Suppose we are given a real and finite-energy (but otherwise arbitrary) DT signal w[n], with associated DTFT W(ejΩ). We want to approximate w[n] by another real, finite-energy DT signal y[n] that is bandlimited to the frequency range |Ω| jΩ) is zero for |Ω| ≥ π/4. Apart from this constraint on its bandwidth, we are free to choose y[n] as needed to get the best approximation. Suppose we measure the quality of approximation by the following sumof-squared-errors criterion:
Our problem is then to minimize E by appropriate choice of the bandlimited y[n], given the signal w[n]. This problem leads you through to the solution.
(a) Express E in terms of a frequency-domain integral on the interval |Ω| jΩ) - Y(ejΩ).
(b) Write your integral from (a) as a sum of integrals, one over each of the ranges -π/4 ≤ Ω ≤ π/4, π/4 ≤ Ω jΩ) needs to be picked in order to minimize E, and what the resulting minimum value of ε is.
(c) Using your result in (b), write down an explicit formula for the y[n] that minimizes ε, expressing this y[n] as a suitable integral involving W(ejΩ).