6 003 Signals and Systems Lecture 14 6 003 Signals and Systems March 30 2010 Mid term Examination 2 Wednesday April 7 7 30 9 30pm 34 101 Fourier Representations No recitations on the day of the exam Coverage Lectures 1 15 Recitations 1 15 Homeworks 1 8 Homework 8 will not collected or graded Solutions will be posted Closed book 2 pages of notes 8 12 11 inches front and back Designed as 1 hour exam two hours to complete Review sessions during open office hours March 30 2010 Conflict Contact freeman mit edu before Friday April 2 5pm Fourier Representations Fourier Series Fourier series represent signals in terms of sinusoids Representing signals by their harmonic components leads to a new representation for systems as filters 0 2 0 3 0 4 0 5 0 6 0 5 6 sixth harmonic 4 fifth harmonic 3 fourth harmonic 2 third harmonic fundamental 1 second harmonic DC 0 harmonic Musical Instruments Musical Instruments Harmonic content is natural way to describe some kinds of signals Harmonic content is natural way to describe some kinds of signals Ex musical instruments http theremin music uiowa edu MIS Ex musical instruments http theremin music uiowa edu MIS piano cello t oboe t horn t piano bassoon t oboe horn bassoon t k altosax t k violin bassoon k k altosax t cello k violin t 1 seconds 252 k 1 k 6 003 Signals and Systems Lecture 14 March 30 2010 Musical Instruments Harmonics Harmonic content is natural way to describe some kinds of signals Harmonic structure determines consonance and dissonance Ex musical instruments http theremin music uiowa edu MIS piano piano octave D D fifth D A 0 1 2 3 4 5 6 7 8 9 101112 0 1 2 3 4 5 6 7 8 9 101112 D E 1 t 0 k violin 1 violin 0 1 2 3 4 5 6 7 8 9 101112 tim e pe rio ds of D t D k bassoon 0 1 2 3 4 5 6 7 A E 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 bassoon t k 0123456789 0123456789 0123456789 D D D h armon ics Harmonic Representations Harmonic Representations What signals can be represented by sums of harmonic components Is it possible to represent ALL periodic signals with harmonics What about discontinuous signals 0 2 0 3 0 4 0 5 0 6 0 2 0 2 T t t t 2 0 0 Fourier claimed YES even though all harmonics are continuous 2 T t Lagrange ridiculed the idea that a discontinuous signal could be written as a sum of continuous signals 0 Only periodic signals all harmonics of 0 are periodic in T 2 0 We will assume the answer is YES and see if the answer makes sense Separating harmonic components Separating harmonic components Underlying properties Assume that x t is periodic in T and is composed of a weighted sum of harmonics of 0 2 T X x t x t T ak e j 0 kt 1 Multiplying two harmonics produces a new harmonic with the same fundamental frequency k Then Z e jk 0 t e jl 0 t e j k l 0 t 2 The integral of a harmonic over any time interval with length equal to a period T is zero unless the harmonic is at DC Z t0 T Z 0 k 6 0 e jk 0 t dt e jk 0 t dt T k 0 t0 T T x t e jl 0 t dt T k Z X ak e j 0 kt e j 0 lt dt T k Z X ak k X e j 0 k l t dt T ak T k l T al k Therefore Z 1 ak x t e j 0 kt dt T T 2 Z 2 1 x t e j T kt dt T T 6 003 Signals and Systems Lecture 14 Fourier Series Check Yourself Determining harmonic components of a periodic signal Let ak represent the Fourier series coefficients of the following square wave Z ak March 30 2010 2 1 x t e j T kt dt T T X x t x t T analysis equation ak e j 2 kt T 1 2 0 synthesis equation 1 t 12 k How many of the following statements are true 1 2 3 4 5 Fourier Series Properties Check Yourself If a signal is differentiated in time its Fourier coefficients are multiplied by j 2 T k Proof Let x t x t T X ak e ak 0 if k is even ak is real valued ak decreases with k 2 there are an infinite number of non zero ak all of the above Let bk represent the Fourier series coefficients of the following triangle wave 1 8 j 2 T kt 0 k then 1 t 18 X 2 2 x t x t T j k ak e j T kt T k How many of the following statements are true 1 2 3 4 5 bk 0 if k is even bk is real valued bk decreases with k 2 there are an infinite number of non zero bk all of the above Fourier Series Fourier Series One can visualize convergence of the Fourier Series by incrementally adding terms One can visualize convergence of the Fourier Series by incrementally adding terms Example triangle waveform Example triangle waveform 5 X k 5 k odd 1 8 0 39 X 1 j2 kt e 2k 2 2 k 39 k odd 1 8 1 t 0 18 1 j2 kt e 2k 2 2 1 t 18 Fourier series representations of functions with discontinuous slopes converge toward functions with discontinuous slopes 3 6 003 Signals and Systems Lecture 14 March 30 2010 Fourier Series Fourier Series One can visualize convergence of the Fourier Series by incrementally adding terms One can visualize convergence of the Fourier Series by incrementally adding terms Example square wave Example square wave 5 X k 5 k odd 1 2 0 39 X 1 j2 kt e jk k 39 k odd 1 2 t 1 0 12 1 j2 kt e jk t 1 12 Fourier Series Fourier Series Summary Partial sums of Fourier series of discontinuous functions ring near discontinuities Gibb s phenomenon Fourier series represent periodic signals as sums of sinusoids valid for an extremely large class of periodic signals 9 valid even for discontinuous signals such as square wave 1 2 However convergence as harmonics increases can be complicated 0 t 1 12 This ringing results because the magnitude of the Fourier coefficients is only decreasing as k1 while they decreased as 12 for the triangle k You can decrease and even eliminate the ringing by decreasing the magnitudes of the Fourier coefficients at higher frequencies Filtering Filtering The output of an LTI system is a filtered version of the input Notion of a filter Input Fourier series sum of complex exponentials X 2 x t x t T ak e …
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