LOCAL THEORETICAL MODELS IN ALGEBRA LEARNING

Vladimir Stojanovic

Description

• leçon - matière potentielle : an unknown
• leçon - matière potentielle : the equations
• cours - matière potentielle : methods
• exposé
• expression écrite
• leçon - matière potentielle : that teenagers

Rojano, T. (2004). Local Theoretical Models in Algebra Learning: a Meeting Point in Mathematics Education. In D. MacDougal (Ed.) Psychology of Mathematics Education - North American Chapter (1) pp 37-56. Toronto. LOCAL THEORETICAL MODELS IN ALGEBRA LEARNING: A MEETING POINT IN MATHEMATICS EDUCATION Teresa Rojano Centre for Research and Advanced Studies (CINVESTAV), Mexico Abstract The need of interpreting unanticipated phenomena, arising from studies on the transition from arithmetic to algebra in the 80s, gave rise to the long-term research program Acquiring Algebraic Language.

• algebraic substitution
• children progress along the list of items
• theoretical models
• word problems
• cognitive processes
• algebra
• line to line
• line for line
• line-line
• line upon line
• line by line
• analysis
• problem

Sujets

• Exposé
• Didacticism
• Algebraic
• Analysis
• Theory

Informations

• Publié par: mevang
• Nombre de lectures: 35

Extrait

Intro to Practical
Digital Communications
Lecture 2
Vladimir Stojanovi ć
6.973 Communication System Design – Spring 2006
Massachusetts Institute of Technology
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Discrete data transmission
Messages are encoded into signal points
signal points
signal waveforms
Signal points are mapped to signal waveforms
Modulation
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6.973 Communication System Design 18‰
Modulation and de-modulation
e.g. Binary Phase-Shift Keying (BPSK) x (t)
0
x (t)
1
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6.973 Communication System Design 19‰
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Vector signal representation
Maps continuous signals to discrete vectors
Significantly simplifies system analysis
signal
signal points
waveforms
basis functions
Modulator
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6.973 Communication System Design 20‰
BPSK example
Signal constellation
What is the information rate (R) of this modulation?
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6.973 Communication System Design 21‰
Manchester modulation example (Ethernet)
Different waveforms can have same vector representations
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6.973 Communication System Design 22More constellations
Quadrature Amplitude Modulation
(QAM)
Pulse Amplitude Modulation
ϕ ()t
2
(PAM)
3
ϕ()t
1
1
ϕ()t
1
-3 -1 1 3 -3 -1 1 3
-1
e.g. PAM4
-3
e.g. 16-QAM
PAM and QAM have pulses as basis functions
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6.973 Communication System Design 23„
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How do we choose basis functions?
Need to be orthonormal – (b/c of demodulation)
Inner products
Continuous
Discrete
Invariant to choice of basis functions
Average energy of the constellation
Invariant to the choice of basis functions
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Constellation energy
Implications of the inner product invariance to
basis functions
If energy is a signal, it is the same regardless of
the mod waveform used
As long as basis functions are orthogonal
Parseval’s identity
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6.973 Communication System Design 25‰
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Correlative demodulator
Demodulator
Modulator
Straightforward demodulator implementation
Use the fact that basis functions are orthogonal
Collect the signal energy
Hard to build in practice
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
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6.973 Communication System Design 26