Least-squares method for multi-dimensional deconvolution

Least-squares method is applied to multi-dimensional deconvolution or estimation of input waveforms to a multi-input multi-output system given the transfer characteristics of the system. Suppose a system accepts n-dimensional input s(t) and it produces m-dimensional output f(t). Let hij(t) be the impulse response of the channel from jth input terminal to ith output terminal. Using an m × n matrix h(t) = [hij(t)], the input-output relation can be written as , where denotes the matrix convolution introduced here. The minimum-norm least-squares estimate for s(t) is expressed as , where ⊕ denotes the generalized convolutional inverse matrix. In the case of m > n, yields the least-squares estimate for s(t). Efficient computation can be performed in the frequency domain. Practical applications are shown as source sound estimation in a multi-source multi-microphone configuration using sinusoidal waves and stationary vowels as source sounds.