1. Basic System Properties

2. Memory Invertibility Causality Stability Time Invariance Linearity

3. Memory Memoryless –output for each value of independent variable at a given time is dependent on the input at only that same time y[n] = 2*x[n] - x^2*[n] memoryless Capacitor is a C-T system with memory In physical systems memory is directly associated with the storage of energy y[n] = x[n-1] memory D-T systems implemented with uP, memory is associated with storage registers Memory typically suggests storing past values but definition covers systems with outputs dependant upon future values of input and output

4. Invertibilty & Inverse Invertible –Distinct inputs lead to distinct outputs y[n] = 2*x[n] inverse system is y[n] = ½*x[n] Noninvertible systems –y[n] = 0 violates distinct outputs –y(t) = x^2(t) can’t tell sign of input from the output Encoding/Decoding Lossless compression

5. Causality Non-anticipative Depends only on present and past values of inputs Non-causal –output has a value before input –output responds to an input that hasn't occured yet Causal y[n] = y[n-1], y[n] = Sk=-inf to n x[k] Non-causal y[n] = x[n] – x[n+1], y(t) = x(t+1) All Memoryless systems are causal – Why? Causality not a constraint in image processing In processing signals recorded previously (speech, geophysical, meterological) we are not constrained to causal processing y[n] = x[-n] causal for n > 0 but what about n < 0? y(t) = x(t)*cos(t+1) causal or noncausal?

6. Stability Stable system –Small inputs lead to responses that do not diverge Stable – pendulum Unstable – inverted pendulum, bank account BIBO – Bounded Input = Bounded Output If we suspect a system is unstable –Look for a specific bounded input that leads to an unbounded output –One example proves unstable –If one example difficult to find use a different method Try unit step on y(t) = tx(t)

7. Time Invariance Behavior and characteristics fixed over time –R C circuit – same results today as tomorrow System is Time Invariant if –A time shift in the input signal results in –Identical time shift in the output signal A system is Time Invariant if – y[n]=x[n] and y[n-n0]=x[n-n0] Examples –y(t) = sin[x(t)] use t-t0 concept to prove –y[n] = nx[n] use x[n]=d[n] & x[n]=d[n-1] to disprove –y(t) = 2x(t) ?

8. Linearity A linear system is a system that possesses the important property of superposition –The response to x1(t) + x2(t) is y1(t) + y2(t) –The response to ax1(t) is ay1(t) where a is any complex constant Systems can be Linear without being Time Invariant Systems can be Time Invariant without being Linear