6.003: Signals and Systems Signals and Systems September 8, 2011 1 - PowerPoint PPT Presentation

6.003: Signals and Systems Signals and Systems September 8, 2011 1

6.003: Signals and Systems Today’s handouts: Single package containing • Slides for Lecture 1 • Subject Information & Calendar Lecturer: Denny Freeman Instructors: Elfar Adalsteinsson Russ Tedrake TAs: Phillip Nadeau Wenbang Xu Website: mit.edu/6.003 Text: Signals and Systems – Oppenheim and Willsky 2

6.003: Homework Doing the homework is essential for understanding the content. • where subject matter is/isn’t learned • equivalent to “practice” in sports or music Weekly Homework Assignments • Conventional Homework Problems plus • Engineering Design Problems (Python/Matlab) Open Office Hours ! • Stata Basement • Mondays and Tuesdays, afternoons and early evenings 3

6.003: Signals and Systems Collaboration Policy Discussion of concepts in homework is encouraged • Sharing of homework or code is not permitted and will be re- • ported to the COD Firm Deadlines Homework must be submitted by the published due date • Each student can submit one late homework assignment without • penalty. Grades on other late assignments will be multiplied by 0.5 (unless • excused by an Instructor, Dean, or Medical Official). 4

6.003 At-A-Glance Tuesday Wednesday Thursday Friday Registration Day: R1: Continuous & L1: Signals and R2: Difference Sep 6 No Classes Discrete Systems Systems Equations L2: Discrete-Time HW1 R3: Feedback, L3: Feedback, Sep 13 R4: CT Systems Systems due Cycles, and Modes Cycles, and Modes L4: CT Operator HW2 Student Holiday: L5: Laplace R5: Laplace Sep 20 Representations due No Recitation Transforms Transforms HW3 L7: Transform R7: Transform Sep 27 L6: Z Transforms R6: Z Transforms due Properties Properties L8: Convolution; Exam 1 L9: Frequency R8: Convolution Oct 4 EX4 Impulse Response No Recitation Response and Freq. Resp. Columbus Day: HW5 R9: Bode Diagrams L10: Bode R10: Feedback and Oct 11 No Lecture due Diagrams Control L11: DT Feedback HW6 R11: CT Feedback L12: CT Feedback R12: CT Feedback Oct 18 and Control due and Control and Control and Control L13: CT Feedback Exam 2 L14: CT Fourier R13: CT Fourier Oct 25 HW7 and Control No Recitation Series Series L15: CT Fourier EX8 R14: CT Fourier L16: CT Fourier R15: CT Fourier Nov 1 Series due Series Transform Transform L17: CT Fourier HW9 R16: DT Fourier L18: DT Fourier Veterans Day: Nov 8 Transform due Transform Transform No Recitation L19: DT Fourier HW10 Exam 3 L20: Fourier R17: Fourier Nov 15 Transform No Recitation Relations Relations EX11 R18: Fourier Thanksgiving: Thanksgiving: Nov 22 L21: Sampling due Transforms No Lecture No-Recitation HW12 Nov 29 L22: Sampling R19: Modulation L23: Modulation R20: Modulation due L25: Applications Dec 6 L24: Modulation EX13 R21: Review Study Period of 6.003 Breakfast with Study Period: Final Exams: Dec 13 EX13 R22: Review Staff No Lecture No-Recitation Dec 20 finals Final Examinations: No Classes finals finals finals finals 5

6.003: Signals and Systems Weekly meetings with class representatives • help staff understand student perspective • learn about teaching Tentatively meet on Thursday afternoon Interested? ... 6

The Signals and Systems Abstraction Describe a system (physical, mathematical, or computational) by the way it transforms an input signal into an output signal . signal signal system in out 7

Example: Mass and Spring x ( t ) y ( t ) x ( t ) y ( t ) mass & spring t t system 8

Example: Tanks r 0 ( t ) h 1 ( t ) r 1 ( t ) h 2 ( t ) r 2 ( t ) r 0 ( t ) r 2 ( t ) tank t t system 9

Example: Cell Phone System sound out sound in sound in sound out cell t phone t system 10

Signals and Systems: Widely Applicable The Signals and Systems approach has broad application: electrical, mechanical, optical, acoustic, biological, financial, ... x ( t ) y ( t ) mass & spring t t system r 0 ( t ) h 1 ( t ) r 1 ( t ) r 0 ( t ) r 2 ( t ) tank h 2 ( t ) t t system r 2 ( t ) sound in sound out cell t phone t system 11

Signals and Systems: Modular The representation does not depend upon the physical substrate. sound out sound in E/M optic E/M sound cell cell sound tower tower out in phone fiber phone focuses on the flow of information , abstracts away everything else 12

Signals and Systems: Hierarchical Representations of component systems are easily combined. Example: cascade of component systems E/M E/M optic sound cell cell sound tower tower in phone fiber phone out Composite system sound sound cell phone system in out Component and composite systems have the same form, and are analyzed with same methods. 13

Signals and Systems Signals are mathematical functions. • independent variable = time • dependent variable = voltage, flow rate, sound pressure x ( t ) y ( t ) mass & t spring t system r 0 ( t ) r 2 ( t ) tank t t system sound in sound out cell t phone t system 14

Signals and Systems continuous “time” (CT) and discrete “time” (DT) x ( t ) x [ n ] n t 0 2 4 6 8 10 0 2 4 6 8 10 Signals from physical systems often functions of continuous time. • mass and spring • leaky tank Signals from computation systems often functions of discrete time. • state machines: given the current input and current state, what is the next output and next state. 15

Signals and Systems Sampling: converting CT signals to DT x ( t ) x [ n ] = x ( nT ) n t 0 T 2 T 4 T 6 T 8 T 10 T 0 2 4 6 8 10 T = sampling interval Important for computational manipulation of physical data. • digital representations of audio signals (e.g., MP3) • digital representations of images (e.g., JPEG) 16

Signals and Systems Reconstruction: converting DT signals to CT zero-order hold x ( t ) x [ n ] n t 0 2 4 6 8 10 0 2 T 4 T 6 T 8 T 10 T T = sampling interval commonly used in audio output devices such as CD players 17

Signals and Systems Reconstruction: converting DT signals to CT piecewise linear x ( t ) x [ n ] n t 0 2 4 6 8 10 0 2 T 4 T 6 T 8 T 10 T T = sampling interval commonly used in rendering images 18

Check Yourself Computer generated speech (by Robert Donovan) f ( t ) t Listen to the following four manipulated signals: f 1 ( t ) , f 2 ( t ) , f 3 ( t ) , f 4 ( t ) . How many of the following relations are true? • f 1 ( t ) = f (2 t ) • f 2 ( t ) = − f ( t ) • f 3 ( t ) = f (2 t ) • f 4 ( t ) = 1 3 f ( t ) 19

Check Yourself Computer generated speech (by Robert Donovan) f ( t ) t Listen to the following four manipulated signals: f 1 ( t ) , f 2 ( t ) , f 3 ( t ) , f 4 ( t ) . How many of the following relations are true? 2 √ • f 1 ( t ) = f (2 t ) • f 2 ( t ) = − f ( t ) X • f 3 ( t ) = f (2 t ) X √ • f 4 ( t ) = 1 3 f ( t ) 20

Check Yourself f ( x, y ) y 250 0 − 250 x − 250 0 250 How many images match the expressions beneath them? y y y 250 250 250 0 0 0 − 250 − 250 − 250 x x x − 250 0 250 − 250 0 250 − 250 0 250 f 1 ( x, y )= f (2 x, y ) ? f 2 ( x, y )= f (2 x − 250 , y ) ? f 3 ( x, y )= f ( − x − 250 , y ) ? 21

Check Yourself y y y y 250 250 250 250 0 0 0 0 − 250 − 250 − 250 − 250 x x x x − 250 0 250 − 250 0 250 − 250 0 250 − 250 0 250 f ( x, y ) f 1 ( x, y ) = f (2 x, y ) ? f 2 ( x, y ) = f (2 x − 250 , y ) ? f 3 ( x, y ) = f ( − x − 250 , y ) ? √ x = 0 → f 1 (0 , y ) = f (0 , y ) x = 250 → f 1 (250 , y ) = f (500 , y ) X √ x = 0 → f 2 (0 , y ) = f ( − 250 , y ) √ x = 250 → f 2 (250 , y ) = f (250 , y ) x = 0 → f 3 (0 , y ) = f ( − 250 , y ) X x = 250 → f 3 (250 , y ) = f ( − 500 , y ) X 22

Check Yourself f ( x, y ) y 250 0 − 250 x − 250 0 250 How many images match the expressions beneath them? y y y 250 250 250 0 0 0 − 250 − 250 − 250 x x x − 250 0 250 − 250 0 250 − 250 0 250 f 1 ( x, y )= f (2 x, y ) ? f 2 ( x, y )= f (2 x − 250 , y ) ? f 3 ( x, y )= f ( − x − 250 , y ) ? 23

The Signals and Systems Abstraction Describe a system (physical, mathematical, or computational) by the way it transforms an input signal into an output signal . signal signal system in out 24

Example System: Leaky Tank Formulate a mathematical description of this system. r 0 ( t ) h 1 ( t ) r 1 ( t ) What determines the leak rate? 25

Check Yourself The holes in each of the following tanks have equal size. Which tank has the largest leak rate r 1 ( t ) ? 1. 2. 3. 4. 26

Check Yourself The holes in each of the following tanks have equal size. Which tank has the largest leak rate r 1 ( t ) ? 2 1. 2. 3. 4. 27

Example System: Leaky Tank Formulate a mathematical description of this system. r 0 ( t ) h 1 ( t ) r 1 ( t ) Assume linear leaking: r 1 ( t ) ∝ h 1 ( t ) What determines the height h 1 ( t ) ? 28

Example System: Leaky Tank Formulate a mathematical description of this system. r 0 ( t ) h 1 ( t ) r 1 ( t ) Assume linear leaking: r 1 ( t ) ∝ h 1 ( t ) dh 1 ( t ) Assume water is conserved: ∝ r 0 ( t ) − r 1 ( t ) dt dr 1 ( t ) ∝ r 0 ( t ) − r 1 ( t ) Solve: dt 29

Check Yourself What are the dimensions of constant of proportionality C ? � � dr 1 ( t ) = C r 0 ( t ) − r 1 ( t ) dt 30