Basic System Properties
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Memory
Invertibility
Causality
Stability
Time Invariance
Linearity
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
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
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
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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?
• Stable system
Stability
– Small inputs lead to responses that do not diverge
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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)
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) ?
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