Lecture 14: Fourier representations - MIT .Fourier Representations October 27, 2011 Fourier...

download Lecture 14: Fourier representations - MIT .Fourier Representations October 27, 2011 Fourier Representations

of 51

  • date post

    31-Jul-2018
  • Category

    Documents

  • view

    215
  • download

    0

Embed Size (px)

Transcript of Lecture 14: Fourier representations - MIT .Fourier Representations October 27, 2011 Fourier...

  • 6.003: Signals and Systems

    Fourier Representations

    October 27, 2011 1

  • Fourier Representations

    Fourier series represent signals in terms of sinusoids.

    leads to a new representation for systems as filters.

    2

  • Fourier Series

    Representing signals by their harmonic components.

    0 20 30 40 50 60

    0 1 2 3 4 5 6 harmonic #

    DC

    fun

    da

    men

    tal

    seco

    nd

    ha

    rmo

    nic

    thir

    dh

    arm

    on

    ic

    fou

    rth

    ha

    rmo

    nic

    fift

    hh

    arm

    on

    ic

    sixth

    ha

    rmo

    nic

    3

  • Musical Instruments

    Harmonic content is natural way to describe some kinds of signals.

    Ex: musical instruments (http://theremin.music.uiowa.edu/MIS )

    piano

    t

    cello

    t

    bassoon

    t

    oboe

    t

    horn

    t

    altosax

    t

    violin

    t

    bassoon

    t1252

    seconds

    4

    .html

    http://theremin.music.uiowa.edu/MIS.html

  • Musical Instruments

    Harmonic content is natural way to describe some kinds of signals.

    Ex: musical instruments (http://theremin.music.uiowa.edu/MIS )

    piano

    k

    cello

    k

    bassoon

    k

    oboe

    k

    horn

    k

    altosax

    k

    violin

    k5

    .html

    http://theremin.music.uiowa.edu/MIS.html

  • Musical Instruments

    Harmonic content is natural way to describe some kinds of signals.

    Ex: musical instruments (http://theremin.music.uiowa.edu/MIS )

    piano

    t

    piano

    k

    violin

    t

    violin

    k

    bassoon

    t

    bassoon

    k6

    .html

    http://theremin.music.uiowa.edu/MIS.html

  • Harmonics

    Harmonic structure determines consonance and dissonance.

    octave (D+D) fifth (D+A) D+E

    t ime(per iods of "D")

    harmonics

    0 1 2 3 4 5 6 7 8 9 101112 0 1 2 3 4 5 6 7 8 9 101112 0 1 2 3 4 5 6 7 8 9 101112

    1

    0

    1

    D

    D'

    D

    A

    D

    E

    0 1 2 3 4 5 6 7 8 9

    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

    0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

    0 1 2 3 4 5 6 7

    7

  • Harmonic Representations

    What signals can be represented by sums of harmonic components?

    0 20 30 40 50 60

    T= 20

    t

    T= 20

    t

    Only periodic signals: all harmonics of 0 are periodic in T = 2/0. 8

  • Harmonic Representations

    Is it possible to represent ALL periodic signals with harmonics? What about discontinuous signals?

    20

    t

    20

    t

    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. 9

  • Separating harmonic components

    Underlying properties.

    1. Multiplying two harmonics produces a new harmonic with the same fundamental frequency:

    jk0t jl0t j(k+l)0t e e = e .

    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: t0+T 0, k = 0

    jk0tdt jk0tdt =e e t0 T T, k = 0

    = T[k]

    10

  • Separating harmonic components

    Assume that x(t) is periodic in T and is composed of a weighted sum of harmonics of 0 = 2/T .

    j0kt x(t) = x(t + T ) = ake

    Then

    =k

    f

    f

    x(t)e jl0tdt = j0kt j0ltdteake T T k=

    fj0(kl)tdt= ak e

    Tk=

    = akT[k l] = Tal k=

    Therefore

    f 1 1 j 2 ktak = x(t)e j0ktdt = x(t)e T dt T TT T

    11

  • Fourier Series

    Determining harmonic components of a periodic signal.

    1 2j ktx(t)e (analysis equation) ak = dt TT T f 2j kt x(t)= x(t + T ) = (synthesis equation) Take

    k=

    12

  • Let ak represent the Fourier series coefficients of the following square wave.

    t

    12

    12

    0 1

    Check Yourself

    How many of the following statements are true?

    1. ak = 0 if k is even 2. ak is real-valued 3. |ak| decreases with k2

    4. there are an infinite number of non-zero ak 5. all of the above

    13

  • Check Yourself

    Let ak represent the Fourier series coefficients of the following square wave.

    t

    12

    12

    0 1

    ak = x(t)e j 2 T ktdt = 1

    2

    0

    1 2 e j2ktdt + 1

    2

    1 2

    0 e j2ktdt

    T

    1 jk e jk = 2 e j4k 1 ; if k is odd = jk 0 ; otherwise

    14

  • Check Yourself

    Let ak represent the Fourier series coefficients of the following square wave. 1 ; if k is odd

    ak = jk 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 k2 X 4. there are an infinite number of non-zero ak 5. all of the above X

    15

  • Let ak represent the Fourier series coefficients of the following square wave.

    t

    12

    12

    0 1

    Check Yourself

    How many of the following statements are true? 2

    1. ak = 0 if k is even 2. ak is real-valued X 3. |ak| decreases with k2 X 4. there are an infinite number of non-zero ak 5. all of the above X

    16

  • Fourier Series Properties

    If a signal is differentiated in time, its Fourier coefficients are multiplied by j 2 k.T Proof: Let

    f 2 x(t) = x(t + T ) = ake

    k=

    j kt T

    then f 2 2 x(t) = x(t + T ) = j j kt k ak e TT

    k=

    17

  • Let bk represent the Fourier series coefficients of the following triangle wave.

    t

    18

    18

    0 1

    Check Yourself

    How many of the following statements are true?

    1. bk = 0 if k is even 2. bk is real-valued 3. |bk| decreases with k2

    4. there are an infinite number of non-zero bk 5. all of the above

    18

  • Check Yourself

    The triangle waveform is the integral of the square wave.

    t

    12

    12

    0 1

    t

    18

    18

    0 1

    Therefore the Fourier coefficients of the triangle waveform are 1

    j2k times those of the square wave.

    1 1 1 = = ; k odd bk jk j2k 2k22 19

  • Check Yourself

    Let bk represent the Fourier series coefficients of the following triangle wave.

    1 bk = ; k odd 2k22

    How many of the following statements are true? 1. bk = 0 if k is even 2. bk is real-valued 3. |bk| decreases with k2 4. there are an infinite number of non-zero bk 5. all of the above

    20

  • Let bk represent the Fourier series coefficients of the following triangle wave.

    t

    18

    18

    0 1

    Check Yourself

    How many of the following statements are true? 5

    1. bk = 0 if k is even 2. bk is real-valued 3. |bk| decreases with k2 4. there are an infinite number of non-zero bk 5. all of the above

    21

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    0k = 0k odd

    12k22

    e j2kt

    18

    18

    0 1

    22

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    1k = 1k odd

    12k22

    e j2kt

    18

    18

    0 1

    23

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    3k = 3k odd

    12k22

    e j2kt

    18

    18

    0 1

    24

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    5k = 5k odd

    12k22

    e j2kt

    18

    18

    0 1

    25

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    7k = 7k odd

    12k22

    e j2kt

    18

    18

    0 1

    26

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    9k = 9k odd

    12k22

    e j2kt

    18

    18

    0 1

    27

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    19k = 19k odd

    12k22

    e j2kt

    18

    18

    0 1

    28

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    29k = 29k odd

    12k22

    e j2kt

    18

    18

    0 1

    29

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: triangle waveform

    t

    39k = 39k odd

    12k22

    e j2kt

    18

    18

    0 1

    Fourier series representations of functions with discontinuous slopes converge toward functions with discontinuous slopes.

    30

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding terms.

    Example: square wave

    t

    0k = 0k odd

    1jk

    e j2kt

    12

    12

    0 1

    31

  • Fourier Series

    One can visualize convergence of the Fourier Series by incrementally adding ter