Sigproc Selfstudy

Digital Signal Processing through Speech, Hearing, and Python

Mel Chua

PyCon 2013

This tutorial was designed to be run on a free pythonanywhere.com Python 2.7 terminal.

If you want to run the code directly on your machine, you'll need python 2.7.x, numpy, scipy, and

matplotlib.

Either way, you'll need a .wav file to play with (preferably 1-2 seconds long).

Agenda

● Introduction● Fourier transforms, spectrums, and spectrograms● Playtime!● SANITY BREAK● Nyquist, sampling and aliasing● Noise and filtering it● (if time permits) formants, vocoding, shifting, etc.● Recap: so, what did we do?

What's signal processing?

● Usually an upper-level undergraduate engineering class

● Prerequisites: circuit theory, differential equations, MATLAB programming, etc, etc...

● About 144 hours worth of work (3 hours per credit per week, 3 credits, 16 weeks)

● We're going to do this in 3 hours (1/48th the time)● I assume you know basic Python and therefore

algebra

Therefore

We'll skip a lot of stuff.

We will not...

● Do circuit theory, differential equations, MATLAB programming, etc, etc...

● Work with images● Write tons of code from scratch● See rigorous proofs, math, and/or definitions

We will...

● Play with audio● Visualize audio● Generate and record audio● In general, play with audio● Do a lot of “group challenge time!”

Side notes

● This is based on a graduate class teaching signal processing to audiology majors

● We've had half a semester to do everything (about 70 hours)

● I'm not sure how far we will get today

Introduction: Trig In One Slide

Sampling

Let's write some code.

Open up the terminal and follow along.

We assume you have a file called 'flute.wav' in the directory you are running the terminal from.

Import the libraries we need...

from numpy import *

...and create some data.Here we're making a signal consisting of 2 sine waves

(1250Hz and 625Hz)sampled at a 10kHz rate.

x = arange(256.0)sin1 = sin(2*pi*(1250.0/10000.0)*x)sin2 = sin(2*pi*(625.0/10000.0)*x)sig = sin1 + sin2

What does this look like?Let's plot it and find out.

import matplotlib.pyplot as pyplotpyplot.plot(sig)pyplot.savefig('sig.png')pyplot.clf() # clear plot

sig.png

While we're at it, let's define a graphing function so we don't need

to do this all again.

def makegraph(data, filename): pyplot.clf() pyplot.plot(data) pyplot.savefig(filename)

Our first plot showed the signal in the time domain. We want to see it

in the frequency domain.A numpy function that implements

an algorithm called the Fast Fourier Transform (FFT) can take us there.

data = fft.rfft(sig)# note that we use rfft because# the values of sig are real

makegraph(data, 'fft0.png')

fft0.png

That's a start.We had 2 frequencies in the signal, and we're seeing 2 spikes here, so

that seems reasonable. But we did get this warning.

>>> makegraph(data, 'fft0.png')/usr/local/lib/python2.7/site-packages/numpy/core/numeric.py:320: ComplexWarning: Casting complex values to real discards the imaginary partreturn array(a, dtype, copy=False, order=order)

That's because the fourier transform gave us a complex output – so we need to take the magnitude of the

complex output...

data = abs(data)makegraph(data, 'fft1.png')

# more detail: sigproc-outline.py# lines 42-71

fft1.png

But this is displaying raw power output, and we usually think of audio

volume in terms of decibels.Wikipedia tells us decibels (dB) are

the original signal plotted on a 10*log10 y-axis, so...

data = 10*log10(data)makegraph(data, 'fft2.png')

fft2.png

We see our 2 pure tones showing up as 2 peaks – this is great. The

jaggedness of the rest of the signal is quantization noise, a.k.a.

numerical error, because we're doing this with approximations.

Question: what's the relationship of the x-axis of the graph and the

frequency of the signal?

Answer: the numpy fft function goes from 0-5000Hz by default.

This means the x-axis markers correspond to values of 0-5000Hz

divided into 128 slices.

5000/128 = 39.0625 Hz per marker

The two peaks are at 16 and 32.

(5000/128)*16 = close to 625Hz(5000/128)*32 = close to 1250Hz

...which are our 2 original tones.

Another visualization: spectrogram

Generate and plot a spectrogram...

from pylab import specgrampyplot.clf()sgram = specgram(sig)pyplot.savefig('sgram.png')

sgram.png

Do you see how the spectrogram is sort of like our last plot, extruded

forward out of the screen, and looked down upon from above?

That's a spectrogram. Time is on the x-axis, frequency on the y-axis,and

amplitude is marked by color.

Now let's do this with a more complex sound. We'll need to use a

library to read/write .wav files.

import scipy

from scipy.io.wavfile import read

Let's define a function to get the data from the .wav file, and use it.

def getwavdata(file): return scipy.io.wavfile.read(file)[1]

audio = getwavdata('flute.wav')

# more detail on scipy.io.wavfile.read# in sigproc-outline.py, lines 117-123

Hang on! How do we make sure we've got the right data? We could write it back to a .wav file and make

sure they sound the same.

from scipy.io.wavfile import writedef makewav(data, outfile, samplerate): scipy.io.wavfile.write(outfile, samplerate, data)

makewav(audio, 'reflute.wav', 44100)# 44100Hz is the default CD sampling rate, and # what most .wav files will use.

Now let's see what this looks like in the time domain. We've got a lot of data points, so we'll only plot the

beginning of the signal here.

makegraph(audio[0:1024], 'flute.png')

flute.png

What does this look like in the frequency domain?

audiofft = fft.rfft(audio)audiofft = abs(audiofft)audiofft = 10*log10(audiofft)makegraph(audiofft, 'flutefft.png')

flutefft.png

This is much more complex. We can see harmonics on the left side.

Perhaps this will be clearer if we plot it as a spectrogram.

pyplot.clf()sgram = specgram(audio)pyplot.savefig('flutespectro.png')

flutespectro.png

You can see the base note of the flute (a 494Hz B) in dark red at the bottom, and lighter red harmonics

above it.http://www.bgfl.org/custom/resources_ftp/client_ftp/ks2/music/piano/flute.htm

http://en.wikipedia.org/wiki/Piano_key_frequencies

Your Turn: Challenge

● That first signal we made? Make a wav of it.● Hint: you may need to generate more samples.● Bonus: the flute played a B (494Hz) – generate

a single sinusoid of that.● Megabonus: add the flute and sinusoid signals

and play them together

Your turn: Challenge 2

● Record some sounds on your computer

● Do an FFT on it

● Plot the spectrum

● Plot the spectrogram

● Bonus: add the flute and your sinusoid and plot their spectrum and spectrogram together – what's the x scale?

● Bonus: what's the difference between fft/rfft?

● Bonus: numpy vs scipy fft libraries?

● Bonus: try the same sound at different frequencies (example: vowels)

Sanity break?

Come back in 20 minutes, OR: stay for a demo of the wave library (aka “why we're using scipy”)

note: wavlibraryexample.py contains thewave library demo (which we didn't get to in the actual workshop)

Things people found during break

Problem #1: When trying to generate a pure-tone (sine wave) .wav file, the sound is not audible.

Underlying reason: The amplitude of a sine wave is 1, which is really, really tiny. Compare that to the amplitude of the data you get when you read in the flute.wav file – over 20,000.

Solution: Amplify your sine wave by multiplying it by a large number (20,000 is good) before writing it to the .wav file.

More things people found

Problem #2: The sine wave is audible in the .wav file, but sounds like white noise rather than a pure tone.

Underlying reason: scipy.io.wavfile.write() expects an int16 datatype, and you may be giving it a float instead.

Solution: Coerce your data to int16 (see next slide).

Coercing to int16

# option 1: rewrite the makewav function# so it includes type coerciondef savewav(data, outfile, samplerate): out_data = array(data, dtype=int16) scipy.io.wavfile.write(outfile, samplerate, out_data)

# option 2: generate the sine wave as int16# which allows you to use the original makewav functiondef makesinwav(freq, amplitude, sampling_freq, \ num_samples):

return array(sin(2*pi*freq/float(sampling_freq) \ *arange(float(num_samples)))*amplitude,dtype=int16)

Post-break: Agenda

● Introduction● Fourier transforms, spectrums, and spectrograms● Playtime!● SANITY BREAK● Nyquist, sampling and aliasing● Noise and filtering it● (if time permits) formants, vocoding, shifting, etc.● Recap: so, what did we do?

Nyquist: sampling and aliasing

● The sample rate matters.● Higher is better.● There is a tradeoff.

Sampling

Aliasing

Nyquist-Shannon sampling theorem (Shannon's version)

If a function x(t) contains no frequencies higher than B hertz, it is completely determined by

giving its ordinates at a series of points spaced 1/(2B) seconds apart.

Nyquist-Shannon sampling theorem (haiku version)

lowest sample rate

for sound with highest freq F

equals 2 times F

Let's explore the effects of sample rate. When you listen to these .wav files, note that doubling/halfing the

sample rate moves the sound up/down an octave, respectively.

audio = getwavdata('flute.wav')

makewav(audio, 'fluteagain44100.wav', 44100)

makewav(audio, 'fluteagain22000.wav', 22000)

makewav(audio, 'fluteagain88200.wav', 88200)

Your turn

● Take some of your signals from earlier● Try out different sample rates and see what

happens● Hint: this is easier with simple sinusoids at first● Hint: determine the highest frequency (your Nyquist

frequency), double it (that's your highest sampling rate) and try sampling above, below, and at that sampling frequency

● What do you find?

What do aliases alias at?

● They reflect around the sampling frequency● Example: 40kHz sampling frequency● Implies 20kHz Nyquist frequency● So if we try to play a 23kHz frequency...● ...it'll sound like 17kHz.

Your turn: make this happen with pure sinusoids

Bonus: with non-pure sinusoids

Agenda

● Introduction● Fourier transforms, spectrums, and spectrograms● Playtime!● SANITY BREAK● Nyquist, sampling and aliasing● Noise and filtering it● (if time permits) formants, vocoding, shifting, etc.● Recap: so, what did we do?

Remember this?

Well, these are filters.

Noise and filtering it

● High pass● Low pass● Band pass● Band stop● Notch● (there are many more, but these basics)

Notice that all these filters work in the frequency domain.

We went from the time to the frequency domain using an FFT.

# get audio (again) in the time domainaudio = getwavdata('flute.wav')

# convert to frequency domainflutefft = fft.rfft(audio)

We can go back from the frequency to the time domain using an inverse

FFT (IFFT).

reflute.wav should sound identical to flute.wav.

reflute= fft.irfft(flutefft, len(audio))

reflute_coerced = array(reflute, \dtype=int16) # coerce to int16

makewav(reflute_coerced, \'fluteregenerated.wav', 44100)

Let's look at flute.wav in the frequency domain again...

# plot on decibel (dB) scalemakegraph(10*log10(abs(flutefft)), \'flutefftdb.png')

What if we wanted to cut off all the frequencies higher than the 5000th

index? (low-pass filter)

Implement and plot the low-pass filter in the frequency domain...

# zero out all frequencies above# the 5000th index# (BONUS: what frequency does this# correspond to?)flutefft[5000:] = 0

# plot on decibel (dB) scalemakegraph(10*log10(abs(flutefft)), \'flutefft_lowpassed.png')

flutefft_lowpassed.png

Going from frequency back to time domain so we can listen

reflute = fft.irfft(flutefft, \len(audio))

reflute_coerced = array(reflute, \dtype=int16) # coerce it

makewav(reflute_coerced, \'flute_lowpassed.wav', 44100)

What does the spectrogram of the low-passed flute sound look like?

pyplot.clf()sgram = specgram(audio)pyplot.savefig('reflutespectro.png')

reflutespectro.png

Compare to flutespectro.png

Your turn

● Take some of your .wav files from earlier, and try making...● Low-pass or high-pass filters● Band-pass, band-stop, or notch filters● Filters with varying amounts of rolloff

Agenda

● Introduction● Fourier transforms, spectrums, and spectrograms● Playtime!● SANITY BREAK● Nyquist, sampling and aliasing● Noise and filtering it● (if time permits) formants, vocoding, shifting, etc.● Recap: so, what did we do?

Formants

Formants

f1 f2 a 1000 Hz 1400 Hzi 320 Hz 2500 Hzu 320 Hz 800 Hze 500 Hz 2300 Hzo 500 Hz 1000 Hz

Vocoding

http://en.wikipedia.org/wiki/Vocoder

Bode plot (high pass)

Another Bode plot...

Credits and Resources

● http://onlamp.com/pub/a/python/2001/01/31/numerically.html

● http://jeremykun.com/2012/07/18/the-fast-fourier-transform/

● http://lac.linuxaudio.org/2011/papers/40.pdf

● Farrah Fayyaz, Purdue University● signalprocessingforaudiologists.wordpress.com● Wikipedia (for images)● Tons of Python library documentation