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Characteristics of a Linear System 2.7

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Topics Memory Invertibility Inverse of a System Causality Stability Time Invariance Linearity

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Memory A system has a memory if its output at t o depends on the input on values of the input other than x(t o )

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Memoryless Circuit

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Invertibility A system is said to be invertible if distinct inputs results in distinct outputs – Y(t)=x 2 (t) is not invertible Output of 4V is attributable to +2 V and -2V. – Y(t)=x 5 (t) is invertible

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Inverse of a System The inverse of a system is a second system that, when cascaded with T, yields the identity system. Gain of 5 Gain of 1/5

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Causality A system is causal if the output at any time to is dependent on the input only for t≤t o Y(t)=x(t-2) is causal. The present output is equal to the input of 2 s ago Y(t)=x(t+2) is not causal since the output at t=0 is equal to the input at t=2s. All physical real-time systems are causal because we can not anticipate the future!

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Non-Causal signal

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Continuous-Time Systems - Stability Stability has Many different definitions Bounded-input-bounded-output Example: – An ideal amplifier y(t) = 10 x(t) B2=10 B1 – Square system: y(t)=x 2 (t) B2=B1 2 A system can be unstable or marginally stable If a system is used responsibly (the input is bounded), the system will behave predictably.

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Time Invariance A system is time invariant if a time shift in the input signal results only in the same time shift in the output signal

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Test for Time Invariance A system is time invariant if a time shift in the input signal results only in the same time shift in the output signal The system is time invariant if

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Example 1 The system is time invariant.

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Example 2 The system is time varying.

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Example 3: Time Reversal

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Continuous-Time Systems – Linearity A linear system must satisfy superposition condition (additive and homogeneity

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