2025-L17su10 - ECE2025 Summer 2010 Lecture 17 Frequency...

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1 7/6/2010 1 ECE2025 Summer 2010 Lecture 17 Frequency Response of Continuous-Time Systems 12 Jul 10 7/6/2010 3 ANNOUNCEMENTS ± HW #8 due July 20 in recitation, or July 21 in lab for L01, L02. ± Do Lab #8 on July 14-15: ± Done entirely in lab, no report ± 50 points instead of 100 ± Quiz 3: Friday July 16 7/6/2010 4 THIRD QUIZ ± Quiz #3, in lecture, Friday, July 16 ± 10% of final grade ± You can use the full 1:20-2:30 time ± Review session Thursday July 15, 6:30 pm. ± Emphasis: ± HWs some of #5 through all of #7 – mainly 6 and 7. ± Lectures #11 through #16 ± No Fourier Transforms ± Closed book, closed notes, except: ± One 8.5”X11” crib sheet allowed, handwritten , OK to write on both sides 7/6/2010 5 LECTURE OBJECTIVES ± Review of convolution ± THE THE operation for LTI LTI Systems ± Complex exponential input signals ± Frequency Response ± Cosine signals ± Real part of complex exponential ± Fourier Series thru H(j ω ) ± These are Analog Filters
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2 7/6/2010 6 LTI Systems ± Convolution defines an LTI system ± Response to a complex exponential gives frequency response H(j ω ) y ( t ) = h ( t ) x ( t ) = h ( τ ) −∞ x ( t ) d 7/6/2010 7 Convolution Properties ± Linear: ± (x 1 (t)+x 2 (t)) * h(t) = x 1 (t)*h(t)+x 2 (t)*h(t) ± Time Invariant: ± If x(t)*h(t)=y(t), then x(t-d)*h(t)=y(t-d) ± Also x(t)*h(t-d)=y(t-d) ± Commutative: ± x(t)*h(t)=h(t)*x(t) 7/6/2010 8 Result: Cascade of LTI Systems δ ( t ) h 1 ( t ) h 1 ( t ) h 2 ( t ) ( t ) h 2 ( t ) h 2 ( t ) h 1 ( t ) h ( t ) = h 1 ( t ) h 2 ( t ) = h 2 ( t ) h 1 ( t ) 7/6/2010 9 Result: Parallel connection of LTI Systems h ( t ) = h 1 ( t ) + h 2 ( t )( t ) ( t ) h 1 ( t ) h 2 ( t ) h 1 ( t ) + h 2 ( t )
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3 7/6/2010 10 Stability ± A system is stable if every bounded input produces a bounded output. ± A continuous-time LTI system is stable if and only if h ( t ) dt <∞ −∞ 7/6/2010 11 Causal Systems ± A system is causal if and only if y(t 0 ) depends only on x( τ ) for τ< t 0 . ± An LTI system is causal if and only if 0 for 0 ) ( < = t t h 7/6/2010 12 Thought Process #1 ± SUPERPOSITION (Linearity) ± Make x(t) a weighted sum of signals ± Then y(t) is also a sum—different weights DIFFERENT OUTPUT SIGNALS usually ± Use SINUSOIDS “SINUSOID IN GIVES SINUSOID OUT” ± Make x(t) a weighted sum of sinusoids ± Then y(t) is also a sum of sinusoids ± Different Magnitudes and Phase ± LTI SYSTEMS : Sinusoidal Response 7/6/2010 13 Thought Process #2
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This note was uploaded on 10/21/2010 for the course ECE 2025 taught by Professor Juang during the Summer '08 term at Georgia Institute of Technology.

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2025-L17su10 - ECE2025 Summer 2010 Lecture 17 Frequency...

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