6.003: Signal Processing Spring 2020 1 6.003: Signal Processing Signal Processing • Overview of Subject • Signals: Definitions, Examples, and Operations • Time and Frequency Representations • Fourier Series February 4, 2020 6.003: Signal Processing Signals are functions that contain and convey information. Examples: • the MP3 representation of a sound • the JPEG representation of a picture • an MRI image of a brain Signal Processing develops the use of signals as abstractions: • identifying signals in physical, mathematical, computation contexts, • analyzing signals to understand the information they contain, and • manipulating signals to modify and/or extract information. 6.003: Signal Processing Signal Processing is widely used in science and engineering to ... • model some aspect of the world, • analyze the model, and • interpret results to gain a new or better understanding. model result world new understanding make model analyze (math, computation) interpret results Signal Processing provides a common language across disciplines. Classical analyses use a variety of maths, especially calculus. We will also use computation to solve real-world problems that are difficult or impos- sible to solve analytically. → strengthens ties to the real world Course Mechanics Schedule Lecture: Tue. and Thu. 2-3pm in 32-141 Recitation: Tue. and Thu. 3-4pm in 24-121 or 26-328 Office Hours: Tue. and Thu. 4-5pm in 24-121 or 26-328 Wed. and Thu. 7-9pm in 36-144 Sun. 4-6pm (room TBD) Homework – issued Tuesdays, due following Tuesday at noon • Drills: focus on facts, definitions, and simple concepts - online with immediate feedback (not graded) • Problems: focus on developing problem solving skills – pencil and paper problems taken from previous exams – simple computational extensions to real-world data – completely specified, unambiguous, self-contained • Labs: focus on applications of 6.003 to authentic problems – more open-ended, multiple approaches, multiple solutions – deepen understanding and demonstrate wide applicability – issued Tuesday, required check-in Thursday, due following Tuesday Two Midterms and a Final Exam Signals Signals are functions that are used to convey information. – may have 1 or 2 or 3 or even more independent variables t sound pressure (t) x y brightness (x, y) Signals Signals are functions that are used to convey information. – dependent variable can be a scalar or a vector x y scalar: brightness at each point (x, y) x y vector: (red,green,blue) at each point (x, y)
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6.003: Signal Processing Spring 2020
1
6.003: Signal Processing
Signal Processing
• Overview of Subject
• Signals: Definitions, Examples, and Operations
• Time and Frequency Representations
• Fourier Series
February 4, 2020
6.003: Signal Processing
Signals are functions that contain and convey information.
Examples:
• the MP3 representation of a sound
• the JPEG representation of a picture
• an MRI image of a brain
Signal Processing develops the use of signals as abstractions:
• identifying signals in physical, mathematical, computation contexts,
• analyzing signals to understand the information they contain, and
• manipulating signals to modify and/or extract information.
6.003: Signal Processing
Signal Processing is widely used in science and engineering to ...
• model some aspect of the world,
• analyze the model, and
• interpret results to gain a new or better understanding.
model result
world new understanding
make model
analyze
(math, computation)
interpret results
Signal Processing provides a common language across disciplines.
Classical analyses use a variety of maths, especially calculus. We will also
use computation to solve real-world problems that are difficult or impos-
sible to solve analytically.
→ strengthens ties to the real world
Course Mechanics
Schedule
Lecture: Tue. and Thu. 2-3pm in 32-141
Recitation: Tue. and Thu. 3-4pm in 24-121 or 26-328
Office Hours: Tue. and Thu. 4-5pm in 24-121 or 26-328
Wed. and Thu. 7-9pm in 36-144
Sun. 4-6pm (room TBD)
Homework – issued Tuesdays, due following Tuesday at noon
• Drills: focus on facts, definitions, and simple concepts
− online with immediate feedback (not graded)
• Problems: focus on developing problem solving skills
– pencil and paper problems taken from previous exams
– simple computational extensions to real-world data
Harmonic structure conveys consonance and dissonance better.
6.003: Signal Processing Spring 2020
5
Fourier Representations of Signals
Fourier series are sums of harmonically related sinusoids.
f(t) =∞∑k=0
(ck cos(kωot) + dk sin(kωot))
where ωo = 2π/T represents the fundamental frequency.
Basis functions:
2πωo
t
cos(
0t)
2πωo
t
sin(0t)
2πωo
t
cos(ωot)
2πωo
tsin
(ωot)
2πωo
t
cos(
2ωot)
...2πωo
t
sin(2ωot)
...
Q1: Under what conditions can we write f(t) as a Fourier series?
Q2: How do we find the coefficients ck and dk.
Fourier Representations of Signals
Under what conditions can we write f(t) as a Fourier series?
Fourier series can only represent periodic signals.
Definition: a signal f(t) is periodic in T if
f(t) = f(t+T )for all t.
Note: if a signal is periodic in T it is also periodic in 2T , 3T , ...
The smallest positive number To for which f(t) = f(t + To) for all t is
sometimes called the fundamental period.
If a signal does not satisfy f(t) = f(t+T ) for any value of T , then the signal
is aperiodic.
Fourier Representations of Signals
Fourier series can only represent periodic signals.
ωωo 2ωo 3ωo 4ωo 5ωo 6ωo
T= 2πωo
t
T= 2πωo
t
All harmonics of ωo (cos(kωot) or sin(kωot)) are periodic in T = 2π/ωo.→ all sums of such signals are periodic in T = 2π/ωo.→ Fourier series can only represent periodic signals.
Calculating Fourier Coefficients
How do we find the coefficients ck and dk for all k?
Key idea: simplify by integrating over the period T of the fundamental.
Start with the general form:
f(t) = f(t+T ) = c0 +∞∑k=1
(ck cos(kωot) + dk sin(kωot))
Integrate both sides over T :∫ T
0f(t) dt =
∫ T
0c0 dt+
∫ T
0
( ∞∑k=1
(ck cos(kωot) + dk sin(kωot)))dt
= Tc0 +∞∑k=1
(ck
∫ T
0cos(kωot) dt+ dk
∫ T
0sin(kωot) dt
)= Tc0
All but the first term integrates to zero, leaving
c0 = 1T
∫ T
0f(t) dt.
This k=0 term represents the average (“DC”) value.
Calculating Fourier Coefficients
Isolate the cl term by multiplying both sides by cos(lωot) before integrating.
f(t) = f(t+T ) = c0 +∞∑k=1
(ck cos(kωot) + dk sin(kωot))
∫ T
0f(t) cos(lωot) dt =
∫ T
0c0 cos(lωot) dt
+∞∑k=1
∫ T
0ck
(12 cos((k−l)ωot) + 1
2 cos((k+l)ωot))dt
+∞∑k=1
∫ T
0dk
(12 sin((k−l)ωot) + 1
2 sin((k+l)ωot))dt
0
T2 cl 0
0 0
If k = l, then sin((k−l)ωot = 0 and the integral is 0.
All of the other dk terms are harmonic sinusoids that integrate to 0.
The only non-zero term on the right side is T2 cl.
We can solve to get an expression for cl as
cl = 2T
∫ T
0f(t) cos(lωot) dt
Calculating Fourier Coefficients
Analogous reasoning allows us to calculate the dk coefficients, but this time
multiplying by sin(lωot) before integrating.
f(t) = f(t+T ) = c0 +∞∑k=1
(ck cos(kωot) + dk sin(kωot))
∫ T
0f(t) sin(lωot) dt =
∫ T
0c0 sin(lωot) dt
+∞∑k=1
∫ T
0ck cos(kωot) sin(lωot) dt
+∞∑k=1
∫ T
0dk sin(kωot) sin(lωot) dt
A single term remains after integrating, allowing us to solve for dl as
dl = 2T
∫ T
0f(t) sin(lωot) dt
6.003: Signal Processing Spring 2020
6
Calculating Fourier Coefficients
Summarizing . . .
If f(t) is expressed as a Fourier series
f(t) = f(t+T ) = c0 +∞∑k=1
(ck cos(kωot) + dk sin(kωot))
the Fourier coefficients are given by
c0 = 1T
∫Tf(t) dt
ck = 2T
∫Tf(t) cos(kωot) dt; k = 1, 2, 3, . . .
dk = 2T
∫Tf(t) sin(kωot) dt; k = 1, 2, 3, . . .
Example of Analysis
Find the Fourier series coefficients for the following triangle wave:
t
f(t) = f(t+2)
0 1 2−1−2
1
T = 2
ωo = 2πT
= π
c0 = 1T
∫ T
0f(t) dt = 1
2
∫ 2
0f(t)dt = 1
2
ck = 2T
∫ T/2
−T/2f(t) cos 2πkt
Tdt = 2
∫ 1
0t cos(πkt) dt =
{− 4π2k2 k odd
0 k = 2, 4, 6, . . .
dk = 0 (by symmetry)
Example of Synthesis
Generate f(t) from the Fourier coefficients in the previous slide.
Start with the Fourier coefficients
f(t) = c0 −∞∑k=1
(ck cos(kωot) + dk sin(kωot)) = 12 −
∞∑k = 1k odd
4π2k2 cos(kπt)
f(t) = 12 −
1∑k = 1k odd
4π2k2 cos(kπt)
t
f(t)
0 1 2−1−2
Example of Synthesis
Generate f(t) from the Fourier coefficients in the previous slide.
Start with the Fourier coefficients
f(t) = c0 −∞∑k=1
(ck cos(kωot) + dk sin(kωot)) = 12 −
∞∑k = 1k odd
4π2k2 cos(kπt)
f(t) = 12 −
3∑k = 1k odd
4π2k2 cos(kπt)
t
f(t)
0 1 2−1−2
Example of Synthesis
Generate f(t) from the Fourier coefficients in the previous slide.
Start with the Fourier coefficients
f(t) = c0 −∞∑k=1
(ck cos(kωot) + dk sin(kωot)) = 12 −
∞∑k = 1k odd
4π2k2 cos(kπt)
f(t) = 12 −
99∑k = 1k odd
4π2k2 cos(kπt)
t
f(t)
0 1 2−1−2
The synthesized function approaches original as number of terms increases.
Two Views of the Same Signal
The harmonic expansion provides an alternative view of the signal.
f(t) =∞∑k=0
(ck cos(kωot) + dk sin(kωot)) =∞∑k=0
mk cos(kωot+φk)
We can view the musical signal as
• a function of time f(t), or
• as a sum of harmonics with amplitudes mk and phase angles φk.
Both views are useful. For example,
• the peak sound pressure is more easily seen in f(t), while
• consonance is more easily analyzed by comparing harmonics.
This type of harmonic analysis is an example of Fourier Analysis,
which is a major theme of this subject.
Next Time: understanding Fourier series and their properties.