time frequency localization

Time frequency localizationTime frequency localization

M-bank filters are used to partition a signal into different frequency channels, with which energy compact regions in the frequency space are located.

Most real world signals exhibit the energy compactness properties where the majority information concentrates in specific regions in the spectrum.

Although we can locate information at different frequency bands in this way, an important point is missing.

WHEN? Briefly speaking, the time at which a particular piece of information exists in a region of the frequency spectrum.

Obviously both localization in time and frequency spaces are important. Lets have some revision first.

Filter , convolution and TransformFilter , convolution and Transform

x(t)

h(t-)

y()

Filter , convolution and TransformFilter , convolution and Transform

x(t) is the input signal

h(t) is the impulse response of the filter.

y() is the filter output

if h(t) is shifted continuously, y() is also continuous

For a particular instance h(t - ),

y() is the projection of x(t) on h(t - )

y() = < x(t) , h(t - ) > (dot product)

h(t) = ejt ---> Fourier Transform

Time Frequency LocalizationTime Frequency Localization

f(t)

h(t-b)Short time Fourier Transform


h(t-b)

ejt

w(t-b)t

t


What is the width of the window ?

What is w(t) ? Should it be a simple on/off function?




If the width is too wide, the time resolution is poor,but frequency resolution is good

i.e., same spectrum will be assumed over a long time duration




If the width is too wide, the time resolution is poor, but frequency resolution is good

i.e., same spectrum will be assumed over a long time duration

If the width is too short, the frequency resolution is poor,but time resolution is good

i.e., not possible to discriminate small difference in frequency

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15t

)t(

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15t

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15t

)t(

)t( 4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15t

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15t

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15t

)t(

)t( 4

2

4t

a

bt

1. shifted by b units.

2. stretched by ‘a’ times if a>1

3. compress by ‘a’ times if a<1

tjea

bt

a

btF

2

1

beaˆa

of TransformFourier theis tˆ

*ˆLet

*aaˆ

0 baˆa

A stretch in time window corresponds to a compression in the frequency window and vice versa

Time window

Frequency window

Long time window:Short frequency window

Short time window:Long frequency window

Time and frequency windows

x(t)

2

th

th

4

th

Time localization Frequency localization

Good

Poor

Poor

Good

Time and frequency windows

To detect low frequency content, a large time window is required for good frequency localization

To detect high frequency content, a small time window is required for good time localization

s

tsts

50.

Consider a waveform scaled by ‘s’ and shifted by ‘’

The larger the value of s, the more the waveform is compressed.

The waveform is shifted to the left for positive .

For discrete signal, let

jkj kt

s

kst

s

t

22

js 2

The family has two groups, one generated from a mother and the other from a father wavelets.

dts

ttfssfW

50.

, 0

dtt

2

501

s

dsd

s

tssfW

Ctf

.,

dC2ˆ

:

An arbitrary can be transformed into the space formed by members of a wavelet family

An inverse transform is also available.

The Harr father wavelet function is a unit step of length 1. It generates a family of scaling functions as shown below:

0 1

k=0j=0

k=0j=1 k=1

k=0j=2

k=1 k=2 k=3

The father

The Harr mother wavelet function is a bipolar [-1,1] unit step of length 0.5. It generates a family of wavelet functions as shown below:

0 1

k=0j=0

k=0j=1 k=1

k=0j=2

k=1 k=2 k=3

The mother

Scaling functions are like Low Pass Filter. Approximating a waveform f(t) with the set of scaling functions for certain value of ‘j’ gives the coarse form of f(t) at the resolution denoted by ‘j’.

f(t)

f0(t)

20 1

20 1

f1(t)

20 1

ktkctkctf jj

kjjk

kjj

22 2 /

‘j’ must be large enough to capture all the details in f(t).

Pure use of scaling function to approximate a waveform is viable but not efficient. Very often the scale has to be very fine to capture all the details, resulting in many coefficients.

ktkdktkc

tkdtkctf

jj

kj

jj

kj

jkk

jjkk

jj

2222 22

//

The wavelet functions are like a set of high pass filter which captures the difference between two resolutions (i.e. two values of j). Let Sj represent the space corresponding to scale ‘j’,

jj Stf

In another words, a high resolution subspace at level ‘j’ can be formed by combining a lower resolution subspaces at ‘j-1’, i.e.,

ktkdktkc

tkdtkctf

jj

kj

M

j

jj

k

M

jjk

kjk

kM

2222 21

0

20

1

000

//

We have

1100

12211

j

jjjjjj

WWWS

WWSWSS

...

An important feature of scaling functions: Each can be derived from translation of double-frequency copies of itself, as

ktkhtk

22 0

It means that a scaling function can be built from higher frequency (resolution) replicas of itself.

Similarly,

ktkhtk

422 0

This is known as Multiresolution analysis.

Similar important feature of wavelet functions: Each can be derived from translation of double-frequency copies of scaling function, as

ktkhtk

22 1

It means that a scaling function can be derived from the father wavelet!

The father wavelet determines the characteristics of all members of the wavelet family.

tscoefficienfunctionwaveletkh

tscoefficienfunctionscalingkh

1

0

:

:

1

1t

t

1

1t

t2

0.5

1

1t

12 t

0.5

ktkhtk

22 0

kthth 212202 00

By inspection,

2

110 00 hh

1

1t

t2

0.5

1

1t

12 t

0.5

ktkhtk

22 1

kthth 212202 11

By inspection,

2

11

2

10 11

hh ,

1

1t

t

-1

kmhmckcm

jj 201

alternatively, an inverse relation is also available, as

kmhmckdm

jj 211

mkhmdmkhmckcm

jm

jj 22 101

khHP

khLP

1

0

LP

HP

2

2

kc j 1

kc j

kd j

LP

HP

2

2

kc j

kd j

kc j 1

khHP

khLP

1

0

LP

HP

2

2

kc j 1

kc j

kd j

LP

HP

2

2

kc j 1

kd j 1

LP

HP

2

2

kc j 2

kd j 2

The signal samples, scaled down by 2j/2, is always used as the first set of coefficients cj[k].

The above decomposition only contains relations between the terms ‘c’ and ‘d’, so where is the signal f(t)?

First, cj+1 is one resolution level higher than cj.

Next for a digital signal, as the level advances, it will ultimately reaches a maximum resolution limited by the sampling lattice, lets call this cmax.

Obviously, cmax is simply the signal itself.

An interesting point. The father and mother wavelets established the formulation, but absent in the decomposition and reconstruction of the signal. It acts like a catalysis!

LP

HP 2

2

NN

HP

LP

2

2

HP

LP

2

2

rows

rows

rows

rows

rows

rows

cols

cols

cols

cols

cols

cols

2

NN

2

NN

22

NN

22

NN

22

NN

22

NN

Diagonal HH

Vertical HL

Horizontal LH

Low pass LL

LP

HP2

2

NN

HP

LP

HP

LP

rows

rows

cols

cols

cols

cols

cols

cols

2

NN

+

+

2

rows

2

rows

2

rows

2

rows

22

NN

22

NN

22

NN

22

NN

diagonal

vertical

horizontal

Low pass

LP

HP2

2

HP

LP

HP

LP

rows

rows

cols

cols

cols

cols

cols

cols

+

+

2

rows

2

rows

2

rows

2

rows

diagonal

vertical

horizontal

Low pass

22

NN

22

NN

22

NN

22

NN2

NN

NN

LP

HP 2

2

HP

LP

2

2

HP

LP

2

2

rows

rows

rows

rows

rows

rows

cols

cols

cols

cols

cols

cols

Diagonal HH

Vertical HL

Horizontal LH

Low pass LL22

NN

22

NN

22

NN

22

NN

NN

2

NN

2

NN

LH HH

LL HL

LH HH

HLLH HH

LL HL

Diagram taken from http://perso.orange.fr/polyvalens/clemens/ezw/ezw.html

In general, wavelet coefficients are smaller in higher subband (more detail resolution).

Thresholding the coefficients will generates lots of continuous zeros which can be represented with runlengths.

If a parent is zero after thresholding, it is very likely that all its descendants will also be zero.

Lets refer to the lab documentation on Embedded Zero-Tree Wavelet (EZW) Coding.

A lot of real world signals exhibit energy compactness in the low frequency band(s). Consequently, a lot of parent nodes and their associate descendant nodes are zero.

This kind of tree with only null members is know as ‘zero tree’. There is no need to transmit or store the content, saving a lot of bit-rate.

time frequency localization

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