COE 341 Data Computer Communications Dr Marwan AbuAmara

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

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

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

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

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

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

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

T Periodic Signals Temporal Period S (t+n. T) = S (t); Where: t is time T is the waveform period n is an integer COE 341 – Dr. Marwan Abu-Amara 8

Sine Wave – s(t) = A sin(2 ft + ) n Peak Amplitude (A) q q n maximum strength of signal unit: volts Frequency (f) q q q rate of change of signal unit: Hertz (Hz) or cycles per second Period = time for one repetition (T) = 1/f n Phase ( ) n relative position in time q unit: radians Angular Frequency (w) q q q w = 2 /T = 2 f unit: radians per second COE 341 – Dr. Marwan Abu-Amara 9

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

Wavelength ( ) n n n Distance occupied by one cycle Distance between two points of corresponding phase in two consecutive cycles Assuming signal velocity v q q q = v. T f = v For an electromagnetic wave, v = speed of light in the medium n In free space, v = c = 3*108 m/s COE 341 – Dr. Marwan Abu-Amara 11

Frequency Domain Concepts n n 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 COE 341 – Dr. Marwan Abu-Amara 12

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

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

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

Signal with a DC Component t 1 V DC Level t 1 V DC Component COE 341 – Dr. Marwan Abu-Amara 16

Bandwidth for these signals: Absolute Effective BW BW fmin fmax 1 f 3 f 2 f 2 f 0 3 f 3 f 3 f 0 1/x ? COE 341 – Dr. Marwan Abu-Amara 17

Bandwidth and Data Rate n n n n 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!!! COE 341 – Dr. Marwan Abu-Amara 18

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

Decibels and Signal Strength n Decibel is a measure of ratio between two signal levels q q q n Nd. B = number of decibels P 1 = input power level P 2 = output power level Example: q q q A signal with power level of 10 m. W inserted onto a transmission line Measured power some distance away is 5 m. W Loss expressed as Nd. B =10 log(5/10)=10(-0. 3)=-3 d. B COE 341 – Dr. Marwan Abu-Amara 20

Decibels and Signal Strength n Decibel is a measure of relative, not absolute, difference q q q n A loss from 1000 m. W to 500 m. W is a loss of 3 d. B A loss of 3 d. B halves the power A gain of 3 d. B doubles the power Example: q q q Input to transmission system at power level of 4 m. W First element is transmission line with a 12 d. B loss Second element is amplifier with 35 d. B gain Third element is transmission line with 10 d. B loss Output power P 2 n n (-12+35 -10)=13 d. B = 10 log (P 2 / 4 m. W) P 2 = 4 x 101. 3 m. W = 79. 8 m. W COE 341 – Dr. Marwan Abu-Amara 21

Relationship Between Decibel Values and Powers of 10 Power Ratio 101 d. B 10 Power Ratio 10 -1 -10 102 20 10 -2 -20 103 30 10 -3 -30 104 40 10 -4 -40 105 50 10 -5 -50 106 60 10 -6 -60 COE 341 – Dr. Marwan Abu-Amara d. B 22

Decibel-Watt (d. BW) n Absolute level of power in decibels Value of 1 W is a reference defined to be 0 d. BW n Example: n q q Power of 1000 W is 30 d. BW Power of 1 m. W is – 30 d. BW COE 341 – Dr. Marwan Abu-Amara 23

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

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

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

Fourier Series n Amplitude-phase representation COE 341 – Dr. Marwan Abu-Amara 27

COE 341 – Dr. Marwan Abu-Amara 28

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

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

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

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

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

Fourier Transform n n 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). q q Spectrum can be defined by Fourier Transform For a signal x(t) with spectrum X(f), the following relations hold COE 341 – Dr. Marwan Abu-Amara 34

COE 341 – Dr. Marwan Abu-Amara 35

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

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

The narrower a function is in one domain, the wider its transform is in the other domain The Extreme Cases COE 341 – Dr. Marwan Abu-Amara 38

Power Spectral Density & Bandwidth n n 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 COE 341 – Dr. Marwan Abu-Amara 39

Signal Power n n A function x(t) specifies a signal in terms of either voltage or current Assuming R = 1 W, 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 COE 341 – Dr. Marwan Abu-Amara 40

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

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

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

Fourier Analysis Example Consider the half-wave rectified cosine signal from Figure A. 1 on page 838: n 1. 2. 3. 4. 5. 6. Write a mathematical expression for s(t) Compute the Fourier series for s(t) Write an expression for the power spectral density function for s(t) Find the total power of s(t) from the time domain Find a value of n such that Fourier series for s(t) contains 95% of the total power in the original signal Determine the corresponding effective bandwidth for the signal COE 341 – Dr. Marwan Abu-Amara 44

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Data Rate and Bandwidth n n n Any transmission system has a limited band of frequencies This limits the data rate that can be carried Example on pages 75 – 77 of textbook COE 341 – Dr. Marwan Abu-Amara 59

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

Bandwidth and Data Rates n Increasing the data rate (bps) with the same BW means working with inferior waveforms at the receiver, which may require: q Better signal to noise ratio at RX (larger signal relative to noise): n n n q q 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 n n Tolerating them Adding more efficient means for error detection and correctionthis also increases overhead!. COE 341 – Dr. Marwan Abu-Amara 61

Bandwidth and Data Rates COE 341 – Dr. Marwan Abu-Amara 62

Analog and Digital Data Transmission n Data q n Signal q n Entities that convey meaning Electric or electromagnetic representations of data Transmission q Communication of data by propagation and processing of signals COE 341 – Dr. Marwan Abu-Amara 63

Analog and Digital Data Transmission n Data q n Signal q n Can be either Analog data or Digital data Can use either Analog signal or Digital signal to convey the data Transmission q Can use either Analog transmission or Digital transmission to carry the signal COE 341 – Dr. Marwan Abu-Amara 64

Analog and Digital Data n Analog q q n Continuous values within some interval e. g. sound, video Digital q q Discrete values e. g. text, integers COE 341 – Dr. Marwan Abu-Amara 65

Acoustic Spectrum (Analog) COE 341 – Dr. Marwan Abu-Amara 66

Analog and Digital Signals n n Means by which data are propagated Analog q q Continuously variable Various media n q q q n wire, fiber optic, space Speech bandwidth 100 Hz to 7 k. Hz Telephone bandwidth 300 Hz to 3400 Hz Video bandwidth 4 MHz Digital q Use two DC components COE 341 – Dr. Marwan Abu-Amara 67

Advantages & Disadvantages of Digital Signals n Advantages: q q n Cheaper Less susceptible to noise Disadvantages: q Greater attenuation n n Pulses become rounded and smaller Leads to loss of information COE 341 – Dr. Marwan Abu-Amara 68

Attenuation of Digital Signals COE 341 – Dr. Marwan Abu-Amara 69

Components of Speech n Frequency range (of hearing) 20 Hz-20 k. Hz q n n n Speech 100 Hz-7 k. Hz 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 q 300 -3400 Hz COE 341 – Dr. Marwan Abu-Amara 70

Conversion of Voice Input into Analog Signal COE 341 – Dr. Marwan Abu-Amara 71

Video Components n USA - 483 lines scanned per frame at 30 frames (scans) per second q n So 525 lines x 30 frames (scans) = 15750 lines per second q n n n 525 lines but 42 lost during vertical retrace 63. 5 s per line n 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 (for black and white video) is 4. 2 MHz COE 341 – Dr. Marwan Abu-Amara 72

Binary Digital Data n n n From computer terminals etc. Two dc components Bandwidth depends on data rate COE 341 – Dr. Marwan Abu-Amara 73

Conversion of PC Input to Digital Signal COE 341 – Dr. Marwan Abu-Amara 74

Data and Signals n n Usually use digital signals for digital data and analog signals for analog data Can use analog signal to carry digital data q n Modem Can use digital signal to carry analog data q Compact Disc audio COE 341 – Dr. Marwan Abu-Amara 75

Analog Signals Carrying Analog and Digital Data COE 341 – Dr. Marwan Abu-Amara 76

Digital Signals Carrying Analog and Digital Data COE 341 – Dr. Marwan Abu-Amara 77

Four Data/Signal Combinations Signal Analog Same spectrum as Analog data (base band): e. g. Digital - Conventional Telephony Different spectrum (modulation): e. g. AM, FM Radio - Data Use a (converter): codec, e. g. for PCM (pulse code modulation) -Simple Digital two signal levels: e. g. NRZ code Use a (converter): modem e. g. with the -Special Encoding: e. g. Manchester code V. 90 standard 78 COE 341 – Dr. Marwan Abu-Amara (Chapter 5)

Analog Transmission n n Analog signal transmitted without regard to content Analog signal may be analog or digital data Attenuated over distance Use amplifiers to boost signal Also amplifies noise COE 341 – Dr. Marwan Abu-Amara 79

Digital Transmission n Concerned with content Integrity endangered by noise, attenuation etc. Repeaters used q q q n n Repeater receives signal Extracts bit pattern Retransmits Attenuation is overcome by a repeater by reconstructing the signal Noise is not amplified COE 341 – Dr. Marwan Abu-Amara 80

Four Signal/Transmission Mode Combinations Transmission mode Analog Uses amplifiers - Not concerned with what data the signal represents - Noise is cumulative - Analog Uses repeaters - Assumes signal represents digital data, recovers it and represents it as a new outbound signal - This way, noise is not cumulative - OK Makes sense only if the analog signal represents digital data Avoid OK Signal Digital COE 341 – Dr. Marwan Abu-Amara 81

Advantages of Digital n Digital technology Transmission q n Data integrity q n q High bandwidth links economical High degree of multiplexing easier with digital techniques Security & Privacy q n Longer distances over lower quality lines Capacity utilization q n Low cost LSI/VLSI technology Encryption Integration q Can treat analog and digital data similarly COE 341 – Dr. Marwan Abu-Amara 82

Transmission Impairments n n Signal received may differ from signal transmitted Analog signal - degradation of signal quality Digital signal - bit errors Caused by q q q Attenuation and attenuation distortion Delay distortion Noise COE 341 – Dr. Marwan Abu-Amara 83

Attenuation n Signal strength falls off with distance Depends on medium (guided vs. unguided) Attenuation affects received signal strength q q q n received signal strength must be enough to be detected received signal strength must be sufficiently higher than noise to be received without error signal strength can be achieved by using amplifiers or repeaters Attenuation is an increasing function of frequency q q Different frequency components of a signal get attenuated differently Signal distortion Particularly significant with analog signals n q for digital signals, strength of signal falls of rapidly with frequency Can overcome signal distortion using equalizers COE 341 – Dr. Marwan Abu-Amara 84

Delay Distortion n n Only in guided media Propagation velocity varies with frequency q q n n Effect: Different frequency components of the signal arrive at slightly different times! (Dispersion) Badly affects digital data due to bit spill-over (intersymbol interference) q n Highest at center frequency (minimum delay) Lower at both ends of the bandwidth (larger delay) major limitation to max bit rate over a transmission channel Can overcome delay distortion using equalizers COE 341 – Dr. Marwan Abu-Amara 85

Noise n n n Additional unwanted signals inserted between transmitter and receiver The most limiting factor in communication systems Noise categories: q q Thermal Intermodulation Crosstalk Impulse COE 341 – Dr. Marwan Abu-Amara 86

Thermal (White) Noise n n n Due to thermal agitation of electrons Uniformly distributed across the bandwidth Can NOT be eliminated! (upper bound on comm. sys. perf. ) 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 107 = -133. 9 d. BW COE 341 – Dr. Marwan Abu-Amara 87

Intermodulation n Occurs when signals at different frequencies share same transmission medium q Produces signals that are the sum and/or the difference of original frequencies sharing the medium (f 1+f 2) and (f 1 -f 2) Caused by nonlinearities in the medium and equipment, e. g. due to overdrive and saturation of amplifiers Resulting frequency components (i. e. f 1+f 2 and f 1 -f 2) may fall within valid signal bands, thus causing interference n f 1, f 2 n n COE 341 – Dr. Marwan Abu-Amara 88

Crosstalk & Impulse n Crosstalk q q n 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 q q q 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 COE 341 – Dr. Marwan Abu-Amara 89

Channel Capacity n n n 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) q q 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 COE 341 – Dr. Marwan Abu-Amara 90

1. Nyquist Bandwidth: (Noise-free, Errorn Idealized, theoretical free) n n n Assumes a noise-free, error-free channel Nyquist: If rate of signal transmission is 2 B 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 2 B 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 2 B 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 = 2 B log 2 M bits/s (bps) Signals/s COE 341 – Dr. Marwan Abu-Amara bits/signal 91

Nyquist Bandwidth: Examples n C = 2 B log 2 M bits/s q q q n n n C = Nyquist Channel Capacity B = Bandwidth M = Number of discrete signal levels (symbols) used Telephone Channel: B = 3400 -300 = 3100 Hz 1 With a binary signal (M = 2): 0 C = 2 B log 2 2 = 2 B = 6200 bps With a quandary signal (M = 4): 01 C = 2 B log 2 4 = 2 B x 2 = 4 B = 12, 400 bps n 11 10 00 Practical limit: larger M makes it difficult for the receiver to operate, particular with noise COE 341 – Dr. Marwan Abu-Amara 92

2. Shannon Capacity Formula: (Noisy, Error. Free) n n n 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 Caution! Log 2 Not Log 10 SNRd. B= 10 log 10 (SNR) Shannon Capacity C = B log 2(1+SNR): Caution! Ratio- Not d. B Highest data rate transmitted error-free with a given noise level n n n 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. COE 341 – Dr. Marwan Abu-Amara 93

n n n Shannon Capacity Formula: Formula says: for data rates calculated C, it is theoretically Comments 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 COE 341 – Dr. Marwan Abu-Amara 94

Shannon Capacity Formula: Comments Contd. n n Formula suggests that changes in B and SNR can be done arbitrarily and independently… but In practice, this may not be the case! q High SNR obtained through excessive amplification may introduce nonlinearities: distortion and intermodulation noise! q High Bandwidth B opens the system for more thermal noise (k. TB), and therefore reduces SNR! COE 341 – Dr. Marwan Abu-Amara 95

Shannon Capacity Formula: Example Spectrum of communication channel extends from 3 MHz to 4 MHz q Also, given SNRd. B = 24 d. B q Then B = 4 MHz – 3 MHz = 1 MHz SNRd. B = 24 d. B = 10 log 10 (SNR) SNR = 251 q Using Shannon’s formula: C = B log 2 (1+ SNR) C = 106 * log 2(1+251) ~ 106 * 8 = 8 Mbps q Based on Nyquist’s formula, determine M that gives the above channel capacity: C = 2 B log 2 M 8 * 106 = 2 * (106) * log 2 M 4 = log 2 M M = 16 q COE 341 – Dr. Marwan Abu-Amara 96

3. Eb/N 0 (Signal Energy per Bit/Noise Power density per Hz) (Noise and Error n Together) Handling both noise and error together n n n Eb/N 0: A standard quality measure for digital communication system performance Eb/N 0 can be independently related to the error rate Expresses SNR in a manner related to the data rate, R Eb = Signal energy per bit (Joules) = Signal power (Watts) x bit interval Tb (second) = S x (1/R) = S/R N 0 = Noise power (watts) in 1 Hz = k. T COE 341 – Dr. Marwan Abu-Amara 97

Eb/N 0 Example 1: q q Given: Eb/No = 8. 4 d. B (minimum) is needed to achieve a bit error rate of 10 -4 Given: n The effective noise temperature is 290 o. K (room temperature) n Data rate is 2400 bps What is the minimum signal level required for the received signal? 8. 4 = S(d. BW) – 10 log 2400 + 228. 6 d. BW – 10 log 290 = S(d. BW) – (10)(3. 38) + 228. 6 – (10)(2. 46) S = -161. 8 d. BW COE 341 – Dr. Marwan Abu-Amara 98

Eb/N 0 (Cont. ) n n Bit error rate for digital data is a decreasing function of Eb/N 0 for a given signal encoding scheme Which encoding scheme is better: A or B? Get Eb/N 0 to achieve a desired error rate, then determine other parameters from formula, e. g. S, SNR, R, etc. (Design) n Error performance of a given system (Analysis) n Effect of S, R, T on error performance Lower Error Rate: larger Eb/N 0 n COE 341 – Dr. Marwan Abu-Amara A B Better Encoding 99

Eb/N 0 (Cont. ) n From Shannon’s formula: C = B log 2(1+SNR) We have: n From the Eb/N 0 formula: With R = C, substituting for SNR we get: n Relates achievable spectral efficiency C/B (bps/Hz) to Eb/N 0 COE 341 – Dr. Marwan Abu-Amara 100

Eb/N 0 (Cont. ) Example 2 n Find the minimum Eb/N 0 required to achieve a spectral efficiency (C/B) of 6 bps/Hz: n Substituting in the equation above: Eb/N 0 = (1/6) (26 - 1) = 10. 5 = 10. 21 d. B COE 341 – Dr. Marwan Abu-Amara 101