Consider a continuous-time system with system function This system is called an...

Consider a continuous-time system with system function

This system is called an integrator, since the output y_c(t) is related to the input x_c(t) by

Suppose a discrete-time system is obtained by applying the bilinear transformation to H_c(s).

(a) What is the system function H(z) of the resulting discrete-time system? What is the impulse response h[n]?

(b) If x[n] is the input and y[n] is the output of the resulting discrete-time system, write the difference equation that is satisfied by the input and output. What problems do you anticipate in implementing the discrete-time system using this difference equation?

(c) Obtain an expression for the frequency response H(e^jω) of the system. Sketch the magnitude and phase of the discrete-time system for 0 ≤ |ω| ≤ π. Compare them with the magnitude and phase of the frequency response of the continuous-time integrator. Under what conditions could the discrete-time “integrator” be considered a good approximation to the continuous-time integrator? Now consider a continuous-time system with system function G_c(s) = s.

This system is a differentiator; i.e., the output is the derivative of the input. Suppose a discrete-time system is obtained by applying the bilinear transformation to G_c(s).

(d) What is the system function G(z) of the resulting discrete-time system? What is the impulse response g[n]?

(e) Obtain an expression for the frequency response G(e^jω) of the system. Sketch the magnitude and phase of the discrete-time system for 0 ≤ |ω| ≤ π. Compare them with the magnitude and phase of the frequency response of the continuous-time differentiator. Under what conditions could the discrete-time “differentiator” be considered a good approximation to the continuous-time differentiator?

( f ) The continuous-time integrator and differentiator are exact inverses of one another. Is the same true of the discrete-time approximations obtained by using the bilinear transformation?