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Question: Draw a diagram of a lattice structure for an all-pole filter and explain it
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Subject: Speech Processing

Topic: LPC and Parametric Speech Coding

Difficulty: Medium

sp(38) • 224 views
 modified 3 months ago by written 4 months ago by awari.swati831 • 160
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(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: