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Memoryless System
A memoryless system is a system for which for any real number $ t_0 $, the output at $ t_0 $ depends only on that value of t.
A system with memory is a system whose output depends on the value $ t_0 $ as well as another value of t for any given $ t_0 $
Causal System
A system is causal if the output at any given time only depends on the input in present and past (not the future)
A system is not causal if the output at any given time depends on input in the future.
Linear System
A system is linear if for any complex constants a and b and for any inputs $ x_1(t) $ and $ x_2(t) $ yielding output $ y_1(t) $ and $ y_2(t) $ respectively, the response is $ a*x_1(t)+b*x_2(t) ==> [SYSTEM] ==> a*y_1(t) + b*y_2(t) $
A system is nonlinear if for any complex constants a and b and for any inputs $ x_1(t) $ and $ x_2(t) $ yielding output $ y_1(t) $ and $ y_2(t) $ respectively, the response is $ a*x_1(t)+b*x_2(t) ==> [SYSTEM] ==> z(t) $ where $ z(t) $ is NOT the constants a and b multiplied by the outputs $ y_1(t) $ and $ y_2(t) $
Time Invariant System
A system is called time invariant if for any input signal x(t) yielding output y(t) and for any time $ t_0 $ the output to the shifted input $ x(t-t_0) $ is the shifted output $ y(t-t_0) $.
A system is not time invariant if shifting the input does not yield the same output shifted by the same amount.
Stable System
1- Give a formal definition of a “memoryless system”. Give a formal definition of a “system with memory”.
2- Give a formal definition of a “causal system”. Give a formal definition of a non-causal system.
3- Give a formal definition of a “linear system”. Give a formal definition of a “non-linear system”.
4- Give a formal definition of a “time invariant system”. Give a formal definition of a “time variant system”.
5- Give a formal definition of a “stable system”. Give a formal definition of an unstable system.
