1. 3.1 Chapter 3 Data and Signals Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2. 3.2 To be transmitted, data must be transformed to electromagnetic signals. Note

3. 3.3 3-1 ANALOG AND DIGITAL Data can be analog or digital. The term analog data refers to information that is continuous; digital data refers to information that has discrete states. Analog data take on continuous values. Digital data take on discrete values.  Analog and Digital Data  Analog and Digital Signals  Periodic and Nonperiodic Signals Topics discussed in this section:

4. 3.4 Analog and Digital Data  Data can be analog or digital.  Analog data are continuous and take continuous values.  Digital data have discrete states and take discrete values.

5. 3.5 Analog and Digital Signals Signals can be analog or digital. Analog signals can have an infinite number of values in a range. Digital signals can have only a limited number of values.

6. 3.6 Figure 3.1 Comparison of analog and digital signals

7. 3.7 3-2 PERIODIC ANALOG SIGNALS In data communications, we commonly use periodic analog signals and nonperiodic digital signals. Periodic analog signals can be classified as simple or composite. A simple periodic analog signal, a sine wave, cannot be decomposed into simpler signals. A composite periodic analog signal is composed of multiple sine waves.  Sine Wave  Wavelength  Time and Frequency Domain  Composite Signals  Bandwidth Topics discussed in this section:

8. 3.8 Figure 3.2 A sine wave

9. 3.9 Figure 3.3 Two signals with the same phase and frequency, but different amplitudes

10. 3.10 Frequency and period are the inverse of each other. Note

11. 3.11 Figure 3.4 Two signals with the same amplitude and phase, but different frequencies

12. 3.12 Table 3.1 Units of period and frequency

13. 3.13 The power we use at home has a frequency of 60 Hz. The period of this sine wave can be determined as follows: Example 3.1

14. 3.14 The period of a signal is 100 ms. What is its frequency in kilohertz? Example 3.2 Solution First we change 100 ms to seconds, and then we calculate the frequency from the period (1 Hz = 10 −3 kHz).

15. 3.15 Frequency Frequency is the rate of change with respect to time. Change in a short span of time means high frequency. Change over a long span of time means low frequency.

16. 3.16 If a signal does not change at all, its frequency is zero. If a signal changes instantaneously, its frequency is infinite. Note

17. 3.17 Phase describes the position of the waveform relative to time 0. Note

18. 3.18 Figure 3.5 Three sine waves with the same amplitude and frequency, but different phases

19. 3.19 A sine wave is offset 1/6 cycle with respect to time 0. What is its phase in degrees and radians? Example 3.3 Solution We know that 1 complete cycle is 360°. Therefore, 1/6 cycle is

20. 3.20 Figure 3.6 Wavelength and period

21. 3.21 Figure 3.7 The time-domain and frequency-domain plots of a sine wave

22. 3.22 A complete sine wave in the time domain can be represented by one single spike in the frequency domain. Note

23. 3.23 The frequency domain is more compact and useful when we are dealing with more than one sine wave. For example, Figure 3.8 shows three sine waves, each with different amplitude and frequency. All can be represented by three spikes in the frequency domain. Example 3.7

24. 3.24 Figure 3.8 The time domain and frequency domain of three sine waves

25. 3.25 Signals and Communication A single-frequency sine wave is not useful in data communications We need to send a composite signal, a signal made of many simple sine waves. According to Fourier analysis, any composite signal is a combination of simple sine waves with different frequencies, amplitudes, and phases.

26. 3.26 Composite Signals and Periodicity If the composite signal is periodic, the decomposition gives a series of signals with discrete frequencies. If the composite signal is nonperiodic, the decomposition gives a combination of sine waves with continuous frequencies.

27. 3.27 Figure 3.9 shows a periodic composite signal with frequency f. This type of signal is not typical of those found in data communications. We can consider it to be three alarm systems, each with a different frequency. The analysis of this signal can give us a good understanding of how to decompose signals. Example 3.4

28. 3.28 Figure 3.9 A composite periodic signal

29. 3.29 Figure 3.10 Decomposition of a composite periodic signal in the time and frequency domains

30. 3.30 Figure 3.11 shows a nonperiodic composite signal. It can be the signal created by a microphone or a telephone set when a word or two is pronounced. In this case, the composite signal cannot be periodic, because that implies that we are repeating the same word or words with exactly the same tone. Example 3.5

31. 3.31 Figure 3.11 The time and frequency domains of a nonperiodic signal

32. 3.32 Bandwidth and Signal Frequency The bandwidth of a composite signal is the difference between the highest and the lowest frequencies contained in that signal.

33. 3.33 Figure 3.12 The bandwidth of periodic and nonperiodic composite signals

34. 3.34 If a periodic signal is decomposed into five sine waves with frequencies of 100, 300, 500, 700, and 900 Hz, what is its bandwidth? Draw the spectrum, assuming all components have a maximum amplitude of 10 V. Solution Let f h be the highest frequency, f l the lowest frequency, and B the bandwidth. Then Example 3.6 The spectrum has only five spikes, at 100, 300, 500, 700, and 900 Hz (see Figure 3.13).

35. 3.35 Figure 3.13 The bandwidth for Example 3.6

36. 3.36 A periodic signal has a bandwidth of 20 Hz. The highest frequency is 60 Hz. What is the lowest frequency? Draw the spectrum if the signal contains all frequencies of the same amplitude. Solution Let f h be the highest frequency, f l the lowest frequency, and B the bandwidth. Then Example 3.7 The spectrum contains all integer frequencies. We show this by a series of spikes (see Figure 3.14).

37. 3.37 Figure 3.14 The bandwidth for Example 3.7

38. 3.38 A nonperiodic composite signal has a bandwidth of 200 kHz, with a middle frequency of 140 kHz and peak amplitude of 20 V. The two extreme frequencies have an amplitude of 0. Draw the frequency domain of the signal. Solution The lowest frequency must be at 40 kHz and the highest at 240 kHz. Figure 3.15 shows the frequency domain and the bandwidth. Example 3.8

39. 3.39 Figure 3.15 The bandwidth for Example 3.8

40. 3.40 An example of a nonperiodic composite signal is the signal propagated by an AM radio station. In the United States, each AM radio station is assigned a 10-kHz bandwidth. The total bandwidth dedicated to AM radio ranges from 530 to 1700 kHz. We will show the rationale behind this 10-kHz bandwidth in Chapter 5. Example 3.9

41. 3.41 Another example of a nonperiodic composite signal is the signal propagated by an FM radio station. In the United States, each FM radio station is assigned a 200- kHz bandwidth. The total bandwidth dedicated to FM radio ranges from 88 to 108 MHz. We will show the rationale behind this 200-kHz bandwidth in Chapter 5. Example 3.10

42. 3.42 Another example of a nonperiodic composite signal is the signal received by an old-fashioned analog black- and-white TV. A TV screen is made up of pixels. If we assume a resolution of 525 × 700, we have 367,500 pixels per screen. If we scan the screen 30 times per second, this is 367,500 × 30 = 11,025,000 pixels per second. The worst-case scenario is alternating black and white pixels. We can send 2 pixels per cycle. Therefore, we need 11,025,000 / 2 = 5,512,500 cycles per second, or Hz. The bandwidth needed is 5.5125 MHz. Example 3.11

43. 3.43 Signal Classifications and Properties

44. 3.44 Fourier analysis is a tool that changes a time domain signal to a frequency domain signal and vice versa. Note Fourier Analysis

45. 3.45 Fourier Series Every composite periodic signal can be represented with a series of sine and cosine functions. The functions are integral harmonics of the fundamental frequency “f” of the composite signal. Using the series we can decompose any periodic signal into its harmonics.

46. 3.46 Fourier Series

47. 3.47 Examples of Signals and the Fourier Series Representation

48. 3.48 Sawtooth Signal

49. 3.49 Fourier Transform Fourier Transform gives the frequency domain of a nonperiodic time domain signal.

50. 3.50 Example of a Fourier Transform

51. 3.51 Inverse Fourier Transform

52. 3.52 Time limited and Band limited Signals A time limited signal is a signal for which the amplitude s(t) = 0 for t > T 1 and t < T 2 A band limited signal is a signal for which the amplitude S(f) = 0 for f > F 1 and f < F 2

53. McGraw-Hill©The McGraw-Hill Companies, Inc., 2000 53 Signals Signals are electric or electromagnetic encoding of data

54. 54 Information, Data and Signals Data - A representation of facts, concepts, or instructions in a formalized manner suitable for communication, interpretation, or processing by human beings or by automatic means Information - The meaning that is currently assigned to data by means of the conventions applied to those data

55. 55 Information, Data and Signals InformationDataSignal 001011101

56. 56 Computers Use Signals for Communcation Computers transmit data using digital signals, sequences of specified voltage levels. Graphically they are often represented as a square wave. Computers sometimes communicate over telephone line using analog signals, which are formed by continuously varying voltage levels.

57. 57 Signal = Function of Time The signal is a function of time. Horizontal axis represents time and the vertical axis represents the voltage level. Signal represents data OR Data is encoded by means of a signal Signal is what travels on a communication medium An understanding of signals is required so that suitable signal may be chosen to represent data

58. 58 Continuous and Discrete Signal Continuous or Analog signals take on all possible values of amplitude Digital or Discrete Signals take on finite set of voltage levels

59. 59 Analog and Digital Signal Continuous/Analog signals take on all possible values of amplitude Digital or Discrete Signals take on finite set of voltage levels

60. 60 Analog and Digital Data Analog data take on all possible values. Voice and video are continuously varying patterns of intensity Digital data take on finite (countable) number of values. Example, ASCII characters, integers

61. 61 Periodic Signals Some signals repeat themselves over fixed intervals of time. Such signals are said to be periodic A signal s(t) is periodic if and only if: s(t+T) = s(t) -  < t < +  where the constant T is the periodic of the signal, otherwise a signal is aperiodic (or non- periodic).

62. 62 Periodic Signal Properties Three important characteristics of a periodic signal are : Amplitude (A): the instantaneous value of a signal at any time measured in volts. Frequency (f): the number of repetitions of the period per second or the inverse of the period; it is expressed in cycles per second or Hertz (Hz). T=1/f Phase (  ): a measure of the relative position in time within a single period of a signal, measured in degrees Wavelength ( ): distance occupied by a signal in one period

63. 63

64. 64 Spectrum and Bandwidth Spectrum of a signal - the range of frequencies it contains Absolute bandwidth - the width of the spectrum Effective bandwidth or just bandwidth - the band of frequencies which contains most of the energy of the signal - half-power bandwidth

65. 65

66. 66 Bandwidth and Data Rate Width of the spectrum of frequencies that can be transmitted if spectrum=300 to 3400Hz, bandwidth=3100Hz Greater bandwidth leads to greater costs Limited bandwidth leads to distortion Analog measured in Hertz, digital measured in baud

67. 67 General Observations about Signals For a signal with multiple frequencies, energy is in the first few frequency components Increasing the bandwidth increases data rate The transmission medium limits the bandwidth Greater the bandwidth, the greater the cost For a given data rate, limiting the bandwidth, increases distortion, and hence the error rate

68. 68 Transmission Impairments Attenuation Delay Noise

69. 69 Attenuation Loss of signal strength over distance Use of amplifiers to boost analog signals; entire signal (including noise or distortion) is amplified Use of repeaters for digital data; data recovered and then transmitted

70. 70 Attenuation Distortion Analog signal is made up of several frequencies Attenuation is different for different frequencies; Different losses at different frequencies More of a problem for analog signals than digital

71. 71 Attenuation – the strength of a signal falls off with distance Attenuation Distortion – attenuation varies as a function of frequency

72. 72 Attenuation is measured in deciBels - dB

73. 73 dB Calculation Example Input power is 1 Watt Output power is 1 mW dB Attenuation is 10 * log (1 W/1 mW) = 10 * (3) = 30 dB

74. 74 Use of Repeaters

75. 75 Delay Distortion The velocity of propagation of a signal through a guided medium varies with frequency. Different frequency components travel at different speeds therefore arrive at a destination at different times. Particularly critical for digital data because bits may spill over causing Inter-Symbol Interference.

76. 76 Transmission Impairments Visually

77. 77 Noise What is Noise? Any unwanted signal Types of Noise Thermal Intermodulation Impulse Crosstalk

78. 78 Thermal Noise Thermal noise, white noise Due to random motion of atoms N = kTW k = Boltzman Constant (1.381 X 10 -23 J/K) T = Absolute Temperature (Kelvin) W = Bandwidth (Hz) Why is it called White Noise?

79. 79 Inter-modulation Noise Inter-modulation noise when two signals at different frequencies are mixed in the same medium, sum or difference of original frequencies or multiples of those frequencies can be produced, which can interfere with the intended signal - occurs when there is some non-linearity in the system

80. 80 Noise Crosstalk when there is an unwanted coupling between signal paths. For example some times talking on the telephone you can hear another conversation. Impulse noise Due to lightning or some other random transient phenomenon

81. 81 Effect of Noise

82. 82 Transmission Impairments in Un-Guided Media Free-Space loss Signal disperses with distance Atmospheric Absorption Attenuation caused by water vapor and oxygen Water vapor: High around 22 GHz, less around 15 GHz Oxygen: High around 60 GHz, less below 30 GHz

83. 83 Transmission Impairments in Un-Guided Media Multipath Receive multiple signals reflected by many obstacles Refraction Radio waves get bent by change in speed with altitude Thermal Noise White noise. Important factor for satellite communications

84. 84 Channel Capacity The rate at which digital data can be transmitted over a given communication channel Two formulations Shannon’s Formulation Nyquist Formulation

85. 85 Shannon’s Law Considers the noise (only white noise) Key parameter is signal-to-noise ratio (S/N, or SNR), which is the ratio of the power in a signal to the power contained in the noise, typically measured at the receiver - often expressed in decibels Maximum theoretical error-free capacity in bits per second C = W log 2 (1+S/N)

86. 86 Signal to Noise Ratio - S/N Signal to Noise ratio: power in signal to power contained in noise Doubling the bandwidth doubles the data rate At a given noise level, higher the data rate, the higher the error rate Increasing signal strength increases intermodulation noise Wider the bandwidth, the more noise is admitted. As W increases, S/N decreases Goal: Get highest data rate with lowest error rate at cheapest cost

87. 87 Example with S/N of 1000 for Telephone LIne Example for voice-grade telephone line: Using Shannon's formulation for channel capacity: C = W * log 2 (1 + S/N) Where log 2 represent logarithm base 2 30 dB S/N = 1000 S/N C = 3100 * log 2 (1 + 1000) C = 30,894 bps Hence the channel capacity is 30,894 bits per second.

88. 88 Nyquist Limit Nyquist limit (in a noise-free environment) C = 2 W log 2 M Given a bandwidth of W, highest signal rate that can be carried is 2W with binary signaling (M=2) For multilevel signaling C = 2W log 2 M where M is the number of discrete signals or voltage levels

89. 89 Example with M-ary Signaling with 3100 Hz Bandwidth C = 2 W log 2 M

90. 90

91. 91 BPS vs. Baud BPS=bits per second Baud= Number of signal changes per second Each signal change can represent more than one bit, through variations on amplitude, frequency, and/or phase

92. 92 Why Study Analog? Telephone system is primarily analog rather than digital (designed to carry voice signals) Low-cost, ubiquitous transmission medium If we can convert digital information (1s and 0s) to analog form (audible tone), it can be transmitted inexpensively Media are inherently analog too, real world is analog

93. 93 Analog Transmission Analog signal transmitted without regard to content May be analog or digital data Attenuated over distance Use amplifiers to boost signal Also amplifies noise

94. 94 Digital Transmission Concerned with content Integrity endangered by noise, attenuation etc. Repeaters are used Repeater receives signal Extracts bit pattern Retransmits Attenuation is overcome Noise is not amplified

95. 95 Use of Repeaters

96. 96 Which Signal/Data is Better Analog or Digital? Digital is better Even Analog data can be converted into digital data and transmitted as digital data Digital data provide the following advantages: Digital technology Data integrity through EDC and ECC Capacity utilization through TDM Security and privacy through encryption Integration of all forms of information

97. McGraw-Hill©The McGraw-Hill Companies, Inc., 2000 Discrete-Time Signals and Systems Quote of the Day Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. Paul Dirac Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.

98. Digital Signal Processing 98 Discrete-Time Signals: Sequences Discrete-time signals are represented by sequence of numbers The n th number in the sequence is represented with x[n] Often times sequences are obtained by sampling of continuous-time signals In this case x[n] is value of the analog signal at x c (nT) Where T is the sampling period

99. 99 Basic Sequences and Operations Delaying (Shifting) a sequence Unit sample (impulse) sequence Unit step sequence Exponential sequences

100. 100 Sinusoidal Sequences Important class of sequences An exponential sequence with complex x[n] is a sum of weighted sinusoids Different from continuous-time, discrete-time sinusoids Have ambiguity of 2  k in frequency Are not necessary periodic with 2  /  o

101. 101 Demo Rotating Phasors Demo

102. 102 Discrete-Time Systems Discrete-Time Sequence is a mathematical operation that maps a given input sequence x[n] into an output sequence y[n] Example Discrete-Time Systems Moving (Running) Average Maximum Ideal Delay System T{.} x[n]y[n]

103. 103 Memory less System A system is memory less if the output y[n] at every value of n depends only on the input x[n] at the same value of n Example Memory less Systems Square Sign Counter Example Ideal Delay System

104. 351M Digital Signal Processing 104 Linear Systems Linear System: A system is linear if and only if Examples Ideal Delay System

105. 351M Digital Signal Processing 105 Time-Invariant Systems Time-Invariant (shift-invariant) Systems A time shift at the input causes corresponding time-shift at output Example Square Counter Example Compressor System

106. 106 Causal System Causality A system is causal it’s output is a function of only the current and previous samples Examples Backward Difference Counter Example Forward Difference

107. 351M Digital Signal Processing 107 Stable System Stability (in the sense of bounded-input bounded-output BIBO) A system is stable if and only if every bounded input produces a bounded output Example Square Counter Example Log