[PPT] - Audio DSP basics Paris Smaragdis paris@illinois.edu PowerPoint Presentation

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U N I V E R S I T Y O F I L L I N O I S @ U R B A N A - C H A M P A I G N

Paris Smaragdis

paris@illinois.edu paris.cs.illinois.edu

CS 498PS – Audio Computing Lab

Audio DSP basics

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U N I V E R S I T Y O F I L L I N O I S @ U R B A N A - C H A M P A I G N

Overview

Basics of digital audio
Signal representations
Time, Frequency, Time/Frequency
Sampling, Quantization
The Fourier transform
DFT and FFT
The Spectogram

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Why digital audio?

Cheaper
Get a smartphone, do anything you want
No burning circuits!
Easier
You can easily rewrite code
But cannot easily rewire circuits
Smaller
Do everything on one chip

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Sound as “numbers”

We treat sound as a series of amplitudes
More on the details later
This is the waveform representation
Encodes instantaneous pressure over time

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PCM format

“Pulse Code Modulation”
Used by CDs, telephones, audio editors, synths, etc.

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1 2 3 4 5 6 7 8 9 10

1
0.5

0.5 1

0, 82, 126, 111, 44, -44, -111, -126, -82, 0

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This is a discrete and digital format

We do not use continuous values
We have finite samples over time
We (usually) encode these samples as signed integers
Common formats
Speech: 16kHz / 16-bit (or 8-bit)
Music: 44.1kHz / 16-bit (or 95kHz / 24-bit)
But how do we pick these numbers?
What do they mean?

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Dynamic range

The choice of bits defines the dynamic range
More bits == more dynamic range == more storage
What is dynamic range?
Ratio of highest and lowest represented pressure value
Usually measured in decibels (dB)
How much dynamic range do we need though?

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It all hinges on how we hear

Outer ear
Sound gets collected at the pinna
The ear canal amplifies (some) sound by ~10dB
The ear drum vibrates according to incoming pressure
Middle ear
The ossicles transfer sound to the oval window
Amplify sound by ~14dB
Also use muscles for damping
Inner ear
Translation to neural signal (more later)

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Perception of sound

The just noticeable sound is:
10-12 W/m2 (cannot hear softer than this)
And the as noticeable as it get is:
1 W/m2 (and then you go deaf!)
Thus our dynamic range is:
10 log10( 1/10-2) = 120 dB
That’s a staggering trillion to one!

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To get you oriented

Weakest detectable sound

~0 dB

Soft breathing

~10dB

Quiet library

~40 dB

Office environment

~60 dB

Food blender

~80 dB

Lawn mower

~90 dB

Car horn at 1m

~110 dB

Military jet at 50ft

~130 dB

Shotgun blast

~165 dB

Loudest possible sound

194 dB

(after which it isn’t “sound” anymore it is a “shock wave”)

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Dangerous levels > 90 dB Pain begins at 125 dB Pain ends at 180 dB (cause your ears just blew up)

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Back to digital sound

How many dB dynamic range to use?
Close to 120 dB ideally
Common ranges (headroom)
16-bit / 96 dB (the industry standard)
12-bit / 72 dB (the cheap standard)
8-bit / 48 db (the 80’s standard! hipsters?)
24-bit / 144 dB (the “I’m charging you extra” standard)
Floating point (what we will use)

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Why worry?

Need headroom to avoid clipping & quantization noise
These happen when the representation is maxed or zero
Very challenging with dynamic content (e.g. classical music)
An audio engineer’s nightmare! (and digital is worse)

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10 20 30 40 50 60 70 80 90 100 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

Hiss Clipping Gone!

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Quantization noise examples

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Clipping examples

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Sampling in time

Also known as A/D conversion
How to we convert real-world sound to a discrete sequence?
The one parameter we care for: the sample rate
i.e. how often do we represent the input sound
Tradeoffs
Sample fast and you waste memory and energy
Sample slow and you risk aliasing

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What is aliasing?

Low sample rates can result in misinterpretations
Sample too low and you will miss some of the action
Rule of thumb: Sample at least at twice the highest frequency

16 100 200 300 400 500 600 700 800 900 1000 −1 1 100 200 300 400 500 600 700 800 900 1000 −1 1 100 200 300 400 500 600 700 800 900 1000 −1 1

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How high should we go?

Highest perceived frequency by humans is 20 kHz
Which goes down as you age (or as you abuse your ears)
We need to represent up to 20 kHz ⟶ sample at > 40 kHz

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Frequency (Hz) Time (sec) 2 4 6 8 10 12 14 16 18 0.5 1 1.5 2 x 10

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1kHz 3kHz 5kHz 7kHz 9kHz 11kHz 13kHz 15kHz 17kHz 19kHz 21kHz

How high can you hear? (or how good are the class speakers?)

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What does aliasing sound like?

Frequencies higher than Nyquist fold over
Upwards movements go downwards and vice-versa
Most noticeable with high-frequency content
How does that sound?

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Chirp @ 44,100 Hz Same chirp @ 22,050 Hz Same chirp @ 11,025 Hz

Frequency ⟶ Time ⟶ Time ⟶ Time ⟶

0 Hz 0 Hz 0 Hz 20 kHz 11 kHz 5.5 kHz

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at 44.1kHz at 11kHz at 4kHz at 22kHz at 5kHz at 3kHz

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What are the usual settings?

“High-quality” music: 44.1 kHz
Why the extra 4.1 kHz?
“Super” high quality music: 96 kHz
Dogs might like it more
Speech coding
High(ish) quality & in research: 16 kHz
Telephony: 8 kHz

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But why do we use the waveform?

Do you see a problem with it?

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What are these signals?

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Waveforms are unintuitive at long scales

Pressure information isn’t that perceptually relevant
We cannot interpret it as a percept
Too much data to parse visually
Is there a better way to represent sound?
How do we start looking for such a way?
What is it that is important when listening?

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Back to hearing …

What happens in the inner ear?
After the oval window there’s the cochlea
Resonates at different lengths with input
Effectively parses sound by frequency
Transmits that vibration to neural code
What we care about is

frequency content!

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What is a frequency component?

You can approximate any waveform by adding sinusoids
They are the elementary building blocks of sounds
Sinusoids have three parameters:
Amplitude, frequency and phase
s(t) = a(t) sin( f t + φ)
Each sinusoid is a “frequency”
Because that is the main

distinguishing parameter

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Approximating a square wave

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Decomposing sounds to sines

For each sound get reconstructing sine parameters
And we’ll be lazy and not bother with frequency
Just get all amplitudes and phases for all integer frequencies
For this we use the Fourier transform
Transforms time samples to the frequency domain, and back

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X[ f ]= FT x[t]

( )

Waveform (time domain) “Spectrum” (frequency domain)

x[t]= FT−1 X[ f ]

( )

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And there are many flavors of it

Fourier transform (Continuous time ⟷ Continuous frequency)
Discrete Time Fourier Transform (DTFT) (Discrete time ⟷ Cont. frequency)
Discrete Fourier Transform (DFT) (Discrete time ⟷ Discrete frequency)

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The one that we will use the most

x t

( )= 1

2π X ω

( )e jωt dω

ω=−∞ ∞

∫

↔ X ω

( )=

x t

( )e− jωt dt

t=−∞ ∞

∫

x n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = 1 2π Xd ω

( )e jωn dω

ω=−π π

∫

↔ Xd ω

( )=

x n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ e− jωt dt

t=−∞ ∞

∫

x n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = 1 N X k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ e

j2πkn N k=0 N−1

∑

↔ X k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = x n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ e

− j2πkn N n=0 N−1

∑

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What really happens here?!?

Each Fourier basis contains a sine and a cosine
The summation estimates how much of each sinusoid

we have in the input time series (inner product)

Thus we get the contribution from each frequency

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X[k]= x[n]e

− j2πkn N n=0 N−1

∑

= x[n] cos 2πkn N ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟− jsin 2πkn N ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟

n=0 N−1

∑

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Getting to the sought-after parameters

The magnitude spectrum is |X[k]|
Tells us how much of each frequency we have (amplitudes)
Adding the sine and cosine terms we make phase-shifted sinusoids
The contribution of each frequency is the amount of sine/cosine present
The phase spectrum ∡X[k]
Gives us each frequency’s phase
Does so by looking at the relative amplitudes of the same-frequency

sine/cosine pairs

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Some examples

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5 10 15 20 25 30

1
0.5

0.5 1

Single sine input

5 10 15 20

Complex spectrum

5 10 15 20

5

5 10 15 20

Polar spectrum

5 10 15 20 5 10 15 20 25 30

1
0.5

0.5 1

Single square input

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20
15
10
5

5

Complex spectrum

5 10 15 20

5

5 10 15 20 25

Polar spectrum

5 10 15 20

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Some extra info

Audio is real-valued
DFT results in a conjugate symmetric transform
Upper half is redundant (real-valued routines will give you lower half)
The first frequency bin is the DC
Offset of the input signal (“zero” frequency)
Doesn’t have a phase value (why?)
The highest frequency bin is the Nyquist
Also as no phase value (why?)

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One problem

Sinusoids extend infinitely on both sides
i.e. what we approximate is assumed to be periodic
So it should transition smoothly from left to right
We need to ensure smoothness here
Discontinuities will result in extra high frequencies

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Windowing

To avoid periodic discontinuities we can “window”
Taper the ends of to zero so that they join better
But too much tapering changes the signal!
Common side effect: “blurring the spectrum”

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20 40 60 80 100 120

0.5

0.5

Input

50 100 150 200 250 300 350

0.5

0.5

Periodic version

2 4 6 8

Fourier transform

10 20 30 40 50 60 70 20 40 60 80 100 120

0.6
0.4
0.2

0.2 0.4 0.6 0.8

Windowed input

50 100 150 200 250 300 350

0.6
0.4
0.2

0.2 0.4 0.6

Periodic version

1 2 3 4 5 6

Fourier transform

10 20 30 40 50 60 70

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Zero-padding

Fourier transforms map N points to N points
What if we want to get more outputs? Zero padding!
Zero-padding interpolates the frequency domain

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Some useful Fourier properties

Additivity:
Shifting:
Parseval’s theorem

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x n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

DFT

← → ⎯⎯ X k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥, y n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

DFT

← → ⎯⎯ Y k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⇒ ax n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ +by n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

DFT

← → ⎯⎯ aX k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ +bY k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

x n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

DFT

← → ⎯⎯ X k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⇒ x ≪ n−no ≫ N ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

DFT

← → ⎯⎯ X k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ e

− j2πk N n0

x n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

DFT

← → ⎯⎯ X k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⇒ x n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

2 n=0 N−1

∑

= 1 N X k ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

2 n=0 N−1

∑

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Frequency domain representation

Representing sounds

by frequency content

Provides a better

glimpse of the input

But provides no

temporal information!

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50 100 150 200 −1 1 20 40 60 80 100 50 100 50 100 150 200 −1 1 20 40 60 80 100 50 100 150 50 100 150 200 −1 1 20 40 60 80 100 10 20

Time series Magnitude spectrum

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On real sounds

We get a better sense of what’s in a signal
But not of the temporal progression

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0.5 1 1.5 2 x 10

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−80 −60 −40 −20 Frequency Power Spectrum Magnitude (dB) First 4 sec 0.5 1 1.5 2 x 10

4

−80 −60 −40 −20 Frequency Power Spectrum Magnitude (dB) Last 4 sec 0.5 1 1.5 2 x 10

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−80 −60 −40 −20 Frequency Power Spectrum Magnitude (dB) Overall

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Adding one more dimension

How about sampling spectra periodically?
Each sound segment will have it’s own spectrum
“Short-time frequency analysis”
Break input into analysis windows and DFT them
Plot all successive spectra side by side
Keeps time info, but also presents frequency content

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Time/frequency representation

Many names/varieties
Spectrogram, sonogram, periodogram,

Short-Time Fourier Transform (STFT), …

A time-ordered series of frequency

compositions

Can help show how things change in

both time and frequency

Most useful representation so far!
Reveals information about the frequency

content without sacrificing the time information

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Time series Time/Frequency

50 100 150 200 −1 −0.5 0.5 1 0.5 1 1.5 2 2.5 3 3.5 10 20 30 50 100 150 200 −1 −0.5 0.5 1 0.5 1 1.5 2 2.5 3 3.5 10 20 30 50 100 150 200 −1 −0.5 0.5 1 0.5 1 1.5 2 2.5 3 3.5 10 20 30

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The details

Get N samples, advance H samples, repeat
N is transform size, H is hop size
On each N-sample frame apply window
Tapers the edges makes for better estimate
On each windowed frame apply DFT
Gets frequency domain of this section only
DFT can also be M-points (M > N)
Input is zero-padded to length M
Collate all spectra in 2d representation

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1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

1
0.5

0.5 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 40 60 80 100 120

…

2 4 6 8 10 12 14 16 18 20 40 60 80 100 120

Input Magnitude spectra Spectrogram

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Pretty picture version

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Make frames DFT the frames Show them better

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Spectrogram parameters

Size of the Discrete Fourier Transform
Determines how fine the frequency resolution is
To get more frequencies we can zero pad
Hop size
Determines how fine the temporal resolution is
Should not be larger than transform size! (why?)
Window
Tradeoff between artifacts and frequency resolution
Stronger window more blurring, weaker window more artifacts

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Time/frequency tradeoff

The more frequencies, the fewer time points
And vice-versa

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Spectral warping

Regular spectrograms are hard to read
Frequency warping can help (e.g. Mel scale, Bark scale, etc)

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Back to a previous example

With the spectrogram we can now see what goes on

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Remember these?

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We can now “see” what goes on

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Frequency (Hz) Time (sec) 0.5 1 1.5 2 100 500 1000 2000 3500 5500 Frequency (Hz) Time (sec) 1 2 3 100 500 1000 2000 3500 8000 Frequency (Hz) Time (sec) 2 4 6 100 500 1000 2000 3500 12000 20000 Frequency (Hz) Time (sec) 2 4 6 100 500 1000 2000 3500 12000 20000

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Very useful diagnostic tool!

Always look at the spectrogram!!
Best way to debug audio glitches!

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Minimum!!

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https:/ /www.youtube.com/watch?v=faBFiEfPxUU

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The inverse spectrogram

We can also go from spectrogram to waveform
Inverting the spectrogram procedure
For each (complex) spectral frame
Convert to time segment using inverse DFT
Optionally apply a window again
To “undo” synthesis window, or to avoid processing artifacts
“Overlap and add” segments that coincided

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Overlap and add

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… …

Spectra Inverse DFT on respective time Waveform

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Careful when inverse windowing!

If you use no windows the output scales with hop size
Number of times you overlap-add segments
If you use windowing you need to satisfy COLA:
Constant OverLap Add:
where H is the hop size

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w(n−mH)=1,∀n ∈ !

m=−∞ ∞

∑

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What does that mean?

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Examples of bad windowing

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Some uses of inverse spectrograms

Useful for spectral editing!
Demo
We will use later for:
Denoising
Time stretching/compression
Spectral manipulations
Fast convolutions
And many more …

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Pictures to sound

Fun applications

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A commercial example

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The Fast Fourier Transform (FFT)

Efficient DFT algorithm
Huge speedup! (always use it!)
Most routines you will find and

use will be FFT routines

These return full complex spectrum
Some are specifically for real inputs
You might have to modify for real inputs

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Recap

Digitizing and discretizing audio
Basic things to remember to represent sound best
Frequency analysis and the DFT
Time-frequency analysis and the spectrogram
Also its inverse

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Reference material

Overview of DSP:
http://www.dspguide.com/pdfcook.htm
Spectral analysis of audio:
http://www.dsprelated.com/dspbooks/sasp/

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Thursday is lab day

First graded lab
Implementing a forward/inverse spectrogram
Examining sounds using your code
Labs administrivia
Released on Thursdays, submit solutions within two weeks
Send your notebooks via email to me (attached or linked)
Use your @illinois email so that I know who you are!!
Use subject: “CS498 Lab #” (where # is the lab’s number)
Late submissions get zero grade (worst two grades thrown out)

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