An analog signal r(t) is the command input to a digital control system, part of which is shown in Figure. The signal r(t) must be sampled and converted to a discrete-time signal for use by the digital...

An analog signal r(t) is the command input to a digital control system, part of which is shown in Figure. The signal r(t) must be sampled and converted to a discrete-time signal for use by the digital controller. The command input consists of a signal component s(t) and a high-frequency (compared to the sampling rate 1/Ts ) noise component n(t). An antialiasing filter is inserted before sampling to eliminate aliasing in rˆ k the input to the controller.

A fourth-order Butterworth low-pass filter is chosen. The transfer function is

a.       The control system sampling rate is 1000 Hz. Find the Nyquist frequency ω_N.

b.      Find ωn, so that the magnitude of G(jω) is −60 db at the Nyquist frequency

c.       The signal and noise components of the command input r(t) are s(t) = 1, t ≥ 0 and n(t)=5 ×10⁻³
sin (2 × 106t), t ≥ 0. Find the filter output rˆ(t) at steady state.

d.      Find G(z), the z-domain transfer function of the discrete-time system approximation to G(s) using explicit Euler integration. Leave your answer in terms of the integration step size T

e.      Comment on the choice of T necessary to simulate the filter response by recursive solution of the difference equation corresponding to G(z).