1 / 42

Chapter 2. Signals and Spectra

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

higginsb
Download Presentation

Chapter 2. Signals and Spectra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related