Approximation of a digital signal using Estimate Wavelet Transform

Archit Yajnik Department of Mathematics

Sikkim Manipal Institute of Technology, SMIT East Sikkim, India

e-mail: archit.yajnik@gmail.com

Abstract—This article presents a general outline of the approximation of digital signal using Discrete Wavelet. The technique used in [1] is a new attitude to a multiresolution digital signal analysis by discrete wavelet transforms. This article demonstrates an approximation of a digital signal using Daubechies D4 wavelets. The present technique exhibits a revised procedure of removing distortion (loss) generated from the approximated double length digital signal presented in [1]. For highly nonlinear digital signal the technique of [1] fails to approximate the signal but the proposed revised technique can approximate the signal. The proposed method not only enables approximating digital signals in a better way but also it can approximate highly nonlinear digital signals. Moreover, the energy conservation is exactly double than the original signal which is not achieved by the approximation technique discussed in [1].

Keywords- Discrete Wavelet Transform (DWT), Multiresolution Analysis (MRA), Estimate Wavelet Transform (EWT)

I. INTRODUCTION Wavelet theory has become popular since last two

decades for many fields of numerical analysis such as data compression or the solution of integral and differential equations. Wavelet-based methods have been proved to yield efficient and fast algorithms.

Multiresolution analysis and digital processing by a wavelet method are nowadays one of the most dynamically developing parts of digital signal processing. Discrete wavelet transform (DWT), can be used to analysis of the signal, to noise reduction and image compression. Fourier analysis, being the most prevalent method of multiresolution analysis, has however a fundamental drawback: we lack the information when the event in frequency domain exactly happened.

Discrete Wavelet Transform and Multi Resolution Analysis using Wavelet Filters are now very widely used in the areas of feature extraction [2, 3, 4, 8]. This article presents a typical application of oversampling a digital signal to double its original signal using wavelets. The new method EWT (to be discussed in section IV) takes Discrete finite length signals with finitely many discontinuities and yields double length approximation and hence fills the gap generated by discontinuities.

The present article is organized as follows. After introducing discrete wavelet transform in the second section, we discuss shortcomings of the approach of approximation presented in [1] in the third section. The revised approximation technique (say, Estimate Wavelet Transform) is introduced in section four along with the numerical examples. The article is summarized in section five followed by references.

II. DISCRETE WAVELET TRANSFORM A spatially localized basis is useful in signal processing

because it provides a local analysis of a signal. Let z(n) represent a signal with frequency n. If a certain coefficient in a series representation of z is large, we can identify the location with which this large coefficient is associated if the coefficients are spatially localized. We could then, for example, focus on this location and analyze it in greater detail. The following lemma [7] gives the discrete wavelet transform (analysis phase) and reconstruction (synthesis phase) of a signal.

Lemma: Suppose { }j

jV ∈Ζ is a multiresolution analysis

with scaling function ϕ and scaling sequence ( ( ))j k zu u k ∈= . Suppose ( ( ))j k zv v k ∈= is

defined by 1( ) ( 1) (1 )kv k u k−= − − , and wavelet function 1)( k

Zkkv ϕψ ∑

∈= , where ( ) ( )kxxk −= 221 ϕϕ .

Suppose 2 ( )f L∈ ℜ , the space of all measurable, square integrable functions and, for each j z∈ , define sequences

( ( ))j j k zx x k ∈= and ( ( ))j j k zy y k ∈= by

( ) , jj kx k f= Ψ where ( ) ( )kxx jjk

j −= 22 2/ ψψ and

( ) , jj ky k f ψ= . Then

1( )j jx D y v+= ∗ (1)

and 1( )j jy D y u+= ∗ , (2)

where downsampling operator (D) and convolution (*) are

2011 International Conference on Computational Science and Its Applications

978-0-7695-4404-5/11 $25.00 © 2011 IEEE

DOI 10.1109/ICCSA.2011.49

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2011 International Conference on Computational Science and Its Applications

978-0-7695-4404-5/11 $25.00 © 2011 IEEE

DOI 10.1109/ICCSA.2011.49

171

on 2 ( )l Z , the space of square summable sequences of complex numbers and ( ) ( ) ( ) ( )u k u k and v k v k= − = − are duals of u (approximation coefficients) and v (detail coefficients) respectively by considering approximation coefficients u = uj and detail coefficients v = vj. Also 1 ( ) ( )j j jy U y u U x v+ = ∗ + ∗ (3)

where U is the upsampling operator on2 ( )l Z . Here

equations (1) and (2) are part of analysis phase while (3) is a part of synthesis phase.

III. APPROXIMATION OF A DIGITAL SIGNAL USING DWT

As shown in [1], the analysis phase of (3) can further be expanded by applying synthesis phase twice yields higher level approximation of a signal as depicted in Fig. 1.

Figure1. Approximation using filter bank of DWT

2 ( ( )* ) *j jw U U y u u+ = + ( ( ) * ) *jU U x v u +( ( )* )*jU U y u v + ( ( )* )*jU U x v v (4)

The output 2jw + contains four parts viz. series 1, series 2, series 3 and series 4. Series 1 is made up of one dual High pass component and two High pass components so we call it as HHH. Similarly series 2, series 3 and series 4 are known as HHL, LLH and LLL respectively. These computations are implemented in Java.

As we know from [2, 3], the low-low coefficients captures an approximation details of a signal. The following expression gives an approximate double length signal by using series LLL.

2 [ ( ( )* )* ]j jr U U y u uμ+ =

(5)

where μ (1 ≤ μ ≤ 2 ) is a scaling parameter (estimated from the observations). In the subsequent examples we

consider μ = 2 . The coefficients obtained using (5) can be separated into

two parts. The first part of the coefficients provides an approximate value of the signal at a particular position

whereas the second part adds a new value between two approximated values obtained from the first part. The new value is nothing but the intermediate value (depends on the behavior of the digital signal) of the two approximated values of the first part.

For example, consider α and β be two consecutive

original values of the given digital signal and α and β be two approximated values as a result of Fig. (1). The new

value (say λ ) between α and β adopts the nature of the given input digital signal and thus it fills in the gap of signal. By this way between every two points one new value gets generated which makes the given signal double in length and dense. And that is why the new value can’t get good approximations in the beginning and ending phase of the output signal of (5). The new value captures a concrete functional behavior in the case of smooth functions (as shown in section III) where as it generates noise also particularly when the function is not smooth [1].

Thus the functions which are smooth in nature can easily be approximated using (5). The functions which are not smooth in nature can’t be treated with (5). Following subsection illustrates the method of handling such functions.

IV. ESTIMATE WAVELET TRANSFORM (EWT)

The previous approach was simply ignoring high pass components and hence could not yield the desired approximation. Various combinations of (4) have been carried out to get good approximation of a signal like considering individual series of Fig. 1, taking algebraic addition of series 1 and 2 (HHH and HHL components) of Fig. 1 etc. In all such cases the resultant double length signal approximations are found noisy. But the content of noise in the case of algebraic addition of HHH and LLH (series 1 and 3) is less as compared to others. In this section, the revised technique of approximation of a digital signal with HHH and LLH series say, Estimate Wavelet Transform is introduced. The expression 2jr + of (5) will take the following form using HHH and LLH components by taking algebraic addition of series 1 and series 3 of Fig. 1.

2jr μ+ = [ ( ( )* )*jU U x v v + ( ( )* )* ]jU U x v u (6)

A. Approximation of Linear lossy signal Consider a linear distorted (lossy) signal f (x) in L2(ℜ),

x∈[1, 20] such that, f (x) ,x x is odd=

Loss is artificially generated by ignoring even values of x hence the resultant lossy signal is as shown in Fig. 2(c).

Upon applying equation (6) with μ = 2 on this signal, what we get is positive and negative values alternatively. These values are symmetric to base line. The negative effect is due to aliasing effect in new values (intermediate values)

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which is opposite in sign as compared to other approximated values. So by multiplying each new values (alternative values) by -1, the double length approximated linear signal can be obtained as shown in Fig. 2(b). Fig. 2(a) depicts the approximation using (5) which shows noise around x = 4 (first four values) and approximate value of y = 14 is 13.36 while Fig. 2(b) violates at the last value only, the approximate values of y = 3 and 14 are 2.63 and 13.63 respectively.

B. Approximation of zigzag lossy signal Consider a zigzag lossy signal f (x), [1, 49]x ∈ ,

f (x) ,x x is odd=

,x x is even= − Here loss is artificially generated by ignoring (15, 15),

(30, -30) and (45, 45) hence the length of the signal is 46. Fig. 3(a) shows lossy original zigzag signal. Fig. 3(b) depicts approximated double length (92) signal using (6) with μ = 1 which yields (29, 15.58), (57, -30.07) and (85, 44.56). So here we get 15.58, -30.07 and 44.56 values instead of 15, -30 and 45. Fig. 3(c) generated by equation (5) simply shows deviations at the places of loss, and hence completely fail in the case of highly nonlinear signal.

C. Approximation of sinusoid lossy signal Consider a zigzag lossy signal f (x) = sin(x),

[ , ]x π π∈ − . Dividing the signal in 127 parts, 128 data points get generated. Loss is artificially generated by considering only even values of f (x) (length = 64) as shown in fig. 4(b). Fig. 4(a) and 4(c) depicts the almost identical approximated double length (128) signal using equations (5) and (6) respectively.

Figure 2. Lossy linear signal and its approximations

Figure 3. Lossy zigzag signal and its approximations

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Figure 4. Lossy sine signal and its approximations

V. ENERGY CONSERVATION AND RECONSTRUCTION

A. Energy Conseration

Energy of signal f = (f1, f2, …, fN) can be computed as εf = f1

2 + f22 + … fN

2 . The following table illustrates the conseration of energy of all the signals discussed in the above sections.

TABLE I. ENERGY CONSERVATION

Original lossy signal

Energy conservation of the signal

Energy conservation

by (5)

Energy conservation

by (6)

Straight line

1330 2560 2660

Zigzag 37275 39772 74550

Sine waves

31.75 63.4983 63.50

Conservation of Energy is playing a vital role in the

signal approximation. The techniques discussed in this article yield the double length approximated signal for the given digital signal as discussed in the previous section. Energy conservation should be almost double for the output signal obtained from (5) or (6) as compared to the original signal. Table I shows that the Energy conservation in the case of sine waves using (5) and (6) is 63.4983 and 63.50

respectively and original lossy sine wave conserves 31.75. The values can be verified by Fig 4. Both techniques give almost same energy. In the case of lossy straight line signal there is some energy loss (40 units) using (5). While (6) conserves exactly double energy (2600) as depicted in Table I. There is a big difference in energy conservation in the case of lossy zigzag signal. Original signal convers 37275 units while (5) and (6) conserve 39772 and 74550 respectively which indicates a big loss of energy from (5) as demonstrated inTable I.

B. Reconstruction Another important parameter in signal approximation is

reconstruction or inverse transformation. Up on applying one more analysis phase on (4), wj+2 is decomposed into two components (High and Low) as discussed in [1]. Among those low components yield the original signal. It is interesting to observe whether (5) or (6) transform into original lossy signal or not when applying one more analysis phase.

Signals shown in Fig 3(b) (EWT) and 3(c) are processed through an analysis phase and results show an identical lossy zigzag signal as in Fig 3(a) in the case of EWT while all the resultant elements of 3(c) are found to be almost zero.

Up on applying the singal of Fig. 2(a) (approximated signal of straight line) for one more analysis phase, all the resultant elements become zero. When the output signal of EWT of lossy straight line is passed through an analysis phase, the resultant low pass components are divided by

2=μ and found to be the same as the original lossy straight line (fig. 2(c)). While in the case of lossy sine signal, low pass components yield an identical lossy sine signal for Fig. 4(a) and 4(c).

VI. CONCLUSION This article demonstrates revised technique of

approximation presented in [1], of a digitized signal to its double length namely Estimate Wavelet Transform. Experiment shows that both the approaches are providing almost identical double length approximation in the case of smooth signal like sinusoid lossy signal as shown in example 3 (Fig. 4) of section IV. In the case of linear lossy signal, the double length approximation obtained by EWT (6) is much better than that of equation (5) as shown in example 2. Highly nonlinear signal like zigzag lossy signal where earlier approach (5) completely fails as shown in Fig. 3(c) can be approximated with significant accuracy in the case of EWT.

As discussed in Table I, the energy conservation is exactly double than that of original signal with the help of EWT. But the same does not happen in all the three cases with the technique discussed in [1]. At same time all the three EWT signals can be constructed but the same is not possible in the case of (5).

The choice of μ plays a major role in the amplitude of the signal for example in Fig. 3(b) of section IV the amplitude of the signal is ranging from -50 to 50 which is suitable to original zigzag signal (Fig. 3(a)) with μ = 1.

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Upon increasing the value of μ from 1 to 2 , the amplitude of the resultant signal exceeds the desired range [-50, 50]. Thus by choosing appropriate value of the parameter μ , the approximations can be made better. This approach has a great potential in the case of lossy two dimensional signals.

REFERENCES [1] Archit Yajnik, S. Rama Mohan, “Approximation of a Signal Using

Discrete Wavelets Transform”, Vol. 2: article S09, International Journal of Artificial Intelligenc, pp. 95-105, 2009.

[2] S. Rama Mohan, Archit Yajnik: “Gujarati Numeral Recognition Using Wavelets and Neural Network” in 2005 Proc. of Indian International Conference on Artificial Intelligence, pp. 397-406.

[3] Archit Yajnik, S. Rama Mohan, “Identification of Gujarati characters using wavelets and neural networks” Artificial Intelligence and Soft Computing 2006, ACTA Press, pp. 150-155

[4] R. Srivastava Rao Kunte, R. D. Sudhakar Samuel, “Online Character /Script Recognition System For Kannada and other Indian Languages Employing Wavelet Features,” in 2002 Proc. Of The Indo-European Conference on Multilingual Communication Technologies, Tata MacGraw – Hill Publishing Company Limited, pp. 26 – 31.

[5] Nhat Nguyen and Peyman Milanfar, “A Wavelet-based Interpolation-Restoration Method for Supperresolution (Wavelet Superresolution),” vol. 19(4), Circuits Systems Signal Process , vol. 19(4), pp. 321-338, 2000.

[6] Alex P. Pentland, “Interpolation Using Wavelet Basis,” IEEE transactions on Pattern Analysis and Machine Intelligence, vol. 16(4), pp. 410-414, Apr. 1994.

[7] Michael W. Frazier, An Introduction to Wavelets Through Linear Algebra, Springer-Verlag, New York, 2000.

[8] Stephane Mallat, A Wavelet tour of signal processing second edition, Academic Press Prentice, California, 2006

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