template-6up1
• Time and Frequency Representations
Examples:
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
model result
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.
Course Mechanics
Recitation: Tue. and Thu. 3-4pm in 5-234 or 36-156
Office Hours: Tue. and Thu. 4-5pm in 5-234 or 36-156
and other times TBD
• Exercises: study aids; not counted in grade
− online with immediate feedback
– pencil and paper problems taken from previous exams
– simple computational extensions to real-world data
– completely specified, unambiguous, self-contained
– more open-ended, multiple approaches, multiple solutions
– deepen understanding and demonstrate wide applicability
– issued Tuesday, required check-in Friday, due following Tuesday
Two Midterms and a Final Exam
Signals
Signals are functions that contain and convey information.
– may have 1 or 2 or 3 or even more independent variables
t
– dependent variable can be a scalar or a vector
x
x
y
6.003: Signal Processing Fall 2021
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Signals
t
1
0
−1
– continuous domain versus discrete domain
t
x(t)
x[n]
Signals from physical systems are often of continuous domain:
• continuous time – measured in seconds
• continuous spatial coordinates – measured in meters
Computations usually manipulate functions of discrete domain:
• discrete time – measured in samples
• discrete spatial coordinates – measured in samples
Signals
computational methods to solve problems that are intrinsically continuous.
Sampling: converting CT signals to DT
t
x(t)
x[n] = x(nT )
T = sampling interval
• digital representations of audio signals (as in MP3)
• digital representations of images (as in JPEG)
Signals
computational methods to solve problems that are intrinsically continuous.
Reconstruction: converting DT signals to CT
zero-order hold
x(t)
T = sampling interval
Signals
computational methods to solve problems that are intrinsically continuous.
Reconstruction: converting DT signals to CT
piecewise linear
x(t)
T = sampling interval
Signals
t
0 N n
• vibrating strings
• planetary motions
3
Signals
right-sided left-sided
Useful for modeling systems that have a well-defined starting point:
• piano note
symmetric antisymmetric
t
f(t)
f1(t), f2(t), f3(t), f4(t).
3f(t)
Signals are functions that contain and convey information.
Example: a musical sound can be represented as a function of time.
t [seconds]
pressure
Although this time function is a complete description of the sound, it does
not expose many of the important properties of the sound.
Musical Sounds as Signals
Even though these sounds have the same pitch, they sound different.
t
piano
t
cello
t
bassoon
t
oboe
t
horn
t
altosax
t
violin
1 262 sec.
It’s not clear how the differences relate to properties of the signals.
(audio clips from from http://theremin.music.uiowa.edu)
Musical Signals as Sums of Sinusoids
One way to characterize differences between these signals is express each
as a sum of sinusoids.
f(t) = ∞∑ k=0
2π ωo
2π ωo
...
Since these sounds are (nearly) periodic, the frequencies of the dominant
sinusoids are (nearly) integer multiples of a fundamental frequency ωo.
6.003: Signal Processing Fall 2021
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Harmonic Structure
The sum of sinusoids describes the distribution of energy across frequencies.
f(t) = ∞∑ k=0
mk cos (kωot+ φk)
where m2 k = c2
ck .
mk
D C →
fu n
d a
m e
n ta
Harmonic Structure
The harmonic structures of notes from different instruments are different.
t
piano
k
piano
t
bassoon
k
bassoon
t
violin
k
violin
Some musical qualities are more easily seen in time, others in frequency.
Consonance and Dissonance
A1
t
A2
t
B1
t
B2
t
C1
t
C2
t
Express Each Signal as a Sum of Sinusoids
f(t) = ∞∑ k=0
mk cos(kωot+ φk)
time
freq
Two views: as a function of time and as a function of frequency
Express Each Signal as a Sum of Sinusoids
f(t) = ∞∑ k=0
mk cos(kωot+ φk)
freq
The signal f(t) can be expressed as a discrete set of frequency components:
ω0: m1, φ1 2ω0: m2, φ2 3ω0: m3, φ3 · · ·
Musical Sounds as Signals
Time functions do a poor job of conveying 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
Harmonic structure conveys consonance and dissonance better.
6.003: Signal Processing Fall 2021
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f(t) = ∞∑ k=0
Basis functions:
2π ωo
2π ωo
t sin
...
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.
T= 2π ωo
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))
0 f(t) dt =
= Tc0 + ∞∑ k=1
c0 = 1 T
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))
2 cos((k+l)ωot) ) dt
2 sin((k+l)ωot) ) dt
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 T 2 cl.
We can solve to get an expression for cl as
cl = 2 T
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))
+ ∞∑ k=1
0 dk sin(kωot) sin(lωot) dt
A single term remains after integrating, allowing us to solve for dl as
dl = 2 T
6
f(t) = f(t+T ) = c0 + ∞∑ k=1
(ck cos(kωot) + dk sin(kωot))
c0 = 1 T
∫ T f(t) dt
ck = 2 T
dk = 2 T
Example of Analysis
Find the Fourier series coefficients for the following triangle wave:
t
1
dk = 0 (by symmetry)
Generate f(t) from the Fourier coefficients in the previous slide.
Start with the Fourier coefficients
f(t) = c0 − ∞∑ k=1
∞∑ k = 1 k odd
4 π2k2 cos(kπt)
Example of Synthesis
Generate f(t) from the Fourier coefficients in the previous slide.
Start with the Fourier coefficients
f(t) = c0 − ∞∑ k=1
∞∑ k = 1 k odd
4 π2k2 cos(kπt)
Example of Synthesis
Generate f(t) from the Fourier coefficients in the previous slide.
Start with the Fourier coefficients
f(t) = c0 − ∞∑ k=1
∞∑ k = 1 k odd
4 π2k2 cos(kπt)
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
mk cos(kωot+φk)
• 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.