the use of the wavelet transformation for simulating certain characteristics of optoelectronic...

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The use of the discrete wavelet transformation for the purpose of simulation and analysis of the characteristics of signals in optoelectronic devices with multi-element radiation detectors for the measurement of the linear motions of objects is discussed. A comparison of the wavelet transformation to the Fourier transformation, a method that is in widespread use in practical applications, is presented and its information efficiency in the solution of analogous problems is demonstrated. Key words: wavelet transformation, simulation of signals, scale parameters, signal regeneration, measurement precision. It is not possible to solve the types of complex problems that arise in the synthesis, analysis, and optimization of dif- ferent optoelectronic devices and units without the use of modern methods of signal processing based on the use of Fourier and wavelet transformations. By comparison with traditional Fourier transformations of optical signals [1], the wavelet trans- formation, which has only begun to be used comparatively recently [2, 3], possesses a number of advantages. The following features are the most significant as regards optoelectronic devices: assuring simultaneous sweeping of a given signal with respect to frequency and coordinate independently over a broad range of variation of frequencies and coordinates, the possi- bility of analysis and investigation of complex nonlinear and nonstationary processes, for example, transient processes, the ability to maintain high resolution at different scales of transpiring processes, good and fast convergence of the approximat- ing functions obtained, where these functions describe processes that occur in the elements and units of optoelectronic devices, by comparison with other methods of approximation of these processes and their regeneration, as well as certain other advantages. The following are the main drawbacks of the Fourier transformation for the theory of optoelectronic devices: 1) the fact that it is not possible to analyze the features (singularities) of nonstationary signals (discontinuities, peaks, phases, etc.), since “blurring” of these signals occurs in the frequency domain over the entire spectral range; 2) the display of information about the frequencies of a particular process and the absence of any representation of the local properties of a signal with rapid variations of its spectral composition over time. For example, the Fourier transfor- mation does not distinguish a signal in the form of a sum of two sine curves from a signal with two successive sine curves with the same frequencies (nonstationary signal), since the spectral density is calculated by means of integration over the entire range of existence of the signal; and 3) the fact that it is not possible to depict signal drops with infinite gradient of their fronts, for example, rectangular pulses, by means of a Fourier transformation, since an infinitely great number of harmonics are needed for this purpose. In the Measurement Techniques, Vol. 52, No. 12, 2009 THE USE OF THE WAVELET TRANSFORMATION FOR SIMULATING CERTAIN CHARACTERISTICS OF OPTOELECTRONIC DEVICES OPTOPHYSICAL MEASUREMENTS V. M. Gorbov, S. M. Gorbov, and V. P. Soldatov UDC 621.035:535.8 Moscow State University of Geodesy and Cartography, Moscow, Russia; e-mail: [email protected]. Translated from Izmeritel’naya Tekhnika, No. 12, pp. 16–18, December, 2009. Original article submitted July 8, 2009. 0543-1972/09/5212-1294 © 2009 Springer Science+Business Media, Inc. 1294

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Page 1: The use of the wavelet transformation for simulating certain characteristics of optoelectronic devices

The use of the discrete wavelet transformation for the purpose of simulation and analysis of the characteristics

of signals in optoelectronic devices with multi-element radiation detectors for the measurement of the linear

motions of objects is discussed. A comparison of the wavelet transformation to the Fourier transformation, a

method that is in widespread use in practical applications, is presented and its information efficiency in the

solution of analogous problems is demonstrated.

Key words: wavelet transformation, simulation of signals, scale parameters, signal regeneration, measurement

precision.

It is not possible to solve the types of complex problems that arise in the synthesis, analysis, and optimization of dif-

ferent optoelectronic devices and units without the use of modern methods of signal processing based on the use of Fourier

and wavelet transformations. By comparison with traditional Fourier transformations of optical signals [1], the wavelet trans-

formation, which has only begun to be used comparatively recently [2, 3], possesses a number of advantages. The following

features are the most significant as regards optoelectronic devices: assuring simultaneous sweeping of a given signal with

respect to frequency and coordinate independently over a broad range of variation of frequencies and coordinates, the possi-

bility of analysis and investigation of complex nonlinear and nonstationary processes, for example, transient processes, the

ability to maintain high resolution at different scales of transpiring processes, good and fast convergence of the approximat-

ing functions obtained, where these functions describe processes that occur in the elements and units of optoelectronic

devices, by comparison with other methods of approximation of these processes and their regeneration, as well as certain

other advantages.

The following are the main drawbacks of the Fourier transformation for the theory of optoelectronic devices:

1) the fact that it is not possible to analyze the features (singularities) of nonstationary signals (discontinuities, peaks,

phases, etc.), since “blurring” of these signals occurs in the frequency domain over the entire spectral range;

2) the display of information about the frequencies of a particular process and the absence of any representation of

the local properties of a signal with rapid variations of its spectral composition over time. For example, the Fourier transfor-

mation does not distinguish a signal in the form of a sum of two sine curves from a signal with two successive sine curves

with the same frequencies (nonstationary signal), since the spectral density is calculated by means of integration over the

entire range of existence of the signal; and

3) the fact that it is not possible to depict signal drops with infinite gradient of their fronts, for example, rectangular

pulses, by means of a Fourier transformation, since an infinitely great number of harmonics are needed for this purpose. In the

Measurement Techniques, Vol. 52, No. 12, 2009

THE USE OF THE WAVELET TRANSFORMATION

FOR SIMULATING CERTAIN CHARACTERISTICS

OF OPTOELECTRONIC DEVICES

OPTOPHYSICAL MEASUREMENTS

V. M. Gorbov, S. M. Gorbov,and V. P. Soldatov

UDC 621.035:535.8

Moscow State University of Geodesy and Cartography, Moscow, Russia; e-mail: [email protected]. Translated

from Izmeritel’naya Tekhnika, No. 12, pp. 16–18, December, 2009. Original article submitted July 8, 2009.

0543-1972/09/5212-1294 ©2009 Springer Science+Business Media, Inc.1294

Page 2: The use of the wavelet transformation for simulating certain characteristics of optoelectronic devices

case of a limitation on the number of harmonics (width of spectrum), there arise oscillations (Gibb’s phenomenon) in neigh-

borhoods of jumps and discontinuities of a regenerated signal.

A so-called window Fourier transformation with window function with a particular width of the window B that suc-

cessively travels along the signal is used in order to produce some weakening of the influence of these drawbacks. The Fourier

transformations are implemented sequentially for each window. Consequently, a transition to a frequency-time or frequency-

space (coordinate-wise) representation of signals is realized. Moreover, the signal must be stationary within the range of each

position of the window. A family of spectra that vary with each shift of the window, i.e., as a function of two variables – fre-

quency and the time (space) position of the window – is the result of such a transformation. The width of the window B must

then correspond to the interval of stationarity of the signal, moreover, it must be known a priori.

The window Fourier transformation of a signal u(t) is described by the relationship [4]

where ω is the frequency of the signal; Bk = k∆B, k = 1, 2, 3, ... are constant coefficients; ∆B, the shift of a window; w*(t – Bk),

a function that is the complex conjugate of the function w(t – Bk), which describes the shift window as a function of time t;

and j, the imaginary unit.

Both the simplest rectangular window of the form w(t) = 1 within the transformation window and equal to 0 outside

the transformation window and special weight functions, such as the Bartlett function, that assure acceptable low distortions

of the spectrum due to the boundary conditions of a truncation of the window segments of a signal and neutralize the Gibb’s

phenomenon may be used as the transformation window.

The effective width of the window function must be kept constant over the entire range of definition of a signal.

Moreover, the resolution with respect to frequency ∆ω is related to the value of B by the formula ∆ω = 2π /B, i.e., a suffi-

ciently wide window in which the signal may often not be considered stationary is necessary to assure high frequency reso-

lution where a window Fourier transformation is used. This is also the principal drawback of this transformation.

Wavelet transformations are free of this drawback. The essential point of such transformations lies in an expansion

with respect to a basis that has been constructed (based on a function, called a wavelet, that possesses certain properties)

through scale variations and transports along the coordinate axis.

A continuous wavelet transformation W(a, b) of a signal u(t) is described by the formula

where a is a scale factor; b, a parameter of a shift of the wavelet function ψ(t); and the discrete wavelet transformation W(m, n)

of a signal u(t) is described by the formula

where m and n are real numbers; a0 > 0 and b0 > 0 are the initial scale factor and shift parameter, respectively.

A discrete wavelet transformation is obtained from continuous wavelet transformation under the assumption that a

and b assume only discrete values: a = a0m, b = nb0a0

m.

The wavelet function must satisfy the following two conditions:

There exist different wavelet functions by means of which a concrete type of signal may be regenerated with a

greater or lesser degree of precision [2–5]. For example, the HAAR wavelet is appropriate for the description of signals with

sharp boundaries.

ψ ψ( ) , ( ) .t dt t dt∫ ∫= < ∞−∞

0 2

W m n a u t a t nbm m( , ) ( ) ,/= −( )− −∑02

0 0ψ

W a b a u t t b a dt( , ) ( ) (( ) / ) ,.= −− ∫0 5 ψ

S B u t w t B t dtk k( , ) ( ) ( ) exp ( ) ,*ω ω= − −−∞

∫ j

1295

Page 3: The use of the wavelet transformation for simulating certain characteristics of optoelectronic devices

Wavelet functions are often constructed on the basis of derivatives of Gauss’s function:

Through the use of derivatives of higher order, it becomes possible to extract information about features of the signal.

Regeneration of a signal is accomplished by an inverse wavelet transformation (continuous and discrete) by means

of the formulas

Here Cψ is a normalizing coefficient while the numbers m and n correspond to the number of coefficients that are

needed to regenerate the function u(t).

The wavelet transformation may be used in optoelectronic devices to solve a number of different problems related to

the measurement of the coordinates of objects, the analysis of sensitivity, precision, and noise immunity, the determination of

optimal relationships between the parameters and characteristics of objects and radiation detectors, etc. Without delving into

the mathematical details of the foundations of the wavelet transformation (these may be found in [5, 6]), let us consider an

example that illustrates its application to the analysis of an asymmetric optical signal described by a one-dimensional spatial

u t W m n tm

a bn

( ) ( , ) ( ).,= ∑ ∑ ψ

u t C W a b t dadb aa b( ) ( , ) ( )( / ) ,,= − ∫∫ψ ψ1 2

ψmmk m k k m* ( ) ( ) exp ( / ) , .= − ≥j 2 2 1

ψmm

m

mt

d t

dt( ) ( )

[exp ( / )];= −

−1

22

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E, rel. units

x, rel. units

Fig. 2. A signal that has been regenerated by means of a wavelet

transformation.

E, rel. units

x, rel. units

Fig. 1. One-dimensional spatial distribution of illuminance.

Page 4: The use of the wavelet transformation for simulating certain characteristics of optoelectronic devices

distribution of illuminance and represented in Fig. 1. The asymmetricity of this distribution may be due to any one of several

factors, in particular, aberrations of the optical system of the optoelectronic device. The signal is transformed by the units of

the device and transmitted by these units to a registration block, where it is regenerated with degree of precision that depends

on its distortions. The procedure involved in processing the signal may be described by means of different transformations.

Let us calculate an approximation of a signal by means of wavelet transformations with the use of the Mathcad 14

program package, after which we will compare their results.

A signal that has been regenerated by means of wavelet transformations is shown in Fig. 2. The LMB wavelet (pro-

posed by Lemarier, Meyer, and Battle) was selected as the wavelet function; with this type of wavelet function, it is possible

to assure a high degree of precision of the reproduction of all of the six scale factors and shift parameters. Other wavelets

produce the same approximation, but require a greater number of scale factors.

Figure 3 shows the Fourier approximation of the signal presented in Fig. 1, with the use of 20 and 6 discrete com-

ponents (harmonics), respectively.

From an analysis of these results, it follows that, assuming the same precision of regeneration by means of a wavelet

transformation, fewer scale factors (less than one-third as many) are required and, consequently, a lesser volume of memory

and lesser machine time are needed for processing than in the case of a Fourier transformation,

In an optoelectronic device, the coordinates of the center of gravity of an object are often calculated in order to obtain

information about the spatial position of an object. In the simplest one-dimensional case, the algorithm used to determine the

coordinates of the energy center of gravity of an image (centroid) in a photodetector device with multi-element radiation

detector is implemented by means of the formula [7]

where xi is the coordinate of the center of the ith element of the detector; Uxi, the signal picked up by its ith element or col-

umn; and N, number of elements covered by the image.

Disregarding constant factors that do not have any effect on the value of xc, the signals Uxi required for calculation

of xc at the output of each element (column) of the radiation detector may be obtained from the relationship

where E(x) is a one-dimensional distribution function of the illuminance of the image of the object in the analysis plane;

su(x), distribution of the voltage sensitivity across the site of an individual ith detector element; ti, integration time of signal

for the detectors with signal accumulation; and X, step between which the elements are situated along the x-axis.

U t E x s x dxxi i uX

= ∫ ( ) ( ) ,

x x U Ui xii

N

xii

N

c == =∑ ∑

1 1

,

1297

E, rel. units

1

2

x, rel. units

Fig. 3. Fourier approximation of signal: 1) with 20 harmonics; 2) with 6 harmonics.

Page 5: The use of the wavelet transformation for simulating certain characteristics of optoelectronic devices

Let us calculate the deviation of the value of the coordinate of the energy center of gravity of a recovered image of

an object that has been obtained in a wavelet transformation, from the value of the coordinate of the center of gravity of the

initial image of the object.

We will assume that an image in the form of a strip occupying 10 elements is projected onto a multi-element radiation

detector (ruler) with cell dimension 12 µm (Fig. 4). We will adopt the distribution shown in Fig. 1 as the initial distribution curve

of the illuminance with respect to coordinate in the image.

Following calculation of the coordinate of the energy center of gravity of the initial image and of the image that has

been reconstructed from 6 scale factors and a shift of the image, we find xc.i = –11.868 µm and xc.r = –11.866 µm, respec-

tively. The difference between these two values ∆xc = –0.002 µm, which has practically no effect on the precision with which

the coordinates of an object is determined in an optoelectronic device.

In recovering an initial signal with the use of a Fourier transformation with the same number of scale factors, the

error in the measured value of the coordinate of the image xc.r = –9.483 µm, which is substantial.

The measurement error associated with the analysis of the precision of an optoelectronic device using a multi-ele-

ment radiation detector for the measurement of the coordinates of objects and marks (theodolites, tachometers, goniometer,

etc.) is usually determined by specifying displacements of the image of an object along one of the coordinate directions, for

example, the x-axis, within a single step of the distribution of the elements of the multi-element radiation detector with a dis-

tribution of the illuminance of the object E(x) as given, for example, in Fig. 1. Moreover, the coordinate xc of its energy cen-

ter of gravity is determined in each position of the image.

A recovery of the function E(x) was performed with the use of the LMB wavelet with number of decomposition

coefficients equal to 6 and the deviations in the shift of the image along X with step ∆x = X/6 were determined.

By means of such a simulation, it is possible to select a required relationship between the width of the image and

the step in the configuration of the elements of a multi-element radiation detector that is adequate to a proposed distribution

of the illuminance of the image of an object or mark such that the partial measurement error of the coordinates of the object

will be minimal.

Thus, with the use of a discrete transformation based on wavelets it becomes possible to increase the precision with

which certain parameters are determined and decrease the machine computation time from that associated with a Fourier

transformation. Methods for processing signals using a wavelet transformation are now being developed at a rapid pace and

have begun to be widely used in practical applications. They are used in image compression, image processing, the extrac-

tion of a useful signal against a noise background, simulation of nonstationary processes, and elsewhere.

1298

E, rel. units

x, µm

1

2

3

4

Fig. 4. Multi-element radiation detector and image transmitted by detector: 1) initial position;

2) initial signal; 3) recovered signal; 4) final position.

Page 6: The use of the wavelet transformation for simulating certain characteristics of optoelectronic devices

REFERENCES

1. G. M. Mosyagin, V. B. Nemtinov, and E. N. Lebedev, Theory of Optoelectronic Devices [in Russian], Mashinostroenie,

Moscow (1990).

2. N. M. Astraf’eva, “Wavelet analysis, foundations of the theory, and application examples,” Usp. Fiz. Nauk, 166, No. 11,

1145 (1996).

3. A. N. Yakovlev, Introduction to Wavelet Transformation [in Russian], Novosibirsk State Technological University,

Novosibirsk (2003).

4. I. Dobeshi, Ten Lectures on Wavelets [in Russian], Research Center for Regular and Chaotic Dynamics, Izhevsk (2001).

5. K. Chui, Introduction to Wavelets [Russian translation], Mir, Moscow (2001).

6. V. P. D’yakonov, Wavelets. From Theory to Application [in Russian], SOLON-R, Moscow (2002).

7. Yu. B. Parvulyusov et al., The Design of Optoelectronic Devices [in Russian], Logos, Moscow (2000).

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