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Time Invariance

Time Invariance A system is said to be time-invariant if a time shift in the input signal causes a corresponding time shift in the output signal To make the concept more precise, for fixed α R, we introduced the shift operator. Q α u(t) = u(t – α ). α. Example. Q α u. t. t.

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Time Invariance

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  1. Time Invariance A system is said to be time-invariant if a time shift in the input signal causes a corresponding time shift in the output signal To make the concept more precise, for fixed α R, we introduced the shift operator Qαu(t) = u(t – α) α Example Qαu t t u(t- α) u(t)

  2. output 1 Input 1 H output 2 Input 2 Definition: A system that is represented by the input output mapping y=H(u) is said to be time-invariant if and only if H Qα(u) = Qα(H(u)) = Qα(y) α R and u  Example

  3. Definition: A system which is not time-invariant is said to be time-varying Example: A system’s Input/Output (I/O) mapping is given by This system is time-varying. In fact

  4. π π Qπ y Qπu Given a linear system modeled by H. Then Question: If H is also assumed to be time-invariant (in addition to linearity), what structure will time-invariance impose on

  5. Answer: By time invariance, Equivalently, As a result, for any given letting We have

  6. Summarizing For a linear time invariant system where If the system is, in addition causal and is relaxed at t = to, then

  7. Linear Time-Invariant Systems in the Frequency Domain The output of a Linear Time-Invariant System (LTI) which is relaxed at is given by Taking the Laplace transform of both sides, we have:

  8. SISO Case: is the system transfer function. It has two interpretations: 1. is the Laplace transform of the system’s impulse response. 2. where y is the output corresponding to the input u when the system is relaxed at MIMO Case:

  9. is the transfer functionmatrix. It is the Laplace transform of the impulse response matrix. Definition: A rational function g(s) is said to be proper if g( ) is finite (zero or nonzero) constant. It is said to be strictlyproper ifg( ) = 0. Example: is not proper is proper is strictly proper

  10. Remark If g(s) =N(s)/D(s) g(s) is proper if and only if deg N(s) < deg D(s)

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