2d Dwt Document

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CHAPTER 1

INTRODUCTION

The fundamental idea behind wavelets is to analyze according to scale.

Indeed, some researchers in the wavelet field feel that, by using wavelets, one is

adopting a perspective in processing data. Wavelets are functions that satisfy certain

mathematical requirements and are used in representing data or other functions. This

idea is not new. Approximation using superposition of functions has existed since the

early !""#s, when $oseph %ourier discovered that he could superpose sines and

cosines to represent other functions. &owever, in wavelet analysis, the scale that we

use to loo' at data plays a special role. Wavelet algorithms process data at different

scales or resolutions.

%ourier Transform (%T) with its fast algorithms (%%T) is an important tool for

analysis and processing of many natural signals. %T has certain limitations to

characterize many natural signals, which are non*stationary (e.g. speech). Though a

time varying, overlapping window based %T namely +T%T (+hort Time %T) is well

'nown for speech processing applications, a time*scale based Wavelet Transform is a

powerful mathematical tool for non*stationary signals.

Wavelet Transform uses a set of damped oscillating functions 'nown as

wavelet basis. WT in its continuous (analog) form is represented as WT. WT with

various deterministic or non*deterministic bases is a more effective representation of

signals for analysis as well as characterization. ontinuous wavelet transform is

powerful in singularity detection. A discrete and fast implementation of WT

(generally with real valued basis) is 'nown as the standard -WT (-iscrete Wavelet

Transform).With standard -WT, signal has a same data size in transform domain and

therefore it is a non*redundant transform. A very important property was ulti*

resolution Analysis (/A) allows -WT to view and process.


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1.1Types of Transforms

%T (%ourier Transform).

-T (-iscrete osine Transform)

+T%(+hort time %ourier Transform)

WT (Wavelet Transform)

1.1.1FT (Fourier Transform):

%ourier transform is a well*'nown mathematical tool to transform time*

domain signal to frequency*domain for efficient extraction of information and it is

reversible also.

Though %T has a great ability to capture signal0s frequency content as long as

x(t) is composed of few stationary components (e.g. sine waves). &owever, any

abrupt change in time for non*stationary signal x(t) is spread out over the whole

frequency axis in 1(f). &ence the time*domain signal sampled with -irac*delta

function is highly localized in time but spills over entire frequency band and vice

versa. The limitation of %T is that it cannot offer both time and frequency localization

of a signal at the same time.

The %T representation for a finite duration sequence is

(.)

dejXnX nj= )(2

)(

N

k

2= (.2)

Where x(n) is a finite duration sequence, 1(34) is periodic with period 25.It is

convenient sample 1(34) with a sampling frequency equal an integer multiple of its

period 6m that is ta'ing 7 uniformly spaced samples between " and 25.

8et (.9)

Therefore

(.:)


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+ince 1(34) is sampled for one period and there are 7 samples 1(34) can be

expressed as

1.1.Dis!re"e Fourier "ransforms

The %ourier transform#s utility lies in its ability to analyze a signal in the time

domain for its frequency content. The transform wor's by first translating a function

in the time domain into a function in the frequency domain. The signal can then be

analyzed for its frequency content because the %ourier coefficients of the transformed

function represent the contribution of each sine and cosine function at each frequency.

An inverse %ourier transform does 3ust what you would expect, transform data from

the frequency domain into the time domain.

Windowed %ourier transform# If f(t) is a non*periodic signal, the summation of

the periodic functions, sine and cosine, does not accurately represent the signal. ;ou

could artificially extend the signal to ma'e it periodic but it would require additional

continuity at the endpoints. The windowed %ourier transform (W%T) is one solution to

the problem of better representing the non periodic signal. The W%T can be used to

give information about signals simultaneously in the time domain and in the

frequency domain.

With the W%T, the input signal f(t) is chopped up into sections, and each

section is analyzed for its frequency content separately. If the signal has sharp

transitions, window uses input data so that the sections converge to zero at the

endpoint. This windowing is accomplished via a weight function that places less

emphasis near the interval#s endpoints than in the middle. The effect of the window is

to localize the signal in time.

%ast %ourier transform *To approximate a function by samples, and to

approximate the %ourier integral by the discrete %ourier transform, requires applying a

matrix whose order is the number sample points n. +ince multiplying an matrix

by a vector costs on the order of arithmetic operations, the problem gets quic'ly

worse as the number of sample points increases. &owever, if the samples are

uniformly spaced, then the %ourier matrix can be factored into a product of 3ust a few


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sparse matrices, and the resulting factors can be applied to a vector in a total of order

arithmetic operations. This is the so*called fast %ourier transform or %%T.

1.1.$ %a&e'e" Transform (%T)

%ixed resolution limitation of +T%T can be resolved by letting the resolution in

time*frequency plane in order to obtain ulti resolution analysis. The Wavelet

Transform (WT) in its continuous (WT) form provides a flexible time*frequency,

which narrows when observing high frequency phenomena and widens when

analyzing low frequency behaviour. Thus time resolution becomes arbitrarily good at

high frequencies, while the frequency resolution becomes arbitrarily good at low

frequencies. This 'ind of analysis is suitable for signals composed of high frequency

components with short duration and low frequency components with long duration,

which is often the case in practical situations.

The -iscrete Wavelet Transform (-WT) refers to wavelet transforms for

which the wavelets are discretely sampled. A transform which localizes a function

both in space and scaling and has some desirable properties compared to the %ourier

transform. The transform is based on a wavelet matrix, which can be computed more

quic'ly than the analogous %ourier matrix. ost notably, the discrete wavelettransform is used for signal coding, where the properties of the transform are

exploited to represent a discrete signal in a more redundant form, often as a

preconditioning for data compression. The discrete wavelet transform has a huge

number of applications in +cience,


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+ignal can be represented by a smaller amount of information than would be

the case if some other transform, such as the more widespread discrete cosine

transform, had been used.

%irst a wavelet transform is applied. This produces as many coefficients as

there are pixels in the image (i.e.> there is no compression yet since it is only a

transform). These coefficients can then be compressed more easily because the

information is statistically concentrated in 3ust a few coefficients.

1. Compara"i&e isua'isa"ion

The time*frequency representation problem is illustrated in figure.2. A

comprehensive visualization of various time*frequency representations shown in

figure ., demonstrates the time*frequency resolution for a given signal in various

transform domains with their corresponding basis functions.


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%ig. omparative visualizations of time*frequency representation of an arbitrary

non*stationary signal in various transform domains

1. *imi'ari"ies +e",een Fourier an- %a&e'e" Transform

The fast %ourier transform (%%T) and the discrete wavelet transform (-WT)

are both linear operations that generate a data structure that contains log n2segments of

various lengths, usually filling and transforming it into a different data vector of

length 2n.

The mathematical properties of the matrices involved in the transforms are

similar as well. The inverse transform matrix for both the %%T and the -WT is the

transpose of the original. As a result, both transforms can be viewed as a rotation in


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function space to a different domain. %or the %%T, this new domain contains basis

functions that are sines and cosines. %or the wavelet transform, this new domain

contains more complicated basis functions called wavelets, mother wavelets, or

analyzing wavelets.

?oth transforms have another similarity. The basis functions are localized in

frequency, ma'ing mathematical tools such as power spectra (how much power is

contained in a frequency interval) and scale grams (to be defined later) useful at

pic'ing out frequencies and calculating power distributions.

1.$ Dissimi'ari"ies +e",een Fourier an- %a&e'e" Transform

The most interesting dissimilarity between these two 'inds of transforms is

that individual wavelet functions are localized in space. %ourier sine and cosine

functions are not. This localization feature, along with wavelets# localization of

frequency, ma'es many functions and operators using wavelets @sparse@ when

transformed into the wavelet domain. This sparseness, in turn, results in a number of

useful applications such as data compression, detecting features in images, and

removing noise from time series.

1. T/esis Or0ania"ion

The 2*- discrete wavelet transforms (-WT) have been widely used in many

applications li'e image compression, signal processing, speech compression because

of their multi*resolution of signals with localization both in time and frequency. In the

past, much architecture have been proposed aimed at providing high speed 2*-

-WT computation with the requirement of utilizing a reasonable amount of hard ware

resources. These architectures can be broadly classified into separable and non

separable architectures. The separable method is the most straight forward

implementation method. In separable method, a 2*- filtering operations, one for

processing the data row*wise and the other column*wise. In this method the

intermediate coefficients stores in a frame memory first. Then it performs *- -WT

in other direction with these intermediate coefficients to complete one*level 2*- -WT

.?ecause the size of this frame memory is usually assumed to off chip. &owever,

the separable method performs *- -WT in both directions simultaneously. Thus inseparable architectures, in which a *- filtering structure is used to perform the 2*-


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-WT, have an additional requirement of transposing the intermediate data between

the two *- filtering processes. &ence the separable method does not require a frame

memory to store the intermediate data. Instead, some internal line buffers are used to

store the intermediate data, and the required size is proportional to the image width.

Bishwanath have proposed a low*storage short*latency separable architecture in

which the row wise operations are performed by systolic filters and the column

operations by parallel filters. In non separable architectures the 2*- transforms are

computed directly by using 2*- filters. ha'rabarti have proposed two non separable

architectures, one using parallel 2*- filters and the other an +I- 2*- array, both

based on a modified /CA. In non separable method internal line buffers are use to

store the boundary data among neighbour bloc's such as to 'eep the required external

frame memory bandwidth as low as the separable method. &owever, the external

memory access would consume the most power and become very sensitive in the case

of system on chip. In addition, the required external memory bandwidth of the non*

separable is more than the double of the separable method. The most existing

separable architectures aim at providing fast computation of the -WT by using

pipeline and a large number of parallel filters and systolic filters. &owever these

existing architectures have large latency and memory size increases because of by

providing additional requirement of transposing the intermediate data between the two

*- filtering processes.

+eparable pipeline architecture for fast computation of the 2*- -WT with a

less memory and low latency is proposed. The low latency and less memory is

achieved by proper designing of two *- -WT filtering processes and also efficiently

transferring the data between the two *- -WT filters.

The documentation is organized as follows. In chapter one, the basics of the

transforms is presented. hapter two explains about the wavelet transform. hapter

three explains about the -iscrete Wavelet Transform and its types, and architecture of

the existing method.

In chapter four a study is conducted to determine the coefficients of 2*- -WT

by using separable 2*- -WT with low latency and less memory.


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CHAPTER

%AE2ET TRAN*FOR3

.1 In"ro-u!"ionThe wavelet transform is computed separately for different segments

of the time*domain signal at different frequencies. ulti*resolution analysis> analyzes

the signal at different frequencies giving different resolutions. ulti*resolution

analysis 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. Dood for signal having high frequency components for short durations

and low frequency components for long duration, e.g. Images and video frames.

The fundamental idea behind wavelets is to analyze according to scale.

Indeed, some researchers in the wavelet field feel that, by using wavelets, one is

adopting a whole new mindset or perspective in processing data.

Wavelets are functions that satisfy certain mathematical requirements and are

used in representing data or other functions. This idea is not new. Approximation

using superposition of functions has existed since the early !""#s, when $oseph%ourier discovered that he could superpose sines and cosines to represent other

functions. &owever, in wavelet analysis, the scale that we use to loo' at data plays a

special role. Wavelet algorithms process data at different scales or resolutions.If we

loo' at a signal with a large @window,@ we would notice gross features. +imilarly, if

we loo' at a signal with a small @window,@ we would notice small features. The result

in wavelet analysis is to see both the forest andthe trees, so to spea'.

This ma'es wavelets interesting and useful. %or many decades, scientists have

wanted more appropriate functions than the sines and cosines which comprise the

bases of %ourier analysis, to approximate choppy signals. ?y their definition, these

functions are non*local (and stretch out to infinity). They therefore do a very poor 3ob

in approximating sharp spi'es. ?ut with wavelet analysis, we can use approximating

functions that are contained neatly in finite domains. Wavelets are well*suited for

approximating data with sharp discontinuities.


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The wavelet analysis procedure is to adopt a wavelet prototype function,

called an analyzing waveletor mother wavelet. Temporal analysis is performed with a

contracted, high*frequency version of the prototype wavelet, while frequency analysis

is performed with a dilated, low*frequency version of the same wavelet. ?ecause the

original signal or function can be represented in terms of a wavelet expansion (using

coefficients in a linear combination of the wavelet functions), data operations can be

performed using 3ust the corresponding wavelet coefficients. And if you further

choose the best wavelets adapted to your data, or truncate the coefficients below a

threshold, your data is sparsely represented. This sparse coding ma'es wavelets an

excellent tool in the field of data compression.

Ether applied fields that are ma'ing use of wavelets include astronomy,

acoustics, nuclear engineering, sub*band coding, signal and image processing,

neurophysiology, music, magnetic resonance imaging, speech discrimination, optics,

fractals, turbulence, earthqua'e*prediction, radar, human vision, and pure

mathematics applications such as solving partial differential equations.

. %a&e'e" Defini"ion

A Fwavelet0 is a small wave which has its energy concentrated in time. It has

an oscillating waveli'e characteristic but also has the ability to allow simultaneous

time and frequency analysis and it is a suitable tool for transient, non*stationary or

time*varying phenomena.

(a) (b)

%ig2. /epresentation of (a) wave (b) wavelet


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. %a&e'e" C/ara!"eris"i!s

The difference between wave (sinusoids) and wavelet is shown in figure ..

Waves are smooth, predictable and everlasting, whereas wavelets are of limited

duration, irregular and may be asymmetric. Waves are used as deterministic basis

functions in %ourier analysis for the expansion of functions (signals), which are time*

invariant, or stationary. The important characteristic of wavelets is that they can serve

as deterministic or non*deterministic basis for generation and analysis of the most

natural signals to provide better time*frequency representation, which is not possible

with waves using conventional %ourier analysis.

.$ %a&e'e" Ana'ysis

The wavelet analysis procedure is to adopt a wavelet prototype function,

called an Fanalyzing wavelet0 or Fmother wavelet0. Temporal analysis is performed

with a contracted, high frequency version of the prototype wavelet, while frequency

analysis is performed with a dilated, low frequency version of the same wavelet.

athematical formulation of signal expansion using wavelets gives Wavelet

Transform (WT) pair, which is analogous to the %ourier Transform (%T) pair.

-iscrete*time and discrete*parameter version of WT is termed as -iscrete Wavelet

Transform (-WT).

ompactly supported wavelets are functions defined over a finite interval and

having an average value of zero. The basic idea of the wavelet transform is to

represent any arbitrary function f(x) as an upper position of a set of such wavelets or

basis functions. These basis functions are obtained from a single prototype wavelet

called the mother wavelet G(x), by dilations or scaling and translations. Wavelet bases

are very good at efficiently representing functions that are smooth except for a smallset of discontinuities.

%or each n, ' H, define (x) by

)2(2 2, kxn

n

kn =

(2.)

onstructing the function G(x), on /, such that (x)Jn,'H is an

orthonormal basis on /. As mentioned before G(x) is a wavelet and the collection


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(x) Jn,'H is a wavelet orthonormal basis on /K this framewor' for constructing

wavelets involves the concept of a multi resolution analysis or /A.

ulti resolution analysis is a device for computation of basis coefficients in

(/) >

f 6 knknC ,, , . It is defined as follows,

6 f(x)Lf(x) 6 22n

g(x), g(x) J, (2.2)

Where

f(x) 6 (f, M(N O n))M(x O n) (2.9)

Then a multi resolution analysis on / is a sequence of subspaces JnH of

functions

on /, satisfying the following properties>

(a) %or all n, ' H, .

(b) If f(x) is on /, then f(x) span0 JnH. That is, given P Q ", there is an

n H and a function g(x) such that LLf gLL R P.

(c) 6 ".

(d) A function f(x) if and only if f( x) .

(e) There exists a function M(x), on /, called the scaling function such that the

collection M(xO n) is an ortho*normal system of translates and

6 span0M(x O n)J.

8et J be an /A with scaling function M(x) which satisfies scaling filter h('),

where

h(') 6 ( M ( )M ( x*')). Then the wavelet filter g(') is defined by

g(') 6 ( (h( O '))0 (2.:)


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and the wavelet by

G(x) 6 g(') M(2x O '). (2.S)

Then (x)J is a wavelet orthonormal basis on /.

The orthogonal pro3ection of an arbitrary function f onto is given by

knzk knn ffP ,, ),( = (2.)

As ' varies, the basis functions are shifted in steps of , so f cannot

represent any detail on a scale smaller than that. We say that the functions in have

the resolution or scale . &ere, f is called an approximation to f at resolution

. %or a given function f, an /A provides a sequence of approximations f of

increasing accuracy.

The difference between the approximations at resolution and is called the

fine detail at resolution which is as follows>

f(x) 6 f(x) O f(x). (2.U)

Er

.),( ,, knknzkn ffQ = (2.!)

is also an orthogonal pro3ection and its range is orthogonal to where the

following holds>

6 fL f 6 fJ (2.V)

6 fL f 6 fJ (2.")

+= nnn VWV (2.)

There are choices of the numbers h and g such that (x)J is a wavelet

orthonormal basis on /.

"= nzn V (2.2)


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and

(2.9)

Then we have L' HJ 6 6 . &ence (x)J is complete if

and only if 6 (/) holds, and this is true.

7ow, as for the orthonormality,

(, )6 )2(2),2(2( 22 xkx nn

n

n

6 () 6 X(' O l). (2.:)

To prove orthonormality between scales, let n, nY H with nY R n, and let ', 'Y H be

arbitrary. +ince G(x) B, (x) B, Then we have . +ince ( ,

) 6 " for all ', l H, it follows that ( , )6 ", for all n, ', l H. Diven

f(x) we 'now that f(x)6 (f, ) (x) &ence for f(x)

( , f) 6 ( (f, ) )

6 (f, )0 ( , ) 6 " (2.S)

+ince, nY R n, and since , nkn Vtt , also.

&ence ( kn, , 6"

Therefore zknkn x ,, )J(I is a wavelet orthonormal basis on /.

. Types of %a&e'e" Transform

The transform of a signal is 3ust another form of representing the signal. It

does not change the information content present in the signal. The Wavelet Transform

provides a time*frequency representation of the signal. It was developed to overcome

the short coming of the +hort Time %ourier Transform (+T%T), which can also be used

to analyze non*stationary signals. While +T%T gives a constant resolution at all

frequencies, the Wavelet Transform uses multi*resolution technique by which

different frequencies are analyzed with different resolutions.

A wave is an oscillating function of time or space and is periodic. In contrast,

wavelets are localized waves. They have their energy concentrated in time or space


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and are suited to analysis of transient signals. While %ourier Transform and +T%T use

waves to analyze signals, the Wavelet Transform uses wavelets of finite energy.

..1 Con"inuous %a&e'e" Transform

A continuous wavelet transform is used to divide a continuous*time function

into wavelets. =nli'e %ourier transform, the continuous wavelet transform possesses

the ability to construct a time frequency represented of a signal that offers very good

time and frequency localization.

The mother wavelet used to generate all the basis functions is designed based

on some desired characteristics associated with that function. The translation

parameter Z relates to the location of the wavelet function as it is shifted through thesignal. Thus, it corresponds to the time information in the Wavelet Transform. The

scale parameter s is defined as L[frequencyL and corresponds to frequency

information. +caling either dilates (expands) or compresses a signal. 8arge scales (low

frequencies) dilate the signal and provide detailed information hidden in the signal,

while small scales (high frequencies) compress the signal and provide global

information about the signal. 7otice that the Wavelet Transform merely performs the

convolution operation of the signal and the basis function. The above analysis

becomes very useful as in most practical applications, high frequencies (low scales)

do not last for a long duration, but instead, appear as short bursts, while low

frequencies (high scales) usually last for entire duration of the signal.

The Wavelet +eries is obtained by discretizing WT. This aids in computation

of WT using computers and is obtained by sampling the time*scale plane. The

sampling rate can be changed accordingly with scale change without violating the

7yquist criterion. 7yquist criterion states that, the minimum sampling rate that allowsreconstruction of the original signal is 24 radians, where 4 is the highest frequency in

the signal. Therefore, as the scale goes higher (lower frequencies), the sampling rate

can be decreased thus reducing the number of computations.

.. Fas" %a&e'e" Transform

The %ast Wavelet Transform is a mathematical algorithmdesigned to turn a

waveformor signal in the time domain into a sequenceof coefficients based on an

orthogonal basis of small finite waves, orwavelets. The transform can be easily
http://en.wikipedia.org/wiki/Time-frequency_representationhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Waveformhttp://en.wikipedia.org/wiki/Time_domainhttp://en.wikipedia.org/wiki/Time_domainhttp://en.wikipedia.org/wiki/Time_domainhttp://en.wikipedia.org/wiki/Sequencehttp://en.wikipedia.org/wiki/Orthogonal_basishttp://en.wikipedia.org/wiki/Waveletshttp://en.wikipedia.org/wiki/Waveletshttp://en.wikipedia.org/wiki/Time-frequency_representationhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Waveformhttp://en.wikipedia.org/wiki/Time_domainhttp://en.wikipedia.org/wiki/Sequencehttp://en.wikipedia.org/wiki/Orthogonal_basishttp://en.wikipedia.org/wiki/Wavelets


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extended to multidimensional signals, such as images, where the time domain is

replaced with the space domain.

.. Dis!re"e %a&e'e" Transform

In numerical analysis and functional analysis, a discrete wavelet transform

(-WT) is any wavelet transformfor which the waveletsare discretely sampled. As

with other wavelet transforms, a 'ey advantage it has over %ourier transforms is

temporal resolution it captures both frequency and location information.

.4 Differen!e +e",een Con"inuous %a&e'e" Transform an- Dis!re"e

%a&e'e" Transform

Wavelet transforms are classified into discrete wavelet transforms (-WTs) and

continuous wavelet transforms (WTs). 7ote that both -WT and WT are

continuous*time (analog) transforms. They can be used to represent continuous*time

(analog) signals. WTs operate over every possible scale and translation whereas

-WTs use a specific subset of scale and translation values or representation grid.

The word wavelet is due to orlet and Drossmann in the early V!"s. They

used the %rench word ondelette, meaning @small wave@. +oon it was transferred to

redundant and non*redundant ones.

The discrete wavelet transform returns a data vector of the same length as theinput is. =sually, even in this vector many data are almost zero. This
http://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Wavelet_transformhttp://en.wikipedia.org/wiki/Wavelethttp://en.wikipedia.org/wiki/Fourier_transformhttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Wavelet_transformhttp://en.wikipedia.org/wiki/Wavelethttp://en.wikipedia.org/wiki/Fourier_transform


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corresponds to the fact that it decomposes into a set of wavelets (functions)

that are orthogonal to its translations and scaling. Therefore we decompose

such a signal to a same or lower number of the wavelet coefficient spectrum as

is the number of signal data points. +uch a wavelet spectrum is very good for

signal processing and compression, for example, as we get no redundant

information here.

The continuous wavelet transform in contrary returns an array one dimension

larger than the input data. %or a - data we obtain an image of the time*

frequency plane. We can easily see the signal frequencies evolution during the

duration of the signal and compare the spectrum with other signals spectra. As

here is used the non*orthogonal set of wavelets, data are correlated highly, so

big redundancy is seen here. This helps to see the results in a more humane

form.

.5 Dis!re"e ,a&e'e" "ransform

The -iscrete Wavelet Transforms (-WT) have been widely used in many

applications li'e image compression, signal processing, speech compression because

of their ulti*resolution of signals with localization both in time and frequency. In the

past, much architecture have been proposed aimed at providing high speed 2*-

-WT computation with the requirement of utilizing a reasonable amount of hard ware

resources. These architectures can be broadly classified into two types

7on +eparable 2*- -WT

+eparable 2*- -WT

.5.1 Non separa+'e #D D%T

In non separable architectures the 2*- transforms are computed directly by

using 2*- filters. The two non separable architectures, one using parallel 2*- filters

and the other an +I- 2*- array, both based on a modified /CA. In non separable

method internal line buffers are use to store the boundary data among neighbour

bloc's such as to 'eep the required external frame memory bandwidth as low as the

separable method. &owever, the external memory access would consume the mostpower and become very sensitive in the case of system on chip. In addition, the


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%ig.2.2.?loc' diagram of the three*stage architecture

%ig. 2.2 gives the bloc' diagram of the pipeline showing all the components

required by the three stages. 7ote that the data flow shown in this figure comprises

only the 88*sub band data necessary for the operations of the stages. The &&, &8

and 8& sub* band data are outputted directly to an external memory. 7ow, we give

details on the structure of the data scanning unit to scan the 2*- data and establish

four distinct sub*windows, as well as on the distribution of the filtering operations to

the processing units in each stage.

%ig 2.9> ?loc' diagram of a processing unit.

In order to obtain the output sample corresponding to a given sub*window, the


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bits of the partial products must be accumulated vertically downward and from right

to left by ta'ing the propagation of the carry bits into consideration. The tas' of

this accumulation can be divided into a sequence of layers. The shortest critical

data path can be achieved by minimizing the number of layers and the delay of

the layers. In each layer, a number of bits consisting of the partial product bits

and[or the carry bits from different rows need to be added. This can be doneby

employing in parallel as many bit*wise adders as needed in each layer. The idea

behind using bit*wise adder is to produce to the extent possible the number of

output bits from a layer is smaller than the number of input bits to that layer.

This canbe done by using full adders and specifically designed double adders, in

which the full adder consumes 9 bits and produces 2 bits (one sum and one carry

bits) whereas the double adder consumes two pairs of bits from neighbouring

columns andproduces 9 bits (one sum and two carry bits[two sum and one carry

bits). The two types of adders have equal delay, and are efficient in generating

carry bits and compressing the number of partial products

CHAPTER

*EPARA62E D%T

.1In"ro-u!"ion

%or many natural signals, the wavelet transform is a more effective tool than

the %ourier transform. The wavelet transform provides a multi resolution

representation using a set of analyzing functions that are dilations and translations of a

few functions (wavelets).

. *epara+'e #D D%T

The separable method is the most straight forward implementation method. In

separable method, 2*- filtering operations, one for processing the data row*wise and


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the other column*wise. In this method the intermediate coefficients stores in a frame

memory first. Then it performs *- -WT in other direction with these intermediate

coefficients to complete one*level 2*- -WT .?ecause the size of this frame memory

is usually assumed to off chip. &owever, the separable method performs *- -WT in

both directions simultaneously. Thus in separable architectures, in which a *-

filtering structure is used to perform the 2*- -WT, have an additional requirement of

transposing the intermediate data between the two *- filtering processes. &ence the

separable method does not require a frame memory to store the intermediate data.

Instead, some internal line buffers are used to store the intermediate data, and the

required size is proportional to the image width. A low*storage short*latency separable

architecture in which the row wise operations are performed by systolic filters and the

column operations by parallel filters. The architecture of2*- -WT is shown in below

figure 9.2.

%ig9.> Architecture of 2*- -WT


23/72

%ig9.2> Architecture of *- -WT

The splitter decomposes the input signal into two sub band signals, even and odd.

The even signal represents the high frequency (or coarse) part of input, while the odd

signal represents the low frequency (or detail) part of input. As shown in the figure,

the output of each register is then delayed by one cloc' pulse to obtain the delayed

signal.By using 1-D DWT architecture the input data divided into

even coefcients and odd coefcients.

..1 A--er

%ig9.9> Adder


24/72

The figure 9.9 shows the bloc' diagram of adder. ?y using adder the two V bit input

signals are added.

A , 2, 9, :J ? S, , U, !J

Eutput s6 , !, ", 2J

.. Re0is"er

- flipflop is used as a register. It is used to provide the delay of signal by one

cloc' cycle. %or every posedge of cloc' cycle if enable is one the input data that is

present in - is transferred to \ else zero is transferred to \.

%ig9.:> - flip flop

. PIPE2INE FOR THE #D D%T CO3PUTATION

In a pipeline structure for the -WT computation, multiple stages are used to

carry out the computations of the various decomposition levels of the transform ]:^.

The computation corresponding to each decomposition level needs to be mapped to a

stage or stages of the pipeline. In order to design a pipeline structure capable of

performing a fast computation of the -WT with low expense on hardware resources

and low design complexity, an optimal mapping of the overall tas' of the -WT

computation to the various stages of the pipeline needs to be determined. Any

distribution of the overall tas' of the -WT computation to stages must consider the

inherent nature of the sequential computations of the decomposition levels that limit


25/72

the computational parallelism of the pipeline stages, and consequently the latency of

the pipeline. %urther, in order to minimise the expense on the hardware resources of

the pipeline, the number of filter units used by each stage ought to be minimum and

proportional to the amount of the tas' assigned to the stage.

%ig9.SCipeline structure with 7 stages

A straight forward of mapping of the overall tas' of the -WT computation to

a pipeline is one*level to one*stage mapping, in which the tas's of $ decomposition

levels are distributed to $ stages of the pipeline. &owever, dividing a stage of the one*

level to one stage pipeline into multiple stages would require a division of the tas'

associated with the corresponding decomposition level into sub*tas's, which in turn,would call for a solution of even a more complex problem of synchronization of the

sub*tas's associated with divided stages. En the other hand, merging multiple small*

size stages of the pipeline into one stage would not create any additional

synchronization problem. As a matter of fact, such a merger could be used to reduce

the overall number of filter units of the pipeline.

+ynchronization of stages#The stages of pipeline need to be synchronized in

such a way that each stage starts the operation at an earliest possible time when the

required data become available for its operation. Ence the operation of a stage is

started, it must continue until the tas' assigned to it is fully completed.

onsider the timing diagram given in %ig. 9. for the operation of the three

stages, where t, t2 and t9 are the times ta'en individually by stages , 2 and 9,

respectively, to complete their assigned tas's, and ta and tb are the times elapsed

between the starting points of the tas's, by stages and 2, and that stages 2 and 9

respectively.


26/72

%ig9.> Timing diagram for the operations of three stages

7ote that the lengths of the times t,t2 and t9 to complete the tas's by

individual stages are approximately the same, since the ratios of the tas's assigned

and the resources made available to the three stages are the same. The average timesto compute one output sample by stages ,2 and 9 are in the ratio >:>!. In %ig. 2 the

relative widths of the slots in the three stages are shown to reflect this ratio. Eur

ob3ective is to minimise the total computation time ta_tb_t9by minimizing t,t and t

individually.


27/72

.$ ADANTA7E* AND APP2ICATION*

.$.1A-&an"a0es

%or many natural signals, the wavelet transform is a more effective tool than the

%ourier transform. The wavelet transform provides a multiresolution representation

using a set of analyzing functions that are dilations and translations of a few functions

(wavelets).

7o need to divide the input coding into non*overlapping 2*- bloc's, it has

higher compression ratios avoid bloc'ing artefacts.

Allows good localization both in time and spatial frequency domain.

Transformation of the whole image *Q introduces inherent scaling.

?etter identification of which data is relevant to human perception *Q higher

compression ratio.

.$. App'i!a"ions

Denerally, an approximation to -WT is used for data compression if signal is

already sampled, and the WT for signal analysis. Thus, -WT approximation is

commonly used in engineering and computer science, and the WT in scientific

research.

Ene use of wavelet approximation is in data compression. 8i'e some other

transforms, wavelet transforms can be used to transform data and then encode the

transformed data, resulting in effective compression. %or example, $C


28/72

vision, and pure mathematics applications such as solving partial differential

equations.

.$. %a&e'e"s for ima0e !ompression

Wavelet transform exploits both the spatial and frequency correlation of data

by dilations (or contractions) and translations of mother wavelet on the input data. It

supports the multi*resolution analysis of data i.e. it can be applied to different scales

according to the details required, which allows progressive transmission and zooming

of the image without the need of extra storage. Another encouraging feature of

wavelet transform is its symmetric nature that is both the forward and the inverse

transform has the same complexity, building fast compression and decompression

routines. Its characteristics well suited for image compression include the ability to

ta'e into account of &uman Bisual +ystem0s (&B+) characteristics, very good energy

compaction capabilities, robustness under transmission, high compression ratio etc.

Wavelet transform divides the information of an image into approximation and

detail sub*signals. The approximation sub*signal shows the general trend of pixel

values and other three detail sub*signals show the vertical, horizontal and diagonal

details or changes in the images. If these details are very small (threshold) then they

can be set to zero without significantly changing the image. The greater the number ofzeros the greater the compression ratio. If the energy retained (amount of information

retained by an image after compression and decompression) is ""` then the

compression is lossless as the image can be reconstructed exactly. This occurs when

the threshold value is set to zero, meaning that the details have not been changed.

Area of app'i!a"ion>

edical application.

+ignal denoising.

-ata compression.


29/72

CHAPTER $

*OFT%ARE AND 2AN7UA7E U*ED

$.1ERI2O7 HD2

Berilog &-8 is a hardware description language that can be used to model a

digital system at many levels of abstraction ranging from the algorithmic*level to the

gate*level to the switch*level. The complexity of the digital system being modeled

could vary from that of a simple gate to a complete electronic digital system, or

anything in between. The digital system can be described hierarchically and timing

can be explicitly modeled within the same description.

The Berilog &-8 language includes capabilities to describe the behavior*al

nature of a design, the dataflow nature of a design, a design#s structural composition,

delays and a waveform generation mechanism including aspects of response

monitoring and verification, all modeled using one single language. In addition, the

language provides a programming language interface through which the internals of a

design can be accessed during simulation including the control of a simulation run.

The language not only defines the syntax but also defines very clear simulation

semantics for each language construct. Therefore, models written in this language

can be verified using a Berilog simulator. The language inherits many of its

operator symbols and constructs from the programming language. Berilog &-8

provides an extensive range of modeling capabilities, some of which are quite difficult

to comprehend initially. &owever, a core subset of the language is quite easy to leam

and use. This is sufficient to model most applications.

A mapping mechanism or a construction mechanism has to be provided that

translates the Berilog &-8 elements into their corresponding hardware elements as

shown in figure :.


30/72

Typical design process

$. *OFT%ARE U*ED

$..18I2IN8

1ilinx software is used by the B&-8[B


31/72

reate a B&-8 +ource formatting all inputs, outputs and buffers if required. which

provides a window to write the B&-8 code, to be synthesized.


32/72


33/72


34/72


35/72

hec' +yntax after finally editing the B&-8 source for any errors.After that perform the /T8 and T


36/72

2. reate a new file[Add existing file in Add items to the Cro3ect window and

reate a new file window.


37/72

9. Then, on the wor' space of the main window, you find your file in which the code

can be written when double*clic'ed.

:. After the code is written, it is saved and then compiled for syntax errors. If no

syntax errors, a Dreen Tic' mar' will be seen in the wor' space, else a /ed

ross. A red message indicates that there is an error in our code. To correct any

errors, 3ust double*clic' on the


38/72

. The appropriate signals are selected and added to the wave window by clic'ing

Add to wave in the tool bar. &ence the obtained signals are assigned with required

values to provide desired outputs.

U. There are different options available in the waveform window to view the output

values in various representations such as ?inary, &exadecimal, +ymbolic, Ectal,

A+II, -ecimal and =nsigned representations.

!. The odelsim can be exited by selecting F%ile *Q quit0 from menu.

V. In this way the odelsim software is used for functional verification or +imulation

of the user0s code.


39/72

CHAPTER

CODE

99CODE for D%T 3o-u'e

module dwt(input cl',rst,input ]V>"^ datain,output reg start9,start,output signed

]V>"^mSoddouttop,m"evenoutdown)K

reg ]V>"^ even ]">U^K

reg ]V>"^ odd ]">U^K

reg ]V>"^tempeven,tempoddK

reg s,loadK

reg ]9>"^ c,cK

wire signed ]V>"^ m2out,m2tempK

wire signed ]V>"^ m9in,m9intempK

wire signed ]V>"^ m:outK

wire signed ]V>"^ mSoutK

wire signed ]V>"^ mUoutKwire signed ]V>"^ m!inK

wire signed ]V>"^ mVoutK

always (posedge cl')

if(rst)

begin

c6"K

s6K

load6"K

end

else if(cR6U)

begin

if(s)

begin

even]c^6datainK


40/72

s6sK

end

else

begin

odd]c^6datainK

s6sK

c6c_K

end

end

else

load6K

always (posedge cl')

if (load)

begin

tempeven 6 "K

tempodd 6 "K

c6"K

end

else if (c R6 U)

begin

tempeven 6 even]c^K

tempodd 6 odd]c^K

c 6 c _ K

end

always (negedge cl')

if (load)

begin

start 6 K

start9 6 K

end


41/72

else if (c 66 2)

begin

start 6 "K

end

else if (c 66 9)

begin

start9 6 "K

end

registers r (.loc'(cl'), .-(tempeven), .en(load),.\(m2out))K

registers r2 (.loc'(cl'), .-(m2out), .en(load),.\(m2temp))K

adderbit m9 (.in(m2out),.in2(tempeven),.out(m9in))K

assign m9intemp 6 m9in]V^, m9in]V>^JK

assign m:out 6 m9intemp _ #bK

registers r9 (.loc'(cl'), .-(tempodd), .en(load),.\(mSout))K

adderbit mS (.in(m:out),.in2(mSout),.out(mSoddouttop))K

registers r: (.loc'(cl'), .-(mSoddouttop), .en(load),.\(mUout))K

adderbit m! (.in(mUout),.in2(mSoddouttop),.out(m!in))K

assign mVout 6 m!inK

adderbit m" (.in(m2temp),.in2(mVout),.out(m"evenoutdown))K

endmodule


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99CODE for TOP#IN 3o-u'e

module topin(input cl',rst,

output signed ]V>"^ mSoddoddouttop, mSoddevenouttop,

mSevenoddouttop, mSevenevenouttop)K

wire ]V>"^dataK

inp in(.cl'(cl'),.rst(rst),.data(data))K

topdwt top

(.cl'(cl'),.rst(rst),.datain(data),.mSoddoddouttop(mSoddoddouttop),.mS

oddevenouttop(mSoddevenouttop),.mSevenoddouttop(mSevenod

douttop),.mSevenevenouttop(mSevenevenouttop))K

endmodule


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99 CODE for TOP#D%T mo-u'e

module topdwt(input cl',rst, input signed ]V>"^datain,

output signed ]V>"^ mSoddoddouttop, mSoddevenouttop,

mSevenoddouttop, mSevenevenouttop)K

wire signed ]V>"^m"evenoutdown,mSoddouttopK

wire start9,startK

dwt e

(.cl'(cl'),.rst(rst),.datain(datain),.start(start),.start9(start9),.mSoddouttop(mS

oddouttop),.m"evenoutdown(m"evenoutdown))K

dwt2 e2

(.cl'(cl'),.start(start),.datain(mSoddouttop),.mSoddoddouttop(mSoddo

ddouttop),.mSoddevenouttop(mSoddevenouttop))K

dwt9 e9

(.cl'(cl'),.start9(start9),.datain(m"evenoutdown),.mSevenoddouttop(

mSevenoddouttop),.mSevenevenouttop(mSevenevenouttop))K

endmodule


44/72

CHAPTER 4

RE*U2T*

.1 %a&e forms

.1.1 Inpu" -a"a

%igS. >Input data

The above figure shows the input data and the input data is in the form of numerical

values.


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.1. *imu'a"ion resu'" of 1#D D%T:

%igS.2> +imulation /esult of -WT* ?loc'

The above figure shows the output of *- -iscrete Wavelet Transform. The *- -WT

contains the low pass coefficients and high pass coefficients.


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.1. *imu'a"ion resu'" of #D D%T

%igS.9> +imulation /esult of -WT*2 ?loc'

The above waveform shows the result of 2*- -iscrete Wavelet Transform. The

2*- -WT contains even*even, even*odd, odd*odd, odd*even coefficients. A *-filtering structure is used to perform the 2*- -WT, have an additional requirement of

transposing the intermediate data between the two *- filtering processes.


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.1.$ *imu'a"ion resu'" of Top#D%T mo-u'e

%igS.:> +imulation /esult of Top*-WT ?loc'

The above waveform shows the result of Top -iscrete Wavelet Transform. The

Top -WT contains even*even, even*odd, odd*odd, odd*even coefficients. A *-

filtering structure is used to perform the 2*- -WT, have an additional requirement of

transposing the intermediate data between the two *- filtering processes to get the

Top module of -iscrete Wavelet Transform.


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.1.$ *imu'a"ion resu'" of Top#D%T mo-u'e

%igS.S> +imulation /esult of Top*in ?loc'

The above waveform shows the result of Top input waveform. The Top input

also contains even*even, even*odd, odd*odd, odd*even coefficients. A *- filtering

structure is used to perform the -WT operation, have an additional requirement of

transposing the intermediate data between the two *- filtering processes to get the

Top module of -iscrete Wavelet Transform.

4. De&i!e u"i'ia"ion summary

De&i!e U"i'ia"ion *ummary (es"ima"e- &a'ues)

2o0i! U"i'ia"ion Use- A&ai'a+'e U"i'ia"ion

7umber of +lices 2"2 :S :`

7umber of +lice %lip %lops 2S V92 2`7umber of : input 8=Ts 9SS V92 9`

7umber of bonded IE?s !2 292 9S`

7umber of D8s 2: :`

Table.> -evice =tilization summary for the device xc9sS""e*:fg 92"


49/72

. RT2 s!/ema"i! of #D D%T

%igS.:> /T8 +chematic

The purpose of /T+ module builder is to build, or acquire from a library of

predefined components, each of the required /T8 bloc's in the user*specified target

technology. The above figure shows the /T8 schematic of 2*- -WT.


50/72

.$ RT2 s!/ema"i! of Top# D%T

%igS.:> /T8 +chematic for Top*-W8 module

The purpose of /T+ module builder is to build, or acquire from a library of


technology. The above figure shows the /T8 schematic of Top*-WT module.


51/72

. RT2 s!/ema"i! of Top# in mo-u'e

%igS.S> /T8 +chematic for Top*in module

The purpose of /T8 module builder is to build, or acquire from a library of


technology. The above figure shows the /T8 schematic of Top input odule.


52/72

4.$ *yn"/esis Repor"

/elease !.2i * xst I.9

opyright (c) VVS*2"" 1ilinx, Inc. All rights reserved.

**Q Carameter TC-I/ set to .[xst[pro3nav.tmp

C= > "."" [ ".22 s L "."" [ "."" s

**Q Carameter xsthdpdir set to .[xst

C= > "."" [ ".22 s L "."" [ "."" s

**Q /eading design> topin.pr3

TA?8< E% E7T mixed

Ignore +ynthesis onstraint %ile > 7E


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**** Target Carameters

Eutput %ile 7ame > @topin@

Eutput %ormat > 7D

Target -evice > xc9s2""*:*ft2S

**** +ource Eptions

Top odule 7ame > topin

Automatic %+ ; lut

/A ;es

/A +tyle > Auto

/E ;es

ux +tyle > Auto

-ecoder ;


54/72

/egister -uplication > ; auto

; +peed

Eptimization

eep &ierarchy > 7E

/T8 Eutput > ;es

Dlobal Eptimization > Allloc'7ets

Write Timing onstraints > 7E

&ierarchy +eparator >

?us -elimiter > RQ

ase +pecifier > maintain

+lice =tilization /atio > ""

+lice =tilization /atio -elta > S

**** Ether Eptions

lso > topin.lso

/ead ores > ; ;es

usesyncset > ;es

usesyncreset > ;es


55/72

&-8 ompilation

ompiling verilog file @adderbit.v@ in library wor' ompiling verilog file @-ff.v@

in library wor'

odule RadderbitQ compiled

ompiling verilog file @dwt9.v@ in library wor'

odule RregistersQ compiled

ompiling verilog file @dwt2.v@ in library wor'

odule Rdwt9Q compiled

ompiling verilog file @dwt.v@ in library wor'

odule Rdwt2Q compiled

ompiling verilog file @top.v@ in library wor'

odule RdwtQ compiled

ompiling verilog file @inputs.v@ in library wor'

odule RtopdwtQ compiled

ompiling verilog file @topin.v@ in library wor'

odule RinpQ compiled

odule RtopinQ compiled

7o errors in compilation

Analysis of file [email protected]@Q succeeded.

-esign &ierarchy Analysis

Analyzing hierarchy for module RtopinQ in library Rwor'Q.

Analyzing hierarchy for module RinpQ in library Rwor'Q.

Analyzing hierarchy for module RtopdwtQ in library Rwor'Q.

Analyzing hierarchy for module RdwtQ in library Rwor'Q.

Analyzing hierarchy for module Rdwt2Q in library Rwor'Q.

Analyzing hierarchy for module Rdwt9Q in library Rwor'Q.

Analyzing hierarchy for module RregistersQ in library Rwor'Q.


56/72

Analyzing hierarchy for module RadderbitQ in library Rwor'Q.

?uilding hierarchy successfully finished.

&-8 Analysis

Analyzing top module RtopinQ.

odule RtopinQ is correct for synthesis.

Analyzing module RinpQ in library Rwor'Q.

odule RinpQ is correct for synthesis.

Analyzing module RtopdwtQ in library Rwor'Q.

odule RtopdwtQ is correct for synthesis.

Analyzing module RdwtQ in library Rwor'Q.

I7%E>1st>:99 * ontents of array RevenQ may be accessed with an index that

exceeds the array size. This could cause simulation mismatch.

&-8 +ynthesis

Cerforming bidirectional port resolution...

+ynthesizing =nit RinpQ.

/elated source file is @inputs.v@.

%ound x2"*bit /E for signal Rjmux"""Q.

%ound 2"*bit register for signal RdataQ.

%ound 92*bit up counter for signal RiQ.

+ummary>

inferred /E(s).

inferred ounter(s).

inferred 2" -*type flip*flop(s).

=nit RinpQ synthesized.

+ynthesizing =nit RregistersQ.

/elated source file is @-ff.v@.

%ound 2"*bit register for signal R\Q.


57/72

+ummary>

inferred 2" -*type flip*flop(s).

=nit RregistersQ synthesized.

+ynthesizing =nit RadderbitQ.

/elated source file is @adderbit.v@.

%ound 2"*bit adder for signal RoutQ.

&-8 +ynthesis /eport

acro +tatistics

k /As >

:x2"*bit dual*port distributed /A > :

!x2"*bit dual*port distributed /A > 2

k /Es >

x2"*bit /E >

k Adders[+ubtractors > S

2"*bit adder > S

k ounters > U

9*bit up counter > :

92*bit up counter >

:*bit up counter > 2

k /egisters > 2U

*bit register > !

2"*bit register > V

k omparators > V

9*bit comparator greater > 2

9*bit comparator lessequal > :


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:*bit comparator greater >

:*bit comparator lessequal > 2

6666666666666666666666666666666666666666666666666666666666666

Advanced &-8 +ynthesis

6666666666666666666666666666666666666666666666666666666666666

8oading device for application /f-evice from file #9s2"".nph# in environment

>1ilinx.

6666666666666666666666666666666666666666666666666666666666666

Advanced &-8 +ynthesis /eport

acro +tatistics

k /As >

:x2"*bit dual*port distributed /A > :

!x2"*bit dual*port distributed /A > 2

k /Es >

x2"*bit /E >

k Adders[+ubtractors > S

2"*bit adder > S

k ounters > U

9*bit up counter > :

92*bit up counter >

:*bit up counter > 2

k /egisters > !

%lip*%lops > !


59/72

k omparators > V

9*bit comparator greater > 2

9*bit comparator lessequal > :

:*bit comparator greater >

:*bit comparator lessequal > 2

%ound area constraint ratio of "" (_ S) on bloc' topin, actual ratio is .

%inal acro Crocessing ..

%inal /egister /eport

acro +tatistics

k /egisters > 9U9

%lip*%lops > 9U9

%inal /esults

/T8 Top 8evel Eutput %ile 7ame > topin.ngr

Top 8evel Eutput %ile 7ame > topin

Eutput %ormat > 7D

Eptimization Doal > +peed

eep &ierarchy > 7E

-esign +tatistics

k IEs > !2

ell =sage >

k ? VS:

k D7- >

k I7B > UU


60/72

k 8=T > 9:

k 8=T2 > 92

k 8=T9 > V"

k 8=T: > !U

k =1; > 2

k =1%S > :

k B >

k 1E/; > 22

k %lip%lops[8atches > 9U9

k %-/ > 2"U

k %-/ > 9

k %-/< > 2

k %-/+ > :

k %-+< > 9

k %-+ 2

k /A+ > V!

k /A1- > V!

k loc' ?uffers >

k ?=%DC >

k IE ?uffers > !

k I?=% >

k E?=% > !"


61/72

6666666666666666666666666666666666666666666666666666666666666

-evice utilization summary>

+elected -evice > 9s2""ft2S*:

7umber of +lices> 92 out of V2" `

7umber of +lice %lip %lops> 9U9 out of 9!:" V`

7umber of : input 8=Ts> out of 9!:" `

7umber used as logic> :2"

7umber used as /As> V

7umber of IEs> !2

7umber of bonded IE?s> !2 out of U9 :U`

7umber of D8s> out of ! 2`

6666666666666666666666666666666666666666666666666666666666666

TII7D /


62/72

Timing +ummary>

+peed Drade > *:

inimum period > :.!S"ns (aximum %requency> U.9:"&z)

inimum input arrival time before cloc'> :.!UVns

aximum output required time after cloc'> 22.::ns

aximum combinational path delay> 7o path found

Timing -etail

All values displayed in nanoseconds (ns)

6666666666666666666666666666666666666666666666666666666666666

Timing constraint> -efault period analysis for loc' #cl'#

loc' period> :.!S"ns (frequency> U.9:"&z)

Total number of paths [ destination ports> S:9V [ ::U

-elay> :.!S"ns (8evels of 8ogic 6 2U)

+ource> top[e[r[\" (%%)

-estination> top[e9[instrammemV (/A)

+ource loc'> cl' rising

-estination loc'> cl' rising

-ata Cath> top[e[r[\" to top[e9[instrammemV

Date 7et

ell>in*Qout fanout -elay -elay 8ogical 7ame (7et 7ame)

%-/>*Q\ 2 ".U2" .2 top[e[r[\" (top[e[r[\")

8=T2>I"*QE ".SS ".""" top[e[m9[addoutlutR"Q (top[e[72U)


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=1;>+*QE ".S"" ".""" top[e[m9[addoutcyR"Q

(top[e[m9[addoutcyR"Q)

1E/;>I*QE 2 ".V": .2 top[e[m9[addoutxorRQ

(top[e[m9inRQ)

8=T>I"*QE ".SS ".""" top[e[m9inRQrt

(top[e[m9inRQrt)

=1;>+*QE ".S"" ".""" top[e[addm:outcyR"Q

(top[e[addm:outcyR"Q)

=1;>I*QE ".": ".""" top[e[addm:outcyRQ

(top[e[addm:outcyRQ)

=1;>I*QE ".": ".""" top[e[addm:outcyR2Q

(top[e[addm:outcyR2Q)



=1;>I*QE ".": ".""" top[e[addm:outcyR:Q

(top[e[addm:outcyR:Q)

=1;>I*QE ".": ".""" top[e[addm:outcyRSQ

(top[e[addm:outcyRSQ)



=1;>I*QE ".": ".""" top[e[addm:outcyRUQ

(top[e[addm:outcyRUQ)

=1;>I*QE ".": ".""" top[e[addm:outcyR!Q

(top[e[addm:outcyR!Q)

=1;>I*QE ".": ".""" top[e[addm:outcyRVQ

(top[e[addm:outcyRVQ)


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=1;>I*QE ".": ".""" top[e[addm:outcyR"Q

(top[e[addm:outcyR"Q)







=1;>I*QE ".": ".""" top[e[addm:outcyR:Q

(top[e[addm:outcyR:Q)

=1;>I*QE ".": ".""" top[e[addm:outcyRSQ

(top[e[addm:outcyRSQ)

1E/;>I*QE ".V": ".VV top[e[addm:outxorRQ

(top[e[m:outRQ)

8=T2>I*QE ".SS ".""" top[e[mS[addoutlutRQ

(top[e[729)

=1;>+*QE ".S"" ".""" top[e[mS[addoutcyRQ

(top[e[mS[addoutcyRQ)

1E/;>I*QE ".V": ."U top[e[mS[addoutxorRUQ

(top[mSoddouttopRUQ)

8=T9>I2*QE " ".SS ".""" top[e[m"[addout

(top[e[m"[addout)

=1;>-I*QE " ".!!V ".""" top[e[m"[addoutaddcyR!Q

(top[e[m"[addoutaddcyR!Q)

1E/;>I*QE 2 ".V": ".""" top[e[m"[addoutaddxorRVQ

(top[m"evenoutdownRVQ)

/A1->- ".:2 top[e9[instrammem9V


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Total :.!S"ns (".9Sns logic, :.:VVns route)

(V.U` logic, 9".9` route)

6666666666666666666666666666666666666666666666666666666666666

Timing constraint> -efault E%%+ :.!UVns (8evels of 8ogic 6 2)

+ource> rst (CA-)

-estination> top[e[instrammem2" (/A)

-estination loc'> cl' rising

-ata Cath> rst to top[e[instrammem2"

Date 7et


I?=%>I*QE :! ".!2 2."9 rstI?=% (rstI?=%)

8=T9>I2*QE V ".SS .2: top[e[and"""" (top[e[and"""")

/A1->W< ".9U" top[e[instrammem9"

Total :.!UVns (.U:2ns logic, 9.9Uns route)

(9S.U` logic, :.9` route)

Timing constraint> -efault E%%+ mSevenevenouttopRVQ (CA-)

+ource loc'> cl' rising


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-ata Cath> top[e9[r9[\" to mSevenevenouttopRVQ

Date 7et


%-/>*Q\ 2 ".U2" .2 top[e9[r9[\" (top[e9[r9[\")

8=T2>I"*QE ".SS ".""" top[e9[m2[addoutlutR"Q (top[e9[72!)

=1;>+*QE ".S"" ".""" top[e9[m2[addoutcyR"Q

(top[e9[m2[addoutcyR"Q)

1E/;>I*QE 2 ".V": .2 top[e9[m2[addoutxorRQ

(top[e9[m9inRQ)

8=T>I"*QE ".SS ".""" top[e9[m9inRQrt (top[e9[m9inRQrt)

=1;>+*QE ".S"" ".""" top[e9[addm:outcyR"Q

(top[e9[addm:outcyR"Q)

=1;>I*QE ".": ".""" top[e9[addm:outcyRQ

(top[e9[addm:outcyRQ)

=1;>I*QE ".": ".""" top[e9[addm:outcyR2Q

(top[e9[addm:outcyR2Q)



=1;>I*QE ".": ".""" top[e9[addm:outcyR:Q

(top[e9[addm:outcyR:Q)

=1;>I*QE ".": ".""" top[e9[addm:outcyRSQ

(top[e9[addm:outcyRSQ)



=1;>I*QE ".": ".""" top[e9[addm:outcyRUQ

(top[e9[addm:outcyRUQ)


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=1;>I*QE ".": ".""" top[e9[addm:outcyR!Q

(top[e9[addm:outcyR!Q)

=1;>I*QE ".": ".""" top[e9[addm:outcyRVQ

(top[e9[addm:outcyRVQ)

=1;>I*QE ".": ".""" top[e9[addm:outcyR"Q

(top[e9[addm:outcyR"Q)







=1;>I*QE ".": ".""" top[e9[addm:outcyR:Q

(top[e9[addm:outcyR:Q)

=1;>I*QE ".": ".""" top[e9[addm:outcyRSQ

(top[e9[addm:outcyRSQ)

1E/;>I*QE ".V": .:" top[e9[addm:outxorRQ

(top[e9[m:outRQ)

8=T2>I"*QE ".SS ".""" top[e9[m29[addoutlutRQ (top[e9[72:)

=1;>+*QE ".S"" ".""" top[e9[m29[addoutcyRQ

(top[e9[m29[addoutcyRQ)

1E/;>I*QE : ".V": .2S top[e9[m29[addoutxorRUQ

(mSevenoddouttopUE?=%)

8=T9>I"*QE ".SS .:" top[e9[m2U[addout

(top[e9[m2U[addout)

8=T:>I"*QE ".SS ".""" top[e9[m2U[addoutaddlutR!Q

(top[e9[7S)


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=1;>+*QE " ".S"" ".""" top[e9[m2U[addoutaddcyR!Q

(top[e9[m2U[addoutaddcyR!Q)

1E/;>I*QE ".V": ".!" top[e9[m2U[addoutaddxorRVQ

(mSevenevenouttopVE?=%)

E?=%>I*QE S.:: mSevenevenouttopVE?=%

(mSevenevenouttopRVQ)

Total 22.::ns (S.VSns logic, .UVns route)

(V.V` logic, 9".` route)

6666666666666666666666666666666666666666666666666666666666666

C= > U.U [ U.:: s L U."" [ U."" s

Total memory usage is U9" 'ilobytes

7umber of errors > " ( " filtered)


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4.CO3PARI*ION 6ET%EEN PRAPO*ED 3ETHOD AND E8I*TIN7

3ETHOD:

De&i!e U"i'ia"ion *ummary (es"ima"e- &a'ues)

2o0i! U"i'ia"ion Use- A&ai'a+'e U"i'ia"ion

7umber of +lices 2"2 :S :`

7umber of +lice %lip %lops 2S V92 2`

7umber of : input 8=Ts 9SS V92 9`

7umber of bonded IE?s !2 292 9S`

7umber of D8s 2: :`

Table.> device utilization summary

Table.2> device utilisation summary of proposed method


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Table.9> Timing summary of proposed method

CHAPTER 4

CONC2U*ION AND FUTURE *COPE

4.1 CONC2U*ION AND FUTURE *COPE

-iscrete wavelet transforms a mathematical procedure in numerical analysis

and functional analysis. In -WT, the most prominent information in the signal

appears in high amplitudes and the less prominent information appears in very low


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amplitudes. The -iscrete Wavelet Transform (-WT) is a linear transformation that

operates on a data vector whose length is an integer power of two, transforming it into

a numerically different vector of the same length. It is a tool that separates data into

different frequency components.

Wavelets find application in speech compression, which reduces transmission

time in mobile applications. They are used in denoising, edge detection, feature

extraction, speech recognition, echo cancellation and others. They are very promising

for real time audio and video compression applications. Wavelets also have numerous

applications in digital communications. Erthogonal %requency -ivision ultiplexing

(E%-) is one of them. Wavelets are used in biomedical imaging. %or example, the


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]:^. C. 7. Cod'ur, En onstructing +ome Types of Wavelets with the +caling

oefficient 7

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