Consider the signal We wish to model this signal using a 2nd-order (p = 2) all-...

Consider the signal

We wish to model this signal using a 2nd-order (p = 2) all-pole model or, equivalently, using 2nd-order linear prediction.

For this problem, since we are given an analytical expression for s[n] and s[n] is the impulse response of an all-pole filter, we can obtain the linear prediction coefficients directly from the z-transform of s[n]. (You are asked to do this in part (a).) In practical situations, we are typically given data, i.e., a set of signal values, and not an analytical expression. In this case, even when the signal to be modeled is the impulse response of an all-pole filter, we need to perform some computation on the data, using methods such as those discussed in Section 11.3, to determine the linear prediction coefficients.

There are also situations in which an analytical expression is available for the signal, but the signal is not the impulse response of an all-pole filter, and we would like to model it as such. In this case, we again need to carry out computations such as those discussed in Section 11.3.

(a) For s[n] as given in Eq. (P11.16-1), determine the linear prediction coefficients a₁, a₂ directly from the z-transform of s[n].

(b) Write the normal equations for p = 2 to obtain equations for a₁, a₂ in terms of r_ss [m].

(c) Determine the values of r_ss [0], r_ss [1], and r_ss [2] for the signal s[n] given in Eq. (P11.16-1).

(d) Solve the system of equations from part (a) using the values you found in part (b) to obtain values for the aks.

(e) Are the values of a_k from part (c) what you would expect for this signal? Justify your answer clearly.

( f ) Suppose you wish to model the signal now with p = 3. Write the normal equations for this case.

(g) Find the value of r_ss [3].

(h) Solve for the values of a_k when p = 3.

( i ) Are the values of a_k found in part (h) what you would expect given s[n]? Justify your answer clearly.

( j ) Would the values of a₁, a₂ you found in (h) change if we model the signal with p = 4?