MMSE Decomposition

We can write this expression for f(n) in matrix form where we consider f to be a column vector of length N, F a column vector of length M, and G is an NxM matrix with each of the M basis functions making up one column of G.
In the special case where M=N (same number of basis functions as original data points), G will be square. If we also require the basis functions (columns of G) be linearly independent, then G will be non-singular and the coefficients, F_m, are easily found:

The basis is then complete or we can say that the basis spans the N-dimensional space in which the signal f(n) resides. This basis can represent any such f(n) or in other words, a set of coefficients exists for all N-point signals.
In general we can find the best estimate of F with respect to the squared error,

where A^T is the pseudo inverse of G. When M=N, this A^T will equal G^-1 and the estimate for F is exact.
Notice that both G and A^T are independent of the actual signal and can be computed off-line. The processes of analysis and synthesis (inverse and forward transforms) are simply matrix multiplications.