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- 1. 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, youll need python 2.7.x, numpy, scipy, and matplotlib. Either way, youll need a .wav file to play with (preferably 1-2 seconds long).
- 2. 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?
- 3. Whats 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)● Were going to do this in 3 hours (1/48th the time)● I assume you know basic Python and therefore algebra
- 4. ThereforeWell skip a lot of stuff.
- 5. 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
- 6. We will...● Play with audio● Visualize audio● Generate and record audio● In general, play with audio● Do a lot of “group challenge time!”
- 7. Side notes● This is based on a graduate class teaching signal processing to audiology majors● Weve had half a semester to do everything (about 70 hours)● Im not sure how far we will get today
- 8. Introduction: Trig In One Slide
- 9. Sampling
- 10. Lets write some code.Open up the terminal and follow along.We assume you have a file called flute.wav inthe directory you are running the terminal from.
- 11. Import the libraries we need...from numpy import *
- 12. ...and create some data. Here were 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
- 13. What does this look like? Lets plot it and find out.import matplotlib.pyplot as pyplotpyplot.plot(sig)pyplot.savefig(sig.png)pyplot.clf() # clear plot
- 14. sig.png
- 15. While were at it, lets define agraphing function so we dont need to do this all again.def makegraph(data, filename): pyplot.clf() pyplot.plot(data) pyplot.savefig(filename)
- 16. Our first plot showed the signal inthe time domain. We want to see it in the frequency domain. A numpy function that implementsan algorithm called the Fast FourierTransform (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)
- 17. fft0.png
- 18. Thats a start.We had 2 frequencies in the signal,and were 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 theimaginary partreturn array(a, dtype, copy=False, order=order)
- 19. Thats 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
- 20. fft1.png
- 21. But this is displaying raw poweroutput, 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)
- 22. fft2.png
- 23. We see our 2 pure tones showing up as 2 peaks – this is great. Thejaggedness of the rest of the signal is quantization noise, a.k.a. numerical error, because were doing this with approximations.Question: whats the relationship of the x-axis of the graph and the frequency of the signal?
- 24. Answer: the numpy fft function goes from 0-5000Hz by default. This means the x-axis markerscorrespond to values of 0-5000Hz divided into 128 slices.5000/128 = 39.0625 Hz per marker
- 25. 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.
- 26. Another visualization: spectrogram
- 27. Generate and plot a spectrogram...from pylab import specgrampyplot.clf()sgram = specgram(sig)pyplot.savefig(sgram.png)
- 28. sgram.png
- 29. 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?Thats a spectrogram. Time is on the x-axis, frequency on the y-axis,and amplitude is marked by color.
- 30. Now lets do this with a morecomplex sound. Well need to use a library to read/write .wav files. import scipy from scipy.io.wavfile import read
- 31. Lets define a function to get thedata 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
- 32. Hang on! How do we make sureweve got the right data? We couldwrite it back to a .wav file and make sure they sound the same. from scipy.io.wavfile import write def 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.
- 33. Now lets see what this looks like inthe time domain. Weve got a lot of data points, so well only plot the beginning of the signal here. makegraph(audio[0:1024], flute.png)
- 34. flute.png
- 35. What does this look like in the frequency domain?audiofft = fft.rfft(audio)audiofft = abs(audiofft)audiofft = 10*log10(audiofft)makegraph(audiofft, flutefft.png)
- 36. flutefft.png
- 37. 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)
- 38. flutespectro.png
- 39. You can see the base note of theflute (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
- 40. 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
- 41. 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 – whats the x scale?● Bonus: whats the difference between fft/rfft?● Bonus: numpy vs scipy fft libraries?● Bonus: try the same sound at different frequencies (example: vowels)
- 42. Sanity break?Come back in 20 minutes, OR: stay for a demoof the wave library (aka “why were using scipy”)note: wavlibraryexample.py contains thewave library demo (which we didnt get to in theactual workshop)
- 43. Things people found during breakProblem #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 is1, which is really, really tiny. Compare that to theamplitude of the data you get when you read in theflute.wav file – over 20,000.Solution: Amplify your sine wave by multiplying it bya large number (20,000 is good) before writing it tothe .wav file.
- 44. More things people foundProblem #2: The sine wave is audible in the.wav file, but sounds like white noise ratherthan a pure tone.Underlying reason: scipy.io.wavfile.write()expects an int16 datatype, and you may begiving it a float instead.Solution: Coerce your data to int16 (see nextslide).
- 45. 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)
- 46. 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?
- 47. Nyquist: sampling and aliasing● The sample rate matters.● Higher is better.● There is a tradeoff.
- 48. Sampling
- 49. Aliasing
- 50. Nyquist-Shannon sampling theorem (Shannons 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.
- 51. Nyquist-Shannon sampling theorem (haiku version) lowest sample rate for sound with highest freq F equals 2 times F
- 52. Lets explore the effects of samplerate. 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)
- 53. 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 (thats your highest sampling rate) and try sampling above, below, and at that sampling frequency● What do you find?
- 54. 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...● ...itll sound like 17kHz. Your turn: make this happen with pure sinusoids Bonus: with non-pure sinusoids
- 55. 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?
- 56. Remember this?
- 57. Well, these are filters.
- 58. Noise and filtering it● High pass● Low pass● Band pass● Band stop● Notch● (there are many more, but these basics)
- 59. Notice that all these filters work in the frequency domain. We went from the time to thefrequency domain using an FFT.# get audio (again) in the time domainaudio = getwavdata(flute.wav)# convert to frequency domainflutefft = fft.rfft(audio)
- 60. We can go back from the frequencyto 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)
- 61. Lets look at flute.wav in the frequency domain again...# plot on decibel (dB) scalemakegraph(10*log10(abs(flutefft)), flutefftdb.png)
- 62. What if we wanted to cut off all thefrequencies higher than the 5000th index? (low-pass filter)
- 63. 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)
- 64. flutefft_lowpassed.png
- 65. Going from frequency back to time domain so we can listenreflute = fft.irfft(flutefft, len(audio))reflute_coerced = array(reflute, dtype=int16) # coerce itmakewav(reflute_coerced, flute_lowpassed.wav, 44100)
- 66. What does the spectrogram of thelow-passed flute sound look like?pyplot.clf()sgram = specgram(audio)pyplot.savefig(reflutespectro.png)
- 67. reflutespectro.png
- 68. Compare to flutespectro.png
- 69. 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
- 70. 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?
- 71. Formants
- 72. Formants f1 f2a 1000 Hz 1400 Hzi 320 Hz 2500 Hzu 320 Hz 800 Hze 500 Hz 2300 Hzo 500 Hz 1000 Hz
- 73. Vocodinghttp://en.wikipedia.org/wiki/Vocoder
- 74. Bode plot (high pass)
- 75. Another Bode plot...
- 76. 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

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One problem: when I get to the step where we check that the wav file sounds the same after we've put it through the getwavdata (slide 32), when I listen to the verification copy, it just comes out like white noise without any hint of the original sound that I can discern. The graphs of the two waveforms look identical, but the copy is just noise.

In an attempt to remedy this, I added a line in my makewav function to coerce the data to int16, like this:

def makewav(data, outfile, samplerate):

scaled = array(data, dtype = int16)

wave.write(outfile, samplerate, scaled)

but it fixes nothing. What could be the problem? I looked everywhere.