve last samples for the next frame saved_td = time_domain[FFT_SAMPLE:] # do not write reserved samples right now time_domain = time_domain[:FFT_SAMPLE] # scale back into WAV 16-bit and write time_domain = [ bound(sample) * 32767 for sample in time_domain ] filtered.writeframes(struct.pack('%dh' % len(time_domain), *time_domain))

There is one twist in this code that is worth mentioning. Since any filtering in frequency domain causes phase distortion, you can't just piggyback audio frames, because the audio waves will not transition smoothly from one frame to the next. A continuous wave in the original audio will have a different phase in every frame generated by the filter. There will be a sharp discontinuity in every frame, very annoying to hear. (If you don't believe me, you can just set OVERLAP to zero, re-run the code, and see what happens.)

Both textbook solutions for this (overlap-save and overlap-add) seem not to solve this problem. I added cross-fading of frames to the overlap-add method, and then it worked. The cross-fade dillutes the phase distortion. It is possible that plain overlap-add works if the filter was a better design (remember what I said: simply cutting off a range of frequencies without a "slope" or transition curve creates a lot of phase distortion).

Another thing that must be remembered by anyone that wants to play with FFT, is that frequency representation is linear. For audio signals, we are accustomed to logarithmic spectrum analyzers that match the octaves in music. But FFT is linear. This means that the musical resolution of FFT is higher for low octaves. Using a FFT with resolution too low will distort treble sounds first.

FFT generates two frequency "lobes": positive and negative frequencies. This sounds very strange, because negative frequencies don't exist in the real world. DFT has this mathematical concept. In our filter, we had two choices: just wipe all negative frequencies (that reduces the audio volume by half), or filter both "lobes" in the same fashion, as we did.

The array generated by FFT consists of complex numbers. This sounds scary but it has a valid purpose: a single complex number can represent both amplitude and phase. For example, if we find 1+0j in the 3000Hz 'bar', it means that audio has a 3000Hz component with amplitiude 1.0 and in phase with cos(x). If the sample were 0.707+0.707j, it means that the 3000Hz component still has amplitude 1.0 (the modulus) but it is 45 degrees ahead of cos(x).

Some applications, for example the spectrum analyzer, care only about the signal strength, not about the signal phase. These applications can employ a "lighter" version of DFT called DCT (discrete cosine transform). The output of DCT is an array of real numbers, and there is an IDCT to transform it back to time domain.