MIT 信号与系统 Lecture 15

1
6.003: Signals and Systems
Check Yourself
Lecture 15
Speech
Vowel sounds are quasi-periodic.
April 1, 2010
How many of the following pairs of functions are orthog­ onal (⊥) in T = 3? 1. cos 2πt ⊥ sin 2πt ? 2. cos 2πt ⊥ cos 4πt ? 3. cos 2πt ⊥ sin πt ? 4. cos 2πt ⊥ e j 2πt ?
T 2π T
ቤተ መጻሕፍቲ ባይዱ
t �
0 −1 2 x(t) = 1 jω0 kt e ; jπk k odd 1 |H (jω )| 0.1 0.01 ∠H (jω )| 0 −π 2 0.01 0.1 0.01 0.1 ω0 =
T 2π T
t
1
10
ω 100 1/RC ω 100 1/RC
1
10
ω 100 1/RC ω 100 1/RC
Example: Low-Pass Filtering with an RC circuit R + vi
+ −
C
vo −
Lowpass Filter
Calculate the frequency response of an RC circuit. R + vi
+ −
Lowpass Filtering
Filtering
Notion of a filter.
April 1, 2010
LTI systems • cannot create new frequencies. • can only scale magnitudes and shift phases of existing components.
2
6.003: Signals and Systems
Speech Production
Vibrations of the vocal cords are “filtered” by the mouth and nasal cavities to generate speech.
Lecture 15
(“analysis” equations)
(“synthesis” equation)
Fourier series: let x(t) represent a signal with harmonic components {a0 , a1 , . . ., ak } for harmonics {e j 0t , e j ak = � 2π 1 x(t)e−j T kt dt T T
Looking down the throat:
Vocal cords open
Glottis
Nasal cavity Hard palate Soft palate (velum) Pharynx Epiglottis Larynx Lips Tongue
Diagram removed due to copyright restrictions.
Last Time: Fourier Series
Determining harmonic components of a periodic signal. � 2π 1 x(t)e−j T kt dt T T
∞ � k=−∞
ak =
(“analysis” equation)
2π kt T
x(t)= x(t + T ) =
Solving: vi (t)
Vi (s) = (1 + sRC )Vo (s) 1 Vo (s) H (s) = = Vi (s) 1 + sRC
0 −1 2 1 jω0 kt ; e jπk k odd 1 |X (jω )| 0.1 0.01 ∠X (jω )| 0 −π 2 0.01 0.1 0.01 0.1 ω0 =
ak e j
(“synthesis” equation)
We can think of Fourier series as an orthogonal decomposition.
Orthogonal Decompositions
Vector representation of 3-space: let r ¯ represent a vector with components {x, y , and z } in the {x ˆ, y ˆ, and z ˆ} directions, respectively. x=r ¯· x ˆ y=r ¯· y ˆ z=r ¯· z ˆ r ¯ = xx ˆ + yy ˆ + zz ˆ
Vocal cords closed
Vocal cords
Esophogus
Vocal cords (glottis) Trachea
G r a y's A n a t o my
Stomach Lungs
Adapted from T.F. Weiss
A dapt ed f r om T. F. Wei ss
Last Time: Describing Signals by Frequency Content
Harmonic content is natural way to describe some kinds of signals. Ex: musical instruments (/MIS) piano piano t k violin violin t k bassoon bassoon t k
X-ray movie showing speech in production.
1 2
Lecture 15
Lowpass Filtering
High frequency square wave: ω0 > 1/RC .
1 2
April 1, 2010
0 −1 2 x(t) = 1 jω0 kt e ; jπk k odd 1 |H (jω )| 0.1 0.01 ∠H (jω )| 0 −π 2 0.01 0.1 0.01 0.1 � ω0 =
∞ � k=−∞
2π t T ,
. . ., e j
2π kt T }
respectively.
(“analysis” equation)
2π kt T
x(t)= x(t + T ) =
ak e j
(“synthesis” equation)
The complex conjugate (∗ ) makes the inner product of the k th and mth components equal to 1 iff k = m: � � � � 2π 2π 2π �∗ � j 2π mt � 1 1 1 if k = m e−j T kt e j T mt dt = e j T kt e T dt = T T T T 0 otherwise
1
10
1
10
Source-Filter Model of Speech Production
Vibrations of the vocal cords are “filtered” by the mouth and nasal cavities to generate speech.
Speech Production
6.003: Signals and Systems
6.003: Signals and Systems
Fourier Series
Lecture 15
Mid-term Examination #2
Wednesday, April 7, 7:30-9:30pm
April 1, 2010
.
No recitations on the day of the exam. Coverage: Lectures 1–15 Recitations 1–15 Homeworks 1–8
T 2π T
t
1
10
ω 100 1/RC ω 100 1/RC
1
10
ω 100 1/RC ω 100 1/RC
1
10
1
10
3
6.003: Signals and Systems
Lowpass Filtering
Still higher frequency square wave: ω0 = 1/RC .
bat t bit t
bait t bite t but t
bet t bought
beet t boat t t
boot t
Speech
Harmonic content is natural way to describe vowel sounds. bat bait bet beet
Speech
Homework 8 will not collected or graded. Solutions will be posted. Closed book: 2 pages of notes (8 1 2 × 11 inches; front and back). Designed as 1-hour exam; two hours to complete. Review sessions during open office hours. April 1, 2010
T 2π T
t
0.1
1
10
ω 100 1/RC ω 100 1/RC