1. Wireless PHY: Frequency-Domain Analysis Y. Richard Yang 09/4/2012

2. Outline • Admin and recap • Frequency domain analysis (Fourier series)

3. Admin • Slides and reading posted to class home page • It is important to read the assigned reading for today’s class

4. 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

5. sender analog signal bit stream source decoding channel coding channel decoding source coding modulation demodulation Overview of Wireless Physical Layer receiver bit stream

6. Wireless Physical Layer Example: Wireless: 802.11

7. Our Objective • Understand key issues and techniques in the design of wireless physical layer • Key approach: identify the problem and then the solution(s).

8. Outline • Recap • Frequency domain analysis (Fourier series)

9. Fourier Series: Decomposing 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 0 0 t t decomposition periodical signal set bk = 0

10. Fourier Series

11. Fourier Series

12. Fourier Series: Example http://en.wikipedia.org/wiki/File:Periodic_identity_function.gif

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

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

15. Fourier Series: Using Euler’s formula

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

17. Making Sense of Complex Numbers

18. Making Sense of Complex Numbers: Conjugate

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

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

21. Defining Decomposition on [0, 1]

22. ej2πft Making Sense of ej2πft

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

24. Two Domain Representations • Two representations: time domain; frequency domain • Knowing one can recover the other

25. Example: Frequency Seriesof sine and cosine

26. Example: Frequency Seriesof sine and cosine

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

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

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

30. From Integral to Computation

31. Discrete Domain Analysis • Transforming a sequence of numbers x0, x1, …, xN-1 to another sequence of numbers X0, X1, …, XN-1 • Inverse DFFT

33. Two Notes on Sampling • How fast to sample a time signal? • According to Nyquist’s Law, if the highest frequency is W, you need to sample at least 2W samples/sec • How to map from FFT output frequency when applied to samples? • Assume discrete FFT applies to Nfft samples • The interpretation is that Nfft samples is 1 sec • Hence, FFT output of X1 is for the base frequency • But Nfft is only Nfft/Nsample sec => X1 is for frequency of Nfft/Nsample

34. Frequency Domain Examples Using GNURadio • spectrum_samples • Observe sample/under sample

35. Frequency Domain Examples Using GNURadio • spectrum_2sin_plus • Audio • FFT Sink • Scope Sink • Noise

36. Backup Slides

37. Implementing Wireless: From Hardware to Software