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A Tutorial on Fourier Analysis


Douglas Eck

University of Montreal

NYU March 2006
http://find/http://goback/

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0 20 40 60 80 100 120 140 160 180 2001

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1A fundamental and three odd harmonics (3,5,7)

fund (freq 100)

3rd harm5th harm

7th harmm

0 20 40 60 80 100 120 140 160 180 200

1

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1Sum of odd harmonics approximate square wave

fund (freq 100)

fund+3rd harm

fund+3rd+5th

fund+3rd+5th+7th
http://goback/http://find/http://goback/

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0 20 40 60 80 100 120 140 160 180 2001

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1

Sum of odd harmonics from 1 to 127
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Linear Combination

In the interval [u1, u2] a function (u) can be written as a linearcombination:

(u) =i=0

ii(u)

where functions i(u) make up a set of simple elementary

functions. If functions are orthogonal (roughly, perpindicular; innerproduct is zero)then coefficients iare independant from oneanother.
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Continuous Fourier Transform

The most commonly used set of orthogonal functions is the Fourierseries. Here is the analog version of the Fourier and Inverse Fourier:

X(w) =

+

x(t)e(2jwt)dt

x(t) = +

X(w)e(2jwt)

dw

A A
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Discrete Fourier and Inverse Fourier Transform

X(n) =

N1k=0

x(k)e2jnk/N

x(k) = 1

N

N1n=0

X(n)e2jnk/N

A T i l F i A l i
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Taylor Series Expansion

f(x) =f(x0)+ f

(x0)1 (xx0)+ f

(x0)1 2 (xx0)2+ f

(x0)1 2 3 (xx0)3+...

or more compactly as

f(x) =

n=0

f(n)(x0)

n! (x x0)n

A T t i l n F i An l sis
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Taylor series expansion of ej

Since ex is its own derivative, the Taylor series expansion forf(x) =ex is very simple:

ex =n=0

xn

n! = 1 +x+

x2

2 +

x3

3! +...

We can define:

ej ==n=0

(j)nn!

= 1 +j 22 j3

3! +...


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Splitting out real and imaginary parts

All even order terms are real; all odd-order terms are imaginary:

reej = 1 2/2 + 4/4! ...

imej = 3/3! + 5/5! ...

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Fourier Transform as sum of sines and cosines

Observe that:

cos() =

n>=0;n is even

(1)n/2

n!

n

sin() =

n>=0;n is odd

(1)(n1)/2

n! n

Thus yielding Eulers formula:

ej = cos() +j sin()


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Fourier transform as kernel matrix

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y

Example

Sum of cosines with frequencies 12 and 9, sampling rate = 120

0 20 40 60 80 100 1200.02

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0

0.01

0.02

signal

0 20 40 60 80 100 1200.5

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1

real part

0 20 40 60 80 100 1200.2

0.1

0

0.1

0.2

imag part

two cosines (freqs=9, 12)

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Example

FFT coefficients mapped onto unit circle

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1FFT projected onto unit circle

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Impulse response

0 10 20 30 40 50 60 70 80 90 1000

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1

signal

0 10 20 30 40 50 60 70 80 90 100

0.01

0.01

0.01

magnitude

impulse response

0 10 20 30 40 50 60 70 80 90 1001

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0

0.5

1

phase
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A look at phase shifting

Sinusoid frequency=5 phase shifted multiple times.

0 10 20 30 40 50 60 70 80 90 10049.5361

49.5361

49.5361

49.5361

magn

itude

number of points shifted

sinusoid freq=5 phase shifted repeatedly

0 10 20 30 40 50 60 70 80 90 1003

2

1

0

1

2

3

anglefreq=6

number of pointsshifted

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Sin period 10 + period 30

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.50

100

200

freq (mag)

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.55

0

5

phase

0 100 200 300 400 500 6001

0

1

component sinusoids

0 100 200 300 400 500 6005

0

5

reconstructed signal using component sinusoids vs original

0 100 200 300 400 500 6002

0

2

reconstructed signalusing ifft vs original

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Aliasing

The useful range is the Nyquist frequency (fs/2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0

1cos(21)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0

1cos(21) sampled at 24 Hz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0

1cos(45)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0

1cos(3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

0


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Leakage

Even below Nyquist, when frequencies in the signal do not alignwell with sampling rate of signal, there can be leakage. Firstconsider a well-aligned exampl (freq = .25 sampling rate)

0 10 20 30 40 50 60 701

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1Sinusoid at 1/4 the Sampling Rate

Time (samples)

Amp

litude

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

Normalized Frequency (cycles per sample))

Mag

nitude(Linear)

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

300

200

100

0

Normalized Frequency (cyclesper sample))

Magnitude(dB)

c)

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Leakage

Now consider a poorly-aligned example (freq = (.25 + .5/N) *sampling rate)

0 10 20 30 40 50 60 701

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1Sinusoid NEAR 1/4 the Sampling Rate

Time (samples)

Amplitude

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30


Magnitude(Linear)

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

0

10

20

30


Magnitude(dB)

c)

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Leakage

Comparison:

0 10 20 30 40 50 60 701

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Time (samples)

Am

plitude

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40


Magnitude(Linear)

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

300

200

100

0


Magnitude(dB)

c)

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Time (samples)

Am

plitude

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30


Magnitude(Linear)

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

0

10

20

30


Magnitude(dB)

c)

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Windowing can help

Can minimize effects by multiplying time series by a window thatdiminishes magnitude of points near signal edge:

0 10 20 30 40 50 60 700

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1Blackman Window

Time (samples)

Amplitude

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

50

0


Magnitude(dB)

b)

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5100

50

0

NormalizedFrequency (cycles per sample))

Magnitude(dB)

c)

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Leakage Reduced

Comparison:

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Time (samples)

Amp

litude

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30


Magnitude(Linear)

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

0

10

20

30


Magnitude(dB)

c)

0 10 20 30 40 50 60 701

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Time (samples)

Amp

litude

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15


Magnitude(Linear)

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 180

60

40

20

0

20


Magnitude(dB)

c)

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Convolution theorem

This can be understood in terms of the Convolution Theorem.

Convolution in the time domain is multiplication in the frequencydomain via the Fourier transform (F).

F(f g) = F(f) F(g)

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Computing impulse response

The impulse response h[n] is the response of a system to the unit

impulse function.

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Using the impulse response

Once computed, the impulse response can be used to filter anysignal x[n] yielding y[n].

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Examples

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Filtering using DFT

Goal is to choose good impulse response h[n]

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Filtering using DFT

Goal is to choose good impulse response h[n]Transform signal into frequency domain

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Filtering using DFT


Modify frequency properties of signal via multiplication

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Filtering using DFT


Modify frequency properties of signal via multiplication

Transform back into time domain

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Difficulties (Why not a perfect filter?)

Youcan

have a perfect filter(!)

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Youcan

have a perfect filter(!)Need long impulse response function in both directions

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You can have a perfect filter(!)

Need long impulse response function in both directions

Very non causal

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You can have a perfect filter(!)

Need long impulse response function in both directions

Very non causal

In generating causal version, challenges arise

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Gibbs Phenomenon

500 400 300 200 100 0 100 200 300 400 5000

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1

ideal lopass filter in frequency domain

500 400 300 200 100 0 100 200 300 400 5000.1

0

0.1

ideal filter coeffs in time domain

0 100 200 300 400 500 600 700 800 900 10000.1

0

0.1

truncated causal filter

500 400 300 200 100 0 100 200 300 400 5000

0.5

1

1.5

Gibbsphenomenon

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Spectral Analysis

Often we want to see spectral energy as a signal evolves overtime

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Spectral Analysis


Compute Fourier Transform over evenly-spaced frames of data

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Spectral Analysis


Compute Fourier Transform over evenly-spaced frames of data

Apply window to minimize edge effects

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Short-Timescale Fourier Transform (STFT)

X(m, n) =

N1

k=0

x(k)w(km)e2jnk/N

Where w is some windowing function such as Hanning or gaussiancentered around zero.The spectrogram is simply the squared magnitude of these STFT

values

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Trumpet (G4)

Time

Frequency

0 1 2 3 4 5 60

1000

2000

3000

4000

5000

[Listen]

http://wavs/trumpet-G4.wavhttp://wavs/trumpet-G4.wavhttp://find/

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Violin (G4)

Time

Frequency

0 0.5 1 1.5 2 2.5 30

1000

2000

3000

4000

5000

[Listen]

http://wavs/violin-G4.wavhttp://wavs/violin-G4.wavhttp://find/

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Flute (G4)

Time

Frequency

0 0.5 1 1.5 2 2.50

1000

2000

3000

4000

5000

[Listen]

http://wavs/flute-G4.wavhttp://wavs/flute-G4.wavhttp://find/

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Piano (G4)

Time

Frequency

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1000

2000

3000

4000

5000

[Listen]

http://wavs/piano-G4.wavhttp://wavs/piano-G4.wavhttp://find/

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Voice

Time

Frequency

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1000

2000

3000

4000

5000

6000

7000

8000

[Listen]

http://wavs/Voice.wavhttp://wavs/Voice.wavhttp://find/

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C Major Scale (Piano)

Time

Frequency

0 1 2 3 4 5 6 7 80

2000

4000

6000

8000

10000

12000

14000

16000

[Listen]

http://wavs/CMajorPianoScale.wavhttp://wavs/CMajorPianoScale.wavhttp://find/

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C Major Scale (Piano)

Log Spectrogram (Constant-Q Transform) reveals low-frequencystructure

Time

Frequency

0 1 2 3 4 5 6 7 8

135

240

427

761

1356

2416

4305

7671

13669

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Time-Space Tradeoff

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.4

0.2

0

0.2

0.4

0.6

Amp.

spoken "Steven Usma"

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Time-Space Tradeoff

Time

Frequency

Narrowband Spectrogram overlap=152 timepts=1633

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1000

2000

3000

4000

Time

Frequency

Wideband Spectrogram overlap=30 timepts=6593

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1000

2000

3000

4000

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Auto-correlation and meter

Autocorrelation long used to find meter in music (Brown 1993)Lag kauto-correlation a(k) is a special case of cross-correlationwhere a signal x is correlated with itself:

a(k) = 1N

N1n=k

x(n)x(n k)

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Auto-correlation and meter

Autocorrelation long used to find meter in music (Brown 1993)Lag kauto-correlation a(k) is a special case of cross-correlationwhere a signal x is correlated with itself:

a(k) = 1N

N1n=k

x(n)x(n k)

Autocorrelation can also be defined in terms of Fourier analysis

a= F

1(|F(x)|)

whereFis the Fourier transform, F1 is the inverse Fouriertransform and || indicates taking magnitude from a complex value.
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Fast Fourier Transform

Fourier Transform O(N2)

Fast version possible O(NlogN)

Size must be a power of two

Strategy is decimation in time or frequency

Divide and conquer

Rearrange the inputs in bit reversed order

Output transformation (Decimation in Time)
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