Draw a diagram of a lattice structure for an all-pole filter and explain it

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written 6.9 years ago by

teamques10 ★ 69k

(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:

Single stage lattice structure for IIR system

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