In the questions below, , , and denote the complex conjugate, the transpose, and the expectation, respectively. and denote the set of all real numbers and the set of all integers, respectively.
Let be a discrete-time signal with the index of , and define the -transform of as
where is a complex variable. Moreover, define the region of convergence of the -transform as a set of such that the series in the right-hand side of Eq. is absolutely convergent. Answer the following questions.
(1) Derive the -transform and its region of convergence of a discrete-time signal
where and .
(2) Derive the discrete-time signal , whose -transform is given by
where the region of convergence is .
(3) Assume that the -transform and its region of convergence of a discrete-time signal are given by and , respectively. Express the -transform of a discrete-time signal for some using . Moreover, answer whether the region of convergence of the -transform of is identical to or not with reasons.
Consider a filter of taps, whose output signal with the index of is given by
where
are the input signal vector and the filter coefficient vector, respectively. Let the input signal and the desired signal be real-valued wide-sense stationary discrete-time random processes. We assume that , , and can be expressed as , , and , respectively.
Moreover, we define the mean-squared error between and as
Answer the following questions.
(1) Express using , and .
(2) Derive the equation that the filter coefficient vector satisfies to minimize (Wiener-Hopf equation).