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1 The Beauty of Mathematics in Communications R. C. T. Lee Dept. of Information Management & Dept. of Computer Science National Chi Nan University

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2 Operating systems and compilers Can be built without mathematics. Most drugs were invented without Mathematics.

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3 Can communications systems be built without mathematics? Ans: Absolutely no. Modern communication systems are totally based upon mathematics.

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4 For computer scientists, data are stored in memory as bits, either 1 or 0. How are the data transmitted? They are transmitted as pulses: A pulse represents a 1 and no pulse represents a 0.

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5 Fig. 1

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6 Is this possible when the transmission is done in a wireless environment? Impossible. Fact: Wireless communication is done every day. How is this possible?

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7 Can we mix together two bits and send out? Impossible if the two bits are represented as pulses. Fact: We often mix 256 bits together and send them at the same time. How is this done?

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8 Is an antenna open-circuited? Yes, it must be. You can easily prove this by looking at your mobile phone antenna.

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9 If an antenna is open-circuited, then there must be no current on it. How can it induce electromagnetic fields without any current?

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10 Can we broadcast our voice signals directly through some antenna? Impossible. Some kind of modulation must be done. Why?

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11 All of these questions can be answered by mathematics and only by mathematics.

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12 Fig. 2

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13 Fig. 3.The Discrete Fourier Transform Spectrum of the Signal in Fig. 2 after Sampling.

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14 Fig. 4 A Signal with Some Noise.

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15 Fig. 5 The Discrete Fourier Transform of the Signal in Fig. 4 after Sampling.

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16 Fig. 6 The Signal Obtained by Filtering Out the Noise.

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17 Fig. 7 A Music Signal Lasting 1 Second.

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18 Fig. 8 A Discrete Fourier Transform Spectrum of the Signal in Fig. 7.

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19 By using Fourier transform, we can see that the frequency components in our human voice are roughly contained in 3k Hertz.

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20 For a signal with frequency f, its wavelength can be found as follows: where v is the velocity of light.

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21 If,.

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22 It can also be proved that the length of an antenna is around. For human voice, this means that the wavelength is 50km. No antenna can be that long.

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23 What can we do? Answer: By amplitude modulation.

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24 Let be a signal. The amplitude modulation is defined as follows: where f c is the carrier frequency?

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25 What is the Fourier transform of ?

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26 Fig. 9

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27 The effect of amplitude modulation is to lift the baseband frequency to the carrier frequency level, a much higher one. Once the frequency becomes higher, its corresponding wavelength becomes smaller. An antenna is now possible.

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28 After we receive, how can we take out of it? Answer: Multiply by.

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29 Thus is recovered.

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30 Fig. 10

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31 Our next question: How is a bit transmitted? Answer: A bit is usually represented by a cosine function.

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32 Let us assume that bit 1 is represented by and bit 0 is represented by. When the receiver receives a bit, how can it detect whether 1, or 0, is sent?

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33 The basic scheme behind the detection is the inner product property of cosine functions: where.

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34 It can be proved that

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35 This inner product property gives us the fundamental mechanism of detecting 1 or 0. Let the sent signal be denoted as. We perform two inner products: and Decision rule: If, say that 1 is sent. If, say that 0 is sent.

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36 Suppose that we have two bits to send. Can we bundle them together and send the bundled result at the same time? Answer: Of course, we can.

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37 Let the two bits be demoted as and. or 0. Let if Let if

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38 The sent signal is Our job is to determine the values of and.

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39 We perform inner product again. and

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40 Can we bundle 256 bits together and send them at the same time? Answer: Yes, as along as the signals are orthogonal to one another. This is the basic principle of ADSL: OFDM (Orthogonal Frequency Division Method).

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41 Can we extend the above idea to two users case? Answer: We can.

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42 Let User 1 use to represent 1 and to represent 0. Let User 2 use to represent 1 and to represent 0. if i=j and if i j. and are orthogonal.

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43 The sent signal is denoted as where and. To determine, we perform inner products:

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44 Fig. 11 Signature Signals Generated from Hadamand matrix H 8. All of the signals are orthogonal to each other.

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45 We can also view the problem as a vector analysis problem. Assume that User 6 sends 1 and User 8 sends 0.

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46 V6=(1,-1,1,-1,-1,,1,-1,1) V8=(1,1,-1,-1,-1,-1,1,1) The inner product of v6 and v8 is =0

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47 The sent signal is 1 is sent. 0 is sent.

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48 This is the principle of CDMA (code division multiple access). It can be extended to more than two users. It was used by the military as an encryption method before.

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49 Suppose we send a signal entirely in digital form, can we say that this signal is an analog signal? Yes, we can because according to Fourier series analysis, a pulse also contains a set of cosine functions.

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52 Obviously, the smaller the pulse-width is, the more frequency components it contains. One may even say that the smaller the pulse-width is, the more information it may contain. Note that if a pulse has a small pulse-width, it means that within a second, a large number of pulses can be sent. This corresponds to high bit rate. Now, we know why a wire which has a high bit rate may be called broadband.

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53 It is important to observe the following: Bits are represented by analog signals. There are no digital signals in the world.

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54 It iIt i Maxwell s Equations Equations Concerning with Electromagnetic Waves

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55 Electric Field Induced by Charges Fig. 12 Coulomb s Law.

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56 Magnetic Field Induced by a Current Segment Fig. 13 Magnetic Flux Density Induced by a Current.

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57 Do the electric field and magnetic field affect each other? No, not in the static field. Yes, if the fields are time-varying.

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58 The curl of a vector.

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59 Faraday s law

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60 Fig. 14 The Voltage Caused by the Movement of a Magnet Inside a Coil The changing of magnetic field with time causes an electric field.

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61 Ampere s law Fig. 15 Magnetic Flux Density Induced by a Current.

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62 The Ampere s law modified by Maxwell The changing of electric field with time will induce a magnetic field. Maxwell modified Ampere s law without performing any experiments.

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63 Differential form Integral form Maxwell s Equations

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64 Plane Electromagnetic Waves With specical boundary conditions, Maxwell s equations reduce to

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65 The speed of the wave: Implication: The electromagnetic waves travel with the speed of light.

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66 Maxwell was not able to prove his theory. Hertz proved the correctness of Maxwell s equations.

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67 Fig. 16 The Traveling of a Wave.

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68 Fig. 17 The Electric and Magnetic Fields in a Plane Electromagnetic Wave.

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69 Fig. 18 The Wavelength. We can prove that.

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70 Transmission Line: Any electric wire which carries currents with high frequency can be considered as a transmission line.

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71 FiFi Fig. 20 Twin-Strip Parallel Plate Transmission Line Fig. 19 A Co-Axial Cable Transmission Line

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72 Fig. 21 An Equivalent Circuit of a Lossless Transmission Line.

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73 The above equations show that there are waves on the transmission line.

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74 It can be proved that the velocity of the waves is roughly the speed of light. Fig. 22 The Waves on a Transmission Line.

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75 Standing Waves Fig. 23 The Case of Open-Circuited Load

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76 Fig. 24 A Half Wave Dipole Antenna

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77 In ancient times, human beings built spectacular buildings. But, modern communications systems were possible only recently. Why? Answer: Modern communication systems can not exist without sophisticated mathematics.

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78 Thank you.

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