Iterative deconvolution using several different distorted versions of an unknown signal

This paper analyses the error behavior of iterative decorivolution algorithms when the distorting system has a frequency response that has negative real part or has a finite number of isolated zeros. The existence of these zeros at a finite number of discrete frequencies results in an inability of the deconvolütion algorithm to restore the lost information at these frequencies with a small number of iterations. A new algorithm is suggested that incorporates multiple distorted versions of the signal and results in a restoration error that approaches zero with a small number of iterations. UNCONSTRAINED ITERATIVE DECONVOLUTION In general an appropriate mathematical representation for a distorting system is y=Dx (la) where x is the unknown input signal, y is the known output signal and D is a known distortion operator or transformation. A standard technique for finding a solution to Eq. (la) is based upon the iteration equation Xk+l A y + (I_XD)xk (ib) where I is the identity operator and A is a convergence parameter that must be chosen. For the class of linear shift invariant distortions y=h*x (2) and x(n) can be found iteratively using the algorithm, * x0(ri) = A h (-n) * y(n) xk+l(n)_xk(n)+xh(_n)*[y(n)_h(n)*xk(n)1 (3) where * denotes convolution, * denotes complex conjugation and h(n) denotes an approximation to the impulse response of the distorting. or blurring system h. This algorithm is henceforth referre to as algorithm #1. The convolution with h (-n) in the above algorithm has been included in order to ensure convergence of the *This work was supported by the Joint Services Electronics Program under Contract #DAAG29-81-K0024. algorithm when the Fourier transform of i(n) has a negative real part [1]. Using frequency domain notation, if X(w) and Xk() represent the Fourier transform of the original and restored signal after k iterations respectively, then it is easily shown that Hk(w) = Xk(W) = {i [1 x 2()]k+l} X () El (w) where Fl(w) and T(w) represent the Fourier transform of the impulse response of the original blurring system and its approximation, respectively. Using the above notation, the spectrum of the restoration error of the k-th iteration can be written as Ek(uX(w)_Xk(o)X(w)[l_Nk(U)] (5) From equation (4) we observe that lim Hk(o)=H(w)/I(w) (6)