A speech signal was sampled with a sampling rate of 8 kHz.A300-sample segment was select...

A speech signal was sampled with a sampling rate of 8 kHz.A300-sample segment was selected from a vowel sound and multiplied by a Hamming window as shown in Figure P11.19. From this signal a set of linear predictors

with orders ranging from i = 1 to i = 11 was computed using the autocorrelation method. This set of predictors is shown in Table 11.1 below in a form suggestive of the Levinson– Durbin recursion.

(a) Determine the z-transform A⁽⁴⁾(z) of the 4th-order prediction error filter. Draw and label the flow graph of the direct form implementation of this system.

(b) Determine the set of k-parameters {k₁, k₂, k₃, k₄} for the 4^th-order prediction error lattice filter. Draw and label the flow graph of the lattice implementation of this system.

(c) The minimum mean-squared prediction error for the 2nd-order predictor is E⁽²⁾ =0.5803.What is the minimum mean-squared prediction error for the 3rd-order predictor? What is the total energy of the signal s[n]?What is the value of the autocorrelation

function r_ss [1]?

(d) The minimum mean-squared prediction errors for these predictors form a sequence {E⁽⁰⁾,E⁽¹⁾,E⁽²⁾, . . . , E⁽¹¹⁾}. It can be shown that this sequence decreases abruptly in going from i = 0 to i = 1 and then decreases slowly for several orders and then

makes a sharp decrease. At what order i would you expect this to occur?

(e) Sketch carefully the prediction error sequence e(11)[n] for the given input s[n] in Figure P11.19. Show as much detail as possible.

( f ) The system function of the 11th-order all-pole model is

State briefly in words where the other six zeros of A(11)(z) are located. Be as precise as possible.

(g) Use information given in the table and in part (c) of this problem to determine the gain parameter G for the 11th-order all-pole model.

(h) Carefully sketch and label a plot of the frequency response of the 11th-order all-pole model filter for analog frequencies 0 ≤ F ≤ 4 kHz.