Spectral Modeling and Signal Processing Intro421

8/15/2019 Spectral Modeling and Signal Processing Intro421

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MUS421/EE367B Lecture 1

Introduction and Overview

Julius O. Smith III ([email protected])Center for Computer Research in Music and Acoustics (CCRMA)

Department of Music, Stanford UniversityStanford, California 94305

April 10, 2007

Course Overview

The focus of this course is spectral modeling andsignal processing using the short-time Fourier

transform (STFT). Applications include musical sound synthesis and

audio signal processing.

1
http://ccrma.stanford.edu/~joshttp://ccrma.stanford.edu/http://www.stanford.edu/group/Music/http://www.stanford.edu/http://www.stanford.edu/http://www.stanford.edu/group/Music/http://ccrma.stanford.edu/http://ccrma.stanford.edu/~jos


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Administrative Information

Units

You may sign up for either 3 or 4 units:

3 units = lectures + assignments + final

4 units adds a final project based on outside readingand/or a software project

Important Pointers The course schedule and outline1 (reachable from the

class home page2) lists the following information:

Assignments

Weekly class schedule

Pointers to all lecture overheads The 421 home page further contains pointers to

Programming examples

Sound examples

Related items of interest online

The MUS421/EE367B Overview3 contains thisadministrative info and more.

1http://ccrma.stanford.edu/~jos/intro421/Schedule_Assignments.html2http://ccrma.stanford.edu/CCRMA/Courses/421/3http://ccrma.stanford.edu/~jos/intro421/

2
http://ccrma.stanford.edu/~jos/intro421/Schedule_Assignments.htmlhttp://ccrma.stanford.edu/CCRMA/Courses/421/http://ccrma.stanford.edu/~jos/intro421/http://ccrma.stanford.edu/~jos/intro421/http://ccrma.stanford.edu/CCRMA/Courses/421/http://ccrma.stanford.edu/~jos/intro421/Schedule_Assignments.html


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Why The Fourier Transform

Natural for visualizing audio signals:The ear performs a kind of Fourier analysis

Spectral models can be very compact and flexible:

MPEG audio coding

Sinusoidal modeling (additive synthesis)

AES talk4 on history of spectral modeling atCCRMA and elsewhere.

Any Linear Time Invariant (LTI) system can beimplemented in the frequency domain by means ofthe Fourier Transform

Efficient FFT implementations exist which make itpossible to implement very large LTI systems in realtime, e.g., room impulse-response convolutions oflength 10,000 to 100,000

4http://ccrma.stanford.edu/~jos/pdf/AES-Heyser.pdf

3
http://ccrma.stanford.edu/~jos/pdf/AES-Heyser.pdfhttp://ccrma.stanford.edu/~jos/pdf/AES-Heyser.pdf


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Applications of theShort-Time Fourier Transform (STFT)

Frequency-domain display of audio signals

Fast convolution

Time-varying linear filtering

Time-varying nonlinear filtering

Fourier analysis, modification, and resynthesis

Musical sound synthesis via spectral modeling:

Additive synthesis using sinusoids

Sines + Noise modeling

Sines + Noise + Transients modeling

Speech synthesis

Vocoders

Time scaling

Pitch shifting

Pitch Detection Noise reduction

Audio compression

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Applications, Contd

Transform coders for audio compression, such as

MPEG AAC (10X common)

MP3 (MPEG-II, Layer III about half thequality of AAC)

Dolby AC-2 and AC-3 (6X, fixed)

Philips DCC, Sony Minidisc (4 or 5X))

Music 422 (EE 367C) is an entire CCRMA coursedevoted to this topic. It is offered only in Winterquarters.

Perfect Reconstruction Filter banks

Computational Auditory Modeling

Filter Design and Implementation

System Identification

Signal Separation (Auditory Scene Analysis)

Non-Parametric Spectral Estimation

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Emerging Applications in Audio Coding

MPEG-4 supports an object oriented approach tocompression algorithms:

First transmitobjects = synthesizer patches = decoder

Next transmitmessages = performance data (like MIDI) =encoded bit stream

Main challenge: to develop classifiers and coders forgeneral purpose audio

Final Project (4th Unit)

Individual or group research project

Must be related to lecture/lab topics (FFT+Audio)

One-page project proposal due by 4th class meeting

Final written report due by the end of finals week

Oral presentation during the last class invited!

Project Types

Programming project and report

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Reading and report

Suggested Outside Reading: Seehttp://ccrma.stanford.edu/~jos/refs421/

(also available in Appendix P of the text.5)

Example Project Topics

Windows

New FFT window types

Explore window types not covered in class

Spectrum Analysis

Short-time spectrum analysis of recorded data

Study of statistical spectrum estimation

Matching spectrum analysis parameters tohuman hearing

Alternative time-frequency representations(Wavelets, Wigner, ...)

Sinusoidal Modeling

Readings in additive synthesis

Implement your own sines+noiseanalysis/synthesis system

Noise reduction based on sinusoidal modeling

Source separation based on sinusoidal modeling5http://ccrma.stanford.edu/~jos/sasp/

7
http://ccrma.stanford.edu/~%7B%7Djos/refs421/http://ccrma.stanford.edu/~jos/sasp/http://ccrma.stanford.edu/~jos/sasp/http://ccrma.stanford.edu/~%7B%7Djos/refs421/


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Transient detection (for sines + noise +transients modeling)

Short-Time Fourier Transform (STFT) based

Analysis, Modification, and Resynthesis Software system development

Noise reduction

Pitch detection/tracking

Time compression/expansion

Transform coding Pitch-synchronous phase vocoder (adaptwindow to pitch period)

Signal reconstruction from the STFT magnitudeonly (phase discarded)

Spectral interpolation schemes to compensate

for lost frames Software Development

Modify course Matlab examples to becompatible with Octave(See http://www.octave.org/.)

Develop missing components of the Signal

Processing Toolbox for Octave (looking only atMatlab help info).

PD FFT-processing patches (seehttp://www.pure-data.org/)

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http://www.pure-data.org/http://www.pure-data.org/


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LADSPA spectral-processing plug-ins (seehttp://www.ladspa.org/

9
http://www.ladspa.org/http://www.ladspa.org/


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Main Pointer

The course schedule and outline6 (reachable from theclass home page7) lists the following information:

Assignments

Weekly class schedule

Pointers to all lecture overheads

6http://ccrma.stanford.edu/~jos/intro421/Schedule_Assignments.html7http://ccrma.stanford.edu/CCRMA/Courses/421/

10
http://ccrma.stanford.edu/~jos/intro421/Schedule_Assignments.htmlhttp://ccrma.stanford.edu/CCRMA/Courses/421/http://ccrma.stanford.edu/CCRMA/Courses/421/http://ccrma.stanford.edu/~jos/intro421/Schedule_Assignments.html


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Notation

Before we begin, we will review the notation we will usein this class. (We will try to stay consistent.)

Frequency and Time:

denotes continuous radian frequency (rad/sec)

f denotes continuous frequency in Hertz (Hz)

= 2f

k denotes discrete frequency, k = 2(k/N)fs

, k R (frequencies are always real)

T = sampling interval (sec)

fs = sampling rate, fs =1

T

tn = nT (discrete time)

n, k Z (integers)

t, tn R (times are always real)

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Introduction to Audio Spectrum Analysis

Spectrum analysis of real-world signals typically occurs inshort segments. We are therefore most interested inshort-time spectrum analysis:

Spectral content typically varies over time.

The human ear uses less than one second of past

sound to form a spectrum. There is a limit to the length of signal we can analyze

at once.

To extract and analyze a sound segment, it is necessaryto apply a window function. An unmodified segment

extraction corresponds to a rectangular window.Everything we look at will be through a window, henceit is important to realize what the window is doing to ourunderlying signal.

Applications well discuss first:

Spectral Analysis for Display

FIR Filter Design

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Example of Windowing

Lets look at a simple example of windowing todemonstrate what happens when we turn an infiniteduration signal into a finite duration signal throughwindowing.

Complex Sinusoid:

x(n) = ejnT, 0 T <

Notes:

real part = cos(nT)

The frequencies present in our signal are only positive.A fancy name for this is an analytic signal

This signal is infinite duration. (It doesnt die out as nincreases.) In order to end up with a signal which diesout eventually (so we can use the DFT), we need tomultiply our signal by a window (which does die out).

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The following is a diagram of a typical window function:

1000 800 600 400 200 0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ZeroPhase Window

Time (samples)

Amplitude

This is loosely called a zero-centered (or zero phase,or even) window function, which means its phase in thefrequency domain is either zero or , as we will see indetail later. (Recall that a real and even function has areal and even Fourier transform.) The window is also

nonnegative, as is typical.

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We might also require that our window be zero fornegative time. Such a window is said to be causal.Causal windows are necessary for real-time processing:

1000 800 600 400 200 0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Linear Phase Window (Causal)

Time (samples)

Amplit

ude

By shifting the original window in time by half its length,we have turned the original non-causal window into acausal window. The Shift property of the FourierTransform tells us that we have introduced a linear phaseterm.

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Putting all this together, we get the following:

Our original signal (unwindowed, infinite duration), is

x(n) = ej0nT, n Z

A portion of the real part, cos(0nT), is plotted below:

2000 1500 1000 500 0 500 1000 1500 20001

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Time (samples)

Amplitude

The imaginary part, sin(0nT), is of course identical butfor a 90-degree phase-shift to the right.

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The Fourier Transform of this infinite duration signal is adelta function at 0: X() = ( 0)

( 0)

0 0

The windowed version is: xw(n) = w(n)ej0nT n Z

(Note carefully the difference between w and .)

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2500 2000 1500 1000 500 0 500 1000 1500 2000 25001

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Time (samples)

Amplitude

The Convolution Theorem tells us that our multiplicationin the time domain results in a convolution in thefrequency domain. Hence, in our case, we will obtain theconvolution of a delta function at frequency 0, and thetransform of the window.

The result of convolution with a delta function is theoriginal function, shifted to the location of the deltafunction. (The delta function is the identity element for

convolution.)

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3 2 1 0 1 2 350

45

40

35

30

25

20

15

10

5

0

Amplitude

dB

main lobe

sidelobes

T 0

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Summary

Windowing in the time domain resulted in asmearing or smoothing in the frequency domain.We need to be aware of this if we are trying to resolvesinusoids which are close together in frequency.

Windowing also introduced side lobes.This is important when we are trying to resolve low

amplitude sinusoids in the presence of higheramplitude signals. When we look at specific windows,we will be looking at this behavior.

A sinusoid at amplitude A, frequency 0, and phase becomes a window transform shifted out to frequency

0, and scaled by Ae

j

.

There are many type of windows which serve variouspurposes and exhibit various properties.

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The Rectangular Window

The rectangular window may be defined as:

wR(n)=

1, |n| M120, otherwise

20 15 10 5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

Time (samples)

Amplitude

ZeroPhase Rectangular Window M = 21

Zero centered definition (even in time domain)

Need M odd in zero-centered case

Scale window by 1/M to obtain unity dc gain

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To see what happens in the frequency domain, we needto look at the DTFT of the window:

WR() = DTFT(wR)

=

n=

wR(n)ejn

=

M12

n=M12

ejn =ej

M12 ej

M+12

1 ej

where we used the closed form of a geometric series:

Un=L

rn =rL rU+1

1 r

We can factor out linear phase terms from the numeratorand denominator of the above expression to get

WR() = ej

1

2

ej12

ej

M

2 ejM

2

ej12 ej

12

=sin

M2

sin

2

= M asincM()where asincM() denotes the aliased sinc function.

asincM()=

sin(M/2)M sin(/2)

(also called the Dirichlet function)

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Rectangular Window Transform (Contd)

Above, we found the rectangular window transform to be

the aliased sinc function:

WR() = M asincM()=

sin

M2

sin

2

6 4 2 0 2 4 64

2

0

2

4

6

8

10

12

freq

ComplexAmplitude

DFT of a Rectangular Window of length M = 11

Amp

This (real) result is for the zero-centered rectangularwindow. For the causal case, a linear phase term appears:

WcR() = ej M12 MasincM()

As the sampling rate goes to infinity, the aliased sincfunction approaches the regular sinc function

sinc(x)=

sin(x)

x

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More generally, we may plot both the magnitude andphase of the window versus frequency:

6 4 2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Magnitude(Linear)

Magnitude of Rectangular Window Transform (M = 11)

/M

6 4 2 0 2 4 60.3927

0

0.3927

0.7854

1.1781

1.5708

1.9635

2.3562

2.7489

3.1416

3.5343

Phase

Phase of Rectangular Window Transform (M = 11)

Main Lobe

/M

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In audio work, we more typically plot the windowtransform magnitude on a decibel (dB) scale:

3 2 1 0 1 2 340

35

30

25

20

15

10

5

0

13 dB down

sidelobes sidelobes

main lobe

nulls nulls

Normalized Frequency (rad/sample)

Magnitude

(dB)

DFT of a Rectangular Window M = 11

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Since the DTFT of the rectangular window approximatesthe sinc function, it should roll off at approximately 6dB per octave, as verified in the log-log plot below:

0.1 0.2 0.4 0.8 1.6 3.2 6.4

54.1854

48.1648

42.1442

36.1236

30.103

24.0824

18.0618

12.0412

6.0206

0

Ideal 6 dB per octave line

Normalized Frequency (rad/sample)

Amplitude(dB

)

DFT of a Rectangular Window M = 20

Partial

Main

Lobe

As the sampling rate approaches infinity, the rectangularwindow transform (asinc) converges exactly to the sincfunction. Therefore, the departure of the roll-off from

that of the sinc function can be ascribed to aliasing in thefrequency domain, due to sampling in the time domain.

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Sidelobe Roll-Off Rate

In general, if the first n derivatives of a continuous

function w(t) exist (i.e., they are finite and uniquelydefined), then its Fourier Transform magnitude isasymptotically proportional to

|W()| constant

n+1(as )

Proof: Look up roll-off rate in text index.

Thus, we have the following rule-of-thumb:n derivatives 6(n + 1) dB per octave roll-off rate

(since 20log10(2) = 6.0205999 . . .).This is also 20(n + 1) dB per decade.

To apply this result, we normally only need to look at the

windows endpoints. The interior of the window is usuallydifferentiable of all orders.

Examples:

Amplitude discontinuity 6 dB/octave roll-off

Slope discontinuity 12 dB/octave roll-off

Curvature discontinuity 18 dB/octave roll-off

For discrete-time windows, the roll-off rate slows down athigh frequencies due to aliasing.

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In summary, the DTFT of the M-sample rectangularwindow is proportional to the aliased sinc function:

asincM(T)

=

sin(MT/2)

sin(T /2)

sin(fMT)

f T= Msinc(f MT)

Some important points:

Zero crossings at integer multiples of M

=2M

where M= 2M = frequency sampling interval used

by a length M DFT

Main lobe width is 2M =4M

As M gets bigger, the mainlobe narrows

(better frequency resolution) M has no effect on the height of the side lobes

(Same as the Gibbs phenomenon for Fourier series)

First sidelobe only 13 dB down from main-lobe peak

Side lobes roll off at approximately 6dB per octave

A phase term arises when we shift the window tomake it causal, while the window transform is real inthe zero-centered case (i.e., when the window w(n) isan even function of n)

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Frequency Resolution

The next series of plots shows the effect that an increased

window length has on our ability to resolve 2 sinusoids.

Two Cosines (In-Phase Case)

2 cosines separated by = 240

Rectangular Windows of lengths: 20, 30, 40, 80

( =

1

2M,

3

4M, M, 2M)

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Magnitude

M = 20

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Magnitude

M = 30

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M

agnitude

M = 40

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M

agnitude

M = 80

Frequency T (rad/sample)Frequency T (rad/sample)


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One Sine and One Cosine(Phase Quadrature Case)

As above, but 1 sine and 1 cosine

Note: least-resolved case appears resolved!

Note: M = 40 case suddenly looks much worse

Only the M = 80 case looks good at all phases

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Magnitude

M = 20

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Magnitude

M = 30

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Magnitude

M = 40

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Magnitude

M = 80



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One Sine and One Cosine(Phase Quadrature Case)

All Four Resolutions Overlaid

Same plots as on previous page, just overlaid

Peak locations are biased in under-resolved cases,both in amplitude and frequency

0 0.5 1 1.5 2 2.5 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

M

agnitude

M=20

M=30M=40M=80

Frequency T (rad/sample)

The preceding figures suggest that, for a rectangularwindow of length M, two sinusoids can be most easily

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resolved when they are separated in frequency by

2M M=

2

MThis implies there must be at least two full cycles of thedifference-frequency under the window. (Well see laterthat this is an overly conservative requirementa morecareful study reveals that 1.44 cycles is sufficient for therectangular window.)

In principle, arbitrarily small frequency separations can beresolved if

there is no noise, and

we are sure we are looking at the sum of two idealsinusoids under the window.

However, in practice, there is almost always some noiseand/or interference, so we prefer to require sinusoidalfrequency separation by at least one main-lobe width (ofthe sinc-function in this case, or the window transformmore generally) whenever possible.

The rectangular window provides an abrupt transition atits edge. We will later look at some other windows whichhave a more gradual transition. This is usually done toreduce the height of the side lobes.

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Resolution Bandwidth (resolving sinusoids)

Our ability to resolve two closely spaced sinusoids isdetermined primarily by the main lobe width of theFourier transform of the window we are using.

Let Bw denote the main lobe width in Hz, with the mainlobe width defined as the width between zero crossings:

-4 -3 -2 -1 0 1 2 3 4-2

-1

0

1

2

3

4

5

6

7

WR

()

2 32

=7

3

2Bw

For the Rectangular Window (length M), we have

WR() = asincM()=

sin(MT/2)

sin(T /2)=

sin(MfT)

sin(f T)

Main lobe width is two sidelobes wide

Bw = 2M2

= 2fsM

(Hz)

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Choosing Window Length to Resolve Sinusoids

A conservative requirement for resolving 2 sinusoids (innoisy conditions) with a spacing of Hz is to choose awindow length M long enough so that their main lobesare clearly discernible. For example, we may require thattheir main lobes meet at the first zero crossings.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Closely spaced sinusoids - Rectangular window - M = 21

Amplitude

Bw

Bw

T (radians per sample)

To obtain the separation shown above, we must haveBw , where Bw is the main lobe width in Hz, and is the sinusoidal frequency separation in Hz.

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For the rectangular window, Bw can be expressed as

Bw = 2fsM

Hence we need:

Bw = 2fsM

M 2fs

or

M 2 fs

|f2 f1|

Thus, to resolve the frequencies f1 and f2 under arectangular window, it is sufficient for the window lengthM to span at least 2 periods of the difference frequencyf2 f1, measured in samples, where 2 is the width of themain lobe, measured in sidelobe-widths.

A rectangular window of length or greater is said toresolve the sinusoidal frequencies f1 and f2.

Documents

Spectral Modeling and Signal Processing Intro421