Signals and Systems EE235 Today’s Cultural Education: Liszt: Von der Wiege bis zum Grabe, Symphonic Poem No. 13 @SeattleSymphony October 6th and 8th Leo Lam © 2010-2011

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 © 2010-2011

Today’s menu Dirac Delta Function (cont’) System properties And prizes… System properties Linearity Time invariance Stability Invertibility Causality Memory Leo Lam © 2010-2011

Recap: Dirac Delta function δ(t) “a spike of signal at time 0” It has height = , width = 0, and area = 1 δ(t) Rules δ(t)=0 for t≠0 Area: If x(t) is continuous at t0, otherwise undefined Leo Lam © 2010-2011

Scaling the Dirac Delta Proof: Suppose a>0 a<0 Leo Lam © 2010-2011

Scaling the Dirac Delta Proof: Generalizing the last result Leo Lam © 2010-2011

Summary: Dirac Delta Function Multiplication of a function that is continuous at t0 by δ(t) gives a scaled impulse. Sifting Properties Relation with u(t) Leo Lam © 2010-2011

Dirac Delta – Another one Evaluate For Prize 1 (20% off Husky Shop) Do it on board. Variable is x, y is treated as a constant. Leo Lam © 2010-2011

Slightly harder Is this function periodic? If so, what is the period? (Sketch to prove your answer) For Prize 2 (20% off U Bookstore) Not periodic – delta function spreads with k2 for t>0 And x(t) = 0 for t<0 Do it on board. Variable is x, y is treated as a constant. Leo Lam © 2010-2011

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 © 2010-2011

Power The power of a signal Definition: A power signal is any signal such that: Physically: this signal has finite average power Leo Lam © 2010-2011

Signal power and energy What is the energy of u(t) Why? Leo Lam © 2010-2011

Signal power and energy What is the power of u(t) Leo Lam © 2010-2011

Summary: Signal energy/power Defined Energy and Power of signals Defined Energy signal/Power signal Leo Lam © 2010-2011

x(t) x(t) x(t) x(t) System delay amplifier integrator sifter Leo Lam © 2010-2011

“System Response is the same no matter when you run the system.” System properties Linearity: A System is Linear if it meets the following two criteria: Time-invariance: A System is Time-Invariant if it meets this criterion If and Then If Then “System Response is the same no matter when you run the system.” “System Response to a linear combination of inputs is the linear combination of the outputs.” If Then Leo Lam © 2010-2011

“The system doesn’t blow up if given reasonable inputs.” System properties Stability: A System is BIBO Stable if it meets this criterion Invertibility: A System is Invertible if it meets this criterion: BIBO = “Bounded input, bounded output” If Then “The system doesn’t blow up if given reasonable inputs.” “If you know the output signal, then you know exactly what the input signal was.” If You can undo the effects of the system. Leo Lam © 2010-2011

System properties Causality: A System is Causal if it meets this criterion Memory: A System is Memoryless if it meets this criterion If T{x(t)}=y(t) then y(t+a) depends only on x(t+b) where b<=a The output depends only on current or past values of the input. “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!) 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.) Leo Lam © 2010-2011

Summary: System properties Leo Lam © 2010-2011