Subject: Speech Processing

Topic: LPC and Parametric Speech Coding

Difficulty: Medium

(i) The system transfer function can be represented as: $$ H(Z) = \frac{1}{1+ \sum_{k=1}^{N} b_k Z^{-k}} $$

The difference equation for this system is given by: $$ y(n) = - \sum_{k=1}^{N} b_k y(n-k) + x(n) $$

If interchanging the role of input and output, that is interchanging x(n) and y(n), we can write that: $$ x(n) = - \sum_{k=1}^{N} b_k x(n-k) + y(n) $$

Rearranging the above equation, we get, $$ y(n) = \sum_{k=1}^{N} b_k x(n-k) + x(n) \\ = \sum_{k=0}^{N} b_k x(n-k) $$

Representing FIR filter: $$ f_0 = f_1(n) - kg_0(n-1) \\ g_1(n) = k_1f_0(n) + g_0(n-1) $$

The equation of first stage lattice becomes:

$$ y(n) = f_0(n) = f_1(n) - k_1 y(n-1) = x(n) - k_1y(n-1) \\ g_1(n) = kf_0(n) + g_0(n-1) = k_1y(n) + y(n-1) $$

Implementation of realization of feedback is shown in the figure: