6 003 Signals and Systems Fourier Series November 3 2009 Fourier Representations Representations based on sinusoids signal in system signal out To date we have focused primarily on time domain techniques especially with transient signals e g impulse response The primary focus for the next few weeks will be frequency domain techniques e g frequency response which concern eternal signals Fourier Series Today Fourier series represent signals in terms of sinusoids This new representation for signals leads to a new representation for systems as filters Harmonics Representing signals by the amplitudes and phases of harmonic components 0 1 2 3 4 5 6 DC fundamental second harmonic third harmonic fourth harmonic fifth harmonic sixth harmonic 0 2 0 3 0 4 0 5 0 6 0 harmonic Musical Instruments Harmonic content is natural way to describe some kinds of signals Ex musical instruments http theremin music uiowa edu MIS piano cello t t oboe bassoon horn t t altosax t violin t bassoon t t 1 seconds 252 Musical Instruments Harmonic content is natural way to describe some kinds of signals Ex musical instruments http theremin music uiowa edu MIS piano cello k k oboe bassoon horn k k altosax k violin k k Musical Instruments Harmonic content is natural way to describe some kinds of signals Ex musical instruments http theremin music uiowa edu MIS piano piano t k violin violin t k bassoon bassoon t k Harmonics Harmonic structure determines consonance and dissonance 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 0 1 0 1 2 3 4 5 6 7 8 9 101112 t im e period s of D D A E 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0123456789 0123456789 0123456789 D D D 0 1 2 3 4 5 6 7 har mo nics Harmonic Representations What signals can be represented by sums of harmonic components 0 2 0 3 0 4 0 5 0 6 0 t 2 T 0 t T 2 0 Only periodic signals all harmonics of 0 are periodic in T 2 0 Harmonic Representations Is it possible to represent ALL periodic signals with harmonics What about discontinuous signals t 2 0 t 2 0 Fourier claimed YES even though all harmonics are continuous Lagrange ridiculed the idea that a discontinuous signal could be written as a sum of continuous signals We will assume the answer is YES and see if the answer makes sense Separating harmonic components Underlying properties 1 Multiplying two harmonics produces a new harmonic with the same fundamental frequency 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 jk 0 t jk 0 t e dt e dt T k 0 t0 T T k Separating harmonic components 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 k Then Z x t e jl 0 t Z dt T 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 Z 2 1 x t e j T kt dt T T Fourier Series Determining harmonic components of a periodic signal ak Z 2 1 x t e j T kt dt T T x t x t T X k ak e j analysis equation 2 kt T synthesis equation Check Yourself Let ak represent the Fourier series coefficients of the following square wave 1 2 0 1 t 12 How many of the following statements are true 1 2 3 4 5 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 Check Yourself Let ak represent the Fourier series coefficients of the following square wave 1 2 0 1 12 Z Z 1 1 0 j2 kt 1 2 j2 kt e dt ak x t e dt e dt 2 1 2 0 T 2 1 2 e j k e j k j4 k 1 if k is odd j k 0 otherwise Z j 2 T kt t Check Yourself Let ak represent the Fourier series coefficients of the following square wave 1 if k is odd ak j k 0 otherwise How many of the following statements are true 1 ak 0 if k is even 2 ak is real valued X 3 ak decreases with k 2 X 4 there are an infinite number of non zero ak 5 all of the above X Check Yourself Let ak represent the Fourier series coefficients of the following square wave 1 2 0 t 1 12 How many of the following statements are true 1 2 3 4 5 2 ak 0 if k is even ak is real valued X ak decreases with k 2 X there are an infinite number of non zero ak all of the above X Fourier Series Properties 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 j 2 kt T k then X 2 2 x t x t T j k ak e j T kt T k Check Yourself Let bk represent the Fourier series coefficients of the following triangle wave 1 8 0 1 t 18 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 Check Yourself The triangle waveform is the integral of the square wave 1 2 0 1 t 12 1 8 0 1 t 18 Therefore the Fourier coefficients of the triangle waveform are times those of the square wave 1 1 1 bk 2 2 k odd jk j2 k 2k 1 j2 k Check Yourself Let bk represent the Fourier series coefficients of the following triangle wave 1 bk 2 2 k odd 2k How many of the following statements are true 1 bk 0 if k is even 2 bk is real valued 3 bk decreases with k 2 4 there are an infinite number of non zero bk 5 all of the above Check Yourself Let bk represent the Fourier series coefficients of the following triangle wave 1 8 0 t 1 18 How many of the following statements are true 1 2 3 4 5 5 bk 0 if k …
View Full Document
Unlocking...