User:Virusys/Temp/3


 * QUESTION THREE:


 * $$ \quad y[n] = w_1[n] + w_2[n] $$
 * $$ \quad w_2[n] = g_1^Kx[n - MK] + g_2w_2[n - M] $$
 * $$ \quad w_1[n] = x[n] + g_1^Kx[n-MK] - g_1w_1[n - M]$$


 * transfer function:
 * $$ \quad H_1(z) = \frac{1 - g_1^{K}z^{-MK}}{1 - g_1z^{-M}} $$
 * $$ \quad H_2(z) = \frac{g_1^Kz^{-MK}}{1 - g_2z^{-M}} $$
 * $$\quad H(z) = H_1(z) + H_2(z) $$
 * $$ \quad H(z) = \frac{1 - g_1^{K}z^{-MK}}{1 - g_1z^{-M}} + \frac{g_1^Kz^{-MK}}{1 - g_2z^{-M}} $$


 * $$ \quad H(z) = \frac{1 - g_2z^{-M} - g_1^{K+1}z^{-MK-M} + g_2g_1^{K}z^{-MK-M}}{1 - g_1^{-M} - g_2z^{-M} + g_1g_2z^{-2M}} $$
 * $$ \quad H(z) = \frac{1 - g_2z^{-M} + (g_2g_1^{K} - g_1^{K + 1})z^{-MK-M}} {1 + (-g_1 - g_2)z^{-M} + g_1g_2z^{-2M}} $$


 * a. For this filter, the feedback coefficients of $$ w_1[n - M] $$ and $$ w_2[n - M] $$ must be $$ < |1| $$.

Therefore, $$ |g_1|, |g_2| < |1| $$.