Audio Signal Processing and Coding

4.3 SHORT-TERM LINEAR PREDICTION

Figure 4.4 presents a typical L-th order FIR linear predictor. During forward linear prediction of, s(n), an estimated value, , is computed as a linear combination of the previous samples, i.e.,

where the weights, a_i, are the LP coefficients. The output of the LP analysis filter, A(z), is called the prediction residual, . This is given by

Because only short-term delays are considered in (4.4), the linear predictor in Figure 4.4 is also referred to as the short-term linear predictor. The linear predictor coefficients, a_i, are estimated using least-square minimization of the prediction error, i.e.,

The minimization of ε in (4.5) with respect to a_i, i.e., ∂ε/∂a_i = 0, for i = 1, 2, …, L, yields a set of equations involving autocorrelations

where r_ss(m) is the autocorrelation sequence of the signal s(n). Equation (4.6) can be written in matrix form, i.e.,