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.

ωω0 2ω0 3ω0 4ω0 5ω0 6ω0

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

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

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

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?

ωω0 2ω0 3ω0 4ω0 5ω0 6ω0

T= 2πω0

t

T= 2πω0

t

Only periodic signals: all harmonics of ω0 are periodic in T = 2π/ω0. 8

Harmonic Representations

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

2πω0

t

2πω0

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:

jkω0t jlω0t j(k+l)ω0t 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

jkω0tdt ≡ jkω0tdt =e e t0 T T, k = 0

= Tδ[k]

10

Separating harmonic components

Assume that x(t) is periodic in T and is composed of a weighted sum of harmonics of ω0 = 2π/T .

∞jω0kt x(t) = x(t + T ) = ake

Then ∞

−∞=k

f

f

x(t)e −jlω0tdt = jω0kt −jω0ltdteake T T k=−∞

∞fjω0(k−l)tdt= ak e

Tk=−∞ ∞

= akTδ[k − l] = Tal k=−∞

Therefore

f

1 1 −j 2π ktak = x(t)e −jω0ktdt = x(t)e T dt T TT T

11

∫

∫∫∫∫

∫

Fourier Series

Determining harmonic components of a periodic signal.

1 2π−j ktx(t)e (“analysis” equation) ak = dt TT T

∞f 2πj 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 2π T ktdt = −

1 2

0

− 1 2

e −j2πktdt + 1 2

1 2

0 e −j2πktdt

T

1 jπk − e −jπk = 2 − e j4πk ⎧ ⎨ 1 ; if k is odd = jπk ⎩ 0 ; otherwise

14

∫ ∫ ∫

Check Yourself

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

ak = jπk 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 multi­

plied by j 2π k.T

Proof: Let ∞f 2π

x(t) = x(t + T ) = ake k=−∞

j kt T

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

T 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

j2πk times those of the square wave.

1 1 −1 = × = ; k odd bk jkπ j2πk 2k2π2

19

Check Yourself

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

−1 bk = ; k odd

2k2π2

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

0∑k = −0k odd

−12k2π2 e

j2πkt

18

−18

0 1

22

Fourier Series

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

Example: triangle waveform

t

1∑k = −1k odd

−12k2π2 e

j2πkt

18

−18

0 1

23

Fourier Series

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

Example: triangle waveform

t

3∑k = −3k odd

−12k2π2 e

j2πkt

18

−18

0 1

24

Fourier Series

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

Example: triangle waveform

t

5∑k = −5k odd

−12k2π2 e

j2πkt

18

−18

0 1

25

Fourier Series

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

Example: triangle waveform

t

7∑k = −7k odd

−12k2π2 e

j2πkt

18

−18

0 1

26

Fourier Series

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

Example: triangle waveform

t

9∑k = −9k odd

−12k2π2 e

j2πkt

18

−18

0 1

27

Fourier Series

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

Example: triangle waveform

t

19∑k = −19k odd

−12k2π2 e

j2πkt

18

−18

0 1

28

Fourier Series

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

Example: triangle waveform

t

29∑k = −29k odd

−12k2π2 e

j2πkt

18

−18

0 1

29

Fourier Series

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

Example: triangle waveform

t

39∑k = −39k odd

−12k2π2 e

j2πkt

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

0∑k = −0k odd

1jkπ

e j2πkt

12

−12

0 1

31

Fourier Series

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

Example: square wave

t

1∑k = −1k odd

1jkπ

e j2πkt

12

−12

0 1

32

Fourier Series

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

Example: square wave

t

3∑k = −3k odd

1jkπ

e j2πkt

12

−12

0 1

33

Fourier Series

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

Example: square wave

t

5∑k = −5k odd

1jkπ

e j2πkt

12

−12

0 1

34

Fourier Series

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

Example: square wave

t

7∑k = −7k odd

1jkπ

e j2πkt

12

−12

0 1

35

Fourier Series

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

Example: square wave

t

9∑k = −9k odd

1jkπ

e j2πkt

12

−12

0 1

36

Fourier Series

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

Example: square wave

t

19∑k = −19k odd

1jkπ

e j2πkt

12

−12

0 1

37

Fourier Series

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

Example: square wave

t

29∑k = −29k odd

1jkπ

e j2πkt

12

−12

0 1

38

Fourier Series

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

Example: square wave

t

39∑k = −39k odd

1jkπ

e j2πkt

12

−12

0 1

39

Fourier Series

Partial sums of Fourier series of discontinuous functions “ring” near discontinuities: Gibb’s phenomenon.

This ringing results because the magnitude of the Fourier coefficients

t

12

−12

0 1

9%

1is only decreasing as k 1 (while they decreased as

k2 for the triangle).

You can decrease (and even eliminate the ringing) by decreasing the magnitudes of the Fourier coefficients at higher frequencies.

40

Fourier Series: Summary

Fourier series represent periodic signals as sums of sinusoids.

• valid for an extremely large class of periodic signals

• valid even for discontinuous signals such as square wave

However, convergence as # harmonics increases can be complicated.

41

Filtering

The output of an LTI system is a “filtered” version of the input.

Input: Fourier series → sum of complex exponentials. ∞

x(t) = x(t + T ) = ake k=−∞

f j 2π kt T

Complex exponentials: eigenfunctions of LTI systems. 2πkt → H(j k)e j

2 2π πj kt T Te T

Output: same eigenfunctions, amplitudes/phases set by system. ∞ ∞ff 2π 2 2π πj kt → y(t) = akH(j k)e j kt x(t) = T Take

T k=−∞ k=−∞

42

Filtering

Notion of a filter.

LTI systems • cannot create new frequencies. • can scale magnitudes and shift phases of existing components.

Example: Low-Pass Filtering with an RC circuit

+−

vi

+

vo

−

R

C

43

0.01

0.1

1

0.01 0.1 1 10 100ω

1/RC

|H(jω

)|

0

−π20.01 0.1 1 10 100

ω

1/RC

∠H

(jω

)|

Lowpass Filter

Calculate the frequency response of an RC circuit.

+−

vi

+

vo

−

R

C

KVL: vi(t) = Ri(t) + vo(t) C: i(t) = Cvo(t) Solving: vi(t) = RC vo(t) + vo(t)

Vi(s) = (1 + sRC)Vo(s) Vo(s) 1

H(s) = = Vi(s) 1 + sRC

44

Lowpass Filtering

x(t) =

Let the input be a square wave.

t

12

−12

0 T

T π2=0ω;kt 0jωe

jπk 1

odd k

f

0.01

0.1

1

0.01 0.1 1 10 100ω

1/RC

|X(jω

)|

0

−π20.01 0.1 1 10 100

ω

1/RC

∠X

(jω

)|

45

Lowpass Filtering

x(t) =

Low frequency square wave: ω0 << 1/RC.

t

12

−12

0 T

T π2=0ω;kt 0jωe

jπk 1

odd k

f

0.01

0.1

1

0.01 0.1 1 10 100ω

1/RC

|H(jω

)|

0

−π20.01 0.1 1 10 100

ω

1/RC

∠H

(jω

)|

46

Lowpass Filtering

x(t) =

Higher frequency square wave: ω0 < 1/RC.

t

12

−12

0 T

T π2=0ω;kt 0jωe

jπk 1

odd k

f

0.01

0.1

1

0.01 0.1 1 10 100ω

1/RC

|H(jω

)|

0

−π20.01 0.1 1 10 100

ω

1/RC

∠H

(jω

)|

47

Lowpass Filtering

x(t) =

Still higher frequency square wave: ω0 = 1/RC.

t

12

−12

0 T

T π2=0ω;kt 0jωe

jπk 1

odd k

f

0.01

0.1

1

0.01 0.1 1 10 100ω

1/RC

|H(jω

)|

0

−π20.01 0.1 1 10 100

ω

1/RC

∠H

(jω

)|

48

Lowpass Filtering

x(t) =

High frequency square wave: ω0 > 1/RC.

t

12

−12

0 T

T π2=0ω;kt 0jωe

jπk 1

odd k

f

0.01

0.1

1

0.01 0.1 1 10 100ω

1/RC

|H(jω

)|

0

−π20.01 0.1 1 10 100

ω

1/RC

∠H

(jω

)|

49

Fourier Series: Summary

Fourier series represent signals by their frequency content.

Representing a signal by its frequency content is useful for many signals, e.g., music.

Fourier series motivate a new representation of a system as a filter.

50

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6.003 Signals and SystemsFall 2011

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