With x[n] denoting the input signal and y[n] denoting the output signal, give the difference equation relating the input signal to the output signal in the discrete-time domain, give the initial conditions and their values, and find the transfer function in the z-domain and the associated region of convergence for the z-transform function, for the following linear time-invariant discrete-time systems:
(a) Causal averaging filter with three coefficients. See lecture slide 3-10. 12 points.
(b) Causal discrete-time approximation to first-order differentiator. See lecture slide 3-21.
12 points.
(c) Causal discrete-time approximation to first-order integrator. See online hints. 12 points.
(d) Causal bandpass filter with center frequency ?0 given by the input-output relationship
y[n] = (2 cos ?0) r y[n-1] – r2 y[n-2] + x[n] - (cos ?0) x[n-1]
where 0 < r < 1. Here, r is the radius of the two pole locations. 12 points.