In this paper, we consider the solution of n-by-n symmetric positive definite Toeplitz systems Tx=b by the preconditioned conjugate gradient (PCG) method. The preconditioner M is defined to be the minimizer of {norm of matrix}T-B{norm of matrix} over all B εH where H is the Hartley algebra. We show that if the generating function f of T is a positive 2π-periodic continuous even function, then the spectrum of the preconditioned system MT will be clustered around 1. Thus, if the PCG method is applied to solve the preconditioned system, the convergence rate will be superlinear.