# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 28/28)

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 DT CT DF DFT FS CF DTFT FT

Because the basis functions of all four Fourier transforms are periodic, the transform of a periodic signal (CT or DT) is a function of discretefrequency. In other words, it is a sequence of series expansion coefficients. If the signal is infinitely long and not periodic, thetransform is a function of continuous frequency and the inverse is an integral, not a sum.

$\text{Periodic}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{time}⇔\text{Discrete}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{frequency}$
$\text{Periodic}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{frequency}⇔\text{Discrete}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{time}$

A bit of thought and, perhaps, referring to appropriate materials on signal processing and Fourier methods will make this clear and show why so manyproperties of Fourier analysis are created by the periodic basis functions.

Also recall that in most cases, it is the Fourier transform, discrete-time Fourier transform, or Fourier series that is needed but it is the DFT thatcan be calculated by a digital computer and that is probably using the FFT algorithm. If the coefficients of a Fourier series drop off fast enoughor, even better, are zero after some harmonic, the DFT of samples of the signal will give the Fourier series coefficients. If a discrete-timesignal has a finite nonzero duration, the DFT of its values will be samples of its DTFT. From this, one sees the relation of samples of asignal to the signal and the relation of the various Fourier transforms.

Now, what is the case for the various wavelet transforms? Well, it is both similar and different. The table that relates the continuous anddiscrete variables is given by where DW indicates discrete values for scale and translation given by $j$ and $k$ , with CW denoting continuous values for scale and translation.

 DT CT DW DTWT DWT CW DTCWT CWT

We have spent most this book developing the DWT, which is a series expansion of a continuous time signal. Because the waveletbasis functions are concentrated in time and not periodic, both the DTWT and DWT will represent infinitely long signals. In most practical cases,they are made periodic to facilitate efficient computation. Chapter: Calculation of the Discrete Wavelet Transform gives the details of how the transform is made periodic. The discrete-time, continuous wavelet transform (DTCWT) is seldom used andnot discussed here.

The naming of the various transforms has not been consistent in the literature and this is complicated by the wavelet transforms having twotransform variables, scale and translation. If we could rename all the transforms, it would be more consistent to useFourier series (FS) or wavelet series (WS) for a series expansion that produced discrete expansion coefficients, Fourier transforms (FT) orwavelet transforms (WT) for integral expansions that produce functions of continuous frequency or scale or translation variable together with DT(discrete time) or CT (continuous time) to describe the input signal. However, in common usage, only the DTFT follows this format!

 Common Consistent Time, Transform Input Output name name C or D C or D periodic periodic FS CTFS C D Yes No DFT DTFS D D Yes Yes DTFT DTFT D C No Yes FT CTFT C C No No DWT CTWS C D Y or N Y or N DTWT DTWS D D Y or N Y or N – DTWT D C N N CWT CTWT C C N N

Recall that the difference between the DWT and DTWT is that the input to the DWT is a sequence of expansion coefficients or a sequence of innerproducts while the input to the DTWT is the signal itself, probably samples of a continuous-time signal. The Mallat algorithm or filter bankstructure is exactly the same. The approximation is made better by zero moments of the scaling function (see Section: Approximation of Scaling Coefficients by Samples of the Signal ) or by some sort of prefiltering ofthe samples to make them closer to the inner products [link] .

As mentioned before, both the DWT and DTWT can be formulated as nonperiodic, on-going transforms for an exact expansion of infinite durationsignals or they may be made periodic to handle finite-length or periodic signals. If they are made periodic (as in Chapter: Calculation of the Discrete Wavelet Transform ), then there is an aliasing that takes place in the transform. Indeed, the aliasinghas a different period at the different scales which may make interpretation difficult. This does not harm the inverse transform whichuses the wavelet information to “unalias" the scaling function coefficients. Most (but not all) DWT, DTWT, and matrix operatorsuse a periodized form [link] .

#### Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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a perfect square v²+2v+_
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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is it 3×y ?
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J, combine like terms 7x-4y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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AMJAD
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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AMJAD
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Stotaw
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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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