Of michigan fall 2012 september 7 2012 19 174 lecture

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Unformatted text preview: (Univ. of Michigan) Fall 2012 September 7, 2012 19 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (1) x (τ ) y (t ) τ h(0 − τ ) τ t x (τ )h(0 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 20 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (2) x (τ ) y (t ) τ h(.5 − τ ) τ t x (τ )h(0.5 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 21 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (3) x (τ ) y (t ) τ h(1 − τ ) τ t x (τ )h(1 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 22 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (4) x (τ ) y (t ) τ h(1.5 − τ ) τ t x (τ )h(1.5 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 23 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (5) x (τ ) y (t ) τ h(2 − τ ) τ t x (τ )h(2 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 24 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (6) x (τ ) y (t ) τ h(2.5 − τ ) τ t x (τ )h(2.5 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 25 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (7) x (τ ) y (t ) τ h(3 − τ ) τ t x (τ )h(3 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 26 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (8) x (τ ) y (t ) τ h(4 − τ ) τ t x (τ )h(4 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 27 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (9) x (τ ) y (t ) τ h(5 − τ ) τ t x (τ )h(5 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 28 / 174 Lecture Notes 2 Linear Systems Example of Graphical Convolution (10) x (τ ) y (t ) τ h(6 − τ ) τ t x (τ )h(6 − τ ) τ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 29 / 174 Lecture Notes 2 Linear Systems Properties of Linear Time-Invariant (LTI) Systems Linearity: If the output of a linear system is y1 (t ) when x1 (t ) is the input and the output of is y2 (t ) when x2 (t ) is the input then the output due to α1 x1 (t ) + α2 x2 (t ) is α1 y1 (t ) + α2 y2 (t ). x1 (t ) → y1 (t ) x2 (t ) → y2 (t ) ⇓ α1 x1 (t ) + α2 x2 (t ) → α1 y1 (t ) + α2 y2 (t ) Time-Invariance: If the output of a linear system is y (t ) when x (t ) is the input then the output due to x (t − τ ) is y (t − τ ). EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 30 / 174 Lecture Notes 2 Linear Systems Example of Linearity, Time Invariance Consider a filter with impulse response that is rectangular of duration T . Find the output due to a sequence of rectangular pulses. Linearity, Time Invariance If the input is shifted in time, the output is shifted in time The output due to the sum of two inputs is the sum of the two outputs h(t ) h(t ) t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 31 / 174 Lecture Notes 2 Linear Systems Example of Linearity, Time Invariance x1 (t ) y1 (t ) t t x2 (t ) y2 (t ) t h(t ) t h(t ) x1 (t ) + x2 (t ) y1 (t ) + y2 (t ) t t EECS 455 (Univ. of Michigan) t Fall 2012 September 7, 2012 32 / 174 Lecture Notes 2 Linear Systems Four rectangular pulses 6 4 x(τ) 2 h(t−τ) 0 −2 x(τ) h(t−τ) −4 conv(x,h) −6 −2 −1 0 1 2 3 4 5 6 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 33 / 174 Lecture Notes 2 Linear Systems Example of Linearity, Time Invariance y1(t) 1 0.5 h(t ) 0 −0.5 −4 −3 −2 −1 0 time 1 2 3 4 0.5 0 −0.5 −4 1.5 −3 −2 −1 0 time 1 2 3 4 1 h(t) 1 1.5 1 x (t) 1.5 0.5 0 −0.5 −4 EECS 455 (Univ. of Michigan) −3 −2 −1 0 time 1 Fall 2012 2 3 4 September 7, 2012 34 / 174 Lecture Notes 2 Linear Systems Example of Linearity, Time Invariance 1.5 1.5 x1(t) 1 y1(t), y2(t) 0.5 2 x (t), x (t) 0.5 1 0 −0.5 −3 −2 −1 0 time 1 2 0 −0.5 h(t ) y2(t) x (t) 2 −1 −1.5 −4 y (t) 1 1 −1 3 4 −1.5 −4 1.5 −3 −2 −1 0 time 1 2 3 4 h(t) 1 0.5 0 −0.5 −4 EECS 455 (Univ. of Michigan) −3 −2 −1 0 time 1 Fall 2012 2 3 4 September 7, 2012 35 / 174 Lecture Notes 2 Linear Systems Example of Linearity, Time Invariance 1 0.5 0.5 2 y1(t)+ y2(t) 1.5 1 x (t)+ x (t) 1.5 1 0 −0.5 −0.5 h(t ) −1 −1.5 −4 0 −1 −3 −2 −1 0 time 1 2 3 4 −1.5 −4 1.5 −3 −2 −1 0 time 1 2 3 4 h(t) 1 0.5 0 −0.5 −4 EECS 455 (Univ. of Michigan) −3 −2 −1 0 time 1 Fall 2012 2 3 4 September 7, 2012 36 / 174 Lecture Notes 2 Linear Systems Total Input EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 37 / 174 Lecture Notes 2 Linear Systems Total Input Input to Filter 2 1.5 1 x(t) 0.5 0 −0.5 −1 −1.5 −2 −5 0 5 10 15 20 time EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 38 / 174 Lecture Notes 2 Linear Systems Total Output Output of Filter 1.5 1 y(t)=x(t)*h(t) 0.5 0 −0.5 −1 −1.5 −5 0 5 10 15 20 time EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 39 / 174 Lecture Notes 2 Linear Systems...
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This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.

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