Leo Lam © Signals and Systems EE235 Lecture 14

Convolution Properties Leo Lam ©

Commutative Leo Lam © Commutative: Doesn’t matter which signal to flip, it’s the same Pick the easier one!

Associative Leo Lam © Associative: Order doesn’t matter h 1 (t)h 2 (t) x(t)y(t) The overall response of two LTI systems in series is given by

Distributive Leo Lam © Distributive: Two types h 1 (t) h 2 (t) x(t) y(t) + The overall response of two systems in parallel is h(t) x 1 (t) y(t) + x 2 (t) “Divide and conquer” for input signals

More Convolution Properties Leo Lam © Convolution of any signal with an impulse, gives the same signal Convolution of any signal with a shifted impulse, shifts the signal

More Convolution Properties Leo Lam © Convolution of the impulse response of an LTI system with a unit step, gives its step response s(t).

Another implication for LTI Leo Lam © Recall: d/dt u(t) h(t) s(t) h(t) d/dt u(t)  (t) “taking the derivative” is an LTI system, and using associative properties: We can find the impulse response of a system from its step response s(t)

More Convolution Properties Leo Lam © Convolution with a time-shifted signal, gives a time shifted output: –If –then

Summary: Leo Lam © Convolution properties –Commutative –Associative –Distributive –Convolve with impulse –Convolve with shifted impulse –Convolve h(t) with u(t) gives s(t)

Echo Properties Leo Lam © Echo properties of impulse * 3 x(t)  (t-3) t t t 3 = What does this system do?

Echo Properties Leo Lam © Multiple echoes (your turn) * 3 x(t) (t) +0.5(t-3)+0.25(t-6) t t 6 3 t 6 = (1) (0.5) (0.25)

Echo Properties Leo Lam © Another example * 2 x(t) h(t)=  (t) +0.5  (t-2) t t (1) (0.5) 1 2 t (0.5) (1.5) (1) 13 Solve and plot? Hint: Distribute

Echo Properties Leo Lam © More… With multiple time shifts, add them all up.

Finding Impulse Response Leo Lam © Example: find h(t) when 1) Plug in  (t) for x(t)

System properties testing given h(t) Leo Lam © Impulse response h(t) fully specifies an LTI system Gives additional tools to test system properties for LTI systems Additional ways to manipulate/simplify problems, too

Causality for LTI Leo Lam © A system is causal if the output does not depend on future times of the input An LTI system is causal if h(t)=0 for t<0 Generally: If LTI system is causal:

Causality for LTI Leo Lam © An LTI system is causal if h(t)=0 for t<0 If h(t) is causal, h( t- )=0 for all ( t- )<0 or all t <  Only Integrate to t for causal systems

Convolution of two causal signals Leo Lam © A signal x(t) is a causal signal if x(t)=0 for all t<0 Consider: If x 2 (t) is causal then x 2 ( t- )=0 for all ( t- )<0 i.e. x 1 (  )x 2 ( t- )=0 for all t<  If x 1 (t) is causal then x 1 (  )=0 for all  <0 i.e. x 1 (  )x 2 ( t- )=0 for all  <0 Only Integrate from 0 to t for 2 causal signals

Step response of LTI system Leo Lam © Impulse response h(t) Step response s(t) For a causal system: T u(t)*h(t) u(t) T h(t)  (t) Only Integrate from 0 to t = Causal! (Proof for causality)

Step response example for LTI system Leo Lam © If the impulse response to an LTI system is: First: is it causal? Find s(t)

Stability of LTI System Leo Lam © An LTI system – BIBO stable Impulse response must be finite Bounded input system Bounded output B 1, B 2, B 3 are constants

Stability of LTI System Leo Lam © Is this condition sufficient for stability? Prove it: abs(sum)≤sum(abs) abs(prod)=prod(abs) bounded input if Q.E.D.

Stability of LTI System Leo Lam © Is h(t)=u(t) stable? Need to prove that

Invertibility of LTI System Leo Lam © A system is invertible if you can find the input, given the output (undo-ing possible) You can prove invertibility of the system with impulse response h(t) by finding the impulse response of the inverse system h i (t) Often hard to do…don’t worry for now unless it’s obvious

LTI System Properties Leo Lam © Example –Causal? –Stable? –Invertible? YES

LTI System Properties Leo Lam © Example –Causal? –Stable? YES

LTI System Properties Leo Lam © How about these? Causal/Stable? Stable, not causal Causal, not stable Stable and causal

LTI System Properties Summary Leo Lam © For ALL systems y(t)=T{x(t)} x-y equation describes system Property tests in terms of basic definitions –Causal: Find time region of x() used in y(t) –Stable: BIBO test or counter-example For LTI systems ONLY y(t)=x(t)*h(t) h(t) =impulse response Property tests on h(t) –Causal: h(t)=0 t<0 –Stable:

Leo Lam © Summary LTI system properties