He reproduced particle velocity: v(x,) ===1 T Tv (x,)wS,television
He reproduced particle velocity: v(x,) ===1 T Television (x,)wS,tv (x|y ,)w SL(four)where Television (x,) = [tv (x|y1 ,), . . . , t(x|y L ,)] T is really a matrix of size L 3. Given that in sound field reproduction applications, we aim to reproduce the desired sound within a particular region of interest, by matching the sound stress and particle velocity at several handle points. That is definitely, we’ve got the matrix kind representation of (3) and (4) as p = T p wS (five) v = T v w S , (six)exactly where offered M handle points, xm and m = 1, . . . , M, p = [ p(x1 ,), . . . , p(x M ,)] T and v = [v T (x1 ,), . . . , v T (x M ,)] T are column vectors of length M and 3M, respectively. The ATF matrix T p = [t p (x1 ,), . . . , t p (x M ,)] T Television = [Tv (x1 ,), . . . , Television (x M ,)] T is of size of M L and 3M L, respectively. According to Equations (five) and (six), and assuming the unit amplitude of the supply audio signal, i.e., S = 1, the price function to get a jointly controlling the reproduced sound pressure and particle velocity is formulated as follows:wmin T p w – pd + (1 -) Tv w – vd 2 (7)exactly where pd and vd represent the desired pressure and particle velocity, respectively. The control tactic is BI-0115 web always to minimize the reproduction error of both components, and [0, 1] may be the parameter to adjust the relative weights for matching of pressure and velocity. Equation (7) is generally known as the weighted least squares difficulty and may be solved employing a Moore enrose pseudoinverse with Tikhonov regularization.Appl. Sci. 2021, 11,four of3. Proposed: Time-Domain Sound Field Reproduction with Joint Manage of Sound Stress and Particle Velocity three.1. Method Formulation Now, we talk about the issue of sound field reproduction within the time domain. Assuming the space impulse responses (RIRs) are pre-calibrated via measurements, the reproduced sound pressure and particle velocity in the mth (1 m M) handle point xm , generated by L loudspeakers located at y1 , . . . , y L , could be expressed as pn (xm ) =sn qn h p,n (xm |y=L)(8)vn (xm ) =sn qn hv,n (xm |y=L),(9)where denotes the linear convolution operator and n denotes the sampling index. sn denotes the input sound signal, qn denotes the control filter for the th loudspeaker, h p,n (xm |y ) and hv,n (xm |y ) denote the RIRs of pressure and velocity elements, SC-19220 web respectively, in the -th loudspeaker towards the m-th manage point. Note that for particle velocity vector, we stick to the convention to define 3 components along x, y and z axes, that is certainly, vn (xm ) = [vn,x (xm ), vn,y (xm ), vn,z (xm )] T andx hv,n (xm |y ) = [hv,n (xm |y ), hv,n (xm |y ), hz (xm |y )] T . v,n yRepresent (eight) and (9) in matrix form, we have pn (xm ) =Lq T H p (xm |y=L)sn = q T H p (xm )sn(10)vn,c (xm ) =qT Hv,c (xm |y=)sn = q T Hv,c (xm )sn , c x, y, z,(11)where offered the K-tap long RIR and J-tap long control filter, sn = sn , sn-1 , . . . , sn-(K + J -2) q = q1 , q2 , . . . , q JT T,,as well as the RIR matrices H p (xm |y ) and Hv,c (xm |y ) are Toeplitz matrices of size J (K + J – 1). The first row vector plus the first column vector of H p (xm |y ) are defined as T h p,1 (xm |y ), . . . , h p,K (xm |y ), 0, . . . , 0 and h p,1 (xm |y ), 0, . . . , 0 , respectively. The sameJ -1 J -formulation is adopted for each and every particle velocity element. Then, we haveT T q = q1 , q2 , . . . , q T L T,TH p ( x m ) = H T ( x m | y1 ), H T ( x m | y2 ), . . . , H T ( x m | y L ) p p pT T T Hv,c (xm ) = Hv,c (xm |y1 ), Hv,c (xm |y2 ), . . . , Hv,c (xm |y L ) T,, c x, y, z,which are of size LJ 1, LJ (.