Leo Lam © Signals and Systems EE235

Leo Lam © Arthur’s knights Who was the largest knight at King Arthur’s round table? Sir Cumfrence, he got his size from eating too much pie.

Leo Lam © Today’s menu Dirac Delta Function (cont’) System properties –Linearity –Time invariance –Stability –Invertibility –Causality –Memory

Recap: Dirac Delta function δ(t) Leo Lam © “a spike of signal at time 0” 0 It has height = , width = 0, and area = 1 δ(t) Rules 1.δ(t)=0 for t≠0 2.Area: 3. If x(t) is continuous at t 0, otherwise undefined

Scaling the Dirac Delta Proof: Suppose a>0 a<0 Leo Lam ©

Scaling the Dirac Delta Proof: Generalizing the last result Leo Lam ©

Multiplication of a function that is continuous at t 0 by δ(t) gives a scaled impulse. Sifting Properties Relation with u(t) Summary: Dirac Delta Function Leo Lam ©

Energy and power The energy of a signal Definition: An energy signal is any signal such that: Physically: this signal has finite energy Leo Lam ©

Power The power of a signal Definition: A power signal is any signal such that: Physically: this signal has finite average power Leo Lam ©

Signal power and energy What is the energy of u(t) Leo Lam © Why?

Signal power and energy What is the power of u(t) Leo Lam ©

Summary: Signal energy/power Defined Energy and Power of signals Defined Energy signal/Power signal Leo Lam ©

System Leo Lam © delay amplifier integrator sifter x(t)

System properties Leo Lam © Linearity: A System is Linear if it meets the following two criteria: Time-invariance: A System is Time-Invariant if it meets this criterion Ifand Then If Then IfThen “System Response to a linear combination of inputs is the linear combination of the outputs.” “System Response is the same no matter when you run the system.”

System properties Leo Lam © Stability: A System is BIBO Stable if it meets this criterion Invertibility: A System is Invertible if it meets this criterion: “If you know the output signal, then you know exactly what the input signal was.” BIBO = “Bounded input, bounded output” If Then “The system doesn’t blow up if given reasonable inputs.” You can undo the effects of the system. If

System properties Leo Lam © Causality: A System is Causal if it meets this criterion Memory: A System is Memoryless if it meets this criterion “The output depends only on the current value of the input.” “The system does not anticipate the input.” (It does not laugh before it’s tickled!) The output depends only on current or past values of the input. If T{x(t)}=y(t) then y(t+a) depends only on x(t+b) where b<=a If T{x(t)}=y(t) then y(t+a) depends only on x(t+a) (If a system is memoryless, it is also causal.)

Summary: System properties Leo Lam ©

Test for Causality Leo Lam © System is causal if output depends only on past and present input signal 1)y(t) = 4x(t) 2)y(t) = x(t –3) 3)y(t) = x(t + 5) 4)y(t) = x(3t) 5)y(t) = (t + 5)x(t) 6)y(t) = x(-t) causal (amplification) causal (delay) non-causal (time-shift forward, y(0)=x(5)) non-causal (speed-up, y(1)=x(3)) causal (ramp times x(t)) non-causal (time reverse, negative time needs future, y(-1)=x(1))

Causality Example Leo Lam © What values of t 0 would make T causal? causal if

Causality Example Leo Lam © Is T causal? YES Depends only on past and present signals

Causality Example Leo Lam © What values of a would make T causal?

Causality Example Leo Lam © NOT causal: x(t)’s include t =t+1 NOT causal: x(t)’s include t =2t Causal: Change variable, y(t) does not depend on future t.