Crit2 Updated

8/12/2019 Crit2 Updated

1/39

Wavelets

Ingrid Daubechies


2/39

Construction of scaling and wavelet

functions


3/39


4/39


5/39

Time and frequency resolution


6/39

Time and frequency perspective for

scaling function

Frequencies of x(t) inside main lobe are

emphasized with respect to frequenciesoutside the main lobe

As we go from V1 to V+1, more and

more frequencies are emphasized as

peak always remains on zero

frequency


7/39

Time and frequency relation for the wavelet function


8/39


9/39

Time and frequency perspective for

wavelet function

Localization in time gets better and better as we go

fromW1

to W+1

Localization in frequency gets poorer and poorer as we

go from W1

to W+1

Along with the bandwidth, the centre frequency also

shifts

Different bands with increasing bandwidth are

emphasized as we go fromW1

to W+1

Haar scaling function aspires to become low pass

filter and wavelet function aspires to be band pass

filter


10/39

Therefore two important observations can be made as

The ratio of bandwidth to center frequency of remains

constant. As we go up the ladder of MRA, we deal with having higher

center frequency and larger bandwidth.

Similarly, as we go down the ladder, possesses lower center

frequency and smaller bandwidth.

(.)

(.)

(.)


11/39

() and () are bounded in both domains in aweaker sense as we focus on main lobe.

Main lobe has certain amount of energy. Then() and () are localized in time andfrequency both.

Variance is important statistical property that

is very useful in calculating spread of a givenfunction, which is indicative of concentration ofenergy of a function within certain band (intime as well as frequency domain).

Is it possible to have finite variances in bothfrequency as well as time domainsimultaneously?


12/39

Haar wavelet, it is somewhat

concentrated in frequency, but well

concentrated in time.

Daubechies function, as we go at higher

order, we get a somewhat better filtering

operation that is better frequency

localization.


13/39

Uncertainty Principle


14/39



15/39



16/39

Th d d it f ti S d it d


17/39

The second density function: Squared magnitude

response


18/39

Why go ahead/ away from Haar?

If we want to get some

meaningful

uncertainty, some

meaningful bound, we

must at least considercontinuous functions.


19/39

Time Bandwidth Product should be minimum for the function to be

compact in both the domains. But for MRA we translate as well as

scale the basis function. Therefore it is necessary to look at the

properties of this product.


20/39

The time bandwidth product is thus a

robust measure of combined time and

frequency spread of a signal. It isessentially a property of the shape of the

waveform. time-bandwidth product of the

Haar scaling function was

What is the minimumvalue of this product?


21/39


22/39

Cauchy Schwarz inequality theorem states that


23/39

For any complex number :


24/39


25/39


26/39

Ideal function to achieve the TBP


27/39

Can the TBP of the Haar function

be improved ?

( )* ( )t t

2

20.13

0.3

t

TBP

Just by cascading the scaling

function once with itself the TBP

has reduced from infinity to 0.3 .

TBP can be further reduced by

cascading multiple times.


28/39

Time frequency tilingOccupancy of x(t) in time-frequency plane can be thought as being

around t0, the center in time, from t0 +tto t0 - ton the

horizontal axis. On the vertical axis we would like to center it at 0,the

frequency center, and we would spread it between0

Thus min area of the signal

spread in both the domains

should be 2 units

Area of the rectangle =


29/39

Time frequency tiling for wavelet transform


30/39

Resolution of Time & Frequency

Time

Frequenc

y

Better time

resolution;

Poor

frequency

resolution

Betterfrequency

resolution;

Poor time

resolutionEach box represents a equal portion

Resolution in STFT is selected once for entireanalysis


31/39

Continuous Wavelet Transform (CWT)

Wavelet Transform :

Find projection on the basis function and its translates

Sum up all these projections to give the piecewise

constant representation of the continuous function x(t).

In the Frequency Domain:


32/39

Continuous Wavelet Transform (CWT)

Continuous Wavelet Transform (CWT):


33/39

Continuous Wavelet Transform (CWT):Reconstruction

We will try to reconstruct the original signal x(t) bysumming the components of CWT along the unit

vector in its direction i.e. its basis function.

According to Parsevals Theorem



34/39


Reconstruction

Substituting back in the earlier integral



35/39


Reconstruction

Let

Let


36/39


37/39


38/39


39/39

Crit2 Updated

Documents