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

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### 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

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