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  • k.S.N.Murthy, m.Suman, v.Sriram, n.Sai Kotewararao, b.Phani Krishna Rao / International Journal of Engineering Research and Applications (IJERA)ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, June-July 2012, pp.631-635 Speech Enhancement Using A Recursive Filter 1 k.S.N.Murthy, 2m.Suman, 3v.Sriram, 4n.Sai Kotewararao,5b.Phani Krishna Rao 1,2 Professors,Department of ECE, K L C E, Guntur DT, AP, India 3,4,5 Students, Department of ECE, K L C E, Guntur DT, AP, IndiaAbstract– one of the major components in speech 2. RECURSIVE FILTERenhancement is noise estimation. In earlier This recursive Filter is an estimator for what ismethods residual noise will be present in the called the “linear quadratic problem”, which focusesenhanced speech signal because of inaccurate on estimating the instantaneous “state” of anoise estimation and not suitable in non-stationary lineardynamic system perturbed by white noise.noise environments. Statistically, this estimator is optimal withrespect toIn this paper noise is estimated using a recursive any quadratic function of estimation errors.filter. In this estimation estimator is recursivelyupdated in each frame so that non-stationary 3. PROCESSnoise is tracked and estimated. In performance After going through some of thecomparison proposed approach we present the of introduction and advantages of using this filter, wesegmental SNR values for different environments. will now take a look at the process. The processThese results shows that proposed approach will commences with the addresses of a general problemproduce enhanced speech with less residual noise. of trying to estimate the state of a discrete-time controlled process that is governed by a linearKeywords-recursive filter, time update stochastic difference equation:measurement update, error co-variance 𝑥 𝑘 =A𝑥 𝑘 −1 +B𝑢 𝑘 +𝑤 𝑘 −1 1. INTRODUCTION ………………… .…………….(1) Speech is a form of communication ineveryday life. It existed since human civilizations with a measurement that isbegan and even till now, speechis applied to high technological telecommunication 𝑧 𝑘 =H𝑥 𝑘 +𝑣 𝑘 ……………………(2)systems. As applications like cellular and satellite The random variables and represent the process andtechnology are getting popular among mankind, measurement noise (respectively). We assume thathuman beings tend to demand more advance they are independent of each other, white, and withtechnology and are in search of improved normal probability distributionsapplications. For this reason, researchers are lookingclosely into the four generic attributes of speech P(w)-N(0,R)…………(3)coding. They are complexity, quality, bit rate anddelay. Other issues like robustness to transmission P(V)-N(0,R)……………(4)errors, multistageencoding/decoding andaccommodation of non-voice signals such as data Ideally, the process noise covariance Q andplay an important role in coding of speech as well. measurement noise covariance R matrices are In order to understand these processes, both assumed to be constant, however in practice, theyhuman and machine speech might change with each time step or measurement.has to be studied carefully on the structures and In the absence of either a driving functionfunctions of spoken language: how we produce and or process noise, the n×n matrix A in the differenceperceive it and how speech technology may assist us equation (1) relates the state at the previous time stepin communication. Therefore in this project, we will k-1 to the state at the current step k . In practice, Abe looking more into speech processing with the aid might change with each time step, however here it isof a recursive Filter. assumed constant. The n×l matrix B relates the optional control input to the state x. H which is a matrix in the measurement equation (2) 631 | P a g e
  • k.S.N.Murthy, m.Suman, v.Sriram, n.Sai Kotewararao, b.Phani Krishna Rao / International Journal of Engineering Research and Applications (IJERA)ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, June-July 2012, pp.631-635which relates the state to the measurement, zk. Inpractice H might change with each time step or Once again, notice how the time updatemeasurement, however we assume it is constant. equations in Table 4.1 project its state, x and covariance, 𝑃 𝑘 estimates forward from time step k-1 4. ALGORITHM to step k. As mentionedearlier, the matrixes A and B This section will begin with a broad are from (1), while is from (3). Initial conditionsoverview, covering the "high-level" operation of one forthe filter are discussed in the earlier section.form of this fillter. After presenting this high-levelview, I will narrow the focus to the specific equations 𝐾 𝑘 = 𝑃 𝑘 𝐻 𝑇 𝐻𝑃 𝑘 𝐻 𝑇 + 𝑅 −1 ………..(7)and their use in this discrete version of the filter.Firstly, it estimates a process by using a form of 𝑥 𝑘 = 𝑥 𝑘 + 𝐾 𝑘 (𝑧 𝑘 − 𝐻𝑥 𝑘 )……….(8)feedback control loop whereby the filter estimates theprocess state at some time and then obtains feedback 𝑃 𝑘 = (𝐼 − 𝐾 𝑘 𝐻)𝑃 𝑘 ………………………(9)in the form of (noisy) measurements. As such, theseequations for this filter fall into two groups: “Time Table 4.2: Measurement update equationsUpdate equations” and “Measurement Updateequations” By referring to above data, it is obvious that The responsibilities of the time update the first task during the measurement update is toequations are for projecting forward (in time) the compute the gain, kk.. By comparing (7) in the tablecurrent state and error covariance estimates to obtain above and the previous section, notice the equationsthe priori estimates for the next time step. The are the same. Next, is to actually measure the processmeasurement update equations are responsible for the in order to obtain zk , and then to generate a posteriorifeedback i.e. for incorporating a new measurement state estimate𝑥 𝑘 by incorporating the measurement asinto the priori estimate to obtain an improved in (8). Once again, notice the repeated equation of (8)posteriori estimate. here for completeness. Finally, the last step is to The time update equations can also be obtain a posteriori error covariance estimate via (9).thought of as “predictor” equations, while the Thus, after each time and measurementmeasurement update equations can be thought of as update pair, this loop process is repeated to project or“corrector” equations. By and large, this loop predict the new time step priori estimates using theprocess of the final estimation algorithm resembles previous time step posteriori estimates. This recursivethat of a predictor-corrector algorithm for solving nature is one of the very appealing features of thisnumerical problems just like the one shown in Fig filter → it makes practical implementations muchbelow. more feasible than (for example) an implementation of a Wiener filterwhich is designed to operate on all of the data directly for each estimate. Instead, this filter recursively conditions the current estimate on all of the past measurements. The high-level diagram of Fig 4.1is combined with the equations from Table 4.1and Table 4.2, in Fig 4.2as shown below, which offers a much more complete and clear picture of the operation of the recursive filter. As the time update projects the current stateestimate ahead in time, the measurement update Time update (“predict”)adjusts the projected estimate from the time updateby an actual measurement at that particular time. Thespecific equations for the “time” and“measurement” updates are presented below inTable 4.1 and Table 4.2𝑥 𝑘 = 𝐴𝑥 𝑘 −1 + 𝐵𝑈 𝑘 …………….(5)𝑃 𝑘 = 𝐴𝑃 𝑘 −1 𝐴 𝑇 + 𝑄……………..(6)Table 4.1:Time update equations 632 | P a g e
  • k.S.N.Murthy, m.Suman, v.Sriram, n.Sai Kotewararao, b.Phani Krishna Rao / International Journal of Engineering Research and Applications (IJERA)ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, June-July 2012, pp.631-635 1. Project the state head In order to apply this filtering to the speech expression shown above, it must be expressed in state 𝑥 𝑘 = 𝑓 𝑥 𝑘 −1 , 𝑢 𝑘 , O space form as 2. Project the error covariance ahead 𝐻 𝑘 = 𝑋𝐻 𝑘−1 + 𝑊 𝑘 ………….(12) 𝑃 𝑘 = 𝐴 𝑘 𝑃 𝑘 −1 𝐴 𝑘𝑇 + 𝑊 𝑘 𝑄 𝑘−1 𝑊 𝑘𝑇 𝑦 𝑘 = 𝑔𝐻 𝑘 …………………….(13) 𝑎1 𝑎2 … 𝑎 𝑁−1 𝑎𝑁 1 0 … 0 0 X= 0 1 … 0 0Measurement update (“correct”) ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 1 0 𝑦𝑘 1. Compute the gain 𝑦 𝑘 −1 𝐻𝑘 = 𝑦 𝑘 −2 𝐾 𝑘 = 𝑃 𝑘 𝐻 𝑘𝑇 (𝐻 𝑘 𝑃 𝑘 𝐻 𝑘𝑇 + 𝑉 𝑘 𝑅 𝑘 𝑉 𝑘𝑇 )−1 ⋮ 𝑦 𝑘 −𝑁+1 2. Update estimate with measurement 𝑤𝑘 𝑥 𝑘 = 𝑥 𝑘 + 𝐾 𝑘 (𝑧 𝑘 − 𝑕 𝑥 𝑘 , 0 ) 0 𝑤𝑘 = 0 3. Update the error covariance ⋮ 0 𝑃 𝑘 = (𝐼 − 𝐾 𝑘 𝐻 𝑘 )𝑃 𝑘 g= 1 0 … 0 X is the system matrix, Hkconsists of the series ofFig: Complete operation of the filter speech samples; 𝑊 𝑘 is the excitation vector and g, the output vector. The reason of (k-N+1)th iteration is due 5. IMPLEMENTATION to the state vector, Hk, consists of N samples, from From a statistical point of view, many the kth iteration back to the (k-N+1)th iteration. Thesignals such as speech exhibit large amounts of above formulations are suitable for this filter. Ascorrelation. From the perspective of coding or mentioned in the previously, this filter functions in afiltering, this correlation can be put to good use. The looping method. Here we denote the following stepsall pole, or autoregressive (AR), signal model is often within the loop of the filter. Define matrix 𝐻 𝑘𝑇−1 asused for speech. The AR signal model is introduced the row vector:as: 𝑻 𝑯 𝒌−𝟏 =-[𝒚 𝒌−𝟏 𝒚 𝒌−𝟐 ……𝒚 𝒌−𝑵 ] ........... (14) Andzk= yk. Then (11) and (14) yield 1𝑦𝑘= 𝑁 𝑊 𝑘 ………(10) 1−∑ 𝑖−1 𝛼𝑍 𝑻Equation (10) can also be written in this form as 𝒛 𝒌 =𝑯 𝒌−𝟏 𝑿 𝒌 +𝑾 𝒌shown below: .............. (15) WhereXkwill always be updated according to the𝑦 𝑘 =𝑎1 𝑦 𝑘 −1 +𝑎2 𝑦 𝑘 −2 ……+𝑎 𝑁 𝑦 𝑘 −𝑁 + 𝑊 𝑘 number of iterations, k. Note that when the k = 0, the matrix Hk-1 is unable to be determined. However, when the time…………………….. (11) zkis detected, the value in matrix Hk-1 is known. Thewhere,k→ Number of iterations; above purpose is thus sufficient enough for definingyk → current input speech signal sample; the recursive filter, which consists of:yk–N→ (N-1)th sample of speech signal; 𝑋 𝑘 = 1 − 𝐾 𝑘 𝐻 𝑘𝑇−1 𝑋 𝑘−1 + 𝐾 𝑘 𝑍 𝑘aN → Nth filter coefficient; and .........…...(16)wk → excitation sequence (white noise). 633 | P a g e
  • k.S.N.Murthy, m.Suman, v.Sriram, n.Sai Kotewararao, b.Phani Krishna Rao / International Journal of Engineering Research and Applications (IJERA)ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, June-July 2012, pp.631-635 1 0 0 0 babble 0.7560 4.8810 5.6148 …. 0 1 0 0 car 2.4997 4.8747 4.3537Where I= ⋮ ⋱ ⋮ exhibition 2.3022 4.8301 4.5720 0 0 1 0 …. restaurant 0.8976 4.9388 5.0451 0 0 0 1 station 0.5629 3.3547 5.1048With 𝐾 𝑘 = 𝑃 𝑘 −1 𝐻 𝑘 −1 [𝐻 𝑘𝑇−1 𝑃 𝑘 −1 𝐻 𝑘 −1 + 𝑅] street 1.8141 3.7160 5.033………………….(17)Where 𝐾 𝑘 𝑖𝑠 𝑡𝑕𝑒 𝑓𝑖𝑙𝑡𝑒𝑟 𝑔𝑎𝑖𝑛 Fig1: Below result is for the input speech𝑃 𝑘 −1 is the priori error covariance matrixR is the measurement noise covariance 𝑃 𝑘 = 𝑃 𝑘−1 − 𝑃 𝑘 −1 𝐻 𝑘−1 𝐻 𝑘−1 𝑃 𝑘 −1 𝐻 𝑘 −1 + 𝑅 𝐻 𝑘𝑇−1 𝑃 𝑘 −1 + 𝑄 𝑇…………………….(18)Where 𝑃 𝑘 is the posteriori error co-varianceMatrix 1 0 0 0 …. 0 1 0 0Q= ⋮ ⋱ ⋮ 0 0 1 0 …. 0 0 0 1 Fig2. Noise addedThereafter the reconstructed speech signal, Ykafterfiltering will be formed in a manner similar to (11):𝒀 𝒌 =𝒂 𝟏 𝒀 𝒌−𝟏 +𝒂 𝟐 𝒀 𝒌−𝟐 +…..𝒂 𝑵 𝒀 𝒌−𝑵 +𝑾 𝒌 Since the value of ykis the input at thebeginning of the process, there will be no problemforming HTk-1. In that case a question rises, how is Yk 𝑁formed? The parameterswk and {𝑎} 𝑖−1 are determinedfrom application of this filter to the input speechsignal yk. That is in order to construct Yk, we willneed matrix X that contains the filtering coefficientsand the white noise, wk which both are obtained fromthe estimation of the input signal. This information isenough to determine HHk-1where 𝑌𝑘−1HH 𝑘−1 = 𝑌𝑘−2 Fig3. Estimated signal 𝑌𝑘−𝑁+1Thus, forming the equation (19)mentioned above.RESULTSThe SNR values for different types of noisesInput O/P SNR O/P SNR O/P SNR for for for input= input=0dB input=5dB 10dBairport 1.1520 4.9623 5.7959 634 | P a g e
  • k.S.N.Murthy, m.Suman, v.Sriram, n.Sai Kotewararao, b.Phani Krishna Rao / International Journal of Engineering Research and Applications (IJERA)ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, June-July 2012, pp.631-635 indeed has the ability to estimate accurately. Furthermore, the results have also shown that this recursive filter could be tuned to provide optimal performance. ACKNOWLEDGEMENT We would like to thank our respected teachers and colleagues for their guidance and support. We are thankful to our HOD, ECE Department. We also thank Principal& Chairman,KoneruLakshmaiah University for providing us most suitable environment for research and development. REFERENCES:Fig 4: combined plot [1] E. Grivel, M. Gabrea, and M. Najim, “Speech Enhancements a Realization Issue,” Signal Processing, vol. 82, pp. 1963–1978, Dec. 2002. [2] M. Gabrea, E. Grivel, and M. Najim, “A single microphone kalman filter-based noise canceller,” IEEESignal Processing Lett., vol. 6, pp. 55–57, Mar. 1999. [3] J. D. Gibson, B. Koo, and S. D. Gray, “Filtering of colored noise for speech enhancement and coding,”IEEE Trans. Signal Processing, vol. 39, pp. 1732–1742, Aug. 1991. [4] H. Morikawa and H. Fujisaki, “Noise reduction of speech signal by adaptive kalman filtering,” Special issue in Signal ProcessingFig 5: mean square error APII-AFCET, vol. 22, pp. 53–68, 1988. [5] M. Gabrea, “Robust adaptive kalman filtering based speech enhancement algorithm,” in Proc.ICASSP’04, 2004. [6] M. Gabrea and D. O’Shaugnessy, “Speech signal recovery in white noise using an adaptive kalman filter, “inProc. EUSIPCO’00, 2000, pp. 159–162. [7] B. A. Anderson and J. B. Moore, Optimal Filtering, NJ:Prentice-Hall, Englewood Cliffs, 1979. [8] B. D. Kovacevic, M. M. Milosavljevic, and M. Dj. Veinovic, “Robust recursive ar speech analysis,” SignalProcessing, vol. 44, pp. 125– 138, 1995. [9] R. K. Mehra, “On the identification ofCONCLUSION variances and adaptive kalman filtering,” IEEE Inthis project, an implementation of Trans. AutomaticControl, vol. AC-15, pp.employing this recursive filtering to speech 175–184, Apr. 1970.processing had been developed. As has been [10] T. Kailath, “An innovations approach to leastpreviously mentioned, the purpose of this approach is squares estimation, part i: Linear filtering into reconstruct an output speech signal by making use additive white noise,” IEEE Tran. Automaticof the accurate estimating ability of this filter. True Control, vol. AC-13, pp. 646–655, Dec. 1968.enough, simulated results had proven that this filter 635 | P a g e