Preconditioned conjugate gradient based fast computation of indirect decision feedback equalizer

We use the preconditioned conjugate gradient (PCG) method to compute rapidly the tap weights of a minimum mean-square error (MMSE) decision feedback equalizer (DFE). The equalizer setting is computed indirectly after channel estimation. According to the Toeplitz block structure of the MMSE DFE equation, Rw=r (R is the system matrix), the preconditioner, P, is chosen to be a block diagonal matrix with circulant blocks along its diagonal. The spectral clustering property of the preconditioner matrix is analyzed. It is shown that the eigenvalues of P^-1R are clustered around unity, except for a small number of outliers when the number of the equalizer taps becomes large. The preconditioner can be inverted via a fast Fourier transform (FFT) with complexity O(N log N). Since the PCG method converges in a small number of steps, the total complexity for computing the DFE setting is proportional to O(N log N). The proposed scheme is also suitable for "smart" initialization which can further reduce the computational burden by decreasing the number of iteration steps. Simulations of a DFE for digital TV channels demonstrates the superior performance of the scheme.