1. COE 341: Data & Computer Communications (T061) Dr. Marwan Abu-Amara Chapter 3: Data Transmission

2. COE 341 (T061) – Dr. Marwan Abu-Amara 2 Agenda Concepts & Terminology Decibels and Signal Strength Fourier Analysis Analog & Digital Data Transmission Transmission Impairments Channel Capacity

3. COE 341 (T061) – Dr. Marwan Abu-Amara 3 Terminology (1) Transmitter Receiver Medium  Guided medium e.g. twisted pair, optical fiber  Unguided medium e.g. air, water, vacuum

4. COE 341 (T061) – Dr. Marwan Abu-Amara 4 Terminology (2) Direct link  No intermediate devices Point-to-point  Direct link  Only 2 devices share link Multi-point  More than two devices share the link

5. COE 341 (T061) – Dr. Marwan Abu-Amara 5 Terminology (3) Simplex  One direction e.g. Television Half duplex  Either direction, but only one way at a time e.g. police radio Full duplex  Both directions at the same time e.g. telephone

6. COE 341 (T061) – Dr. Marwan Abu-Amara 6 Frequency, Spectrum and Bandwidth Time domain concepts  Analog signal Varies in a smooth way over time  Digital signal Maintains a constant level then changes to another constant level  Periodic signal Pattern repeated over time  Aperiodic signal Pattern not repeated over time

7. COE 341 (T061) – Dr. Marwan Abu-Amara 7 Analogue & Digital Signals

8. COE 341 (T061) – Dr. Marwan Abu-Amara 8 Periodic Signals

9. COE 341 (T061) – Dr. Marwan Abu-Amara 9 Sine Wave Peak Amplitude (A)  maximum strength of signal  volts Frequency (f)  Rate of change of signal  Hertz (Hz) or cycles per second  Period = time for one repetition (T)  T = 1/f Phase (  )  Relative position in time

10. COE 341 (T061) – Dr. Marwan Abu-Amara 10 Varying Sine Waves s(t) = A sin(2  ft +  )

11. COE 341 (T061) – Dr. Marwan Abu-Amara 11 Wavelength Distance occupied by one cycle Distance between two points of corresponding phase in two consecutive cycles Assuming signal velocity v  = vT  f = v  c = 3*10 8 m/sec (speed of light in free space)

12. COE 341 (T061) – Dr. Marwan Abu-Amara 12 Frequency Domain Concepts Signal usually made up of many frequencies Components are sine waves Can be shown (Fourier analysis) that any signal is made up of component sine waves Can plot frequency domain functions

13. COE 341 (T061) – Dr. Marwan Abu-Amara 13 Addition of Frequency Components (T=1/f) Fundamental Frequency

14. COE 341 (T061) – Dr. Marwan Abu-Amara 14 Frequency Domain Representations

15. COE 341 (T061) – Dr. Marwan Abu-Amara 15 Spectrum & Bandwidth Spectrum  range of frequencies contained in signal Absolute bandwidth  width of spectrum Effective bandwidth  Often just bandwidth  Narrow band of frequencies containing most of the energy DC Component  Component of zero frequency

16. COE 341 (T061) – Dr. Marwan Abu-Amara 16 Signal with a DC Component t 1V DC Level 1V DC Component t Courtesy of Dr. Radwan Abdel-Aal

17. COE 341 (T061) – Dr. Marwan Abu-Amara 17 Bandwidth for these signals: fminfmax Absolute BW Effective BW 1f3f2f 03f 0  1/x ? Courtesy of Dr. Radwan Abdel-Aal

18. COE 341 (T061) – Dr. Marwan Abu-Amara 18 Bandwidth and Data Rate Any transmission system supports only a limited band of frequencies for satisfactory transmission “system” includes: TX, RX, and Medium Limitation is dictated by considerations of cost, number of channels, etc. This limited bandwidth degrades the transmitted signals, making it difficult to interpret them at RX For a given bandwidth: Higher data rates More degradation This limits the data rate that can be used with given signal and noise levels, receiver type, and error performance More about this later!!! Courtesy of Dr. Radwan Abdel-Aal

19. COE 341 (T061) – Dr. Marwan Abu-Amara 19 Bandwidth Requirements f 3f f 5f f 3f 5f 7f f 3f 5f 7f ……  BW = 2f BW = 4f BW = 6f BW =  … Larger BW needed for better representation More difficult reception with more limited BW 1,3 1,3,5 1,3,5,7,9,…  1 2 3 4 Fourier Series for a Square Wave Courtesy of Dr. Radwan Abdel-Aal

20. COE 341 (T061) – Dr. Marwan Abu-Amara 20 Decibels and Signal Strength Decibel is a measure of ratio between two signal levels  N dB = number of decibels  P 1 = input power level  P 2 = output power level Example:  A signal with power level of 10mW inserted onto a transmission line  Measured power some distance away is 5mW  Loss expressed as N dB =10log(5/10)=10(-0.3)=-3 dB

21. COE 341 (T061) – Dr. Marwan Abu-Amara 21 Decibels and Signal Strength Decibel is a measure of relative, not absolute, difference  A loss from 1000 mW to 500 mW is a loss of 3dB  A loss of 3 dB halves the power  A gain of 3 dB doubles the power Example:  Input to transmission system at power level of 4 mW  First element is transmission line with a 12 dB loss  Second element is amplifier with 35 dB gain  Third element is transmission line with 10 dB loss  Output power P2 (-12+35-10)=13 dB = 10 log (P 2 / 4mW) P 2 = 4 x 10 1.3 mW = 79.8 mW

22. COE 341 (T061) – Dr. Marwan Abu-Amara 22 Relationship Between Decibel Values and Powers of 10 Power Ratio dB dB 10 1 1010 -1 -10 10 2 2010 -2 -20 10 3 3010 -3 -30 10 4 4010 -4 -40 10 5 5010 -5 -50 10 6 6010 -6 -60

23. COE 341 (T061) – Dr. Marwan Abu-Amara 23 Decibel-Watt (dBW) Absolute level of power in decibels Value of 1 W is a reference defined to be 0 dBW Example:  Power of 1000 W is 30 dBW  Power of 1 mW is –30 dBW

24. COE 341 (T061) – Dr. Marwan Abu-Amara 24 Decibel & Difference in Voltage Decibel is used to measure difference in voltage. Power P=V 2 /R Decibel-millivolt (dBmV) is an absolute unit with 0 dBmV equivalent to 1mV.  Used in cable TV and broadband LAN

25. COE 341 (T061) – Dr. Marwan Abu-Amara 25 Fourier Analysis Signals Periodic (f o ) Aperiodic Discrete Continuous DFSFS DTFT FT DFT Infinite time Finite time FT: Fourier Transform DFT: Discrete Fourier Transform DTFT: Discrete Time Fourier Transform FS: Fourier Series DFS: Discrete Fourier Series

26. COE 341 (T061) – Dr. Marwan Abu-Amara 26 Fourier Series Any periodic signal of period T (f 0 = 1/T) can be represented as sum of sinusoids, known as Fourier Series If A 0 is not 0, x(t) has a DC component DC Component fundamental frequency

27. COE 341 (T061) – Dr. Marwan Abu-Amara 27 Fourier Series Amplitude-phase representation

28. COE 341 (T061) – Dr. Marwan Abu-Amara 28

29. COE 341 (T061) – Dr. Marwan Abu-Amara 29 Fourier Series Representation of Periodic Signals - Example 1 1/2-1/213/2-3/22 T x(t) Note:(1) x(– t)=x(t)  x(t) is an even function (2) f 0 = 1 / T = ½

30. COE 341 (T061) – Dr. Marwan Abu-Amara 30 Fourier Series Representation of Periodic Signals - Example Replacing t by –t in the first integral sin(-2  nf t)= - sin(2  nf t)

31. COE 341 (T061) – Dr. Marwan Abu-Amara 31 Fourier Series Representation of Periodic Signals - Example Since x(– t)=x(t) as x(t) is an even function, then B n = 0 for n=1, 2, 3, … Cosine is an even function

32. COE 341 (T061) – Dr. Marwan Abu-Amara 32 Another Example 1 1 2 T -2 x1(t) Note that x1(-t)= -x1(t)  x(t) is an odd function Also, x1(t)=x(t-1/2) x(t)

33. COE 341 (T061) – Dr. Marwan Abu-Amara 33 Another Example Sine is an odd function Where:

34. COE 341 (T061) – Dr. Marwan Abu-Amara 34 Fourier Transform For a periodic signal, spectrum consists of discrete frequency components at fundamental frequency & its harmonics. For an aperiodic signal, spectrum consists of a continuum of frequencies (non-discrete components).  Spectrum can be defined by Fourier Transform  For a signal x(t) with spectrum X(f), the following relations hold

35. COE 341 (T061) – Dr. Marwan Abu-Amara 35

36. COE 341 (T061) – Dr. Marwan Abu-Amara 36 Fourier Transform Example x(t) A

37. COE 341 (T061) – Dr. Marwan Abu-Amara 37 Fourier Transform Example A   1/     f Study the effect of the pulse width  Sin (x) / x “sinc” function

38. COE 341 (T061) – Dr. Marwan Abu-Amara 38 The narrower a function is in one domain, the wider its transform is in the other domain The Extreme Cases Courtesy of Dr. Radwan Abdel-Aal

39. COE 341 (T061) – Dr. Marwan Abu-Amara 39 Power Spectral Density & Bandwidth Absolute bandwidth of any time-limited signal is infinite However, most of the signal power will be concentrated in a finite band of frequencies Effective bandwidth is the width of the spectrum portion containing most of the signal power. Power spectral density (PSD) describes the distribution of the power content of a signal as a function of frequency

40. COE 341 (T061) – Dr. Marwan Abu-Amara 40 Signal Power A function x(t) specifies a signal in terms of either voltage or current Assuming R = 1  Instantaneous signal power = V 2 = i 2 = |x(t)| 2 Instantaneous power of a signal is related to average power of a time-limited signal, and is defined as For a periodic signal, the averaging is taken over one period to give the total signal power

41. COE 341 (T061) – Dr. Marwan Abu-Amara 41 Power Spectral Density & Bandwidth For a periodic signal, power spectral density is where  (f) is C n is as defined before on slide 27, and f 0 being the fundamental frequencyslide 27

42. COE 341 (T061) – Dr. Marwan Abu-Amara 42 Power Spectral Density & Bandwidth For a continuous valued function S(f), power contained in a band of frequencies f 1 < f < f 2 For a periodic waveform, the power through the first j harmonics is

43. COE 341 (T061) – Dr. Marwan Abu-Amara 43 Power Spectral Density & Bandwidth - Example Consider the following signal The signal power is

44. COE 341 (T061) – Dr. Marwan Abu-Amara 44 Fourier Analysis Example Consider the half-wave rectified cosine signal from Figure B.1 on page 793: 1. Write a mathematical expression for s(t) 2. Compute the Fourier series for s(t) 3. Write an expression for the power spectral density function for s(t) 4. Find the total power of s(t) from the time domain 5. Find a value of n such that Fourier series for s(t) contains 95% of the total power in the original signal 6. Determine the corresponding effective bandwidth for the signal

45. COE 341 (T061) – Dr. Marwan Abu-Amara 45 Example (Cont.) 1. Mathematical expression for s(t): -T/4 -3T/4 +3T/4 +T/4 Where f 0 is the fundamental frequency, f 0 = (1/T)

46. COE 341 (T061) – Dr. Marwan Abu-Amara 46 Example (Cont.) 2. Fourier Analysis: f 0 = (1/T)

47. COE 341 (T061) – Dr. Marwan Abu-Amara 47 Example (Cont.) 2. Fourier Analysis (cont.): f 0 = (1/T)

48. COE 341 (T061) – Dr. Marwan Abu-Amara 48 Example (Cont.) 2. Fourier Analysis (cont.):

49. COE 341 (T061) – Dr. Marwan Abu-Amara 49 Example (Cont.) 2. Fourier Analysis (cont.): Note: cos 2  = ½(1 + cos 2  )

50. COE 341 (T061) – Dr. Marwan Abu-Amara 50 Example (Cont.) 2. Fourier Analysis (cont.):

51. COE 341 (T061) – Dr. Marwan Abu-Amara 51 Example (Cont.) 2. Fourier Analysis (cont.):

52. COE 341 (T061) – Dr. Marwan Abu-Amara 52 Example (Cont.) 2. Fourier Analysis (cont.):

53. COE 341 (T061) – Dr. Marwan Abu-Amara 53 Example (Cont.) 3. Power Spectral Density function (PSD): Or more accurately:

54. COE 341 (T061) – Dr. Marwan Abu-Amara 54 Example (Cont.) 3. Power Spectral Density function (PSD):

55. COE 341 (T061) – Dr. Marwan Abu-Amara 55 Example (Cont.) 4. Total Power: Note: cos 2  = ½(1 + cos 2  )

56. COE 341 (T061) – Dr. Marwan Abu-Amara 56 Example (Cont.) 5. Finding n such that we get 95% of total power:

57. COE 341 (T061) – Dr. Marwan Abu-Amara 57 Example (Cont.) 5. Finding n such that we get 95% of total power:

58. COE 341 (T061) – Dr. Marwan Abu-Amara 58 Example (Cont.) 5. Finding n such that we get 95% of total power: 6. Effective bandwidth with 95% of total power: B eff = f max – f min = 2f 0 – 0 = 2f 0 0 f0f0 2f 0 3f 0 …  f B eff

59. COE 341 (T061) – Dr. Marwan Abu-Amara 59 Data Rate and Bandwidth Any transmission system has a limited band of frequencies This limits the data rate that can be carried Example on pages 65 & 66

60. COE 341 (T061) – Dr. Marwan Abu-Amara 60 Bandwidth and Data Rates 2B = 4f B = 4f f 3f f 5f Period T = 1/f 1 0 1 0 Data Element, Signal Element T/2 Data rate = 1/(T/2) = 2/T bits per sec = 2f Given a bandwidth B, Data rate = 2f = 2(B/4) = B/2 Two ways to double the data rate… To double the data rate you need to double f 1. Double the bandwidth, with the same receiver and error rate (same received waveform) f3f 5f New bandwidth: 2B, Data rate = 2f = 2(2B/4) = B 2. Same bandwidth, B, with a better receiver, higher S/N, or by tolerating more error (poorer received waveform) B = 2f Bandwidth: B, Data rate = 2f = 2(B/2) = B 1 1 11 00 00 1 1 11 0 0 00 B 2B B Courtesy of Dr. Radwan Abdel-Aal

61. COE 341 (T061) – Dr. Marwan Abu-Amara 61 Bandwidth and Data Rates Increasing the data rate (bps) with the same BW means working with inferior waveforms at the receiver, which may require:  Better signal to noise ratio at RX (larger signal relative to noise): Shorter link spans Use of more repeaters/amplifiers Better shielding of cables to reduce noise, etc.  More sensitive (& costly!) receiver  Dealing with higher error rates Tolerating them Adding more efficient means for error detection and correction- this also increases overhead!. Courtesy of Dr. Radwan Abdel-Aal

62. COE 341 (T061) – Dr. Marwan Abu-Amara 62 Analog and Digital Data Transmission Data  Entities that convey meaning Signals  Electric or electromagnetic representations of data Transmission  Communication of data by propagation and processing of signals

63. COE 341 (T061) – Dr. Marwan Abu-Amara 63 Analog and Digital Data Analog  Continuous values within some interval  e.g. sound, video Digital  Discrete values  e.g. text, integers

64. COE 341 (T061) – Dr. Marwan Abu-Amara 64 Acoustic Spectrum (Analog)

65. COE 341 (T061) – Dr. Marwan Abu-Amara 65 Analog and Digital Signals Means by which data are propagated Analog  Continuously variable  Various media wire, fiber optic, space  Speech bandwidth 100Hz to 7kHz  Telephone bandwidth 300Hz to 3400Hz  Video bandwidth 4MHz Digital  Use two DC components

66. COE 341 (T061) – Dr. Marwan Abu-Amara 66 Advantages & Disadvantages of Digital Cheaper Less susceptible to noise Greater attenuation  Pulses become rounded and smaller  Leads to loss of information

67. COE 341 (T061) – Dr. Marwan Abu-Amara 67 Attenuation of Digital Signals

68. COE 341 (T061) – Dr. Marwan Abu-Amara 68 Components of Speech Frequency range (of hearing) 20Hz-20kHz  Speech 100Hz-7kHz Easily converted into electromagnetic signal for transmission Sound frequencies with varying volume converted into electromagnetic frequencies with varying voltage Limit frequency range for voice channel  300-3400Hz

69. COE 341 (T061) – Dr. Marwan Abu-Amara 69 Conversion of Voice Input into Analog Signal

70. COE 341 (T061) – Dr. Marwan Abu-Amara 70 Video Components USA - 483 lines scanned per frame at 30 frames per second  525 lines but 42 lost during vertical retrace So 525 lines x 30 scans = 15750 lines per second  63.5  s per line  11  s for retrace, so 52.5  s per video line Max frequency if line alternates black and white Horizontal resolution is about 450 lines giving 225 cycles of wave in 52.5  s Max frequency of 4.2MHz

71. COE 341 (T061) – Dr. Marwan Abu-Amara 71 Binary Digital Data From computer terminals etc. Two dc components Bandwidth depends on data rate

72. COE 341 (T061) – Dr. Marwan Abu-Amara 72 Conversion of PC Input to Digital Signal

73. COE 341 (T061) – Dr. Marwan Abu-Amara 73 Data and Signals Usually use digital signals for digital data and analog signals for analog data Can use analog signal to carry digital data  Modem Can use digital signal to carry analog data  Compact Disc audio

74. COE 341 (T061) – Dr. Marwan Abu-Amara 74 Analog Signals Carrying Analog and Digital Data

75. COE 341 (T061) – Dr. Marwan Abu-Amara 75 Digital Signals Carrying Analog and Digital Data

76. COE 341 (T061) – Dr. Marwan Abu-Amara 76 Four Data/Signal Combinations Signal AnalogDigital Data Analog - Same spectrum as data (base band): e.g. Conventional Telephony - Different spectrum (modulation): e.g. AM, FM Radio Use a (converter): codec, e.g. for PCM (pulse code modulation) Digital Use a (converter): modem e.g. with the V.90 standard - Simple two signal levels: e.g. NRZ code - Special Encoding: e.g. Manchester code (Chapter 5) Courtesy of Dr. Radwan Abdel-Aal

77. COE 341 (T061) – Dr. Marwan Abu-Amara 77 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

78. COE 341 (T061) – Dr. Marwan Abu-Amara 78 Digital Transmission Concerned with content Integrity endangered by noise, attenuation etc. Repeaters used  Repeater receives signal  Extracts bit pattern  Retransmits Attenuation is overcome Noise is not amplified

79. COE 341 (T061) – Dr. Marwan Abu-Amara 79 Four Signal/Transmission Mode Combinations Transmission mode Analog - Uses amplifiers - Not concerned with what data the signal represents - Noise is cumulative Digital - Uses repeaters - Assumes signal represents digital data, recovers it and represents it as a new outbound signal - This way, noise is not cumulative Signal Analog OK Makes sense only if the analog signal represents digital data Digital Avoid OK Courtesy of Dr. Radwan Abdel-Aal

80. COE 341 (T061) – Dr. Marwan Abu-Amara 80 Advantages of Digital Transmission Digital technology  Low cost LSI/VLSI technology Data integrity  Longer distances over lower quality lines Capacity utilization  High bandwidth links economical  High degree of multiplexing easier with digital techniques Security & Privacy  Encryption Integration  Can treat analog and digital data similarly

81. COE 341 (T061) – Dr. Marwan Abu-Amara 81 Transmission Impairments Signal received may differ from signal transmitted Analog - degradation of signal quality Digital - bit errors Caused by  Attenuation and attenuation distortion  Delay distortion  Noise

82. COE 341 (T061) – Dr. Marwan Abu-Amara 82 Attenuation Signal strength falls off with distance Depends on medium (guided vs. unguided) Received signal strength:  must be enough to be detected  must be sufficiently higher than noise to be received without error Attenuation is an increasing function of frequency  Different frequency components of a signal get attenuated differently  Signal distortion  Particularly significant with analog signals

83. COE 341 (T061) – Dr. Marwan Abu-Amara 83 Delay Distortion Only in guided media Propagation velocity varies with frequency  Highest at center frequency (minimum delay)  Lower at both ends of the bandwidth (larger delay) Effect: Different frequency components of the signal arrive at slightly different times! (Dispersion) Badly affects digital data due to bit spill-over (timing is more important than for analog data)

84. COE 341 (T061) – Dr. Marwan Abu-Amara 84 Noise (1) Additional unwanted signals inserted between transmitter and receiver The most limiting factor in communication systems Thermal  Due to thermal agitation of electrons  Uniformly distributed  White noise

85. COE 341 (T061) – Dr. Marwan Abu-Amara 85 More on Thermal (White) Noise Power of thermal noise present in a bandwidth B (Hz) is given by T is absolute temperature in kelvin and k is Boltzmann’s constant (k = 1.38  10 -23 J/K) Example: at t = 21  C (T = 294  K) and for a bandwidth of 10 MHz: N = -228.6 + 10 log 294 + 10 log 10 7 = - 133.9 dBW Courtesy of Dr. Radwan Abdel-Aal

86. COE 341 (T061) – Dr. Marwan Abu-Amara 86 Noise (2) Intermodulation  Signals that are the sum and difference of original frequencies sharing a medium  f1, f2  (f1+f2) and (f1-f2)  Caused by nonlinearities in the medium and equipment, e.g. due to overdrive and saturation of amplifiers  Resulting frequency components may fall within valid signal bands, thus causing interference Courtesy of Dr. Radwan Abdel-Aal

87. COE 341 (T061) – Dr. Marwan Abu-Amara 87 Noise (3) Crosstalk  A signal from one channel picked up by another channel  e.g. Coupling between twisted pairs, antenna pick up, leakage between adjacent channels in FDM, etc. Impulse  Irregular pulses or spikes  Short duration  High amplitude  e.g. External electromagnetic interference  Minor effect on analog signals but major effect on digital signals, particularly at high data rates

88. COE 341 (T061) – Dr. Marwan Abu-Amara 88 Channel Capacity Channel capacity: Maximum data rate usable under given communication conditions How BW, signal level, noise and other impairments, and the amount of error tolerated limit the channel capacity? Max data rate = Function (BW, Signal wrt noise, Error rate allowed)  Max data rate: Max rate at which data can be communicated, bits per second (bps)  Bandwidth: BW of the transmitted signal as constrained by the transmission system, cycles per second (Hz)  Signal relative to Noise, SNR = signal power/noise power ratio (Higher SNR  better communication conditions)  Error rate: bits received in error/total bits transmitted. Equal to the bit error probability Courtesy of Dr. Radwan Abdel-Aal

89. COE 341 (T061) – Dr. Marwan Abu-Amara 89 1. Nyquist Bandwidth: (Noise-free, Error-free) Idealized, theoretical Assumes a noise-free, error-free channel Nyquist: If rate of signal transmission is 2B then a signal with frequencies no greater than B is sufficient to carry that signalling rate In other words: Given bandwidth B, highest signalling rate possible is 2B signals/s Given a binary signal (1,0), data rate is same as signal rate  Data rate supported by a BW of B Hz is 2B bps For the same B, data rate can be increased by sending one of M different signal levels (symbols): as a signal level now represents log 2 M bits Generalized Nyquist Channel Capacity, C = 2B log 2 M bits/s (bps) Signals/s bits/signal Courtesy of Dr. Radwan Abdel-Aal

90. COE 341 (T061) – Dr. Marwan Abu-Amara 90 Nyquist Bandwidth: Examples C = 2B log 2 M bits/s  C = Nyquist Channel Capacity  B = Bandwidth  M = Number of discrete signal levels (symbols) used Telephone Channel: B = 3400-300 = 3100 Hz With a binary signal (M = 2): C = 2B log 2 2 = 2B = 6200 bps With a quandary signal (M = 4): C = 2B log 2 4 = 2B x 2 = 4B = 12,400 bps Practical limit: larger M makes it difficult for the receiver to operate, particular with noise 0 1 00 01 10 11 Courtesy of Dr. Radwan Abdel-Aal

91. COE 341 (T061) – Dr. Marwan Abu-Amara 91 2. Shannon Capacity Formula: (Noisy, Error-Free) Assumes error-free operation with noise Data rate, noise, error: A given noise burst affects more bits at higher data rates, which increases the error rate So, maximum error-free data rate increases with reduced noise Signal to noise ratio SNR = signal / noise levels SNR dB = 10 log 10 (SNR) dBs Shannon Capacity C = B log 2 (1+SNR): Highest data rate transmitted error-free with a given noise level For a given BW, the larger the SNR the higher the data rate I can use without errors C/B: Spectral (bandwidth) efficiency, BE, (bps/Hz) (>1) Larger BEs mean better utilizing of a given B for transmitting data fast. Caution! Log 2 Not Log 10 Caution! Ratio- Not log Courtesy of Dr. Radwan Abdel-Aal

92. COE 341 (T061) – Dr. Marwan Abu-Amara 92 Shannon Capacity Formula: Comments Formula says: for data rates  calculated C, it is theoretically possible to find an encoding scheme that gives error-free transmission. But it does not say how… It is a theoretical approach based on thermal (white) noise only However, in practice, we also have impulse noise and attenuation and delay distortions So, maximum error-free data rates obtained in practice are lower than the C predicted by this theoretical formula However, maximum error-free data rates can be used to compare practical systems: The higher that rate the better the system is Courtesy of Dr. Radwan Abdel-Aal

93. COE 341 (T061) – Dr. Marwan Abu-Amara 93 Shannon Capacity Formula: Comments Contd. Formula suggests that changes in B and SNR can be done arbitrarily and independently… but In practice, this may not be the case!  High SNR obtained through excessive amplification may introduce nonlinearities: distortion and intermediation noise!  High Bandwidth B opens the system for more thermal noise (kTB), and therefore reduces SNR! Courtesy of Dr. Radwan Abdel-Aal

94. COE 341 (T061) – Dr. Marwan Abu-Amara 94 Shannon Capacity Formula: Example  Spectrum of communication channel extends from 3 MHz to 4 MHz  SNR = 24dB  Then B = 4MHz – 3MHz = 1MHz SNR dB = 24dB = 10 log 10 (SNR) SNR = 251  Using Shannon’s formula: C = B log 2 (1+ SNR) C = 10 6 * log 2 (1+251) ~ 10 6 * 8 = 8 Mbps  Based on Nyquist’s formula, determine M that gives the above channel capacity: C = 2B log 2 M 8 * 10 6 = 2 * (10 6 ) * log 2 M 4 = log 2 M M = 16 Courtesy of Dr. Radwan Abdel-Aal

95. COE 341 (T061) – Dr. Marwan Abu-Amara 95 3. E b /N 0 (Signal Energy per Bit/Noise Power density per Hz) (Noise and Error Together) Handling both noise and error together E b /N 0 : A standard quality measure for digital communication system performance E b /N 0 Can be independently related to the error rate Expresses SNR in a manner related to the data rate, R E b = Signal energy per bit (Joules) = Signal power (Watts) x bit interval T b (second) = S x (1/R) = S/R N 0 = Noise power (watts) in 1 Hz = kT Courtesy of Dr. Radwan Abdel-Aal

96. COE 341 (T061) – Dr. Marwan Abu-Amara 96 E b /N 0 Example 1:  Given: E b /N o = 8.4 dB (minimum) is needed to achieve a bit error rate of 10 -4  Given: The effective noise temperature is 290 o K (room temperature) Data rate is 2400 bps  What is the minimum signal level required for the received signal?  8.4 = S(dBW) – 10 log 2400 + 228.6 dBW – 10 log290 = S(dBW) – (10)(3.38) + 228.6 – (10)(2.46) S = -161.8 dBW Courtesy of Dr. Radwan Abdel-Aal

97. COE 341 (T061) – Dr. Marwan Abu-Amara 97 E b /N 0 (Cont.) Bit error rate for digital data is a decreasing function of E b /N 0 for a given signal encoding scheme Which encoding scheme is better: A or B?  Get E b /N 0 to achieve a desired error rate, then determine other parameters from formula, e.g. S, SNR, R, etc. (Design) Error performance of a given system (Analysis) Effect of S, R, T on error performance Lower Error Rate: larger E b /N 0 A B Better Encoding Courtesy of Dr. Radwan Abdel-Aal

98. COE 341 (T061) – Dr. Marwan Abu-Amara 98 E b /N 0 (Cont.) From Shannon’s formula: C = B log 2 (1+SNR) We have: From the E b /N 0 formula: With R = C, substituting for SNR we get: Relates achievable spectral efficiency C/B (bps/Hz) to E b /N 0 Courtesy of Dr. Radwan Abdel-Aal

99. COE 341 (T061) – Dr. Marwan Abu-Amara 99 E b /N 0 (Cont.) Example 2 Find the minimum E b /N 0 required to achieve a spectral efficiency (C/B) of 6 bps/Hz: Substituting in the equation above: E b /N 0 = (1/6) (2 6 - 1) = 10.5 = 10.21 dB Courtesy of Dr. Radwan Abdel-Aal