introduction to fourier transform and time-frequency analysis speaker: li-ming chen advisor:...

Introduction to Fourier Transform and Time-Frequency Analysis

Speaker: Li-Ming Chen

Advisor: Meng-Chang Chen, Yeali S. Sun

2009/10/9 Speaker: Li-Ming Chen 2

Outline

Periodic Phenomena and Fourier Series

Non-periodic Phenomena and Fourier Transform

Why Needs Time-Frequency Analysis?

Wavelet Transform and its Applications


History of Fourier Series

Fourier series Named in honor of Joseph Fourier (1768-1830) Originally used to solve “heat equation” Initially, the paper submitted in 1807 However, the theory published in 1822

A Fourier series decomposes a periodic function (or periodic signal) into a sum of simple oscillating functions, namely sines and cosines (or complex exponentials) (Wikipedia)


Example Fourier series model a general periodic phenomena by using basic building blocks;

Time (ms)50001000 2000 3000 4000

1

1

1

3

0

Am

plitu

de

-3

0sin(2*pi*t)

sin(2*pi*2*t)

sin(2*pi*5*t)

sin(2*pi*t) + sin(2*pi*2*t) + sin(2*pi*5*t)


Fourier Transform & Fourier Series We can consider:

Fourier transform as a limiting case of Fourier series in concerned with analysis of non-periodic phenomena

Applications of Fourier analysis: Physics, partial differential equations, signal processing, imaging,

acoustics, cryptography… Why so popular, so applicable? due to some properties:

Transforms are linear operators and “usually” invertible Using complex exponential (computational convenience) Convenient to compute convolution operation Has fast Fourier transform (FFT) algorithm


What is a Periodic Phenomenon? Periodic:

Some pattern that repeated and repeat regularly

Periodic

Periodic in Time

Periodic in Space

Called “Frequency”(e.g., number of repetitions of patterns in a second)

Called “Period”(e.g., Heat) The temperature

on a ring is periodic(depend on position)

※ Periodic in Time and Space!? e.g., wave motion


Using Periodic Function to Model Periodic Phenomena sin and cos are periodic fun

ctions Can configure their frequency,

amplitude, and phase

“One period, many frequencies” Let’s consider with period 1 also has perio

d 1 !! We can modify and combine th

ese building blocks to model very general periodic phenomena

)**2sin(* tkA

)*2sin( t

Time (ms)0 1

Am

plitu

de1

1

1

)**2sin( tk


Fourier Series

A periodic function f(t) can be represented as

Using 和角公式

Such that, “partial sums” of the Fourier series for ƒ

N

kkk tkAtf

1

)**2sin(*)(

N

kkkk tktkAtf

1

)sin(*)**2cos()cos(*)**2sin(*)(

these are constants

N

kkk tkbtkatf

1

)**2sin(*)**2cos(*)(

an and bn called Fourier coefficients of f.(include the info. of amplitude and phase)


Using Complex Exponential Euler’s formula

Fourier Series (General Form)

But, how to get ck ? (how to compute Fourier coefficients?)

)**2sin()**2cos(2 tkitke ikt

k

iktk ectf 2*)(

2)**2cos(

22 iktikt eetk

i

eetk

iktikt

2)**2sin(

22

Ck is also a complex number


How to Compute ck ?

Say, isolate cn and then solve cn

…(utilizing the property of complex exponential), we can get

nk

iktk

intn ectfec

!

22 *)(*

1

0

2*)( dtetfc intn

andnnn cca

)( nnn ccib

k

iktk ectf 2*)(

(Fourier series)

(Fourier coefficient)

f(t) is the function we observed[Analysis]

[Synthesis]


Outline






Fourier Transform (FT)

Not all phenomena are periodic, and periodic phenomena will die out eventually… Let’s view non-periodic function as limiting case of

period function as period ∞

Fourier transform is invertible FT is the generalization of Fourier coefficient Inverse FT is the generalization of Fourier series


Fourier Transform (FT) (cont’d) Definition

FT:

Inverse FT:

dtetfsF ist2*)()(

dsesFtf ist2*)()(


Example (periodic function)

FT

f(t) = cos(2pi*5t) + cos(2pi*10t) + cos(2pi*20t) + cos(2pi*50t)

5 10 20 50

all the amplitudes are 1

Magnitude/amplitude of the Freq.is half of the original amplitude

Time (ms) Frequency (Hz)


Example (non-periodic function)

FT

2

*)*6cos()( tettf

Compute c3 Compute c5

Freq. (Hz)

Integrals of the function in GREEN

Integrals of the function in Red

3 5Time (s)


Outline






Example

What if the periodic components occur at different time? Non-stationary

FT

5 10 20 50Time (ms) Frequency (Hz)

cos(2pi*5t) | cos(2pi*10t) | cos(2pi*20t) | cos(2pi*50t)

The noise is due tothe sudden change between the freq.

※ FT can still find the frequencies (4 peaks)


Why Needs Time-Frequency Analysis? Drawback of Fourier Transform

Time information is lost !! unable to tell when in time a particular event (Freq.)

took place Not a problem for signals which are stationary But when we have a signal which changes with time (non-

stationary), we need more information about the signal behavior

(Idea) can we assume that, some portion of a non-stationary signal is stationary? If so, we can do time-frequency analysis


Short Term Fourier Transform (STFT) In STFT, the signal is divided into small enough

segments, where these segments (portions) of the signal can be assumed to be stationary. Assume the signal is NOT changed for that particular

period Using window function (mask function) to cover those

periods

dtettxfXtxSTFT ift

2)()(),()}({

frequencytime


STFT

dtettxfXtxSTFT ift

2)()(),()}({

τ= t1’

τ= t2’τ= t3’

The length of this window function is pre-assigned.(the length will affect the results)

Window function could be:Rectangular function, Gaussian function, …


Example (STFT)

0~300ms: 300Hz,300~600ms: 200Hz,600~800ms: 100Hz,800~1000ms: 50Hz.

if we use FFT to analyze this signal,we might have good frequency resolution but poor time resolution.

300Hz 200Hz 100Hz 50Hz


4 Gaussian window Func. with different length

Poor freq. resolution,Good time resolution

Good freq. resolution,Poor time resolution

τf


Drawback of STFT

Heisenberg uncertainty principle ( 海森堡測不準原理 ) One can not know the exact time-frequency representati

on of a signal i.e., One can not know what spectral components exist at what i

nstances of times. What one can know are the time intervals in which certain band

of frequencies exist, which is a resolution problem.

Drawback of STFT is due to the constant length windows STFT: single-resolution for complete signal !! Need multi-resolution


Outline






Multi-resolution Analysis (MRA) MRA is designed to give

“good time resolution and poor frequency resolution” at high frequencies

and “good frequency resolution and poor time resolution” at low frequencies.

This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations. Common practical applications are often of this type


Continuous Wavelet Transform (CWT) 2 main differences between STFT and CWT:

1.) The Fourier transforms of the windowed signals are not taken, and therefore single peak will be seen corresponding to a sinusoid, i.e., negative frequencies are not computed.

2.) The width of the window is changed as the transform is computed for every single spectral component, which is probably the most significant characteristic of the wavelet transform.


Continuous Wavelet Transform Forward wavelet transform:

Inverse wavelet transform:

1,

t aX a b x t dt

bb

(t): mother wavelet

Scale (~ 1/freq)

Location (time, translation)

energy normalization能量守恆

,, a ba b

x t X a b t a,b(t) is dual orthogonal to (t)

output

Fourier transform X(f) or F(s), f, s: frequency (spectrum)

time-frequency analysis X(t, f), t: time, f: frequency

wavelet transform X(a, b), a: time, b: scale


Mother Wavelet, (t)

(t) is a prototype for generating the other window functions (t)(t/b) is the function with different scale derived from the moth

er wavelet b < 1, compresses the signal, (low scale detailed view) b > 1, dilates the signal, (high scale non-detailed global view)

0 10 20 300

0.5

1

0 10 20 300

0.5

1

0 10 20 300

0.5

1

0 10 20 300

0.5

1

1.5

0 10 20 300

0.2

0.4

0.6

0.8

0 10 20 300

0.2

0.4

0.6

0.8

a = 8, b=1 a = 15, b=1 a = 22, b=1

a = 8, b=0.5 a = 8, b=2 a = 8, b=3

)(1

b

at

b

能量守恆，故面積一樣

(a: 調整位置 )

(b: 調整寬度 )


Example30Hz 20Hz 10Hz 5Hz

CWT

Unlike the STFT which has a constant resolution at all times and frequencies, the WT has a good time and poor frequency resolution at high frequencies, and good frequency and poor time resolution at low frequencies


The discrete wavelet transform is very different from the continuous wavelet transform. It is simpler and more useful than the continuous one. ( 較為實用 )

x1,L[n]

x1,H[n]

g[n]

x[n]

h[n]

2

2

x[n] 的低頻成份

x[n] 的高頻成份

lowpass filter

highpass filter

down samplingxL[n]

xH[n]

1, 2Lk

x n x n k g k

1, 2Hk

x n x n k h k

down sampling

Discrete Wavelet Transform


原影像

2-D DWT

的結果

100 200 300 400 500

100

200

300

400

500

100 200 300 400 500

100

200

300

400

500

x1,L[m, n]

x1,H1[m, n]

x1,H2[m, n]

x1,H3[m, n]

Example:ImageCompression

( 低頻部分 ) (n 軸高頻 )

(m 軸高頻 )

2009/10/9 Speaker: Li-Ming Chen 3250 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Example: Image Compression

重複對低頻部分進行 DWT

和原圖類似，但資料量僅 1/64


Example: Traffic Volume Anomaly Detection Jianning Mai, Chen-Nee Chuah Ashwin Sridharan, Tao Y

e, Hui Zang, “Is Sampled Data Sufficient for Anomaly Detection?,” IMC 2006

Discrete wavelet transform (DWT) based detection An off-line algorithm 3 steps:

Decomposition Re-synthesis Detection


(Volume Anomaly Detection)

DWT-based Detection Decomposition

Input signal X will be separate into detail information D1 or approximation information A1 (level 1)

Repeated using Ai as an input to generate Di+1 and Ai+1 at the next level

, each level j represents the strength of a particular frequency in the signal Higher value of j indicating a lower frequency

Re-synthesis Aggregate the various frequency levels into low, mid and high

bands ,


(Volume Anomaly Detection)

DWT-based Detection (cont’d) Detection

Compute local variability of the high and mid bands Compute the variance of data within a sliding window

Compute deviation score The ratio between the local variance within the window and

the global variance Windows with deviation scores higher than a threshold

are marked as volume anomalies


Reference

Wikipedia (keyword search :p) The Wavelet Tutorial (by Robi Polikar):

http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html Standford Video Course (The Fourier Transform and its Applications) (by Br

ad G. Osgood): http://academicearth.org/courses/the-fourier-transform-and-its-applicatio

ns Wavelets and Time-Frequency Analysis (by Mudasir)

http://hubpages.com/hub/wavelets1

introduction to fourier transform and time-frequency analysis speaker: li-ming chen advisor:...

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