New Developments in Robotics, Automation and Control 2009 Part 2 doc

New Developments in Robotics, Automation and Control 24 terms of optimal control techniques. All the constraints introduced by kinematics and dynamic limits on mobility of the moving elements as well as by communications limits (network connectivity) have been considered. A global approach has been followed making use of time and space discretization, so getting a suboptimal solution. Some simulation results show the behaviour and the effectiveness of the proposed solution. 8. References Acar, Choset, and Lee, J. Y. (2006). Sensor-based coverage with extended range detectors. IEEE Transactions on Robotics and Automation, 22(1):189–198. Akyildiz, I., Su, W., Sankarasubramaniam, Y., and Cayirci, E. (2002). A survey on sensor networks. IEEE Communications Magazine, 40(8):102–114. Cardei, M. and Wu, J. (2006). Energy-efficient coverage problems in wireless ad hoc sensor networks. Computer communications, 29(4):413–420. Cecil and Marthler (2004). 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ICCCN 2004. Proceedings. 13th International Conference on, pages 373–378. 2 Multichannel Speech Enhancement Lino García and Soledad Torres-Guijarro Universidad Europea de Madrid, Universidad de Vigo Spain 1. Introduction 1.1 Adaptive Filtering Review There are a number of possible degradations that can be found in a speech recording and that can affect its quality. On one hand, the signal arriving the microphone usually incorporates multiple sources: the desired signal plus other unwanted signals generally termed as noise. On the other hand, there are different sources of distortion that can reduce the clarity of the desired signal: amplitude distortion caused by the electronics; frequency distortion caused by either the electronics or the acoustic environment; and time-domain distortion due to reflection and reverberation in the acoustic environment. Adaptive filters have traditionally found a field of application in noise and reverberation reduction, thanks to their ability to cope with changes in the signals or the sound propagation conditions in the room where the recording takes place. This chapter is an advanced tutorial about multichannel adaptive filtering techniques suitable for speech enhancement in multiple input multiple output (MIMO) very long impulse responses. Single channel adaptive filtering can be seen as a particular case of the more complex and general multichannel adaptive filtering. The different adaptive filtering techniques are presented in a common foundation. Figure 1 shows an example of the most general MIMO acoustical scenario. Fig. 1. Audio application scenario. ( ) ns 2 ( ) ns I ( ) nx 1 ( ) nx 2 ( ) nx P () ns 1 ( ) nr () ny 1 () ny 2 () ny O W V New Developments in Robotics, Automation and Control 28 The box, on the left, represents a reverberant room. V is a L I P × matrix that contains the acoustic impulse responses (AIR) between the I sources and P microphones (channels); L is a filters length. Sources can be interesting or desired signals (to enhance) or noise and interference (to attenuate). The discontinuous lines represent only the direct path and some first reflections between the ( ) ns 1 source and the microphone with output signal () nx 1 . Each () n pi v vector represents the AIR between Ii K1 = and Pp K1 = positions and is constantly changing depending on the position of both: source or microphone, angle between them, radiation pattern, etc. ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = PIPP I I vvv vvv vvv V L MOMM L L 21 22221 11211 , v pi =[ v pi1 v pi2 ··· v piL ]. (1) () nr is an additive noise or interference signal. ( ) nx p , Pp K1= is a corrupted or poor quality signal that wants to be improved. The filtering goal is to obtain a W matrix so that () () nsny io ˆ ≈ corresponds to the identified signal. The signals in the Fig. 1 are related by ( ) ( ) ( ) nrnn += Vsx , (2) y(n) = Wx(n). (3) () ns is a 1×LI vector that collects the source signals, () () () () [ ] T T I TT nnnn ssss L 21 = , (4) () ( ) ( ) ( ) [ ] T iiii Lnsnsnsn 11 +−−= Ls . () nx is a 1×P vector that corresponds to the convolutive system output excited by () ns and the adaptive filter input of order LPO × . ( ) nx p is an input corresponding to the channel p containing the last L samples of the input signal x , () () () () [ ] T T P TT nnnn xxxx L 21 = , (5) x p (n)=[ x p (n) x p (n-1) ··· x p (n-L+1)] T . Multichannel Speech Enhancement 29 W is an LPO× adaptive matrix that contains an AIRs between the P inputs andO outputs ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = OPOO P P www www www W L MOMM L L 21 22221 11211 , w op = [w op1 w op2 ··· w opL ]. (6) For a particular output Oo K1 = , normally matrix W is rearranged as column vector [ ] T P wwww L 21 = . (7) Finally, () ny is an 1 × O target vector, ( )() ( ) ( ) [ ] T O nynynyn L 21 =y . The used notation is the following: a or α is a scalar, a is a vector and A is a matrix in time-domain a is a vector and A is a matrix in frequency-domain. Equations (2) and (3) are in matricial form and correspond to convolutions in a time-domain. The index n is the discrete time instant linked to the time (in seconds) by means of a sample frequency s F according to s nTt = , ss FT 1 = . s T is the sample period. Superscript T denotes the transpose of a vector or a matrix, ∗ denotes the conjugate of a vector or a matrix and superscript H denotes Hermitian (the conjugated transpose) of a vector or a matrix. Note that, if adaptive filters are 1×L vectors, L samples have to be accumulated per channel (i.e. delay line) to make the convolutions (2) and (3). The major assumption in developing linear time-invariant (LTI) systems is that the unwanted noise can be modeled by an additive Gaussian process. However, in some physical and natural systems, noise can not be modelled simply as an additive Gaussian process, and the signal processing solution may also not be readily expressed in terms of mean squared errors (MSE) 1 . From a signal processing point of view, the particular problem of noise reduction generally involves two major steps: modeling and filtering. The modelling step generally involves determining some approximations of either the noise spectrum or the input signal spectrum. Then, some filtering is applied to emphasize the signal spectrum or attenuate/reject the noise spectrum (Chau, 2001). Adaptive filtering techniques are used largely in audio applications where the ambient noise environment has a complicated spectrum, the statistics are rapidly varying and the filter coefficients must automatically change in order to maintain a good intelligibility of the speech signal. Thus, filtering techniques must be 1 MSE is the best estimator for random (or stochastic) signals with Gaussian distribution (normal process). The Gaussian process is perhaps the most widely applied of all stochastic models: most error processes, in an estimation situation, can be approximated by a Gaussian process; many non-Gaussian random processes can be approximated with a weighted combination of a number of Gaussian densities of appropriated means and variances; optimal estimation methods based on Gaussian models often result in linear and mathematically tractable solutions and the sum of many independent random process has a Gaussian distribution (central limit theorem) (Vaseghi, 1996). New Developments in Robotics, Automation and Control 30 powerful, precise and adaptive. Most non-referenced noise reduction systems have only one single input signal. The task of estimating the noise and/or signal spectra must then make use of the information available only from the single input signal and the noise reduction filter will also have only the input signal for filtering. Referenced adaptive noise reduction/cancellation systems work well only in constrained environments where a good reference input is available, and the crosstalk problem is negligible or properly addressed. 2. Multichannel Adaptive Filters In a multichannel system ( 1>P ) it is possible to remove noise and interference signals by applying sophisticated adaptive filtering techniques that use spatial or redundant information. However there are a number of noise and distortion sources that can not be minimized by increasing the number of microphones. Examples of this are the surveillance, recording, and playback equipment. There are several classes of adaptive filtering (Honig & Messerschmitt, 1984) that can be useful for speech enhancement, as will be shown in Sect. 4. The differences among them are based on the external connections to the filter. In the estimator application [see Fig. 2(a)], the internal parameters of the adaptive filter are used as estimate. In the predictor application [see Fig. 2(b)], the filter is used to filter an input signal, () nx , in order to minimize the output signal, ( ) ( ) ( ) nynxne − = , within the constrains of the filter structure. A predictor structure is a linear weighting of some finite number of past input samples used to estimate or predict the current input sample. In the joint-process estimator application [see Fig. 2(c)] there are two inputs, ( ) nx and ( ) nd . The objective is usually to minimize the size of the output signal, ( ) ( ) ( ) nyndne − = , in which case the objective of the adaptive filter itself is to generate an estimate of ( ) nd , based on a filtered version of () nx , () ny (Honig & Messerschmitt, 1984). Fig. 2. Classes of adaptive filtering. (a) (b) (c) Adaptive filter Adaptive filter Adaptive filter Parameters ( ) nx ( ) nx ( ) nx ( ) ny ( ) ne ( ) ne ( ) ny ( ) nd Multichannel Speech Enhancement 31 2.1 Filter Structures Adaptive filters, as any type of filter, can be implemented using different structures. There are three types of linear filters with finite memory: the transversal filter, lattice predictor and systolic array (Haykin, 2002). 2.1.1 Transversal The transversal filter, tapped-delay line filter or finite-duration impulse response filter (FIR) is the most suitable and the most commonly employed structure for an adaptive filter. The utility of this structure derives from its simplicity and generality. The multichannel transversal filter output used to build a joint-process estimator as illustrated in Fig. 2(c) is given by () ( ) () () 11 1 1, , PL P pl p p p pl p yn wx n l n n == = =−+= = ∑∑ ∑ wx wx . (8) Where ( ) nx is defined in (5) and w in (7). Equation (8) is called finite convolution sum. Fig. 3. Multichannel transversal adaptive filtering. 2.1.2 Lattice The lattice filter is an alternative to the transversal filter structure for the realization of a predictor (Friedlander, 1982). ( ) nd ( ) ne ( ) ny ( ) ny 1 ( ) ny P ( ) nx 1 ( ) nx P 1− z 1− z 1− z 1− z 1− z 1− z 11 w 12 w L w 1 1 P w 2P w PL w P w 1 w New Developments in Robotics, Automation and Control 32 Fig. 4. Multichannel adaptive filtering with lattice-ladder joint-process estimator. The multichannel version of lattice-ladder structure (Glentis et al., 1999) must consider the interchannel relationship of the reflection coefficients in each stage l . ( ) ( ) ( )() ( ) 111 1, ∗ −− =+ − = ll ll nn nnnff Kb fx , (9) ( ) ( ) ( ) ( ) ( ) 111 1, ll ll nn nnn −− =−+ =bb Kfbx. (10) Where () () ( ) ( ) [] T Pllll nfnfnfn L 21 =f , ( ) ( ) ( ) ( ) [] T Pllll nbnbnbn L 21 =b , () () () ( ) [] T P nxnxnxn L 21 =x , and T PPllPlP Plll Plll l kkk kkk kkk ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = L MOMM L L 21 22221 11211 K . The joint-process estimation of the lattice-ladder structure is especially useful for the adaptive filtering because its predictor diagonalizes completely the autocorrelation matrix. The transfer function of a lattice filter structure is more complex than a transversal filter because the reflexion coefficients are involved, () nx 1 () nx P ( ) ne ( ) nd ( ) ny ( ) ny 1 ( ) ny P () nf 11 () nb 11 1− z 11 w ( ) nf 12 ( ) nb 12 1− z 12 w () ( ) nf L 11 − () ( ) nb L 11 − ( ) nf L1 ( ) nb L1 1− z () 11 −L w L w 1 () nf P1 () nb P1 1− z 1P w ( ) nf P2 ( ) nb P2 1− z 2P w () ( ) nf LP 1− () ( ) nb LP 1− ( ) nf PL ( ) nb PL 1− z () 1−LP w PL w [...]... derivatives) defined in (45, 52) while stochastic recursive methods replace these functions by impartial estimations ⎡ ∂J ( w ) ∇J ( w ) = ⎢ ⎣ ∂w 1 T ∂J ( w ) ∂J ( w ) ⎤ L ⎥ , ∂w 2 ∂w L ⎦ (28 ) New Developments in Robotics, Automation and Control 38 ⎡ 2 J (w ) ⎢ ⎢ ∂w 1 ∂ w 1 ⎢ 2 J (w ) ⎢ 2 ∇ J ( w ) = ⎢ ∂w 2 ∂w 1 ⎢ M ⎢ ⎢ 2 J (w ) ⎢ ⎣ ∂ w L ∂w 1 (29 ) T 2 J (w ) ∂ w 1 ∂w 2 L 2 J (w ) ∂w 2 ∂ w 2 M L O ∂... The subband adaptive filtering approach splits the spectra of the signal in a number of subbands that can be adapted independently and afterwards the filtering can be carried out in a fullband The frequencydomain adaptive filtering partitions the signal in time-domain and projects it into a transformed domain (i.e frequency) using better properties for adaptive processing In both cases the input signals... employed in an attempt to separate the deterministic ˆ ˆ y (n ) ≈ s (n ) and stochastic part e(n ) ≈ r (n ) assuming that the noise and interference signal has a broadband spectra ALP corresponds to single-input and single-ouput (SISO) predictor application (class b, Fig 2) with a single microphone, P = 1 Most signals, such as speech and music, are partially predictable and partially random The random input... v P ]T of size N = LP × 1 and initially partitioned in a reasonable number Q of equally-sized blocks vq , q = 1KQ , of length K Each of these blocks is treated as a 48 New Developments in Robotics, Automation and Control separate impulse response, and convolved by a standard overlap -and- save process, using T operator (FFT windows of length L ) All input data are processed in overlapped blocks of L... lattice structure that uses two cost functions: instantaneous squared error for the tranversal part and prediction MSE for the lattice-ladder part, ( 2 Bl ( n ) = β Bl ( n − 1 ) + ( 1 − β ) fl ( n ) + bl ( n − 1 ) 2 ) , where α and σ are relaxation factors New Developments in Robotics, Automation and Control 42 Method Definition Comments α μl ( n ) = bl ( n ) 2 gl ( n) = bl ( n) e∗ ( n) GAL λl ( n ) =... Developments in Robotics, Automation and Control primary input d(n ) = s(n ) + r (n ) collects the sum of unwanted noise r (n ) and speech signal s(n ) , and the auxiliary or reference input measures the noise signal x(n ) = r (n ) ANC corresponds to multiple-input and single-output (MISO) joint-process estimator application (class c, Fig 2) with at least two microphones, P = 2 4.4 Beamforming Beamforming... by each new set of input/output samples In general, most of the adaptive algorithms turn a stochastic optimization problem into a deterministic one and the obtained solution is an approximation to the one of the original problem g = ∇J ( w ) = ∂J ( w ) = −2Xd∗ + 2 XX H w , can be estimated by means of ∂w g = 2 ( r + Rw ) , or by the equivalent one g = − Xe∗ , considering R and r according to (24 ) The... spectra and subtracts it from the whole signal in the frequency-domain The Wiener filter estimator is the result of y( n) − s( n) estimating y (n ) from s(n ) that minimizes the MSE 2 given by y = Qx , x = s + r , and that results x q ≅ Q = diag {[ q1 2 − x d 2 ( 52) 2 , q2 L qM ]} is a diagonal matrix which contains the spectral gain in the frequency-domain; normally T is a short-time Fourier transform... ⎢X X R = XX H = ⎢ 2 1 ⎢ M ⎢ ⎣ X P X1 r = Xd∗ = ⎡ X1d∗ ⎣ (22 ) X1X 2 L X1X P ⎤ X2 X2 L X2 X P ⎥ ⎥, M O M ⎥ ⎥ X P X2 L X P X P ⎦ (23 ) T ∗ X 2d ∗ L X P d ⎤ ⎦ H For each i = 1K I input source, P (P − 1) 2 relations are obtained: x H w q = x q w p for p p , q = 1K P , with p ≠ q Given vector u = [∑ P p =2 wT p − wT 1 ] T L − wT , due to the nearness 1 with which microphones are placed in scenario of Fig... wp = ⎡wp1 ⎣ obtained by means of the T operator as ⎧M 2 w p = ℜ ⎨ ∑ h m ↓ K ∗ w pm ⎩m =1 ( ) ↑K T wp2 L wPL ⎤ ⎦ is (44) ⎫ ∗ gm ⎬ , ⎭ from the subband adaptive filters per each channel w pm , p = 1K P , m = 1K M 2 (Reilly et al., 20 02) The subband filters are very short, of length C = ⎡ L + N − 1 ⎤ − ⎡ N ⎢ ⎥ ⎢K K ⎢ ⎥ ⎢ ⎤ ⎥+1 ⎥ , which 44 New Developments in Robotics, Automation and Control allows to . 1 1 − ∗− ⎡⎤ == ⎣⎦ H wXX XdRr. (21 ) R is a correlation matrix and r is a cross-correlation vector defined by New Developments in Robotics, Automation and Control 36 11 12 1 21 22 2 12 P P H PP PP ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ == ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ XX. graphs. In Decision and Control, 20 05 and 20 05 European Control Conference. CDC-ECC ’05. 44th IEEE Conference on. Zhang, H. and Hou, J. C. (20 05). Maintaining sensing coverage and connectivity in. New Developments in Robotics, Automation and Control 24 terms of optimal control techniques. All the constraints introduced by kinematics and dynamic limits on mobility of the moving