Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 14.
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Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 14
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Transcript of Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 14.
- Slide 1
- Slide 2
- Leo Lam 2010-2012 Signals and Systems EE235 Lecture 14
- Slide 3
- Convolution Properties Leo Lam 2010-2011 2
- Slide 4
- Commutative Leo Lam 2010-2011 3 Commutative: Doesnt matter which signal to flip, its the same Pick the easier one!
- Slide 5
- Associative Leo Lam 2010-2011 4 Associative: Order doesnt matter h 1 (t)h 2 (t) x(t)y(t) The overall response of two LTI systems in series is given by
- Slide 6
- Distributive Leo Lam 2010-2011 5 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
- Slide 7
- More Convolution Properties Leo Lam 2010-2011 6 Convolution of any signal with an impulse, gives the same signal Convolution of any signal with a shifted impulse, shifts the signal
- Slide 8
- More Convolution Properties Leo Lam 2010-2012 7 Convolution of the impulse response of an LTI system with a unit step, gives its step response s(t).
- Slide 9
- Another implication for LTI Leo Lam 2010-2012 8 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)
- Slide 10
- More Convolution Properties Leo Lam 2010-2012 9 Convolution with a time-shifted signal, gives a time shifted output: If then
- Slide 11
- Summary: Leo Lam 2010-2012 10 Convolution properties Commutative Associative Distributive Convolve with impulse Convolve with shifted impulse Convolve h(t) with u(t) gives s(t)
- Slide 12
- Echo Properties Leo Lam 2010-2012 11 Echo properties of impulse * 3 x(t) (t-3) t t t 3 = What does this system do?
- Slide 13
- Echo Properties Leo Lam 2010-2012 12 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)
- Slide 14
- Echo Properties Leo Lam 2010-2012 13 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
- Slide 15
- Echo Properties Leo Lam 2010-2012 14 More With multiple time shifts, add them all up.
- Slide 16
- Finding Impulse Response Leo Lam 2010-2012 15 Example: find h(t) when 1) Plug in (t) for x(t)
- Slide 17
- System properties testing given h(t) Leo Lam 2010-2012 16 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
- Slide 18
- Causality for LTI Leo Lam 2010-2012 17 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