. The analysis of multichannel two-dimensional random signals. gions of support. The system will be recursively computable if there exists anordering for processing of the points such that values of the signal y needed to computethe signals current value at the point (n1,n2) are always available. Whether such anordering exists depends on the shape of the output region a. The output for a multichannel 2-D system can be equivalently represented by the2-D convolution operation in either of the forms CO ,oo y(n1,n2)= ]T H(lx, l2)x(nx, -lx,n2 - l2) () (.1 ,(.2 = — oo , — CO CO ,co y(n1)n2)= J2
. The analysis of multichannel two-dimensional random signals. gions of support. The system will be recursively computable if there exists anordering for processing of the points such that values of the signal y needed to computethe signals current value at the point (n1,n2) are always available. Whether such anordering exists depends on the shape of the output region a. The output for a multichannel 2-D system can be equivalently represented by the2-D convolution operation in either of the forms CO ,oo y(n1,n2)= ]T H(lx, l2)x(nx, -lx,n2 - l2) () (.1 ,(.2 = — oo , — CO CO ,co y(n1)n2)= J2 H(nx -lx,n2 -4)x(£l54) () ti t2 = - CO ,- CO where H{-,-) is a matrix function representing the 2-D multichannel impulse response.(Each term H(£x,£2) in Eq. () is an M x M matrix.) A 2-D multichannel system * Although it is tempting to use vector notation for the arguments, this tends tohide the essential 2-D nature of the signals. Therefore we use the longer but moreexplicit notation involving the two indices nx and n2. - 3- Channelm X,(n1,n2). X(nrn2) = X2 (n^n ) X^n^n ) ^•| X3(nrn2) Wn2} Figure Discrete Multichannel 2-D Signal. 4- will be called a Finite Impulse Response (FIR) system if the support of H(i., m) is finiteand an Infinite Impulse Response (IIR) system otherwise. Clearly for an FIR systemall of the Ailia in Eq. () are zero and #(^,4) = Blllrt for {l,,l2) G /?• For purposes of this report, we will be dealing with statistical properties of randomsignals. Thus the mean of a signal x(n1 ,n2) is a vector quantity defined by mI(n1,n2) = £[x(n1?n2)] () and the correlation and covariance are M x M matrix functions defined (respectively)by R{nl,n2 ; mx, m2) = E [x(nj, n2)xT (mi, m2)] () C{nl,n2\ml,m2) = E (x(nl5n2) - mse(n1 ,n2)) (x(m1,m2) - mx(m1,m2)) ()When the random process representing the signal is homogenous or stationary the meanis constant and the correlation and covariance are functions only of the vector distancebetween the
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