1. CS434/534: Topics in Networked (Networking) Systems Network OS Abstraction: From Data to Function Store; Wireless Foundation: Frequency-Domain Analysis Yang (Richard) Yang Computer Science Department Yale University 208A Watson

2. Outline Admin and recap High-level datapath programming
Network OS supporting programmable networks
overview
OpenDaylight
distributed network OS (Paxos, Raft)
from data store to function store
Programmable wireless communications and networking
Background

3. Admin
Hardcopies of PS1 can be turned into my mail box (208A) on the first floor
A talk on mobile systems
Time: Tuesday 10:30 am
Location: AKW

4. Recap Distributed data store using Raft
Programming complexities of data store
Programmers need to manually record data access and register subscriptions
Programmers need to handle cleanup of any changes made in previous execution
Programmers need to make sure that no data-change loops are triggered, to avoid instability
From data store to function store
Function store API
add(func, [meta]), remove(handler)
Data access API
read(xpath), update(xpath, op, val), test(xpath, booleanExpr, ??

5. Recap: From API to Realization
Re-exec a function using collected dependency.
R: {b, c} f2 f4
R: {h}
R: {h} f1 f6 f3 f5
R: {a, f}
R: {a, b, c}
Assume exec’d the seq f1-f6. Then a changes. How to recover? Which functions must re-execute? Which do not need re-exec?

6. From API to Realization: Design 1
UpdateAlg1( a )
Fupdate = a.DEPENDENCY_CHANGE Ffixed = executed before Fupdate in total order
restore2( Ffixed )
while ( ! Fupdate.empty() ) {
pick f as a root in Fupdate
Fupdate -= f; Ffixed += f execute f
add functions: not in Ffixed ^ depends on f updates }

7. Question Is UpdateAlg1 correct? R: {b, c} f2 f4 R: {h} R: {h} f1 f6 f3
R: {a, f}
R: {a, b, c}
Is UpdateAlg1 correct?

8. Function Store There are multiple directions where one can explore:
How to best utilize test()?
How to best utilize past computation to learn dependency (e.g., learning)?
How to utilize multi-cores in recovery?
Can we distribute the functions at multiple servers (e.g., running Raft)?
A good topic as a course project.

9. Roadmap High-level networking system programming
Networking operating system
overview
OpenDaylight
distributed networking OS (Paxos, Raft)
from data store to function store
Programmable wireless communications and networking
background
software defined wireless networking systems
Network function virtualization

10. Recap: Wireless and Mobile Computing
Driven by infrastructure and device technology
global infrastructures
device miniaturization and capabilities
software development platforms
Challenges:
wireless channel: unreliable, open access
mobility
portability
changing environment
heterogeneity

11. Wireless Networking Systems: Objectives
Avoid a full course on wireless communications, but cover enough to understand the main issues and main emerging techniques.

12. Implementing Wireless Systems: From Hardware to Software

13. Overview of Wireless Communicatinos/Networking
source coding
bit
stream
channel coding
analog
signal
sender
modulation
receiver
bit
stream
source decoding
channel decoding
demodulation

14. Wireless Example: Wireless: 802.11

15. Basic Question: Why Not Send Digital Signal in Wireless Communications?1
digital signal t

16. Outline Admin and recap
Programmable wireless communications and networking
background
Frequency domain analysis (Fourier series)

17. Fourier Series: Decomposing a function into a Collection of Harmonics
A periodic real function g(t) on [-π, π] can be decomposed as a set of harmonics (cos, sin):
Time domain 1 1 t t
decomposition
periodical signal
set bk = 0

18. Fourier Series

19. Fourier Series

20. Fourier Series: Example

21. Fourier Series: An Alternative Representation
A problem of the expression
It contains both cos() and sin() functions, and hence is somehow complex to manipulate.

22. Fourier Series: Using Euler’s formula
Applying Euler’s formula
We have

23. Fourier Series: Using Euler’s formula

24. Making Sense of Complex Numbers
What is the effect of multiplying c by ejπ/2?
What is the effect of multiplying c by j?

25. Making Sense of Complex Numbers

26. Making Sense of Complex Numbers: Conjugate

27. Summary of Progress: Fourier Series of Real Function on [-π, π]

28. Defining Decomposition on a General Interval
A periodic function g(t) with period T on [a, a+T] can be decomposed as:

29. Defining Decomposition on [0, 1]

30. ej2πft Making Sense of ej2πft

31. Making Sense of ej2πft G[f]ej2πft ej2πft ϕ=2πft ϕ=-2πft e-j2πft

32. For those who are curious, we do not need it formally
Fourier Transform
For those who are curious, we do not need it formally
Problem: Fourier series for periodic function g(t), what if g(t) is not periodical?
Approach:
Truncate g(t) beyond [-L/2, L/2] (i.e., set = 0) and then repeat to define gL(t)

33. Fourier Transform
Define
Let L grow to infinity, we derive Fourier Transform:

34. Fourier Series vs Fourier Transform
For periodical functions, e.g., [0, 1]
Fourier transform
For non periodical functions

35. Two Domain Representations
See examples: spectrum_2in.m
Two representations: time domain; frequency domain
Knowing one can recover the other

36. Example: Frequency Series of sine and cosine
See examples: spectrum_2in.m

37. Example: Frequency Series of sine and cosine

38. Example: Frequency Domain’s View of Euler’s Formula

39. Example: Frequency Domain’s View of Euler’s Formula

40. Example: Frequency Domain’s View of Euler’s Formula

41. From Integral to Computation

42. Discrete Domain Analysis
FFT: Transforming a sequence of numbers x0, x1, …, xN-1 to another sequence of numbers X0, X1, …, XN-1
Interpretation: consider x0, x1, …, xN-1 as sampled values of a periodical function defined on [0, 1]
Xk is the coefficient (scaled by N) for k Hz harmonics if the FFT N samples span one sec
Draw on blackboard [Nfft, Nfft, …] = Ns (defined for 1 sec)

43. Discrete Domain Analysis
FFT:
Inverse DFFT