Eliminate Signal Noise With Discrete Wavelet Transformation

7/26/2019 Eliminate Signal Noise With Discrete Wavelet Transformation

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Eliminate Signal Noise With Discrete WaveletTransformation

Modern DSP and communications applications are beginning to use wavelet transforms incritical algorithms.

The wavelet transform is a mathematical tool that's becoming quite useful for analyzing

many types of signals. It has been proven especially useful in data compression, as well

as in adaptive equalizer and transmultiplexer applications.

A wavelet is a small, localized wave of a particular shape and finite duration. everal

families, or collections of similar types of wavelets, are in use today. A few go by the

names of !aar, "aubechies, and #iorthogonal. $avelets within each of these families

share common properties. %or instance, the #iorthogonal wavelet family exhibits linear

phase, which is an important characteristic for signal and image reconstruction.

$avelet analysis is simply the process of decomposing a signal into shifted and scaled

versions of a particular wavelet. An important property of wavelet analysis is perfect

reconstruction, which is the process of reassembling a decomposed signal or image into

its original form without loss of information. #y examining wavelet theory as it applies to

three specific applications, we find that it wor&s so well because these examples rely on

perfect reconstruction for their fundamental operation.

There are no set rules for the choice of the mother wavelet used in wavelet analysis. The

choice depends on the properties of the mother wavelet, the properties of the signal to be

examined, and the requirements of the analysis. %or this reason, it's convenient to have

tools that let you easily explore and experiment with many different wavelets and input

signals. The following examples use AT(A#, the $avelet Toolbox, and imulin& to

ma&e exploration of wavelet concepts convenient.

In this article, the wavelet we use as an example )called the *mother* wavelet+ is the

"aubechies wavelet, db. The in the name represents the order of the filter, whichcorresponds to eight coefficients.

The "iscrete $avelet Transform )"$T+ is commonly employed using dyadic multirate

filter ban&s, which are sets of filters that divide a signal frequency band into subbands.

These filter ban&s are comprised of low-pass, high-pass, or bandpass filters. If the filter

ban&s are wavelet filter ban&s that consist of special low-pass and high-pass wavelet

filters, then the outputs of the low-pass filter are the approximation coefficients. Also,

the outputs of the high-pass filter are the detail coefficients.


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The process of obtaining the approximation and detail coefficients is called

decomposition. Termed multilevel decomposition, this process can be repeated, with

successive approximations )the output of the low-pass filter in the first ban&+ being

decomposed in turn, so that one signal is bro&en down into a number of components.

A two-level decomposition is shown in%igure .In this illustration, a/represents the

approximation coefficients, while d/and drepresent the detail coefficients resulting

from the two-level decomposition. After each decomposition, we employ decimation by

two to remove every other sample and, therefore, reduce the amount of data present.

The Inverse "iscrete $avelet Transform )I"$T+ reconstructs a signal from the

approximation and detail coefficients derived from decomposition. The I"$T differs

from the "$T in that it requires upsampling and filtering, in that order. 0psampling,

also &nown as interpolating, means the insertion of zeros between samples in a signal.

The right side of the figure shows an example of reconstruction.

Another way to interpret the figure is that the analysis filter ban& on the left reduces the

rate of an input signal and produces multiple output signals with varying rates. The

analysis filter ban& performs the "$T represented by the decomposition. The synthesis

filter ban& on the right increases the rates of multiple input signals while combining

them into a single output signal. It performs the I"$T represented by the

reconstruction.

The %ilters Are The 1ey

2ow one might as&, what's unique about wavelet filter ban&s3 The magic is in the filters

themselves. #y choosing filters that are intimately related for both decomposition and

reconstruction processes, the effects of aliasing, which can be introduced by the

decimation, are removed.

$hen the signal is reconstructed, it doesn't exhibit any aliasing or distortion (right side

of Fig. 1). As a result, the output is said to be a perfect reconstruction.

$avelet filters have finite length. They aren't truncated versions of infinitely long filter

re-sponses. #ecause of this property, wavelet filter ban&s can perform local analysis, or

the examination of a localized area of a larger signal. (ocal analysis is an important

consideration when dealing with signals that have discontinuities. $avelet transforms

can be applied to these &inds of signals with excellent results. This is due to their ability

to locate short-time )local+ high-frequency features of a signal and resolve low-frequency

behavior at the same time.
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As stated earlier, perfect reconstruction is an important property of wavelet filter ban&s.

$hen the analysis filter ban& output is connected to the synthesis filter ban& input and

the proper delays for alignment are used, as in %igure , then the output of the entire

system is identical to the input. If a threshold operation is applied to the output of the

"$T and wavelet coefficients that are below a specified value are removed, then the

system will perform a *de-noising* function.

Two different threshold operations can be viewed in %igure /.In the first, hard

thresholding, coefficients whose absolute values are lower than the threshold are set to

zero. !ard thresholding is extended by the second technique, soft thresholding, by

shrin&ing the remaining nonzero coefficients toward zero.

The de-noising process consists of decomposing the original signal, thresholding the

detail coefficients, and reconstructing the signal. The decomposition portion of our de-

noising example is accomplished via the "$T. The $avelet Toolbox provides various

parameters from which one must pic& in order to decompose the signal. These

parameters include loading the original signal, selecting the wavelet family, and

specifying the level of decomposition.

$e have pic&ed "aubechies )db+ as our analysis wavelet, a three-level decomposition.

$e could have elected to perform more levels of decomposition, as the more levels we

chose to decompose our signal, the more detail coefficients we get. #ut for de-noising our

signal in this example, a three-level decomposition provides sufficient noise reduction.

#y employing the $avelet -" "iscrete $avelet Analysis Tool from the $avelet Toolbox,

one can calculate the "$T by clic&ing on the Analyze button.The results of the

decomposition are illustrated by%igure 4. The original noisy bloc& signal is shown in the

s trace. The a4trace represents the third-level approximation coefficients, which are the

high-scale, low-frequency components. 2ote how the approximation a4is similar to the

original signal s. The other three waveforms )d4, d/, and d+ are the detail coefficients,

which are low-scale, high-frequency components.

After the decomposition was complete, we opened the $avelet -" "e-2oising Tool so

that we could define our de-noising parameters (Fig. 4)$e chose to implement the

defaults of the %ixed %orm oft Threshold and 0nscaled $hite 2oise for our soft

thresholding method. The threshold values were fixed at exactly .555 for each

decomposition level, which gave us good noise suppression. #oth the de-noised and

original signals are shown in %igure 6.

imulation 7f 8eal-Time "e-2oisingThe previous example explored the theory of wavelet decomposition and thresholding in
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a data-flow environment. $hile this theoretical exploration is an important part of the

de-noising implementation process, the implementation of a real-time system has other

aspects that must be considered as well. This is best performed with a simulation tool

li&e imulin&, which allows the use of individual subsytem components in a multirate

time-flow environment.

The first and most familiar technique for implementing multirate "9 systems is to

propagate scalar data samples with varying rates within a system model. A imulin&

model containing analysis and synthesis filter ban&s propagating scalar data samples is

shown in %igure :.This example uses the same input signal s from %igure 4.

%or simulation in a data-flow environment, such as AT(A#, processing signals of

differing lengths due to changes in sample rates needs to be addressed. The data

alignment would be carried out using the appropriate indexing schemes. In a multirate

time-flow environment, li&e imulin&, the data is conceptually infinite in length, and

indexing cannot perform the data alignment. Instead, delay elements are introduced

after the analysis filter ban& to achieve data alignment. This is accomplished by the

*"elay Alignment* subsystem (Fig. 6, again).

%urthermore, to compare the output signal with the input, additional delays are

introduced into the input signal path. "ata alignment is a significant aspect of a

practical, real-time implementation. The input, output, and residual signals shown in

%igure : can be viewed in the scope display in %igure ;.

The wavelet transmultiplexer )$T+ provides an interesting example of the perfect

reconstruction property of the "$T. The transmultiplexer combines two source signals

for transmission over a single lin&, then separates the two signals at the receiving end of

the channel (Fig. 8). The inputs are assumed to be baseband signals.

The ability of wavelets to provide perfect reconstruction of independent signals,

transmitted over a single communications lin&, is demonstrated in%igure demonstrates a two-channel transmultiplexer. #ut the

method can be extended to an arbitrary number of channels. 2ote that the total data rate

is still limited by the 2yquist rate of the high-speed data lin&.

imilarities $ith %" 7peration

The operation of a $T is analogous to a frequency-domain multiplexer )%"+ in

several respects. In an %", baseband input signals are filtered and modulated into
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ad?acent frequency bands, summed together, and then transmitted over a single lin&. 7n

the receiving end, the transmitted signal is filtered to separate the two ad?acent

frequency channels. The signals are then demodulated bac& to baseband.

The filters need to pass the desired signal through the filter passband with as littledistortion as possible. In addition, the filters must strongly attenuate the ad?acent signal

to provide a sharp transition from the filter passband to its stopband. This process limits

the amount of crosstal&, or signal lea&age, from one frequency band to the next. These

constraints generally require longer and more expensive filters.

7ften, %" employs an unused frequency band, &nown as a guard band, between the

two modulated frequency bands to relax the requirements on the %" filters. This

decreases spectral efficiency, thereby reducing the usable bandwidth for each input

signal.

In a $T, the filtering performed by the synthesis and analysis wavelet filters is

analogous to the filtering steps in the %". 9lus, the interpolation in the I"$T is

equivalent to frequency modulation. %rom a frequency-domain perspective, the wavelet

filters are fairly poor spectral filters, exhibiting slow transitions from passband to

stopband, and providing significant distortion in their response.

$hat ma&es the $T special, though, is that the analysis and synthesis filters together

completely cancel the filter distortions and signal aliasing. That produces perfect

reconstruction of the input signals and, thus, perfect extraction of the multiplexed

inputs.

Ideal spectral efficiency can be achieved with the $T, because no guard band is

required. 9ractical limitations of implementing the channel filter create out-of-band

lea&age and distortion. In the conventional %" approach, every channel within the

same communications system requires its own filter and is susceptible to crosstal& from

neighboring channels. #ut the $T method only requires a single bandpass filter for the

entire communications channel, and the channel-to-channel interference is eliminated.

1eep in mind that a noisy lin& can cause imperfect reconstruction of the input signals.

%urthermore, the effects of channel noise and other impairments on the recovered

signals can differ in %"- and $T-based systems.

Image compression is becoming increasingly important as the efficient use of available

transmission bandwidth becomes more complex. As complexity increases, system

resources must be optimized to use minimal bandwidth and memory. 7ne way to

optimize these resources is to employ image compression. The method and amount of


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compression needs to be such that it's still possible to achieve a reasonable

reconstruction of the image. $avelet transforms have this capability.

The compression procedure is similar to that of de-noising used in an earlier example.

The only difference lies in the thresholding applied to the detail coefficients. Two ap-proaches are available in the $avelet Toolbox for thresholding detail coefficients when

compressing two-dimensional data. These are global thresholding and level

thresholding.

$ith the global-thresholding ap-proach, we define a global-threshold method, a

compression performance factor, and a relative square norm recovery performance

factor. The $avelet Toolbox derives a global threshold from an equal balance between

the percentages of retained energy and number of zero coefficients. $ith the level-

thresholding approach, one would need to visually determine level-dependent

thresholds.

In this example, we allow the $avelet Toolbox to derive a global threshold for our

example image. The image shown in%igure 5was decomposed using the two-

dimensional discrete wavelet analysis tool )similar to the one-dimensional tool found in

%igure 4+. %or this example, we decided to perform a two-level decomposition using the

biorthogonal spline wavelet bior4.;, which specifies a third-order reconstruction filter

and a seventh-order decomposition filter.

The compression tools available in the $avelet Toolbox perform only the thresholding

portion of the compression process. Its performance is measured by the percentage of

remaining nonzero elements in the wavelet decomposition. $hen implementing a real-

world compression scheme, one would need to further consider quantization and bit-

allocation factors.

The two-dimensional wavelet compression tool automatically generates a threshold

based on the thresholding method selected (Fig. 10, again). $e pic&ed *8emove near 5,*

which sets this global threshold to . $hen we clic& on the =ompress button, all

coefficients whose values are less than )in this case, @+ are forced to zero. In spite

of this case, .@ of the original image energy is retained. ee the $avelet Toolbox

0ser's uide for more information on how these percentages were calculated.

$avelet analysis is a new and promising tool which complements traditional signal

processing techniques. It can offer significant advantages for real-time systems, and it

opens the door to new and exciting communications applications.

%or %urther InformationB
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. =. Taswell, *The $hat, !ow, and $hy of $avelet hrin&age "enoising,* Computing InScience And ngineering, vol. /, no. 4, ayCDune /555, p. /-, no. 6, ay/555, p. -5.

. . trang and T. 2guyen, 'a#e"ets And Fi"ter ans, $ellesley-=ambridge 9ress,

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Eliminate Signal Noise With Discrete Wavelet Transformation