We study methods for solving the constrained and weighted least squares problem minX 1/2 (b -Ax)W(b -Ax) by the preconditioned conjugate gradient (PCG) method. Here W = diag (ω,⋯,ω) with ω ≥ ⋯ > ω > 0, and A = [T, ⋯, T] with Toeplitz blocks T ∈ R×, l = 1,⋯,k. It is well-known that this problem can be solved by solving an augmented linear 2×2 block linear system Mλ+Ax = b, Aλ = 0, where M = W. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.