COMPARING NOISE REMOVAL IN THE WAVELET AND FOURIER DOMAINS

Transcription

1 COMPARING NOISE REMOVAL IN THE WAVELET AND FOURIER DOMAINS Robert J. Barsant, and Jordon Glmore Department of Electrcal and Computer Engneerng The Ctadel Charleston, SC, e-mal: Key Words: Wavelet Analyss, Flterng, Dscrete Fourer Transform Abstract-- Ths paper compares tme seres decomposton n the frequency doman va the dscrete Fourer transform to tme seres decomposton n the wavelet doman va the Wavelet transform for the purpose of sgnal smoothng and nose removal. The nformaton cost of the sgnal s computed as a predctor of the performance of the flterng process. Smulatons are conducted comparng the frequency doman flter to wavelet doman flters on a varety of sgnals corrupted wth addtve Gaussan nose. I. INTRODUCTION The flterng of tme seres data for the purpose of removng unwanted sgnal components s of nterest to a wde varety of engneerng and scence dscplnes. The process of mappng a tme seres nto another doman va a lnear transform s used to separate the sgnal characterstcs from that of the nose. The wavelet thresholdng technques used n ths study are a combnaton of the methods found throughout the sgnal processng lterature. We apply a varaton of the wavelet based de-nosng technques of [1] and [2] to sgnals n the presence of addtve whte Gaussan nose. Addtonally, we extend these methods to the Fourer frequency doman usng the dscrete Fourer transform (DFT). The concept of nformaton cost and sgnal entropy can be used to predct the nose removal performance of a decomposng bass. Wckerhauser proposed the use of nformaton cost functons for the selecton of a wavelet packet bass[3]. In partcular, we use a smlar method to predct the flterng performance of the Fourer or wavelet bass on dfferent sgnals. The remander of ths paper s dvded nto the followng sectons; Secton II descrbes the dscrete wavelet transform (DWT), the DFT, and the bascs of transform doman flterng. Secton III ntroduces sgnal entropy and nformaton cost functons. Secton IV explans the smulatons and presents results, and secton V contans a summary. II. TRANSFORM DOMAIN FILTERING A. The Dscrete Wavelet Transform The dscrete wavelet transform (DWT) of a sequence x(n) s gven by [3], 1 n - b W(J,b)= x(n) * Ψ ( ), (1) J n J 2 2 where Ψ represents a wavelet functon, whch s dlated and contracted by the nteger scale factor J, and delayed n tme by parameter b. For an N pont sequence the scale factor J assumes the values J = 0,1, log 2 (N), producng a mult-resoluton decomposton of the nput nto octave bands. The delay values b are related to the scale by b = K 2 J for K an nteger. Thus the DWT output s decmated by a factor of two at each successve octave J. The DWT requres an nput sequence length that s an even power of two,.e., N = 2 p, and produces an equal number of wavelet coeffcents. The DWT s a lnear but tme varant transformaton. [3,4,5,6] The nner product of Eq. 1 produces a DWT output W whch s a set of N coeffcents that represent the data n the wavelet doman. Ths set contans the nformaton necessary to reconstruct the orgnal sgnal from the correspondng wavelet functon va the nverse wavelet transform (IDWT). The magntude of the coeffcents represent the correspondence between the nput sgnal and the decomposng wavelet functon at each partcular delay b, and scale J. For smplcty and ease of dsplay, the dscrete wavelet coeffcents can be represented as a vector (W) by summng over the scales. [4,5] W = [ w0, w2, w3,... wn 1 ]. (2) Ths formulaton allow for plottng of the DWT output as shown n the bottom plot of Fg. 1.

2 Fg 1: Sgnal and DFT and DWT. B. The Dscrete Fourer Transform The well known dscrete Fourer transform (DFT) of a sequence x(n) s gven by [7], j2πkn X ( k) = x(n) exp( ), (3) n N where the sequence [X(0), X(2),, X(N-1)] s the transform sequence. The frequency ndex k = 0,, N-1 s related to the analog frequency f and the samplng frequency fs by k =N f/fs. A sample sgnal along wth the magntude of ts DFT and DWT coeffcents are dsplayed n Fg 1. C. Nose Flterng Fg. 2 shows the sgnal of Fg. 1 wth added Gaussan nose, along wth ts DFT and DWT. Observaton of Fg. 1 and Fg. 2, shows that the nose s dstrbuted as small coeffcents throughout both transform domans. The separaton of sgnal and nose nto large and small coeffcents permts the applcaton of a nose threshold to remove the smaller coeffcents of the decomposton, those presumably assocated wth the nose. The general method for calculatng a threshold s based on the statstcal propertes of the transform coeffcents [1]. Estmaton of the nose standard devaton can be performed n a number of ways. An often used approach s to compute the absolute medan devaton of the coeffcents[8]. Once the nose level of the transformed data s establshed a threshold value can be set. A popular choce for the threshold level s the unversal threshold, whch s defned as [1]: T = σ 2 log( N), (4) and thus s a multple of σ (the nose standard devaton) Fg 2: Sgnal wth nose and DFT and DWT based on the number of coeffcents N [1]. Ths threshold s optmum for very large N. In our smulatons N = 1024, and T was too large. Usng a tral and error approach, a threshold of 2σ worked well. After threshold selecton varous methods of applcaton are dscussed n the lterature. Two of the most popular are hard thresholdng and soft thresholdng [1,2]. Hard thresholdng sets all coeffcents below the threshold value to zero and retans the remanng coeffcents unchanged. Soft thresholdng sets all coeffcents below the threshold to zero and also reduces the magntude of remanng coeffcents by the threshold value. For ths paper hard thresholdng was appled n all cases [4,5]. The transform nose removal technque can be descrbed by three steps as dsplayed n the block dagram of Fg 3. (1) transform the nosy sgnal x(n) nto the transform doman va the DFT or DWT, (2) threshold the transform coeffcents (to remove nose), and (3) perform the nverse transform on modfed coeffcents to produce the fltered sgnal y(n) [3,4,5,6]. x(n) DWT or DFT Threshold Denose Fg 3: Block dagram of the three step Transform doman Flterng. III. BASIS SELECTION IDWT or IDFT A. The Sgnal Entropy and Informaton Cost y(n) The method of nose removal employed n ths study depends on the ablty to transform the data nto a relatvely few large coeffcents representng the sgnal

3 and smaller coeffcents representng the nose. For ths applcaton the best bass s the one that most compactly represents the data by concentratng the sgnal nformaton nto the fewest sgnfcant coeffcents. An nformaton cost functon Q, can be used to quantfy ths dea. The functon should be addtve, so that the nformaton cost of the sequence s the sum of the cost of ts elements, e.g., Q(x) for a sequence x = { x } should be constructed such that Q(0) = 0, and Q(x) = 3 Q(x ) [6, 10]. A popular cost functon s the Shannon Entropy whch s defned as [10]: 2 2 Q( x) = x log(1/ x ). (5) In sgnal processng the nformaton ganed from observng a sngle element x of a sgnal x = {x } can be found from the expresson [9]: I ( x ) log(1/ p ), I = 0 for p = 0, = where p = x 2 / x 2 s the normalzed energy of the th element of the sgnal. The quantty p s a probablty dstrbuton functon n that, 0# p # 1 and 3 p = 1. The entropy of the sgnal x s then defned as the expected value of I(x) over the length of the sgnal and s gven by [9]: H ( x) = E{ I( x )} = p I( x ) = p log(1/ p ). (7) H(x) s the entropy of the sgnal, and s a measure of the average nformaton content per symbol of the sequence x [9]. H(x) provdes a measure of the concentraton of the energy n the sgnal. For example, f two sgnals contan equal energy but dfferent entropy, the sgnal wth the lower entropy has ts energy concentrated n fewer elements[9]. It s ths attrbute of H(x) that makes t useful for comparng flterng domans. Note that H(x) does not possess the addtve property snce H(x) 3 H(x ), however Q(x) of Eq. 5 s addtve, and t s related to H(x) by H(x) = x 2 Q(x) + log( x 2 ). So mnmzng the Shannon entropy Q(x), also mnmzes the entropy H(x) [6]. Note also that H(x) s ndependent of rearrangement of the sequence elements, whch justfes the smpler vector formulaton of W gven by Eq. 2. B. The Best Bass for Flterng The entropy H(x) s bounded such that; 0 H ( x) log( N) (8) where N s number of elements of the sequence x. Note that, H(x) = 0 only f the probablty p = 1 for one, and all remanng probabltes are zero. In other words, f all the sgnal energy s concentrated n one coeffcent. Also, H(x) = log(n), only f p = 1/N for all. Thus the upper (6) bound s acheved when the sgnal energy s evenly dstrbuted among the coeffcents [9]. Thus the best bass for flterng s determned by comparson of the sgnal entropy n there correspondng doman. The decomposton wth the smaller entropy corresponds to the better bass for threshold flterng. IV. SIMULATIONS AND RESULTS The DFT and DWT thresholdng technques were compared on a varety of sgnals usng computer smulatons and Matlab software. The performance comparson was conducted va Monte Carlo smulatons. The measure used to compare the relatve performance of the flters s the mean squared error (MSE) defned as MSE 1 = ( ( ) ( )) 2 x n y n. (9) N n The symbol x(n) represents the nose corrupted sgnal and y(n) represents the fltered sgnal, as shown n Fg. 3. Three smulated sgnal waveforms were generated usng 2^10 ponts. In each smulaton tral one of the sgnals was subjected to added whte Gaussan nose (AWGN) to produce the smulated nose corrupted sgnal. Many trals usng dfferent nstances of AWGN were conducted at sgnal to nose ratos rangng from -5 db to 10 db. A suffcent number of trals were conducted to produce a representatve MSE curve. Smulatons for the all the flters used the same nose scale. The transform coeffcents of the nosy sgnal were computed and the denosng technque of secton II was used. The wavelet flterng was performed usng the Symmlet 4 wavelet. Both flters used the hard thresholdng method wth a threshold value of 2σ. Fg. 4, 6, and 8 dsplay the orgnal sgnal, the nose corrupted sgnal at 10 db SNR, and the flterng process usng the doman (DFT or DWT) that performed the best. After flterng, the nverse transform was appled, and the MSE of Eq. 9 was computed. The average MSE for each SNR was then plotted n Fg. 5, 7, and 9. The plots nclude flterng wth a seven pont medan flter (labeled Flter on the plots), for comparson purposes. The entropy of each sgnal was computed per Eq. 7 n each of the three domans, and s tabulated n table 1. Comparson of table 1 wth Fg. 5, 7, and 9, reveals that n each case the doman wth the lowest entropy also produced the best flterng. Therefore, the sgnal entropy can be used to predct whch doman flter wll perform the best. For sgnal 1, a pure snusod, the DFT flter performs best as expected snce t best concentrates the sgnal 1 energy n ts doman. For sgnal 2 and 3, the DWT provdes a small entropy advantage and thus a slghtly better flterng MSE.

4 Fg 4. Flterng sgnal 1 at 10 db usng the DFT Fg 5. MSE vs. SNR for sgnal 1. Fg 6. Flterng sgnal 2 at 10 db usng the DWT. Fg 7. MSE vs. SNR for sgnal 2. Fg 8. Flterng sgnal 3 at 10 db usng the DWT. Fg 9. MSE vs. SNR for sgnal 3.

5 Doman Sgnal 1 Sgnal 2 Sgnal 3 Tme Fourer Wavelet Table 1. Computed values of the Informaton cost H(x) for sgnals 1, 2, and 3 n each doman usng Eq.7. V. SUMMARY Ths paper compared the performance of the DFT and DWT threshold nose removal for sgnals n the presence of AWGN. The detals of method were developed, and smulatons of the performance of the proposed flters were presented. The paper also dscussed sgnal entropy n the Fourer and wavelet doman and related t to the flter performance. Future work wll nclude the performance on more realst sgnals e.g., audo, or communcatons sgnals, and nclude study of the flter performance n other types of nose corrupton. VI. ACKNOWLEDGEMENT The authors would lke to acknowledge The Ctadel Foundaton for the grant that supported ths research. VII. REFERENCES [1] D. Donoho and I. Johnston, Adaptng to Unknown Smoothness va Wavelet Shrnkage, Journal of the Amercan Statstcal Assocaton, Dec 1995, Vol. 90, No. 432, Theory and Methods [2] A. Bruce, H. Gao, WaveShrnk: Shrnkage Functons and Thresholds, Techncal Report, StatSc Dvson, MathSoft Inc., 1995 [3] V. Wckerhauser, Adapted Wavelet Analyss from Theory to Software, A.K. Peters, Ltd., Masschusetts, 1994 [4] R. Barsant, E. Spencer, J. Cares, L. Parobek, Feature Matchng and Sgnal Recognton Usng Wavelet Analyss, Proceedngs of 38 th Southeastern Symposum on System Theory, Cookvlle, TN, 2006 [5] R. Barsant, T. Smth, R. Lee, Performance of a Wavelet-Based Recever for BPSK and QPSK Sgnals n Addtve Whte Gaussan Nose Channels, Proceedngs of 39 th Southeastern Symposum on System Theory, Macon, GA, 2007 [6] R. Barsant, Denosng of Ocean Acoustc Sgnals Usng Wavelet Based Technques, MSEE Thes, NPS, 1996 [7] J. Proaks, Dgtal Communcatons, McGraw-Hll, Inc., New York, 1995 [8] C. Sten, Estmaton of the Mean of a Multvarate Normal Dstrbuton, The Annals of Statstcs, Vol 9, pp , 1981 [9] S. Haykn, Communcaton Systems, John Wleys & Sons, Inc., New York, 1994 [10] The Mathworks Inc., Wavelet Toolbox User Gude, Massachussetts, 1996.