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  1. 1. C. Madhu, P. Suresh Babu, S. Jayachandranath / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.1352-1357 Denoising of Spectral Data Using Complex Wavelets * ** C. Madhu, P. Suresh Babu, ***S. Jayachandranath * Assistant Professor, SV College of Engineering, Tirupati, **Assistant Professor, SV College of Engineering, Tirupati, ***Assistant Professor, SV College of Engineering, Tirupati.Abstract Atmospheric signal processing is of peaks of the same range gate and tries to extract theinterest to many scientists, where there is scope best peak, which satisfies the criteria chosen for thefor the development of new and efficient tools for adaptive method of estimation. The method,cleaning the spectrum, detection, and estimation however, has failed to give consistent results. Hence,of parameters like zonal (U), meridional (V ), wind there is a need for development of better algorithmsspeed (W), etc. This paper deals with a signal for efficient cleaning of spectrum.processing technique for the cleansing ofspectrum, based on the complex wavelets withcustom thresholding, by analyzing theMesosphere–Stratosphere–Troposphere radardata that are backscattered from the atmosphereat high altitudes and severe weather conditionswith low signal-to-noise ratio. The proposedalgorithm is self-consistent in detecting windspeeds up to a height of 18 km, in contrast to theexisting method which estimates the Dopplermanually and fails at higher altitudes. The resultshave been validated using the Global PositioningSystem sonde data.Keywords:- Signal Processing, ComplexWavelets, MST RADARS, Denoising.I. INTRODUCTION RADAR can be employed, in addition to thedetection and characterization of hard targets, toprobe the soft or distributed targets such as theEarth‟s atmosphere. Atmospheric radars, of interestto the current study, are known as clear air radars,and they operate typically in very high frequency(30–300 MHz) and ultrahigh-frequency (300 MHz–3GHz) bands. The turbulent fluctuations in therefractive index of the atmosphere serve as a targetfor these radars. The present algorithm used in The National Atmospheric Researchatmospheric signal processing called “classical” Laboratory, Andhra Pradesh, India, has developed aprocessing can accurately estimate the Doppler package for processing the atmospheric data. Theyfrequencies of the backscattered signals up to a refer to it as the atmospheric data processor [4]. Incertain height. However, the technique fails at higher this paper, it is named as existing algorithm (EALG).altitudes and even at lower altitudes when the data The flowchart of this algorithm is given in Fig. 1.are corrupted with noise due to interference, clutter, Coherent integration of the raw data (I and Q)etc. Bispectral-based estimation algorithm has been collected by radar is performed. It improves thetried to eliminate noise [1]. However, this algorithm process gain by a factor of number of inter-pulsehas the drawback of high computational cost. period. The presence of a quadrature component ofMultitaper spectral estimation algorithm has been the signal improves the signal-to-noise ratio (SNR).applied for radar data [2]. This method has the The normalization process will reduce the chance ofadvantage of reduced variance at the expense of data overflow due to any other succeeding operation.broadened spectral peak. The fast Fourier transform The data are windowed to reduce the leakage and(FFT) technique for power spectral estimation and picket fence effects. The spectrum of the signal is“adaptive estimates technique” for estimating the computed using FFT. The incoherent integrationmoments of radar data has been proposed in [3]. This improves the detectability of the Doppler spectrum.method considers a certain number of prominent 1352 | P a g e
  2. 2. C. Madhu, P. Suresh Babu, S. Jayachandranath / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.1352-1357The radar echoes may be corrupted by ground clutter, needed to accurately describe the energy localizationsystem bias, interference, etc. of oscillating functions (wavelet basis). The Fourier transform is based on complex-valued oscillating sinusoids e jt  cos (t )  j sin(t ) The corresponding complex-valued scaling function and complex-valued wavelet is given as  c (t )   r (t )  j i (t ) where  r (t ) is real and even, j i (t ) is imaginary and odd. Gabor introduced the Hilbert transform into signal theory in [9], by defining a complex extension of a real signal f (t ) as: x(t )  f (t )  j g (t ) where, g (t ) is the Hilbert transform of f (t ) and denoted as H { f (t )} and j  (1)1/2 . ο The signal g (t ) is the 90 shifted version of f (t ) as shown in figure (3.1 a).The real part f (t ) and imaginary part g (t ) of the analytic signal x (t ) are also termed as the „Hardy Space‟ projections of original real signal f (t ) in Hilbert space. Signal g (t ) is orthogonal to f (t ) . In the time domain, g (t ) can be represented as [7]  1 f (t ) 1 The data is to be cleaned from theseproblems before going for analysis. After performing g (t )  H { f (t )}    t  d  f (t )*  t power spectrum cleaning, one has to manually select If F ( ) is the Fourier transform of signala proper window size depending upon the wind shear[6], etc., from which the Doppler profile is estimated f (t ) and G ( ) is the Fourier transform of signalby using a maximum peak detection method [3]. The g (t ) , then the Hilbert transform relation betweenEXISTING METHOD is able to detect the Doppler f (t ) and g (t ) in the frequency domain is given byclearly up to 11 km as the noise level is very low.Above 11 km, noise is dominating, and, hence, the G( )  F{H{ f (t )}}   j sgn() F ()accuracy of the Doppler estimated using theEXISTING METHOD is doubtful as is discussed in where,  j sgn( ) is a modified „signum‟ function.the subsequent sections. To overcome the effect of This analytic extension provides thenoise at high altitudes, a wavelet-based denoising estimate of instantaneous frequency and amplitude ofalgorithm is applied to the radar data before the given signal x (t ) as:computing its spectrum [5]. This paper gives better Magnitude of x(t )  ( f (t )  g (t ) 2 2results compared to [4], but this method fails toextract the exact frequency components after 1denoising at higher altitudes. To overcome this effect Angle of x(t )  tan [ g (t ) / f (t )]we proposed a new method, where spectrum is The other unique benefit of this quadratureestimated prior to denoising and then denoised using representation is the non-negative spectralComplex Wavelet Transform (CWT) with the help of representation in Fourier domain [7] and [8], whichCustom thresholding method. leads toward half the bandwidth utilization. The reduced bandwidth consumption is helpful to avoidIII. Complex Wavelet Transform (CWT) aliasing of filter bands especially in multirate signal Complex wavelet transforms (CWT) uses processing applications. The reduced aliasing of filtercomplex-valued filtering (analytic filter) that bands is the key for shift-invariant property of CWT.decomposes the real/complex signals into real and In one dimension, the so-called dual-tree compleximaginary parts in transform domain. The real and wavelet transform provides a representation of aimaginary coefficients are used to compute amplitude signal x(n) in terms of complex wavelets, composedand phase information, just the type of information of real and imaginary parts which are in turn wavelets 1353 | P a g e
  3. 3. C. Madhu, P. Suresh Babu, S. Jayachandranath / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.1352-1357themselves. Figure 3 shows the Analysis and customized thresholding function is introduced. TheSynthesis of Dual tree complex wavelet transform for Custom thresholding function is continuous aroundthree levels. the threshold, and which can be adapted to the characteristics of the input signal. Based on extensiveIII. CUSTOM THRESHOLDING experiments, we could see that soft-thresholding To overcome the pulse broadening effect inimproved thresholding function a new function called 1354 | P a g e
  4. 4. C. Madhu, P. Suresh Babu, S. Jayachandranath / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.1352-1357outperforms hard-thresholding in general. However, 8) Then variance is calculated usingthere were also cases where hard-thresholding 2 1 N 1  N 1 yielded a much superior result, and in those cases thequality of the estimate could be improved by using a   Y (k )   Y (k )  N k 0  k 0 custom thresholding function which is similar to thehard-thresholding function but with a smooth V. RESULTStransition around the threshold  . Based on these In this section, we present the results for theobservations, we defined a new custom thresholding cleansing of spectrum for complex test data based onfunction as follows: the proposed Method. The data are generated by applying a Gaussian random input to a complex  system. As the true spectrum is not available variance  is taken as the performance measure, instead of  x  sgn( x )( 1   ) if x    Mean-square error.f c ( x )  0 if x   The variance is calculated as    x     2  x   2   1 N 1  N 1       3    4    otherwise  Y (k )   Y (k )               N k 0  k 0  where 0     and 0    1 . This idea A complex signal is generated and Fourieris similar to that of the semisoft or firm shrinkage transform is calculated for that signal. To test theproposed by Gao and Bruce [10], and the non- performance of the algorithm in the presence ofnegative garrote thresholding function suggested by additive white Gaussian noise, a new signal y1(n) isGao [10], in the sense that they are continuous at  generated asand can adapted to the signal characteristics. In our y1 (n)  y (n)  (n) for n  0,1, 2,......N  1;definition of f c ( x ) ,  s the cut-off value, below where (n) is a complex Gaussian noise with zerowhich the wavelet coefficients are set zero, and  is mean and unit variance and α is the amplitudethe parameter that decides the shape of the associated with (n) . The number of samples inthresholding function f c ( x ) . This function can be each realization is assumed to be 512, i.e., N = 512. The result based on the Monte Carlo simulation usingviewed as the linear combination of the hard- proposed method is shown in fig.2. The same usingthresholding function and the soft-thresholding existing method is shown in the fig.3. Table 1 givesfunction  . f h ( x)  (1  ). f s ( x) that is made the improved SNR and Variance of denoised signalscontinuous around the threshold  . by processing time domain data and frequency Note that, domain data using complex wavelets. From the table 6.1 it is clear that the denoised data obtained bylim fc ( x)  f s ( x) and lim  f h ( x) processing0 1, This shows that the custom thresholding function canbe adapted to both the soft- and hard-thresholding 1200 (a) 1200 (b)functions. 1000 1000IV. PROPOSED METHOD 800 800The algorithm for cleansing the spectrum using CWT 600 600with custom thresholding is as follows. 400 4001) Let f(n) be the noisy spectral data, for n =0, 200 200 1,...,N-1. Amplitude 0 02) Generate noisy signal y(n) using 1200 0 100200300400500600 0 100 200 300 400 500 600 x(n)  f (n)   z(n)n  1, 2,...., N 1000 (c) (d) 23) Spectrum is calculated for the above signal y(n) 800   y[n] e 600 j  j n as Y (e )  . 400 n  200 04) Input x(n) to the two DWT trees with one tree 0 uses the filters h0, h1 and the other tree with 0 100200300400500600 0 100 200 300 400 500 600 filters g0, g1. Frequency5) Apply custom thresholding to wavelet coefficients in the two trees. Fig. 2 (a) original Spectrum (b) Noisy Spectrum (c)6) Compute IDWT using these thresholded wavelet Denoised Spectrum (d) variance coefficients.7) The coefficients from the two trees are then averaged to obtain the denoised original signal. 1355 | P a g e
  5. 5. C. Madhu, P. Suresh Babu, S. Jayachandranath / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.1352-1357 Now I processed same test signal and MST radar signal using complex wavelet transform with 4 (a) improved and custom thresholding techniques. 2 0 -2 -4 Spectral data -6 Time domain data 0 100 200 300 400 500 600 4500 12 10 (b) 32 8 6 4 4000 2 0 30 -2 -4 -6 -8 28 Improved SNR -10 -12 3500 -14 Variance -16 Amplitude 0 100 200 300 400 500 600 26 6 3000 4 (c) 24 2 0 2500 -2 22 -4 -6 20 2000 -8 0 100 200 300 400 500 600 6 18 1500 (d) 4 16 1000 2 14 0 500 12 0 100 200 300 400 500 600 10 0 Time 8 -500 -10 -5 0 5 10 -12 -8 -4 24 6 810 -10 -6 -20 12Fig. 3 (a) original Signal (b) Noisy Signal (c) Input SNRDenoised Signal (d) variancefrequency data using CWT has got betterperformance than processing the time-domain data. Fig. 4 (a) Input SNR vs. Output SNR (b) Input SNRThe same is represented graphically in figure 4. vs. Variance Among these two methods with differentTable 1 Performance of Proposed Method With thresholding techniques, the processing of spectralrespect to input SNR. data using complex wavelet transform with customDomain Input Output thresholding method is giving better results thanType SNR SNR Variance processing the time domain signal. The proposed -10 12.9608 2.8997e+003 method may be used effectively in detection, image -5 21.5316 800.6487 compression and image denoising and also it isFrequency giving better results at low signal-to-noise ratio cases 0 24.3023 231.3500Domain (even at -15 dB) 5 26.6103 78.7414 10 30.9121 25.2987 -10 10.3287 3.9843e+003 References -5 17.6794 1.7855e+003 [1] V. K. Anandan, G. Ramachandra Reddy,Time and P.B. Rao, “Spectral analysis of 0 22.1051 498.1208Domain atmospheric signal using higher orders 5 24.9066 101.4611 10 28.9177 57.5245 spectral estimation technique,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 9, pp. 1890–1895, Sep. 2001.VI. CONCLUSION [2] V. K. Anandan, C. J. Pan, T. Rajalakshmi, The wavelet transform allows processing of and G. Ramchandra Reddy, “Multitapernon-stationary signals such as MST radar signal. This spectral analysis of atmospheric radaris possible by using the multi resolution decomposing signal,” Ann. Geophys., vol. 22, no. 11, pp.into sub signals. This assists greatly to remove the 3995–4003, Nov. 2004.noise in the certain pass band of frequency. At first I [3] V. K. Anandan, P. Balamuralidhar, P. B.have processed test signal (time and frequencydomains) with -10 dB SNR using complex wavelet Rao, and A. R. Jain, “A method for adaptivetransform (CWT) by designing sub band filters using moments estimation technique applied to MST radar echoes,” in Proc. Prog.Hilbert transform maintaining perfect-reconstruction Electromagn. Res. Symp., 1996, pp. 360–property, with the help of custom thresholding. The 365.main idea in implementing this technique is to [4] V. K. Anandan, Atmospheric Dataovercome the limitations like shift sensitivity, poor Processor—Technical and User Referencedirectionality, and absence of phase information Manual. Tirupati, India: NMRF, DOSwhich occurs in DWT. Using this technique we can Publication, 2001.extract the signal even-though input SNR is -15 dB. 1356 | P a g e
  6. 6. C. Madhu, P. Suresh Babu, S. Jayachandranath / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.1352-1357[5] T.Sreenivasulu Reddy, Dr G.Ramachandra Reddy, and Dr S.Varadarajan, “MST Radar signal processing using Wavelet based denoising”, IEEE Geoscience and Remote Sensing Letters, Volume: 6, No.4, pp 752- 756, October 2009.[6] M. Venkat Ratnam, A. Narendra Babu, V. V. M. Jagannadha Rao, S. Vijaya Bhaskar Rao, and D. Narayana Rao, “MST radar and radiosonde observations of inertia- gravity wave climatology over tropical stations: Source mechanisms,” J. Geophys. Res., vol. 113, no. D7, p. D07 109, Apr. 2008. DOI: 10.1029/2007jd008986.[7] S. Hahn, „Hilbert Transforms in Signal Processing‟, Artech House, Boston, MA, 1996.[8] Adolf Cusmariu, „Fractional Analytic Signals‟, Signal Processing, Elsevier, 82, 267 – 272, 2002.[9] D Gabor, „Theory of Communication‟, Journal of the IEE, 93, 429-457, 1946.[10] A Brice, D L Donoho, and H Y GAO, „Wavelets Analysis‟, IEEE Spectrum, 33(10), 26-35, 1996. 1357 | P a g e