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Periodic or cyclic discrete wavelet transform

If f ( t ) has finite support, create a periodic version of it by

f ˜ ( t ) = n f ( t + P n )

where the period P is an integer. In this case, f , φ j , k and f , ψ j , k are periodic sequences in k with period P 2 j (if j 0 and 1 if j < 0 ) and

d ( j , k ) = 2 j / 2 f ˜ ( t ) ψ ( 2 j t - k ) d t
d ( j , k ) = 2 j / 2 f ˜ ( t + 2 - j k ) ψ ( 2 j t ) d t = 2 j / 2 f ˜ ( t + 2 - j ) ψ ( 2 j t ) d t

where = < k > P 2 j ( k modulo P 2 j ) and l { 0 , 1 , ... , P 2 j - 1 } . An obvious choice for J 0 is 1. Notice that in this case given L = 2 J 1 samples of the signal, f , φ J 1 , k , the wavelet transform has exactly 1 + 1 + 2 + 2 2 + + 2 J 1 - 1 = 2 J 1 = L terms. Indeed, this gives a linear, invertible discrete transform which can be considered apart fromany underlying continuous process similar the discrete Fourier transform existing apart from the Fourier transform or series.

There are at least three ways to calculate this cyclic DWT and they are based on the equations [link] , [link] , and [link] later in this chapter. The first method simply convolves the scalingcoefficients at one scale with the time-reversed coefficients h ( - n ) to give an L + N - 1 length sequence. This is aliased or wrapped as indicated in [link] and programmed in dwt5.m in Appendix 3. The second method creates a periodic c ˜ j ( k ) by concatenating an appropriate number of c j ( k ) sections together then convolving h ( n ) with it. That is illustrated in [link] and in dwt.m in Appendix 3. The third approach constructs a periodic h ˜ ( n ) and convolves it with c j ( k ) to implement [link] . The Matlab programs should be studied to understand how these ideas are actually implemented.

Because the DWT is not shift-invariant, different implementations of the DWT may appear to give different results because of shifts of the signaland/or basis functions. It is interesting to take a test signal and compare the DWT of it with different circular shifts of the signal.

Making f ( t ) periodic can introduce discontinuities at 0 and P . To avoid this, there are several alternative constructions of orthonormalbases for L 2 [ 0 , P ] [link] , [link] , [link] , [link] . All of these constructions use (directly or indirectly) the concept of time-varyingfilter banks. The basic idea in all these constructions is to retain basis functions with support in [ 0 , P ] , remove ones with support outside [ 0 , P ] and replace the basis functions that overlap across the endpoints with special entry/exit functions that ensure completeness . These boundary functions are chosen so that the constructed basis isorthonormal. This is discussed in Section: Time-Varying Filter Bank Trees . Another way to deal with edges or boundaries uses “lifting" as mentioned in Section: Lattices and Lifting .

Filter bank structures for calculation of the dwt and complexity

Given that the wavelet analysis of a signal has been posed in terms of the finite expansion of [link] , the discrete wavelet transform (expansion coefficients) can be calculated using Mallat's algorithm implemented by afilter bank as described in Chapter: Filter Banks and the Discrete Wavelet Transform and expanded upon in   Chapter: Filter Banks and Transmultiplexers . Using the direct calculations described by the one-sided tree structure of filters and down-samplers in Figure: Three-Stage Two-Band Analysis Tree allows a simple determination of the computational complexity.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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Stoney Reply
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Adin Reply
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what school?
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Damian Reply
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Introduction about quantum dots in nanotechnology
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Anassong Reply
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Damian Reply
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s. Reply
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Sanket Reply
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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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