6.003: Signals and Systems
Fourier Series
November 1, 2011
1
Last Time: Describing Signals by Frequency Content
Harmonic content is natural way to describe some kinds of signals.
Ex: musical instruments (http://theremin.music.uiowa.edu/MIS .html )
piano
piano
t
k
violin
violin
t
k
bassoon
bassoon
t
k
2
Last Time: Fourier Series
Determining harmonic components of a periodic signal.
ak =
�
2π
1
x(t)e
−j T kt dt
T T
x(t)= x(t + T ) =
∞
0
ak e j
(“analysis” equation)
2π kt
T
(“synthesis” equation)
k=−∞
We can think of Fourier series as an orthogonal decomposition.
3
Orthogonal Decompositions
Vector representation of 3-space: let r̄ represent a vector with
components {x, y, and z} in the {x̂, ŷ, and ẑ} directions, respectively.
x = r̄ · x̂
y = r̄ · ŷ
z = r̄ · ẑ
(“analysis” equations)
r̄ = xx̂ + yŷ + zẑ
(“synthesis” equation)
Fourier series: let x(t) represent a signal with harmonic components
{a0 , a1 , . . ., ak } for harmonics {e j0t , e
j
Z�
2π
1
ak =
x(t)e
−j T kt dt
T T
x(t)= x(t + T ) =
∞
0
ak e j
2π t
T ,
. . ., e
j
2π kt
T
}
respectively.
(“analysis” equation)
2π kt
T
(“synthesis” equation)
k=−∞
4
Orthogonal Decompositions
Vector representation of 3-space: let r̄ represent a vector with
components {x, y, and z} in the {x̂, ŷ, and ẑ} directions, respectively.
x = r̄ · x̂
y = r̄ · ŷ
z = r̄ · ẑ
(“analysis” equations)
r̄ = xˆ
x + y ŷ + z zˆ
(“synthesis” equation)
Fourier series: let x(t) represent a signal with harmonic components
{a0 , a1 , . . ., ak } for harmonics {e j0t , e
j
Z�
2π
1
ak =
x(t)e
−j T kt dt
T T
x(t)= x(t + T ) =
∞
0
ak e j
2π t
T ,
. . ., e
j
2π kt
T
}
respectively.
(“analysis” equation)
2π kt
T
(“synthesis” equation)
k=−∞
5
Orthogonal Decompositions
Vector representation of 3-space: let r̄ represent a vector with
components {x, y, and z} in the {x̂, ŷ, and ẑ} directions, respectively.
x = r̄ · x̂
y = r̄ · ŷ
z = r̄ · ẑ
(“analysis” equations)
r̄ = xx̂ + yŷ + zẑ
(“synthesis” equation)
Fourier series: let x(t) represent a signal with harmonic components
{a0 , a1 , . . ., ak } for harmonics {e j0t , e
j
Z
�
2π
1
x(t)e−j T kt dt
ak =
T T
x(t)= x(t + T ) =
∞
0
ak e j
2π t
T ,
. . ., e
j
2π kt
T
}
respectively.
(“analysis” equation)
2π kt
T
(“synthesis” equation)
k=−∞
6
Orthogonal Decompositions
Integrating over a period sifts out the k th component of the series.
Sifting as a dot product:
x = r̄ · x̂ ≡ |r̄||x̂| cos θ
Sifting as an inner product:
Z�
2π
1
j 2Tπ kt
ak =
e
· x(t) ≡
x(t)e−j T kt dt
T T
Z
where
�
1
a(t) · b(t) =
a∗ (t)b(t)dt .
T T
The complex conjugate (∗ ) makes the inner product of the k th and
mth components equal to 1 iff k Z
= m:
Z
�
�
2π
2π
1
1
1
if k = m
j 2Tπ kt ∗
j 2Tπ mt
e
e
dt =
e
−j T kt e
j T mt dt =
T T
T T
0
otherwise
7
Check Yourself
How many of the following pairs of functions are
orthogonal (⊥) in T = 3?
1. cos 2πt ⊥ sin 2πt ?
2. cos 2πt ⊥ cos 4πt ?
3. cos 2πt ⊥ sin πt ?
4. cos 2πt ⊥ e j2πt ?
8
Check Yourself
How many of the following are orthogonal (⊥) in T = 3?
cos 2πt ⊥ sin 2πt ?
cos 2πt
t
1
2
3
1
2
3
sin 2πt
t
cos 2πt sin 2πt = 12 sin 4πt
t
2
1
Z
� 3
dt = 0 therefore YES
0
9
3
Check Yourself
How many of the following are orthogonal (⊥) in T = 3?
cos 2πt ⊥ cos 4πt ?
cos 2πt
t
1
2
3
1
2
3
cos 4πt
t
cos 2πt cos 4πt = 12 cos 6πt + 21 cos 2πt
t
2
1
Z 3
�
dt = 0 therefore YES
0
10
3
Check Yourself
How many of the following are orthogonal (⊥) in T = 3?
cos 2πt ⊥ sin πt ?
cos 2πt
t
1
2
3
1
2
3
sin πt
t
cos 2πt sin πt = 12 sin 3πt − 12 sin πt
t
2
1
Z� 3
dt = 0 therefore NO
0
11
3
Check Yourself
How many of the following are orthogonal (⊥) in T = 3?
cos 2πt ⊥ e2πt ?
e2πt = cos 2πt + j sin 2πt
cos 2πt ⊥ sin 2πt but not cos 2πt
Therefore NO
12
Check Yourself
How many of the following pairs of functions
are orthogonal (⊥) in T = 3? 2
√
1. cos 2πt ⊥ sin 2πt ?
√
2. cos 2πt ⊥ cos 4πt ?
3. cos 2πt ⊥ sin πt ?
X
4. cos 2πt ⊥ e j2πt ?
X
13
Speech
Vowel sounds are quasi-periodic.
bat
bait
t
bit
bet
t
t
bought
bite
t
beet
boat
t
t
but
boot
t
t
14
t
t
Speech
Harmonic content is natural way to describe vowel sounds.
bat
bait
k
bit
bet
k
k
bought
bite
k
beet
but
boat
k
k
boot
k
k
15
k
k
Speech
Harmonic content is natural way to describe vowel sounds.
bat
bat
t
k
beet
beet
t
k
boot
boot
t
k
16
Speech Production
Speech is generated by the passage of air from the lungs, through
the vocal cords, mouth, and nasal cavity.
Nasal
cavity
Hard palate
Soft palate
(velum)
Tongue
Lips
Pharynx
Epiglottis
Larynx
Vocal cords
(glottis)
Esophogus
Trachea
Stomach Lungs
A dapt ed f rom T. F. W eis s
17
Speech Production
Controlled by complicated muscles, vocal cords are set in vibration
by the passage of air from the lungs.
Loo ki ng d ow n th e th ro at :
Vocal cords open
Glottis
Vocal cords closed
Vocal
cords
Gr a y 's A na t o m y
Ada p t e d f ro m T. F. We i s s
18
Speech Production
Vibrations of the vocal cords are “filtered” by the mouth and nasal
cavities to generate speech.
19
Filtering
Notion of a filter.
LTI systems
• cannot create new frequencies.
• can only scale magnitudes & shift phases of existing components.
Example: Low-Pass Filtering with an RC circuit
R
+
vi
+
−
C
vo
−
20
Lowpass Filter
Calculate the frequency response of an RC circuit.
R
+
vi (t)
= Ri(t) + vo (t)
C:
i(t)
= Cv̇o (t)
Solving: vi (t)
vo
C
= RC v̇o (t) + vo (t)
Vi (s) = (1 + sRC)Vo (s)
1
Vo (s)
H(s) =
=
1 + sRC
Vi (s)
−
1
|H(jω)|
+
−
0.1
0.01
∠H(jω)|
vi
KVL:
0.01
0.1
1
10
ω
100 1/RC
10
ω
100 1/RC
0
− π2
0.01
0.1
1
21
Lowpass Filtering
Let the input be a square wave.
1
2
0
− 12
1 jω0 kt
e
;
jπk
k odd
1
|X(jω)|
0
ω0 =
2π
T
0.1
0.01
∠X(jω)|
x(t) =
t
T
0.01
0.1
1
10
ω
100 1/RC
10
ω
1/RC
100
0
− π2
0.01
0.1
1
22
Lowpass Filtering
Low frequency square wave: ω0 << 1/RC.
1
2
0
− 12
1 jω0 kt
e
;
jπk
k odd
1
|H(jω)|
0
ω0 =
2π
T
0.1
0.01
∠H(jω)|
x(t) =
t
T
0.01
0.1
1
10
ω
100 1/RC
10
ω
1/RC
100
0
− π2
0.01
0.1
1
23
Lowpass Filtering
Higher frequency square wave: ω0 < 1/RC.
1
2
0
− 12
1 jω0 kt
e
;
jπk
k odd
1
|H(jω)|
0
ω0 =
2π
T
0.1
0.01
∠H(jω)|
x(t) =
t
T
0.01
0.1
1
10
ω
100 1/RC
10
ω
1/RC
100
0
− π2
0.01
0.1
1
24
Lowpass Filtering
Still higher frequency square wave: ω0 = 1/RC.
1
2
0
− 12
1 jω0 kt
e
;
jπk
k odd
1
|H(jω)|
0
ω0 =
2π
T
0.1
0.01
∠H(jω)|
x(t) =
t
T
0.01
0.1
1
10
ω
100 1/RC
10
ω
1/RC
100
0
− π2
0.01
0.1
1
25
Lowpass Filtering
High frequency square wave: ω0 > 1/RC.
1
2
0
− 12
1 jω0 kt
e
;
jπk
k odd
1
|H(jω)|
0
ω0 =
2π
T
0.1
0.01
∠H(jω)|
x(t) =
t
T
0.01
0.1
1
10
ω
100 1/RC
10
ω
1/RC
100
0
− π2
0.01
0.1
1
26
Source-Filter Model of Speech Production
Vibrations of the vocal cords are “filtered” by the mouth and nasal
cavities to generate speech.
buzz from
vocal cords
throat and
nasal cavities
27
speech
Speech Production
X-ray movie showing speech in production.
Courtesy of Kenneth N. Stevens. Used with permission.
28
Demonstration
Artificial speech.
buzz from
vocal cords
throat and
nasal cavities
29
speech
Formants
Resonant frequencies of the vocal tract.
amplitude
F1
F2
F3
frequency
Men
Women
Children
Formant
F1
F2
F3
F1
F2
F3
F1
F2
F3
heed
270
2290
3010
310
2790
3310
370
3200
3730
head
530
1840
2480
610
2330
2990
690
2610
3570
had
660
1720
2410
860
2050
2850
1010
2320
3320
hod
730
1090
2440
850
1220
2810
1030
1370
3170
http://www.sfu.ca/sonic-studio/handbook/Formant.html
30
haw’d
570
840
2410
590
920
2710
680
1060
3180
who’d
300
870
2240
370
950
2670
430
1170
3260
Speech Production
Same glottis signal + different formants → different vowels.
glottis signal
vocal tract filter
vowel sound
ak
bk
ak
bk
We detect changes in the filter function to recognize vowels.
31
Singing
We detect changes in the filter function to recognize vowels
... at least sometimes.
Demonstration.
“la” scale.
“lore” scale.
“loo” scale.
“ler” scale.
“lee” scale.
Low Frequency: “la” “lore” “loo” “ler” “lee”.
High Frequency: “la” “lore” “loo” “ler” “lee”.
http://www.phys.unsw.edu.au/jw/soprane.html
32
Speech Production
We detect changes in the filter function to recognize vowels.
lo w
i n t er me dia t e
hig h
33
Speech Production
We detect changes in the filter function to recognize vowels.
lo w
i n t er me dia t e
hig h
34
Continuous-Time 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.
Representing a system as a filter is useful for many systems, e.g.,
speech synthesis.
35
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6.003 Signals and Systems
Fall 2011
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