Consider a continuous-time system with system function
This system is called an integrator, since the output yc(t) is related to the input xc(t) by
Suppose a discrete-time system is obtained by applying the bilinear transformation to Hc(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(ejω) 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 Gc(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 Gc(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(ejω) 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?
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