# 0.9 Calculation of the discrete wavelet transform  (Page 3/5)

 Page 3 / 5

## Periodic or cyclic discrete wavelet transform

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

$\stackrel{˜}{f}\left(t\right)=\sum _{n}f\left(t+Pn\right)$

where the period $P$ is an integer. In this case, $⟨f,{\phi }_{j,k}⟩$ and $⟨f,{\psi }_{j,k}⟩$ are periodic sequences in $k$ with period $P\phantom{\rule{0.166667em}{0ex}}{2}^{j}$ (if $j\ge 0$ and 1 if $j<0$ ) and

$d\left(j,k\right)={2}^{j/2}\int \stackrel{˜}{f}\left(t\right)\psi \left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt$
$d\left(j,k\right)={2}^{j/2}\int \stackrel{˜}{f}\left(t+{2}^{-j}k\right)\psi \left({2}^{j}t\right)\phantom{\rule{0.166667em}{0ex}}dt={2}^{j/2}\int \stackrel{˜}{f}\left(t+{2}^{-j}\ell \right)\psi \left({2}^{j}t\right)\phantom{\rule{0.166667em}{0ex}}dt$

where $\ell =}_{P\phantom{\rule{0.166667em}{0ex}}{2}^{j}}$ ( $k$ modulo $P\phantom{\rule{0.166667em}{0ex}}{2}^{j}$ ) and $l\in \left\{0,1,...,P\phantom{\rule{0.166667em}{0ex}}{2}^{j}-1\right\}$ . An obvious choice for ${J}_{0}$ is 1. Notice that in this case given $L={2}^{{J}_{1}}$ samples of the signal, $⟨f,{\phi }_{{J}_{1},k}⟩$ , the wavelet transform has exactly $1+1+2+{2}^{2}+\cdots +{2}^{{J}_{1}-1}={2}^{{J}_{1}}=L$ 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 $h\left(-n\right)$ 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 ${\stackrel{˜}{c}}_{j}\left(k\right)$ by concatenating an appropriate number of ${c}_{j}\left(k\right)$ sections together then convolving $h\left(n\right)$ with it. That is illustrated in [link] and in dwt.m in Appendix 3. The third approach constructs a periodic $\stackrel{˜}{h}\left(n\right)$ and convolves it with ${c}_{j}\left(k\right)$ 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 $f\left(t\right)$ periodic can introduce discontinuities at 0 and $P$ . To avoid this, there are several alternative constructions of orthonormalbases for ${L}^{2}\left[0,P\right]$ [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 $\left[0,P\right]$ , remove ones with support outside $\left[0,P\right]$ 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

how do we prove the quadratic formular
Seidu Reply
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? Kala Reply lim x to infinity e^1-e^-1/log(1+x) given eccentricity and a point find the equiation Moses Reply 12, 17, 22.... 25th term Alexandra Reply 12, 17, 22.... 25th term Akash College algebra is really hard? Shirleen Reply Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table. Carole I'm 13 and I understand it great AJ I am 1 year old but I can do it! 1+1=2 proof very hard for me though. Atone Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily. Vedant hi vedant can u help me with some assignments Solomon find the 15th term of the geometric sequince whose first is 18 and last term of 387 Jerwin Reply I know this work salma The given of f(x=x-2. then what is the value of this f(3) 5f(x+1) virgelyn Reply hmm well what is the answer Abhi If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10 Augustine how do they get the third part x = (32)5/4 kinnecy Reply make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be AJ how Sheref A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?

 By By By By By