6.003: Signals and Systems Lecture 1 February 2, 201016.003: Signals and SystemsSignals and SystemsFebruary 2, 20106.003: Signals and SystemsToday’s handouts: Single package containing• Slides for Lecture 1• Subject Information & CalendarLecturer: Denny Freeman ([email protected])Instructors: Peter Hagelstein ([email protected])Rahul Sarpeshkar ([email protected])TAs: Sefa Demirtas ([email protected])Ulric Ferner ([email protected])Alison Laferriere ([email protected])Website: mit.edu/6.003Text: Signals and Systems – Oppenheim and Willsky6.003: HomeworkDoing the homework is essential for understanding the content.• where subject matter is/isn’t learned• equivalent to “practice” in sports or musicWeekly Homework Assignments• Conventional Homework Problems plus• Engineering Design Problems (Python/Matlab)Open Office Hours !• Stata Basement (32-044)• Mondays and Tuesdays, afternoons and early evenings6.003: Signals and SystemsCollaboration Policy•Discussion of concepts in homework is encouraged• Sharing of homework or code is not permitted and will be re-ported to the CODFirm Deadlines• Homework must be submitted in recitation on due date• Each student can submit one late homework assignment withoutpenalty.• Grades on other late assignments will be multiplied by 0.5 (unlessexcused by an Instructor, Dean, or Medical Official).6.003 At-A-GlanceFeb 2L1: Signals andSystemsR1: Continuous &Discrete SystemsL2: Discrete-TimeSystemsR2: DifferenceEquationsFeb 9L3: Feedback,Cycles, and ModesHW1dueR3: Feedback,Cycles, and ModesL4: CT OperatorRepresentationsR4: CT SystemsFeb 16Presidents Day:Monday ScheduleHW2dueR5: CT OperatorRepresentationsL5: Second-OrderSystemsR6: Second-OrderSystemsFeb 23L6: Laplace and ZTransformsHW3dueR7: Laplace and ZTransformsL7: TransformPropertiesR8: TransformPropertiesMar 2L8: Convolution;Impulse ResponseEX4Exam 1no recitationL9: FrequencyResponseR9: Convolutionand Freq. Resp.Mar 9L10: BodeDiagramsHW5dueR10: BodeDiagramsL11: DT Feedbackand ControlR11: Feedback andControlMar 16L12: CT Feedbackand ControlHW6dueR12: CT Feedbackand ControlL13: CT Feedbackand ControlR13: CT Feedbackand ControlMar 23spring spring spring spring springMar 30L14: CT FourierSeriesHW7R14: CT FourierSeriesL15: CT FourierSeriesR15: CT FourierSeriesApr 6L16: CT FourierTransformEX8dueExam 2no recitationL17: CT FourierTransformR16: CT FourierTransformApr 13L18: DT FourierTransformHW9dueR17: DT FourierTransformL19: DT FourierTransformR18: DT FourierTransformApr 20Patriots DayVacationHW10R19: FourierTransformsL20: FourierRelationsR20: FourierRelationsApr 27L21: SamplingEX11dueExam 3no recitationL22: Sampling R21: SamplingMay 4L23: ModulationHW12dueR22: Modulation L24: Modulation R23: ModulationMay 11L25: Applicationsof 6.003EX13 R24: ReviewBreakfast withStaffStudy PeriodMay 18finals finals finals finals finalsTuesday Wednesday Thursday FridaySpring WeekFinal Examination Period6.003: Signals and SystemsWeekly meetings with class representatives• help staff understand student perspective• learn about teachingOne representative from each section (4 total)Tentatively meet on Thursday afternoonInterested? ... Send email to [email protected]: Signals and Systems Lecture 1 February 2, 20102The Signals and Systems AbstractionDescribe a system (physical, mathematical, or computational) bythe way it transforms an input signal into an output signal.systemsignalinsignaloutExample: Mass and Springx(t)y(t)mass &springsystemx(t) y(t)t tExample: Tanksr0(t)r1(t)r2(t)h1(t)h2(t)tanksystemr0(t) r2(t)t tExample: Cell Phone Systemsound insound outcellphonesystemsound in sound outt tSignals and Systems: Widely ApplicableThe Signals and Systems approach has broad application: electrical,mechanical, optical, acoustic, biological, financial, ...mass &springsystemx(t) y(t)t tr0(t)r1(t)r2(t)h1(t)h2(t)tanksystemr0(t) r2(t)t tcellphonesystemsound in sound outt tSignals and Systems: ModularThe representation does not depend upon the physical substrate.sound insound outcellphonetower towercellphonesoundinE/MopticfiberE/Msoundoutfocuses on the flow of information, abstracts away everything else6.003: Signals and Systems Lecture 1 February 2, 20103Signals and Systems: HierarchicalRepresentations of component systems are easily combined.Example: cascade of component systemscellphonetower towercellphonesoundinE/MopticfiberE/MsoundoutComposite systemcell phone systemsoundinsoundoutComponent and composite systems have the same form, and areanalyzed with same methods.Signals and SystemsSignals are mathematical functions.• independent variable = time• dependent variable = voltage, flow rate, sound pressuremass &springsystemx(t) y(t)t ttanksystemr0(t) r2(t)t tcellphonesystemsound in sound outt tSignals and Systemscontinuous “time” (CT) and discrete “time” (DT)tx(t)0 246 8 10nx[n]0 246 8 10Many physical systems operate in continuous time.• mass and spring• leaky tankDigital computations are done in discrete time.• state machines: given the current input and current state, whatis the next output and next state.Signals and SystemsSampling: converting CT signals to DTtx(t)0T 2T 4T 6T 8T 10Tnx[n] = x(nT )0 246 8 10T = sampling intervalImportant for computational manipulation of physical data.• digital representations of audio signals (e.g., MP3)• digital representations of pictures (e.g., JPEG)Signals and SystemsReconstruction: converting DT signals to CTzero-order holdnx[n]0 246 8 10tx(t)02T 4T 6T 8T 10TT = sampling intervalcommonly used in audio output devices such as CD playersSignals and SystemsReconstruction: converting DT signals to CTpiecewise linearnx[n]0 246 8 10tx(t)02T 4T 6T 8T 10TT = sampling intervalcommonly used in rendering images6.003: Signals and Systems Lecture 1 February 2, 20104Check YourselfComputer generated speech (by Robert Donovan)tf(t)Listen to the following four manipulated signals:f1(t), f2(t), f3(t), f4(t).How many of the following relations are true?• f1(t) = f(2t)• f2(t) = −f(t)• f3(t) = f(2t)• f4(t) = 2f(t)Check Yourself−250 0 250−2500 250yxf1(x, y)=f(2x, y) ?−250 0 250−2500 250yxf2(x, y)=f(2x− 250, y) ?−250 0 250−2500 250yxf3(x, y)=f(−x− 250, y) ?−250 0 250−2500 250yxf(x, y)How many images match the expressions beneath them?The Signals and Systems AbstractionDescribe a system (physical, mathematical, or computational) bythe way it transforms an input signal into an output signal.systemsignalinsignaloutExample System: Leaky
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