Chapter 2. Signals and Spectra

1. Chapter 2. Signals and Spectra • This chapter reviews one of the two pre-requisites for communications research. • Signals and Systems • Probability, Random Variables, and Random Processes • We use linear, particularly LTI, systems to develop the theory for communications. • Outline • 2.1 Line Spectra and Fourier Series • 2.2 Fourier Transform and Continuous Spectra • 2.3 Time and Frequency Relations • 2.4 Convolution • 2.5 Impulses and Transforms in the Limit • 2.6 Discrete Time Signals and the Discrete Fourier Transform

2. Communication Engineering 통신공학 Step 1. Given a communication medium, we first analyze the channel and build a mathematical model. 주어진 통신 매체에 따라 Channel 을 분석하고 모형을 만든다. Step 2. Using the model, we design the pair of a transmitter and a receiver that best exploits the channel characteristic. Channel 에 가장 효과적 신호처리를 할 수 있도록 Transmitter 와 Receiver를 설계한다. ex) Modulation (변조)과 Demodulation (복조) Encoding 과 Decoding Multiplexing 과 Demultiplexing

3. Mathematical Tool for Signal Processing: Fourier Analysis time domain frequency domain analysis, synthesis, design • 2.1 Line Spectra and Fourier Series • Linear Time-Invariant system

4. < 정현파 신호 (Sinusoidal Signal ) 의 표현 >

5. 대한민국 1호 라디오 (금성 A-501) 1959년, 금성사 김해수가 설계와 생산을 담당. –대한민국 역사 박물관

6. Line spectrum of periodic signals • 복소지수 (Complex exponential)에 의한 sinusoidal wave정현파 신호의 표현 복소수? Euler’s theorem/identity Amplitude A phase

7. 따라서

8. Phasor를 이용한 정현파 신호의 표현 Phasor representation is useful when sinusoidal signal is processed by real-in real-out LTI systems. 허수축 실수축

9. Q1왜 frequency domain 표현이 중요한가? (여러 가지 정현파형이 선형적으로 결합된 신호)

10. A1 Line Spectrum “왜 Phase는 Amplitude보다 덜 중요한가? (phase time delay ) “모든 주기적 신호는 정현파 신호의 선형적 결합으로 표현될 수 있다.” Phase Amplitude 90 5 40 3 2 10 35 0 0 10 35 Frequency content

11. Periodic Signals (주기 신호) Rectangular pulse train Figure 2.1-7

12. Fourier Series 어떠한 periodic signal 정현파 신호의 선형적 집합 Where Phasor표현 two-sided line spectrum

13. 주기함수의 주파수 특성 (Spectrum of periodic signals) 1. harmonics of fundamental frequency . 2. 3. 실함수 는

14. Spectrum of rectangular pulse train with ƒ0 = 1/4 (a) Amplitude (b) Phase Figure 2.1-8

15. trigonometric Fourier series for real signals 매우 중요한 함수

16. Fourier-series reconstruction of a rectangular pulse train Figure 2.1-9

17. Fourier-series reconstruction of a rectangular pulse train Figure 2.1-9c

18. Gibbs phenomenon at a step discontinuity Figure 2.1-10

19. Average Power of Periodic Signal

20. Parseval’s Power Theorem

21. 2.2 FourierTransforms and Continuous Spectra • Fourier Transform 비주기 신호 or Energy signal called the analysis equation. Definition

22. Inverse Fourier Transform called the synthesis equation.

23. Ex1 Rectangular pulse

24. Rectangular pulse spectrum V(ƒ) = A sinc ƒ Figure 2.2-2

25. Rayleigh’s Energy Theorem Generally Also called Parseval’s relation/theorem.

26. Duality Theorem

27. 2.3 Time and Frequency Relations • Superposition Property • Time Delay • Time Scale Change useful tool for linear systems linear phase Slow Playback Fast Playback Low Tone High Tone

28. Frequency Translation/Shift and Modulation

29. continued (a) RF pulse (b) Amplitude spectrum Figure 2.3-3

30. Differentiation and Integration In general Example. Triangular pulse Principle of FM demodulator differentiator

31. 2.4 Convolution • Convolution Integral Graphical interpretation of convolution Figure 2.4-1

32. Result of the convolution in Fig. 2.4-1 Figure 2.4-2 In general, convolution is a complicated operation in the TD.

33. Convolution Theorems

34. 2.5 Impulses and Transforms in the Limit • Dirac delta function Thus

35. Two functions that become impulses as  0 Figure 2.5-2

36. Properties

37. 실제적 함수 (Practical Impulses)

38. Fourier Transform of Power Signals infinite energy

39. From Fourier Series , Other periodic signals

40. 2.6 Discrete Time Signals and Discrete Fourier Transform • DT signal • DT periodic signal and DFTS • Analysis equation • Synthesis equation • DFT, IDFT • Periodic extension and Fourier Series • DTFT • Analysis equation • Synthesis equation

41. Convolution using the DFT • Q. We are given a convolution sum of two finite-length DT signals. Each signal has support N_1, N_2. Find the finite-length (at most N_1+N_2-1) output of the convolution using DFT. • A. Choose N>= N_1+N_2-1. Compute DFT(x) and DFT(h). Perform entry-by-entry multiplication. Apply the inverse DFT. Done.

42. HW #1 (Due on Next Tuesday 9/22. Please turn in handwritten solutions.) • 2.7 Questions • 3 • 4 • 6 • 2.1-9, 13 • 2.2-7, 10 • 2.3-8, 14 • 2.4-8, 15 • 2.5-10 • 2.6-4, 6