Question

Question 2 - Signal and System Properties. State whether each of the statements is true or false. Note that a statement is true if it is always true, without further qualifications. If the statement is false, produce a counterexample to it. (a) If a continuous-time signal is periodic with period T (where T > 0), then it is also periodic with period 2T. (b) Let y(t) be the output of a continuous-time linear system for the input x(t). Then the output of the system for the input 2x(t) is 2y(t). (c) Let y(t) be the output of a continuous-time linear system for the input x(t). Then the output of the system for the input x(t + 1) is y(t + 1). (d) If the input x(t) of a stable continuous-time linear system satisfied |x(t)| < 1 for all t, then the output y(t) satisfies |y(t)| < 1 for all t.

Answer 1

Answer:

  (a) true

  (b) true

  (c) false; {y = x, t < 1; y = 2x, t ≥ 1}

  (d) false; y = 200x for .005 < |x| < 1

Step-by-step explanation:

(a)  "s(t) is periodic with period T" means s(t) = s(t+nT) for any integer n. Since values of n may be of the form n = 2m for any integer m, then this also means ...

  s(t) = s(t +2mt) = s(t +m(2T)) . . . for any integer m

This equation matches the form of a function periodic with period 2T.

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(b) A system being linear means the output for the sum of two inputs is the sum of the outputs from the separate inputs:

  s(a) +s(b) = s(a+b) . . . . definition of linear function

Then if a=b, you have

  2s(a) = s(2a)

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(c) The output from a time-shifted input will only be the time-shifted output of the unshifted input if the system is time-invariant. The problem conditions here don't require that. A system can be "linear continuous time" and still be time-varying.

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(d) A restriction on an input magnitude does not mean the same restriction applies to the output magnitude. The system may have gain, for example.