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

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

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

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

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

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 COE 341 – Dr. Marwan Abu-Amara

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 COE 341 – Dr. Marwan Abu-Amara

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

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

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

Varying Sine Waves s(t) = A sin(2ft +)

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

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 COE 341 – Dr. Marwan Abu-Amara

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

Frequency Domain Representations

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 COE 341 – Dr. Marwan Abu-Amara

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

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

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!!! COE 341 – Dr. Marwan Abu-Amara

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

Decibels and Signal Strength
Decibel is a measure of ratio between two signal levels NdB = number of decibels P1 = input power level P2 = 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 NdB =10log(5/10)=10(-0.3)=-3 dB COE 341 – Dr. Marwan Abu-Amara

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 ( )=13 dB = 10 log (P2 / 4mW) P2 = 4 x mW = 79.8 mW COE 341 – Dr. Marwan Abu-Amara

Relationship Between Decibel Values and Powers of 10
Power Ratio dB 101 10 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

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 COE 341 – Dr. Marwan Abu-Amara

Decibel & Difference in Voltage
Decibel is used to measure difference in voltage. Power P=V2/R Decibel-millivolt (dBmV) is an absolute unit with 0 dBmV equivalent to 1mV. Used in cable TV and broadband LAN COE 341 – Dr. Marwan Abu-Amara

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

Fourier Series (Appendix A)
Any periodic signal of period T (i.e. f0 = 1/T) can be represented as sum of sinusoids, known as Fourier Series fundamental frequency DC Component If A0 is not 0, x(t) has a DC component COE 341 – Dr. Marwan Abu-Amara

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

Fourier Series Representation of Periodic Signals - Example
x(t) 1 -3/2 -1 -1/2 1/2 1 3/2 2 -1 T Note: (1) x(– t)=x(t)  x(t) is an even function (2) f0 = 1 / T = ½ COE 341 – Dr. Marwan Abu-Amara

Replacing t by –t in the first integral sin(-2pnf t)= - sin(2pnf t) COE 341 – Dr. Marwan Abu-Amara

Since x(– t)=x(t) as x(t) is an even function, then Bn = 0 for n=1, 2, 3, … Cosine is an even function COE 341 – Dr. Marwan Abu-Amara

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

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

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 COE 341 – Dr. Marwan Abu-Amara

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

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

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

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 COE 341 – Dr. Marwan Abu-Amara

Signal Power A function x(t) specifies a signal in terms of either voltage or current Assuming R = 1 W, Instantaneous signal power = V2 = i2 = |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

For a periodic signal, power spectral density is where (f) is Cn is as defined before on slide 27, and f0 being the fundamental frequency COE 341 – Dr. Marwan Abu-Amara

For a continuous valued function S(f), power contained in a band of frequencies f1 < f < f2 For a periodic waveform, the power through the first j harmonics is COE 341 – Dr. Marwan Abu-Amara

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

Fourier Analysis Example
Consider the half-wave rectified cosine signal from Figure A.1 on page 838: 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

Example (Cont.) Mathematical expression for s(t): Where f0 is the fundamental frequency, f0 = (1/T) -T/4 -3T/4 +3T/4 +T/4 COE 341 – Dr. Marwan Abu-Amara

Example (Cont.) Fourier Analysis: f0 = (1/T) COE 341 – Dr. Marwan Abu-Amara

Example (Cont.) Fourier Analysis (cont.): f0 = (1/T) COE 341 – Dr. Marwan Abu-Amara

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

Example (Cont.) Fourier Analysis (cont.): Note: cos2q = ½(1 + cos 2q) COE 341 – Dr. Marwan Abu-Amara

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

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

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

Example (Cont.) Total Power: Note: cos2q = ½(1 + cos 2q) COE 341 – Dr. Marwan Abu-Amara

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

Example (Cont.) Finding n such that we get 95% of total power: Effective bandwidth with 95% of total power: Beff = fmax – fmin = 2f0 – 0 = 2f0 Beff …  f f0 2f0 3f0 COE 341 – Dr. Marwan Abu-Amara

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 75 – 77 of textbook COE 341 – Dr. Marwan Abu-Amara

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

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!. COE 341 – Dr. Marwan Abu-Amara

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

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

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

Acoustic Spectrum (Analog)

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 COE 341 – Dr. Marwan Abu-Amara

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

Attenuation of Digital Signals

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 Hz COE 341 – Dr. Marwan Abu-Amara

Conversion of Voice Input into Analog Signal

Video Components USA lines scanned per frame at 30 frames (scans) per second 525 lines but 42 lost during vertical retrace So 525 lines x 30 frames (scans) = 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 (for black and white video) is 4.2MHz COE 341 – Dr. Marwan Abu-Amara

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

Conversion of PC Input to Digital Signal

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 COE 341 – Dr. Marwan Abu-Amara

Analog Signals Carrying Analog and Digital Data

Digital Signals Carrying Analog and Digital Data

Four Data/Signal Combinations
Analog Digital Data 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) 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) COE 341 – Dr. Marwan Abu-Amara

Analog Transmission 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

Digital Transmission Concerned with content Integrity endangered by noise, attenuation etc. Repeaters used 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

Four Signal/Transmission Mode Combinations
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 OK Makes sense only if the analog signal represents digital data Avoid COE 341 – Dr. Marwan Abu-Amara

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 COE 341 – Dr. Marwan Abu-Amara

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

Attenuation Signal strength falls off with distance Depends on medium (guided vs. unguided) Attenuation affects received signal strength 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 Different frequency components of a signal get attenuated differently  Signal distortion Particularly significant with analog signals for digital signals, strength of signal falls of rapidly with frequency Can overcome signal distortion using equalizers COE 341 – Dr. Marwan Abu-Amara

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 (intersymbol interference) major limitation to max bit rate over a transmission channel Can overcome delay distortion using equalizers COE 341 – Dr. Marwan Abu-Amara

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

Thermal (White) Noise 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 = log log 107 = dBW COE 341 – Dr. Marwan Abu-Amara

Intermodulation Occurs when signals at different frequencies share same transmission medium Produces signals that are the sum and/or the difference of original frequencies sharing the 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 (i.e. f1+f2 and f1-f2) may fall within valid signal bands, thus causing interference COE 341 – Dr. Marwan Abu-Amara

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

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 COE 341 – Dr. Marwan Abu-Amara

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 log2M bits Generalized Nyquist Channel Capacity, C = 2B log2M bits/s (bps) Signals/s bits/signal COE 341 – Dr. Marwan Abu-Amara

Nyquist Bandwidth: Examples
C = 2B log2M bits/s C = Nyquist Channel Capacity B = Bandwidth M = Number of discrete signal levels (symbols) used Telephone Channel: B = = 3100 Hz With a binary signal (M = 2): C = 2B log2 2 = 2B = 6200 bps With a quandary signal (M = 4): C = 2B log2 4 = 2B x 2 = 4B = 12,400 bps Practical limit: larger M makes it difficult for the receiver to operate, particular with noise 1 11 10 01 00 COE 341 – Dr. Marwan Abu-Amara

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 SNRdB= 10 log10 (SNR) Shannon Capacity C = B log2(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! Log2 Not Log10 Caution! Ratio- Not dB COE 341 – Dr. Marwan Abu-Amara

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 COE 341 – Dr. Marwan Abu-Amara

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 intermodulation noise! High Bandwidth B opens the system for more thermal noise (kTB), and therefore reduces SNR! COE 341 – Dr. Marwan Abu-Amara

Shannon Capacity Formula: Example
Spectrum of communication channel extends from 3 MHz to 4 MHz Also, given SNRdB = 24dB Then B = 4MHz – 3MHz = 1MHz SNRdB = 24dB = 10 log10 (SNR) SNR = 251 Using Shannon’s formula: C = B log2 (1+ SNR) C = 106 * log2(1+251) ~ 106 * 8 = 8 Mbps Based on Nyquist’s formula, determine M that gives the above channel capacity: C = 2B log2 M 8 * 106 = 2 * (106) * log2 M 4 = log2 M M = 16 COE 341 – Dr. Marwan Abu-Amara

3. Eb/N0 (Signal Energy per Bit/Noise Power density per Hz) (Noise and Error Together) Handling both noise and error together Eb/N0: A standard quality measure for digital communication system performance Eb/N0 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 N0 = Noise power (watts) in 1 Hz = kT COE 341 – Dr. Marwan Abu-Amara

Eb/N0 Example 1: Given: Eb/No = 8.4 dB (minimum) is needed to achieve a bit error rate of 10-4 Given: The effective noise temperature is 290oK (room temperature) Data rate is 2400 bps What is the minimum signal level required for the received signal? 8.4 = S(dBW) – 10 log dBW – 10 log290 = S(dBW) – (10)(3.38) – (10)(2.46) S = dBW COE 341 – Dr. Marwan Abu-Amara

Eb/N0 (Cont.) Bit error rate for digital data is a decreasing function of Eb/N0 for a given signal encoding scheme Which encoding scheme is better: A or B?  Get Eb/N0 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 Eb/N0 B A Better Encoding COE 341 – Dr. Marwan Abu-Amara

Eb/N0 (Cont.) From Shannon’s formula: C = B log2(1+SNR) We have: From the Eb/N0 formula: With R = C, substituting for SNR we get: Relates achievable spectral efficiency C/B (bps/Hz) to Eb/N0 COE 341 – Dr. Marwan Abu-Amara

Eb/N0 (Cont.) Example 2 Find the minimum Eb/N0 required to achieve a spectral efficiency (C/B) of 6 bps/Hz: Substituting in the equation above: Eb/N0 = (1/6) (26 - 1) = 10.5 = dB COE 341 – Dr. Marwan Abu-Amara

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

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