Signals and Systems was developed in 1987 as a distance-education course for engineers. An introduction to analog and digital signal processing, including discrete- and continuous-time signals, linear time-invariant systems, feedback, and data processing. http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures 2013-01-12T07:16:51+05:00 MIT OpenCourseWare http://ocw.mit.edu en-US Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong>&nbsp;Course format and overview; Demonstration of a feedback system used to stabilize an inverted pendulum; Demonstration of digital signal processing used to remove distortions and background noise from a musical recording. Mathematical representation of signals and systems.</p><p><strong>Instructor:</strong>&nbsp;Prof. Alan V. Oppenheim</p>Thumbnail - <a href= http://img.youtube.com/vi/KJnAy6hzetw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec01_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-1-introduction/id458320213?i=113398881>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/KJnAy6hzetw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-1-introduction Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Mathematical representation of signals and systems; Transformation of the independent variable of a signal (reflection, time shifts, and scaling); Sinusoidal signals: real and complex exponentials; Periodicity properties of discrete-time complex exponentials.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: sinusoidal signals, periodicity, systems, discrete-time, Mathematical representation of signals and systems, Transformation of the independent variable of a signal (reflection, time shifts, and scaling), Sinusoidal signals: real and complex exponentials, Periodicity properties of discrete-time complex exponentials<br><br>Thumbnail - <a href= http://img.youtube.com/vi/6xaaeop7gJ8/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec02_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-2-signals-systems/id458320213?i=96641910>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/6xaaeop7gJ8>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-2-signals-and-systems-part-i Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US sinusoidal signals periodicity systems discrete-time Mathematical representation of signals and systems Transformation of the independent variable of a signal (reflection, time shifts, and scaling) Sinusoidal signals: real and complex exponentials Periodicity properties of discrete-time complex exponentials MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Unit step and unit impulse signals; Block-diagram representations and interconnections of systems; System properties: memory, invertibility and inverse systems, causality, stability, time invariance, linearity.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: invertibility, time-invariance, linearity, Unit step, unit impulse signals, Block-diagram representations, interconnections of systems, System properties, memory, inverse systems, causality, stability<br><br>Thumbnail - <a href= http://img.youtube.com/vi/jGk3w1b7UXQ/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec03_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-3-signals-systems/id458320213?i=96641908>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/jGk3w1b7UXQ>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-3-signals-and-systems-part-ii Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US invertibility time-invariance linearity Unit step unit impulse signals Block-diagram representations interconnections of systems System properties memory inverse systems causality stability MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Representation of signals in terms of impulses; Convolution sum representation for discrete-time linear, time-invariant (LTI) systems: convolution integral representation for continuous-time LTI systems; Properties: commutative, associative, and distributive.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: impulses, system representation, Convolution sum, discrete-time linear, time-invariant (LTI) systems, convolution integral representation, continuous-time LTI systems, Properties, commutative, associative, distributive<br><br>Thumbnail - <a href= http://img.youtube.com/vi/_vyke3vF4Nk/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec04_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-4-convolution/id458320213?i=96641907>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/_vyke3vF4Nk>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-4-convolution Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US impulses system representation Convolution sum discrete-time linear time-invariant (LTI) systems convolution integral representation continuous-time LTI systems Properties commutative associative distributive MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Properties of linear, time-invariant systems, including the commutative, associative, and distributive properties. Also covers operational definition of impulses; cascade systems; parallel combinations; properties of convolution; discrete-time accumulator; first-order continuous-time system.</p><p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p><p>&nbsp;</p>Thumbnail - <a href= http://img.youtube.com/vi/mC3TiBJiCsY/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec05_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-5-properties-linear/id458320213?i=113398880>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/mC3TiBJiCsY>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-5-properties-of-linear-time-invariant-systems Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> First-order differential and difference equations; Solution as a sum of particular and homogeneous terms; Auxiliary conditions and relation to system linearity, causality, and time-invariance; Block-diagram representations of LTI systems described by difference equations and differential equations using adders, coefficient multipliers, and delay elements (discrete-time) or integrators (continuous-time).</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p> <p>&nbsp;</p>Keywords: difference equations, differential equations, delay elements, coefficient multipliers, delay elements, integrators, Auxiliary conditions and relation to system linearity, causality, time-invariance, Block-diagram representations of LTI systems, adders<br><br>Thumbnail - <a href= http://img.youtube.com/vi/c6jKux_RkqI/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec06_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-6-systems-represented/id458320213?i=96547370>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/c6jKux_RkqI>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-6-systems-represented-by-differential-equations Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US difference equations differential equations delay elements coefficient multipliers delay elements integrators Auxiliary conditions and relation to system linearity causality time-invariance Block-diagram representations of LTI systems adders MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Response of continuous-time LTI systems to complex exponentials: the eigenfunction property; Representation of periodic signals as linear combinations of harmonically related complex exponentials; Fourier series for continuous time: the analysis and synthesis equations; Example: symmetric and anti-symmetric periodic square wave; Approximation of periodic signals; Fourier series convergence.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: complex exponentials, periodic signals, Fourier series, square wave, Fourier series convergence, linear combinations of harmonically related complex exponentials, analysis and synthesis equations, symmetric and anti-symmetric periodic square wave, Approximation of periodic signals<br><br>Thumbnail - <a href= http://img.youtube.com/vi/nuzA75DpSuw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec07_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-7-continuous-time/id458320213?i=96547367>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/nuzA75DpSuw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-7-continuous-time-fourier-series Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US complex exponentials periodic signals Fourier series square wave Fourier series convergence linear combinations of harmonically related complex exponentials analysis and synthesis equations symmetric and anti-symmetric periodic square wave Approximation of periodic signals MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Derivation of the Fourier transform representation of aperiodic signals; Synthesis equation (inverse Fourier transform) and analysis equation (Fourier transform); Periodic signals: coefficients as samples of the transform of one period; Relationships between Fourier series and Fourier transforms; Fourier transform representation of periodic signals; Examples.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Fourier transform, aperiodic signals, Fourier series, synthesis equation, inverse Fourier transform, analysis equation, Periodic signals: coefficients as samples of the transform of one period, Relationships between Fourier series and Fourier transforms<br><br>Thumbnail - <a href= http://img.youtube.com/vi/3UkGd3LK2NY/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec08_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-8-continuous-time/id458320213?i=96547366>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/3UkGd3LK2NY>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-8-continuous-time-fourier-transform Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Fourier transform aperiodic signals Fourier series synthesis equation inverse Fourier transform analysis equation Periodic signals: coefficients as samples of the transform of one period Relationships between Fourier series and Fourier transforms MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Linearity, symmetry, time shifting, differentiation and integration, time and frequency scaling, duality, Parseval's relation; Convolution and modulation properties and the basis they provide for filtering, modulation, and sampling; Polar representation, magnitude and phase, Bode plots; Use of transform methods to analyze LTI systems characterized by differential and difference equations; Cascade and parallel form realization: first- and second-order systems.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Parseval's relation, Polar representation, time shifting, time and frequency scaling, duality, Convolution and modulation properties, magnitude and phase, Bode plots, Cascade and parallel form realization: first- and second-order systems<br><br>Thumbnail - <a href= http://img.youtube.com/vi/D1WF9YKqf3o/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec09_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-9-fourier-transform/id458320213?i=96641909>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/D1WF9YKqf3o>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-9-fourier-transform-properties Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Parseval's relation Polar representation time shifting time and frequency scaling duality Convolution and modulation properties magnitude and phase Bode plots Cascade and parallel form realization: first- and second-order systems MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Response of discrete-time LTI systems to complex exponentials; Representation of periodic signals: linear combinations of harmonically related complex exponentials; Similarities and differences with continuous time; Analysis and synthesis equations; Approximation of periodic signals and convergence; Discrete-time Fourier representation of aperiodic signals: the discrete-time Fourier transform; Fourier transform representation of periodic signals.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: complex exponentials, aperiodic signals, Analysis and synthesis equations, convergence, Fourier transform representation, periodic signals, harmonically related complex exponentials<br><br>Thumbnail - <a href= http://img.youtube.com/vi/TkMsVwzd1C0/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec10_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-10-discrete-time-fourier/id458320213?i=96547369>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/TkMsVwzd1C0>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-10-discrete-time-fourier-series Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US complex exponentials aperiodic signals Analysis and synthesis equations convergence Fourier transform representation periodic signals harmonically related complex exponentials MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Representation of aperiodic signals; Discrete-time Fourier transform properties: periodicity, linearity, symmetry, time shifting and frequency shifting, differencing and summation, time and frequency scaling, differentiation in frequency, Parseval's relation; Convolution and modulation, duality, polar representation; Calculation of frequency and impulse responses; Summary of relationships between continuous-time and discrete-time Fourier series and Fourier transforms.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Parseval's relation, polar representation, Discrete-time Fourier transform properties, periodicity, linearity, symmetry, time shifting, frequency shifting, differencing and summation, time and frequency scaling, Convolution and modulation, impulse responses, Representation of aperiodic signals<br><br>Thumbnail - <a href= http://img.youtube.com/vi/HKMY-8BqWWw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec11_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-11-discrete-time-fourier/id458320213?i=96547368>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/HKMY-8BqWWw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-11-discrete-time-fourier-transform Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Parseval's relation polar representation Discrete-time Fourier transform properties periodicity linearity symmetry time shifting frequency shifting differencing and summation time and frequency scaling Convolution and modulation impulse responses Representation of aperiodic signals MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Relation to the convolution property of Fourier transform; Ideal and non ideal frequency-selective filters: frequency-domain and time-domain characteristics; Continuous-time frequency-selective filters described by differential equations; RC low-pass and high-pass filters; Discrete-time frequency-selective filters described by difference equations; Moving average filters; Recursive discrete-time filters; Demonstration: a look at filtering in a commercial audio control room.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: filtering, frequency-selective filters, moving average filters, Ideal and non ideal frequency-selective filters, RC low-pass filters, high-pass filters, recursive discrete-time filters<br><br>Thumbnail - <a href= http://img.youtube.com/vi/P5Ce9tbK86M/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec12_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-12-filtering/id458320213?i=96547360>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/P5Ce9tbK86M>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-12-filtering Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US filtering frequency-selective filters moving average filters Ideal and non ideal frequency-selective filters RC low-pass filters high-pass filters recursive discrete-time filters MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Sinusoidal amplitude modulation; Synchronous and asynchronous demodulation; Implementation of frequency-selective filters with variable-center frequencies; Communication applications (frequency-division multiplexing and single-side band modulation).</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Sinusoidal amplitude modulation, Synchronous demodulation, asynchronous demodulation, variable-center frequencies, Implementation of frequency-selective filters with variable-center frequencies<br><br>Thumbnail - <a href= http://img.youtube.com/vi/UIgA0czNj5g/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec13_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-13-continuous-time/id458320213?i=96547708>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/UIgA0czNj5g>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-13-continuous-time-modulation Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Sinusoidal amplitude modulation Synchronous demodulation asynchronous demodulation variable-center frequencies Implementation of frequency-selective filters with variable-center frequencies MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Demonstration: time-domain and frequency-domain signals with signal generator, oscilloscope, and spectrum analyzer; Demonstration: time and frequency scaling with musical tones; Demonstration: amplitude modulation with sinusoidal, triangular, and square-wave carrier signals; Demonstration: effect of changing the percent of modulation in both time and frequency domains; Demonstration: amplitude modulation signals in an AM radio receiver.</p> <p><strong>Instructors:</strong> Prof. Alan V. Oppenheim; Prof. Sandy Hill</p>Keywords: Oscilloscope, amplitude modulation, RC audio generator, spectrum analyzer, radio, speech signal, frequency modulation<br><br>Thumbnail - <a href= http://img.youtube.com/vi/KT3yNuY_FPM/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec14_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-14-demonstration-amplitude/id458320213?i=96547362>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/KT3yNuY_FPM>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-14-demonstration-of-amplitude-modulation Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Oscilloscope amplitude modulation RC audio generator spectrum analyzer radio speech signal frequency modulation MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Amplitude modulation with complex exponential and sinusoidal carriers; Use of amplitude modulation to convert a fixed filter to a variable filter; Pulse amplitude modulation for continuous-time signals; Preview of sampling.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: discrete-time modulation, sinusoidal carriers, carrier signals, Amplitude modulation, complex exponentials, sinusoidal carriers, fixed filters, variable filters, sampling<br><br>Thumbnail - <a href= http://img.youtube.com/vi/GrnYlDAsmuA/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec15_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-15-discrete-time-modulation/id458320213?i=96547706>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/GrnYlDAsmuA>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-15-discrete-time-modulation Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US discrete-time modulation sinusoidal carriers carrier signals Amplitude modulation complex exponentials sinusoidal carriers fixed filters variable filters sampling MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> The sampling theorem: representation of a continuous-time signal by its samples; Aliasing: the effect of undersampling; Demonstration: sampling and reconstruction of the output of a sinusoidal oscillator; Demonstration: some practical uses of aliasing&mdash;a visit with Dr. Harold Edgerton at the MIT Strobe Laboratory.</p> <p><strong>Instructors:</strong> Prof. Alan V. Oppenheim; Dr. Harold Edgerton</p>Keywords: sampling theorem, aliasing, undersampling, reconstruction of sinusoidal oscillator output, representation of a continuous-time signal by its samples, some practical uses of aliasing, Dr. Harold Edgerton, MIT Strobe Laboratory<br><br>Thumbnail - <a href= http://img.youtube.com/vi/P3eLer1edx8/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec16_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-16-sampling/id458320213?i=96641905>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/P3eLer1edx8>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-16-sampling Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US sampling theorem aliasing undersampling reconstruction of sinusoidal oscillator output representation of a continuous-time signal by its samples some practical uses of aliasing Dr. Harold Edgerton MIT Strobe Laboratory MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Reconstruction of a signal from its samples as a process of interpolation; Band limited interpolation; Approximate interpolation: zero-order hold, first-order hold (linear interpolation); Illustration of sampling and interpolation for pictures; The use of sampling in computer processing of signals.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Band limited interpolation, approximate interpolation, reconstruction of signal from samples, zero-order hold, first-order hold, linear interpolation, Illustration of sampling and interpolation for pictures<br><br>Thumbnail - <a href= http://img.youtube.com/vi/_WV4JlBOQro/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec17_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-17-interpolation/id458320213?i=96547702>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/_WV4JlBOQro>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-17-interpolation Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Band limited interpolation approximate interpolation reconstruction of signal from samples zero-order hold first-order hold linear interpolation Illustration of sampling and interpolation for pictures MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Conversion of continuous-time to discrete-time signals through sampling; Digital differentiaion, half-sample delay; Demonstration: sampling a continuous-time signal, filtering sequences with a low-pass filter, reconstructing a continuous-time signal using bandlimited interpolation.</p><p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Thumbnail - <a href= http://img.youtube.com/vi/Q7aZNgY18b4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec18_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-18-discrete-time-processing/id458320213?i=113398882>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/Q7aZNgY18b4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-18-discrete-time-processing-of-continuous-time-signals Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Comparison of discrete-time and continuous-time sampling; Down-sampling (decimation) and up-sampling (reconstruction of original sequence); Use of decimation/interpolation for sampling rate conversion; Sampling in the frequency domain.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Down-sampling, decimation, up-sampling, sequence reconstruction, interpolation, sampling rate conversion, frequency domain<br><br>Thumbnail - <a href= http://img.youtube.com/vi/mmkOAMOw73U/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec19_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-19-discrete-time-sampling/id458320213?i=96547700>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/mmkOAMOw73U>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-19-discrete-time-sampling Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Down-sampling decimation up-sampling sequence reconstruction interpolation sampling rate conversion frequency domain MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Relationship to the Fourier transform; Class of rational transforms and the concept of poles and zeroes; Region of convergence (ROC); Inverse transforms using partial fraction expansion.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Laplace transform, rational transforms, Fourier transform, poles, Region of convergence, ROC, Inverse transforms, partial fraction expansion<br><br>Thumbnail - <a href= http://img.youtube.com/vi/pSN7t79RxC4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec20_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-20-the-laplace-transform/id458320213?i=96547704>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/pSN7t79RxC4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-20-the-laplace-transform Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Laplace transform rational transforms Fourier transform poles Region of convergence ROC Inverse transforms partial fraction expansion MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Geometric evaluation of frequency responses from pole-zero plots; Difference equation and system function for first-order and second-order systems; Effect of properties; Overdamped and underdamped systems; Analysis and characteristics of second-order systems; Demonstration of use in speech synthesis.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Geometric evaluation, overdamped, underdamped, frequency responses, pole-zero plots, system function for second-order systems, Analysis of second-order systems, characteristics of second-order systems, speech synthesis<br><br>Thumbnail - <a href= http://img.youtube.com/vi/8g4UudyOetE/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec21_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-21-continuous-time/id458320213?i=96641903>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/8g4UudyOetE>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-21-continuous-time-second-order-systems Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Geometric evaluation overdamped underdamped frequency responses pole-zero plots system function for second-order systems Analysis of second-order systems characteristics of second-order systems speech synthesis MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Relationship to the discrete-time Fourier transform; Region of convergence (ROC); The inverse z-transform; Geometric evaluation of the Fourier transform from the pole-zero plot, first-order and second-order systems; Analysis and characterization of LTI systems using z-transforms.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: z-transform, LTI systems, inverse z-transform, Fourier transform, Region of convergence, ROC, pole-zero plot, first-order and second-order systems<br><br>Thumbnail - <a href= http://img.youtube.com/vi/0Gat_aSzi5Y/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec22_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-22-the-z-transform/id458320213?i=96547699>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/0Gat_aSzi5Y>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-22-the-z-transform Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US z-transform LTI systems inverse z-transform Fourier transform Region of convergence ROC pole-zero plot first-order and second-order systems MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Properties of the z-transform; Analyzing systems characterized by linear constant-coefficient difference equations; Transformations between continuous-time and discrete-time systems, impulse invariance, backward difference approximation to differential equations.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: z-transform, impulse invariance, backward difference approximation, linear constant-coefficient difference equations<br><br>Thumbnail - <a href= http://img.youtube.com/vi/z8sXXQxcmN4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec23_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-23-mapping-continuous/id458320213?i=96641904>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/z8sXXQxcmN4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-23-mapping-continuous-time-filters-to-discrete-time-filters Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US z-transform impulse invariance backward difference approximation linear constant-coefficient difference equations MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Parameters, cutoff frequency and filter order; Distribution of poles of continuous-time Butterworth filters; Design of a discrete-time Butterwoth filter using impulse invariance; The bilinear transformation; Design of a discrete-time Butterworth filter using the bilinear transformation.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: Butterworth filters, bilinear transformation, impulse invariance, parameters, cutoff frequency, filter order, distribution of poles, design of Butterworth filters<br><br>Thumbnail - <a href= http://img.youtube.com/vi/4Q1fWMxVDZY/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec24_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-24-butterworth-filters/id458320213?i=96547703>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/4Q1fWMxVDZY>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-24-butterworth-filters Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US Butterworth filters bilinear transformation impulse invariance parameters cutoff frequency filter order distribution of poles design of Butterworth filters MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Applications and consequences: inverse system design, compensation for non ideal elements, stabilization of unstable systems, tracking, destabilization caused by feedback; Basic feedback equation for continuous-time and discrete-time systems; Root-locus analysis (equation for closed-loop poles, end points, angle criterion, properties); Gain and phase margins.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: feedback, root-locus analysis, destabilization, inverse system design, non ideal elements, unstable systems, tracking, destabilization caused by feedback, feedback equation, closed-loop poles, end points, angle criterion, gain and phase margins<br><br>Thumbnail - <a href= http://img.youtube.com/vi/S7MG1hgn0dY/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec25_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-25-feedback/id458320213?i=96547701>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/S7MG1hgn0dY>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-25-feedback Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US feedback root-locus analysis destabilization inverse system design non ideal elements unstable systems tracking destabilization caused by feedback feedback equation closed-loop poles end points angle criterion gain and phase margins MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics covered:</strong> Analysis of the open loop system and exploration of choices for the feedback dynamics; Behavior description through a second-order linear constant-coefficient differential equation; Root-locus analysis; Combination of proportional and derivative feedback to achieve pendulum stability; Demonstration: inverted pendulum on a track, effect of modifying dynamics, effect of modifying damaging characteristics.</p> <p><strong>Instructor:</strong> Prof. Alan V. Oppenheim</p>Keywords: feedback dynamics, inverted pendulum, stability, open loop system, root-locus analysis, proportional feedback, derivative feedback, effect of modifying dynamics, effect of modifying damaging characteristics<br><br>Thumbnail - <a href= http://img.youtube.com/vi/D3bblng-Kcc/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.6.007S11/MITRES_6-007S11lec26_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-26-feedback-example/id458320213?i=96641906>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/D3bblng-Kcc>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-26-feedback-example-the-inverted-pendulum Oppenheim, Alan V. 2011-06-06T09:35:04+05:00 en-US feedback dynamics inverted pendulum stability open loop system root-locus analysis proportional feedback derivative feedback effect of modifying dynamics effect of modifying damaging characteristics MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm