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Digital ModulationDavid TipperAssociate ProfessorDepartment of Information Science andTelecommunicationsUniversity of ://www.tele.pitt.edu/tipper.htmlTypical Communication ource1

About Channel Capacity Channel Capacity (C)– the maximum rate at which data can be transmitted over a givencommunication path, or channel, under given conditions Data rate (bps)– rate at which data can be communicated , impairments, such asnoise, limit data rate that can be achieved Bandwidth (B)– the bandwidth of the transmitted signal as constrained by thetransmitter and the nature of the transmission medium (Hertz) Noise (N)– impairments on the communications path Error rate - rate at which errors occur (BER)– Error transmit 1 and receive 0; transmit 0 and receive 1Reasons for Choosing Encoding Techniques Digital data, digital signal– Equipment less complex and expensive than digital-to-analogmodulation equipment Analog data, digital signal– Permits use of modern digital transmission and switching equipment Digital data, analog signal– Some transmission media will only propagate analog signals– E.g., unguided media (air) Analog data, analog signal– Analog data in electrical form can be transmitted easily and cheaply– E.g., AM Radio2

Signal Encoding Criteria What determines how successful a receiverwill be in interpreting an incoming signal?– Signal-to-noise ratio (SNR)– Data rate– Bandwidth (B)– Inter-related quantities Increase in SNR decreases bit error rate Increase in data rate increases bit error rate Increase in bandwidth allows an increase in data rate Shannon Bound for AWGN non fadingchannelConcepts Related to Channel Capacity Shannon Bound for AWGN non fading channelC B log2 (1 S / N ) Nyquist Bandwidth– For binary signals (two voltage levels) C 2B– With multilevel signaling (M-ary signalling) C 2B log2 MM number of discrete signal or voltage levelsN number of bitsM 2N3

Example of Nyquist and Shannon Formulations Spectrum of a channel between 3 MHz and 4 MHz ; SNRdB 24 dBB 4 MHz 3 MHz 1 MHzSNR dB 24 dB 10 log 10 (SNR )SNR 251 Using Shannon’s formulaC 10 6 log 2 (1 251) 10 6 8 8Mbps How many signaling levels are required?C 2B log 2 M( )8 10 6 2 10 6 log 2 M4 log 2 MM 16Digital Transmission Why Digital ?– Increase System Capacity compression, more efficient modulation– Error control coding, equalizers,etcequalizers,etc. possible to combatnoise and interference lower power needed– Reduce cost and simplify designs– Improve Security (encryption possible) Digital Modulation– Analog signal carrying digital data4

Digital Modulation and aldigitalmodulationanalogmodulationradio 1001radio receiverradiocarrierModulation Review Modulation– Converting digital or analog information to a waveform suitablefor transmission over a given medium– Involves varying some parameter of a carrier wave (sinusoidalwaveform) at a given frequency as a function of the messagesignal– General sinusoid A cos (2πfCt ϕ)PhaseAmplitudeFrequency– If the information is digital changing parameters is called“keying” (e.g. ASK, PSK, FSK)5

Modulation Motivation– Smaller antennas (e.g., λ /4 typical antenna size) λ wavelength c/f , where c speed of light, f frequency. 3000Hz baseband signal 15 mile antenna, 900 MHz 8 cm–––– Frequency Division Multiplexing – provides separation of signalsmedium characteristicsInterference rejectionSimplifying circuitryModulation– shifts center frequency of baseband signal up to the radio carrier Basic schemes– Amplitude Modulation (AM)– Frequency Modulation (FM)– Phase Modulation (PM)Amplitude Shift Keying (ASK)Frequency Shift Keying (FSK)Phase Shift Keying (PSK)Digital modulation Amplitude Shift Keying (ASK):101– change amplitude with each symbol– frequency constant– low bandwidth requirements– very susceptible to interferencet1 01Frequency Shift Keying (FSK):– change frequency with each symbol– needs larger bandwidth Phase Shift Keying (PSK):– Change phase with each symbol– More complex– robust against interferencet101t6

Basic Encoding TechniquesAmplitude-Shift Keying One binary digit represented by presence ofcarrier, at constant amplitude Other binary digit represented by absence ofcarrier A cos(2πf c t )binary 1s (t ) 0binary 0 where the carrier signal is Acos(2pf ct) Very Susceptible to noise Used to transmit digital data over opticalfiber7

Binary Frequency-Shift Keying (BFSK) Two binary digits represented by two differentfrequencies near the carrier frequency A cos( 2πf t )1s (t ) A cos(2πf t )2binary 1binary 0– where f1 and f2 are offset from carrier frequencyfc by equal but opposite amounts– B 2([f2 – f1]/2 fb) Where fb input bit ratePhase-Shift Keying (PSK) Two-level PSK (BPSK)– Uses two phases to represent binary digits A cos(2πf t )binary 1cs (t ) A cos( 2πf c t π ) binary 0B fb A cos(2πf c t ) A cos(2πf ct )binary 1binary 08

Selection of Encoding/Modulation Schemes Performance in an AWGN channel– How does the bit error rate vary with the energy per bitavailable in the system when white noise present Performance in fading multipath channels– Same as above, but add multipath and fading Bandwidth requirement for a given data rate– Also termed spectrum efficiency or bandwidth efficiency– How many bits/sec can you squeeze in one Hz of bandwidthfor a given error rate Cost– The modulation scheme needs to be cost efficient Circuitry should be simple to implement and inexpensive(e.g. detection, amplifiers)Signal Constellation Given any modulation scheme, it is possible toobtain its signal constellation.– Represent each possible signal as a vector in aEuclidean space spanned by an orthonormal basis. If we know the signal constellation, we canestimate the performance in terms of theprobability of symbol error or probability of biterror given the noise parameters. Probability of error depends on the minimumdistance between the constellation points.9

BPSK Signal CostellationTomasiElectronic Communications Systems, 5eSymbol Detection The receiver implementation can affect theperformance.– Coherent detection receiver will exploit the exact knowledge of the phase ofthe carrier to detect the signal better.– Non-coherent detection involves making some approximations to the phaseinformation that results in a loss in performance. However,it simplifies the circuitry. In symbol detection – decode incoming signalas closest symbol in the signal constellationspace10

Example of BPSKA binary 1 is represented by:s1 ( t ) 2 EbTcos (2πf c t ) , 0 t T , f c nTA binary 0 is represented by:s2 (t ) 2 EbTcos(2πf ct ) , 0 t Ts1 ( t ) We can writeEb Ψ ( t )s2 (t ) Eb Ψ ( t )Ψ (t) where2Tcos(2πf c t ),0 t TWhat is the energy of Ψ(t)?EΨ 2TT0Ψ 2 (t )dt 2T T0cos 2 (2π f c t )dt [1 cos (4π f t )]dt 1T012cNote that the energy in one bit of BPSK is Eb.The constellation of BPSK isZ2Z1 EbEbs2s111

DetectionWhen do errors occur in BPSK? S1 is transmitted and the received point fallsin Z1 region. S2 is transmitted and the received point fallsin Z2 region.Why will a point fall elsewhere? The received signal point is si n r , i 1,2 n is the noise vector that is normallydistributed with mean zero and variance N0/2.This can shift the transmitted signal to someother value.Bit Error Rate What is the probability that the signal point r falls in Z1given s2(t) was transmitted? (Conditional probability) r s2 n is a normally distributed random variablewith mean - Eb and variance N0 /2.Pe 0(x E )exp N 0 b 21N022π dx Eb12

Normal Distribution ReviewLetZ (x Eb)N02, dZ When x 0, Z Pe 2 EbN012πdxN022 EbN0exp( )dz Q( Z 222 EbN0) Q (γ b )PSK Error PerformanceNote coherent symbol detection out performs non-coherentPeDifferential PSKCoherent PSK10-510Eb/N0(dB)13

Remarksa) We use Eb/N0 as a figure of merit because itprovides a good comparison of the “powerefficiency” or “energy efficiency” of amodulation scheme. Sometimes called SNRper bit.b) We will not derive the bit error rateperformance of different modulationschemes but we will only use the results.Some Remarks (cont.)c) The constellation of orthogonal FSK lookslike thisPe ( FSK ) Qψ2 (t) Eb( )EbN0(a 3dB reduction in performanc e)d 2EbPeFSK Ebψ1 (t)BPSK10-510 13 Eb/N014

Performance Bandwidth of modulated signal (BT)– ASK, PSK– FSKBT (1 r)RBT 2DF (1 r)R R bit rate 0 r 1; related to how signal is filtered DF f 2 -fc fc-f 1D Phase-Shift Keying (PSK) Differential PSK (DPSK)– Phase shift with reference to previous bit Binary 0 – signal burst of same phase as previous signal burst Binary 1 – signal burst of opposite phase to previous signal burst15

Performance in AWGN channelsBinary modulation schemes0 Similar to BPSKanalysis have Pefor FSK, and /N0101214M-ary Signaling/Modulation What is M-ary signaling?– The transmitter considers ‘k’ bits at a times. Itproduces one of M signals where M 2 k .Example: QPSK (k 2)Input:Signal :002ETcos (2 π f c t ) ,012ETcos (2 π f c t 112ETcos (2 π f c t π ) , 0 t T102ETcos (2π f c t 0 t Tπ2), 0 t T3π2), 0 t T16

QPSK Constellationsψ2 (t)ψ2 (t)ψ1 (t)ψ1 (t)Rotated by π/4π/4 QPSK17

π/4 –QPSK ModulationCan use simple AM balanced modulatorTomasiElectronic Communications Systems, 5eCopyright 2004 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.π/4 QPSK Coherent DemodulatorTomasiElectronic Communications Systems, 5eCopyright 2004 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.18

8 –PSKΙncreasing thenumber of levelsincreases the datarate – butIncreases thesymbol error rateas the symbols arecloser together inthe constellationspace8-PSK Output19

M-ary Error Performance MPSK, as M increases– the bandwidth remains constant,– the minimum distance between signals reduces increase in symbol error rate MFSK, as M increases– the bandwidth increases– the performance improves but the minimumdistance between signals remains the samePerformance Bandwidth of modulated signal (BT) 1 r 1 r RB R T– MPSK L log2 M – MFSK (1 r )M RBT logM 2 L number of bits encoded per signal element M number of different signal elements20

Quadrature Amplitude Modulation QAM is a combination of ASK and PSK– Two different signals sent simultaneously onthe same carrier frequency –– Change phase and amplitude as function ofinput data– Simple case 8 QAM (two amplitudes – 4phases)s(t ) d1(t ) cos 2πf ct d 2 (t ) sin 2πf c t8 - QAM21

8-QAM OutputQuadrature Amplitude Modulation22

Matched Filter In order to detect a signal at the receiver, a linearfilter that is designed to provide the maximum outputSNR in AWGN for a given symbol waveform is used.This filter is called a “matched filter” (section 3.2.2)r(t)y(t)(SNR)maxMatched FilterSample att T If the transmitted signal is s(t), the impulse responseof the matched filter can be shown to be k s (T t ) , 0 t Th (t ) , outside 0This assumes that s(t) exists only for a duration of T seconds.Let us look at the output for k 1.y (t ) r (t ) h (t ) r (τ )s (T r (τ )s (T r (τ )s (τ (t τ))d τ t τ )d τ T t )d τCompare with cross-correlation: R rs (τ ) r (t )s (t τ )dtThe output of the matched filter is the cross-correlation of thereceived signal and the time shifted transmitted signal.At t T , y (T ) r (τ ) s (τ ) dτ R sr (0)If s(t ) r (t ), y (t ) E s or E s23

Correlation Implementation of MatchedFilterr(t)y(t)s(T-t)t TT 0s(t)M-ary Error PerformanceA received symbol is decodedinto the closest the symbol inthe signal constellationBPSKAs the number of symbols inthe signal space increases thedecoding region for eachsymbol decreasesQPSK24

M-ary Error PerformanceM-PSK Error Rate PerformanceIncreasing M increases error rate and data rateIncreasing M25

QAM Error PerformancePerformance Comparison of Various Digital Modulation Schemes (BER 10-6)26

Tradeoffs between BER, power and bandwidth (1) Trade BERperformance forpower – fixed datarate (2) Trade data rate forpower – fixed BER (3) Trade BER fordata rate – fixedpower31Pe2Eb/N027

modulation equipment Analog data, digital signal – Permits use of modern digital transmission and switching equipme nt Digital data, analog signal – Some transmission media will only propagate analog signals – E.g., unguided media (air) Analog data, analog signal – Analog data in

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