A Tutorial on Fourier Analysis

A Tutorial on Fourier Analysis

Douglas Eck

University of Montreal

NYU March 2006

A Tutorial on Fourier Analysis

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

fund (freq 100)3rd harm5th harm7th harmm

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1Sum of odd harmonics approximate square wave

fund (freq 100)fund+3rd harmfund+3rd+5thfund+3rd+5th+7th

A Tutorial on Fourier Analysis

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A Tutorial on Fourier Analysis

Linear Combination

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

Θ(u) =∞∑i=0

αiψi (u)

where functions ψi (u) make up a set of simple elementaryfunctions. If functions are orthogonal (roughly, perpindicular; innerproduct is zero)then coefficients αi are independant from oneanother.

A Tutorial on Fourier Analysis

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(−2πjwt)dt

x(t) =

∫ +∞

−∞X (w)e(2πjwt)dw

A Tutorial on Fourier Analysis

Discrete Fourier and Inverse Fourier Transform

X (n) =N−1∑k=0

x(k)e−2πjnk/N

x(k) =1

N

N−1∑n=0

X (n)e2πjnk/N

A Tutorial on Fourier Analysis

Taylor Series Expansion

f (x) = f (x0)+f ′(x0)

1(x−x0)+

f ′′(x0)

1 ∗ 2(x−x0)

2+f ′′′(x0)

1 ∗ 2 ∗ 3(x−x0)

3+...

or more compactly as

f (x) =∞∑

n=0

f (n)(x0)

n!(x − x0)

n

A Tutorial on Fourier Analysis

Taylor series expansion of e jΘ

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:

e jΘ ==∞∑

n=0

(jΘ)n

n!= 1 + jΘ− Θ2

2− j

Θ3

3!+ ...

A Tutorial on Fourier Analysis

Splitting out real and imaginary parts

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

ree jΘ = 1−Θ2/2 + Θ4/4!− ...

ime jΘ = Θ−Θ3/3! + Θ5/5!− ...

A Tutorial on Fourier Analysis

Fourier Transform as sum of sines and cosines

Observe that:

cos(Θ) =∞∑

n>=0;n is even

(−1)n/2

n!Θn

sin(Θ) =∞∑

n>=0;n is odd

(−1)(n−1)/2

n!Θn

Thus yielding Euler’s formula:

e jθ = cos(Θ) + j sin(Θ)

A Tutorial on Fourier Analysis

A Tutorial on Fourier Analysis

Fourier transform as kernel matrix

A Tutorial on Fourier Analysis

Example

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

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signal

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real part

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two cosines (freqs=9, 12)

A Tutorial on Fourier Analysis

Example

FFT coefficients mapped onto unit circle

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1FFT projected onto unit circle

A Tutorial on Fourier Analysis

Impulse response

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signal

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magnitude

impulse response

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phase

A Tutorial on Fourier Analysis

Impulse response

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signal

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magnitude

delayed impulse response

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phase

A Tutorial on Fourier Analysis

A look at phase shifting

Sinusoid frequency=5 phase shifted multiple times.

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49.5361

49.5361

49.5361

mag

nitu

de

number of points shifted

sinusoid freq=5 phase shifted repeatedly

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1

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angl

e fr

eq=

6

number of points shifted

A Tutorial on Fourier Analysis

Sin period 10 + period 30

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freq (mag)

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phase

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component sinusoids

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reconstructed signal using component sinusoids vs original

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reconstructed signal using ifft vs original

A Tutorial on Fourier Analysis

Aliasing

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

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1cos(21)

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1cos(21) sampled at 24 Hz

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1cos(45)

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1cos(45) sampled at 24 Hz

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1cos(−3)

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1cos(−3) sampled at 24 Hz

A Tutorial on Fourier Analysis

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)

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Time (samples)

Am

plitu

de

a)

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Normalized Frequency (cycles per sample))

Mag

nitu

de (

Line

ar)b)

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Normalized Frequency (cycles per sample))

Mag

nitu

de (

dB)

c)

A Tutorial on Fourier Analysis

Leakage

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

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1Sinusoid NEAR 1/4 the Sampling Rate

Time (samples)

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a)

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Normalized Frequency (cycles per sample))

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nitu

de (

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ar)

b)

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nitu

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c)

A Tutorial on Fourier Analysis

Leakage

Comparison:

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1Sinusoid at 1/4 the Sampling Rate

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plitu

de

a)

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Normalized Frequency (cycles per sample))

Mag

nitu

de (

Line

ar)b)

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Normalized Frequency (cycles per sample))

Mag

nitu

de (

dB)

c)

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1Sinusoid NEAR 1/4 the Sampling Rate

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plitu

de

a)

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nitu

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c)

A Tutorial on Fourier Analysis

Windowing can help

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

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1Blackman Window

Time (samples)

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a)

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nitu

de (

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b)

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nitu

de (

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c)

A Tutorial on Fourier Analysis

Leakage Reduced

Comparison:

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1Sinusoid NEAR 1/4 the Sampling Rate

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plitu

de

a)

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10

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30

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Mag

nitu

de (

Line

ar)

b)

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30

Normalized Frequency (cycles per sample))

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nitu

de (

dB)

c)

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Time (samples)

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plitu

de

a)

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ar)

b)

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A Tutorial on Fourier Analysis

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)

A Tutorial on Fourier Analysis

Computing impulse response

The impulse response h[n] is the response of a system to the unitimpulse function.

A Tutorial on Fourier Analysis

Using the impulse response

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

A Tutorial on Fourier Analysis

Examples

A Tutorial on Fourier Analysis

Filtering using DFT

Goal is to choose good impulse response h[n]

Transform signal into frequency domain

Modify frequency properties of signal via multiplication

Transform back into time domain

A Tutorial on Fourier Analysis

Filtering using DFT

Goal is to choose good impulse response h[n]

Transform signal into frequency domain

Modify frequency properties of signal via multiplication

Transform back into time domain

A Tutorial on Fourier Analysis

Filtering using DFT

Goal is to choose good impulse response h[n]

Transform signal into frequency domain

Modify frequency properties of signal via multiplication

Transform back into time domain

A Tutorial on Fourier Analysis

Filtering using DFT

Goal is to choose good impulse response h[n]

Transform signal into frequency domain

Modify frequency properties of signal via multiplication

Transform back into time domain

A Tutorial on Fourier Analysis

Difficulties (Why not a perfect filter?)

You can have a perfect filter(!)

Need long impulse response function in both directions

Very non causal

In generating causal version, challenges arise

A Tutorial on Fourier Analysis

Difficulties (Why not a perfect filter?)

You can have a perfect filter(!)

Need long impulse response function in both directions

Very non causal

In generating causal version, challenges arise

A Tutorial on Fourier Analysis

Difficulties (Why not a perfect filter?)

You can have a perfect filter(!)

Need long impulse response function in both directions

Very non causal

In generating causal version, challenges arise

A Tutorial on Fourier Analysis

Difficulties (Why not a perfect filter?)

You can have a perfect filter(!)

Need long impulse response function in both directions

Very non causal

In generating causal version, challenges arise

A Tutorial on Fourier Analysis

Gibbs Phenomenon

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ideal lopass filter in frequency domain

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ideal filter coeffs in time domain

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truncated causal filter

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Gibbs phenomenon

A Tutorial on Fourier Analysis

Spectral Analysis

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

Compute Fourier Transform over evenly-spaced frames of data

Apply window to minimize edge effects

A Tutorial on Fourier Analysis

Spectral Analysis

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

Compute Fourier Transform over evenly-spaced frames of data

Apply window to minimize edge effects

A Tutorial on Fourier Analysis

Spectral Analysis

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

Compute Fourier Transform over evenly-spaced frames of data

Apply window to minimize edge effects

A Tutorial on Fourier Analysis

Short-Timescale Fourier Transform (STFT)

X (m, n) =N−1∑k=0

x(k)w(k −m)e−2πjnk/N

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

A Tutorial on Fourier Analysis

Trumpet (G4)

Time

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quen

cy

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[Listen]

A Tutorial on Fourier Analysis

Violin (G4)

Time

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quen

cy

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[Listen]

A Tutorial on Fourier Analysis

Flute (G4)

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quen

cy

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[Listen]

A Tutorial on Fourier Analysis

Piano (G4)

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quen

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[Listen]

A Tutorial on Fourier Analysis

Voice

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A Tutorial on Fourier Analysis

C Major Scale (Piano)

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A Tutorial on Fourier Analysis

C Major Scale (Piano)

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

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13669

A Tutorial on Fourier Analysis

Time-Space Tradeoff

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Am

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A Tutorial on Fourier Analysis

Time-Space Tradeoff

Time

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A Tutorial on Fourier Analysis

Auto-correlation and meter

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

a(k) =1

N

N−1∑n=k

x(n) x(n − k)

Autocorrelation can also be defined in terms of Fourier analysis

a = F−1(|F(x)|)

where F is the Fourier transform, F−1 is the inverse Fouriertransform and || indicates taking magnitude from a complex value.

A Tutorial on Fourier Analysis

Auto-correlation and meter

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

a(k) =1

N

N−1∑n=k

x(n) x(n − k)

Autocorrelation can also be defined in terms of Fourier analysis

a = F−1(|F(x)|)

where F is the Fourier transform, F−1 is the inverse Fouriertransform and || indicates taking magnitude from a complex value.

A Tutorial on Fourier Analysis

0 5 10 15 20 25 30time (seconds)

0 5 10 15 20 25 30time (seconds)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000lags (ms)

Time series (top), envelope (middle) and autocorrelation (bottom) of

excerpt from ISMIR 2004 Tempo Induction contest

(Albums-Cafe Paradiso-08.wav). A vertical line marks the actual tempo

(484 msec, 124bpm). Stars mark the tempo and its integer multiples.

Triangles mark levels in the metrical hierarchy.

A Tutorial on Fourier Analysis

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 orderOutput transformation (Decimation in Time)