Outline Chapter 3: Principles of Digital Communications

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1 Outline Chapter 3: Principles of Digital Communications Structure of a Data Transmission System Up- and Down-Conversion Lowpass-to-Bandpass Conversion Baseband Presentation of Communication System Basic Principles of Digital Transmission Impulse Filter, Spectrum of Transmit Signal and Interference Free Transmission st Nyquist Condition Raised Cosine Filter, Eye Pattern, nd Nyquist Condition Bandwidth Efficiency Impact of the Channel: Additive White Gaussian Noise Matched Filter Bit Error Probability Outline Chapter 3 (continued) Linear Digital Modulation Methods Definition of PSK, QAM Gray Coding Transmitter Structures Receiver Structures Coherent Receivers Noncoherent Receivers Error Probability Simplified System Model Coherent and noncoherent (D)PSK QAM

2 Digital source Structure of a Data Transmission System Equivalent channel encoder Channel encoder Modulator Upconverter Digital sink Digital Transmission Analogue Transmission Physical Channel Sink decoder Channel decoder Demodulator Downconverter transmits signals (e.g. speech) encoder samples, quantizes and compresses analog signal Channel encoder adds redundancy to enable error detection or Rx Modulator maps discrete symbols onto analog waveform Up-converter moves analog waveform into the transmission frequency band Physical channel represents transmission medium: multipath propagation, time varying fading, additive noise, Down-converter: moves signal back into baseband Demodulator: performs lowpass filtering, sampling, quantization Channel decoder: Estimation of info sequence out of code sequence error correction decoder: Reconstruction of analog signal 3 Digital Simplified Transmission System d[k] Modulator s(t) Upconverter Physical Channel Digital Sink ˆd[k] Downconverter Demodulator r(t) k: discrete time index Objective: Transmitting sequence of discrete random values d[k] over analog channel, e.g. d[k] {, +} or d[k] { 3,, +, +3} Modulator transforms sequence of discrete values d[k] into a form that can be transmitted over analog channel, e.g. sinusoidal waveform Demodulator aims to estimate the values d[k] on basis of the received signals Up-/Down-converter shifts signal to transmission band 4

3 Outline Chapter 3: Principles of Digital Communications Structure of a Data Transmission System Up- and Down-Conversion Lowpass-to-Bandpass Conversion Baseband Presentation of Communication System Basic Principle of Digital Transmission Impulse Filter, Spectrum of Transmit Signal and Interference Free Transmission st Nyquist Condition Raised Cosine Filter, Eye Pattern, nd Nyquist Condition Bandwidth Efficiency Impact of the Channel: Additive White Gaussian Noise Matched Filter Bit Error Probability Linear Digital Modulation Methods Error Probability for lin Modulationforms 5 Up-/Down-Converter In typical wireless applications communication takes place in the bandpass domain [f -B/, f +B/] of bandwidth B around a center frequency f (and [-f -B/, -f +B/]) Example: For cellular communications three basic frequency bands around.9 GHz,.9 GHz and 5.8 GHz are used due to regulatory authorities B B f -f f However, most processing like coding/decoding, modulation/demodulation, synchronization, etc. is done in the baseband (lowpass) (around f = ) At transmitter last stage of operation is to up-convert the s BP (t) signal s(t) to the carrier frequency f and transmit s BP (t) s(t) Up via the antenna converter Similarly, the first step at the receiver is to down-convert Physical the RF (radio-frequency) signal r BP (t) to the baseband before further processing r(t) Equivalent description of communication system in the baseband is desired Equivalent Basisband Model r(t) Downconverter Channel r BP (t) 6

4 Up-Converter: Lowpass-to-Bandpass Conversion Bandpass spectrum S(jω) SBP (jω) =S(jω jω )+S (jω + jω ) S(jω jω ) conjugate complex S BP (jω) ω ω ω ω ω ω ω ω ω Calculation of the real-valued bandpass signal in time-domain s BP (t) =F {S BP (jω)} = s(t) e jω t + s (t) e jω t = Re s(t) e jω t ª s(t) complex e jωt s(t) e jωt Re { } s BP (t) real 7 Up-Converter: Lowpass-to-Bandpass Conversion Bandpass signal s BP (t) transmitted over the antenna is real-valued s BP (t) = Re s(t) e jω t ª = Re {s(t)} cos(ω t) Im {s(t)} sin(ω t)] = s I (t) cos(ω t) s Q (t) sin(ω t) Corresponding lowpass signal s(t) is complex-valued, i.e. consists of two orthogonal components Inphase phase s I (t) and Quadrature phase s Q (t) s(t) =s I (t)+j s Q (t) s I (t) s(t) e jωt Re { } s BP (t) s Q (t) -π/ cos(ωo t) sin(ω o t) s BP (t) 8

5 Equivalent Baseband Model of Transmission System s I (t) Up-Converter Down-Converter r I (t) s Q (t) -π/ cos(ωo t) sin(ω o t) s BP (t) h BP (t) r BP (t) -π/ cos(ωo t) sin(ω o t) r Q (t) h(t) Received bandpass signal (channel impulse response h BP (t) ) r BP (t) =s BP (t) h BP (t)+n BP (t) Compact description in baseband (lowpass) domain r(t) =s(t) h(t)+n(t) Actual carrier frequency ω is used only within derivation of h(t) 9 Baseband Presentation of Communication System Digital d[k] Modulator s(t) Channel Demodulator Digital Sink Why is equivalent lowpass (baseband) presentation preferred? Elegant system theory for bandpass signals and systems Compact and simple presentation of bandpass signals and channels (compare complex representation of AC electricity) Advantages for the simulation of communications systems (f has not to be considered) Simple technical realization of transmitter and receiver in lowpass domain cost efficient Simple correction of non-ideal bandpass channels by complex-valued equalizer in lowpass The subsequent presentation of communication systems is based on equivalent basisband r(t) ˆd[k]

6 Outline Chapter 3: Principles of Digital Communications Structure of a Data Transmission System Up- and Down-Conversion Lowpass-to-Bandpass Conversion Baseband Presentation of Communication System Basic Principle of Digital Transmission Impulse Filter, Spectrum of Transmit Signal and Interference Free Transmission st Nyquist Condition Raised Cosine Filter, Eye Pattern, nd Nyquist Condition Bandwidth Efficiency Impact of the Channel: Additive White Gaussian Noise Matched Filter Bit Error Probability Linear Digital Modulation Methods Error Probability for lin Modulationforms Basic Principle of Digital Transmission Digital d[k] P T s d[k] δ(t kt s ) k= g Tx (t) s(t) Digital Sink ˆd[k] Decision r[k] k T s r(t) g Rx (t) r a (t) Channel Modulator transforms sequence of discrete values d[k] into a from that can be transmitted over analog channel, e.g. sinusoidal waveform Weighting time-shifted analog impulses g Tx (t-kt s ) with d[k] " # X X s(t) = T s d[k] δ(t kt s ) g Tx (t) = T s d[k] g Tx (t kt s ) k= k=

7 Rectangle Transmit Filter For d[k] = [ ] and rectangular impulse filter Ts Ts <t< g Tx (t) = rect Ts (t) = t = T s else the modulator output signal s(t) is given by d[k] s(t) -T s / t T s / g Tx (t) s(t) = T s P d[k] rect Ts (t kt s ) k d[k] rect Ts (t kt s ) t T s 3 Spectrum of Transmit Signal and Impulse Filter Spectrum of transmit signal In general the spectrum depends on statistic of source data For uncorrelated data the power spectral density is determined by the transmit filter S ss (jω) = σ d T s G Tx (jω) Spectral shaping by design of transmit filter g Tx (t) (Not) possible choice: Direct transmission of impulse sequence g Tx (t) = δ(t) G Tx (jω) = Transmit signal occupies infinite bandwidth Rectangular impulse leads also to large bandwidth for transmit signal sinc(jω) Receive filter g Rx (t) is used to limit the input bandwidth (especially to limit the effective bandwidth of the noise) How to design the transmit and the receive filter? Finite transmission bandwidth is required lowpass shape Allow error-free reconstruction of the data st Nyquist condition 4

8 Interference Free Transmission Assumption: ideal channel (no noise and no fading) Impulse response of entire transmission system g(t) =g Tx (t) g Rx (t) Sampled received signal r[k] = r(kt s ) = r(t) t=kts = T s Superposition of infinite impulse responses shifted by T s Condition for interference-free reception (up to tolerable delay k ) r[k] T s! = d[k k ] P `= d[`] δ(t `T s ) X g(t) `= Impulse response that enables interference free-transmission and meets spectral requirements is desired r(t) k T s d[`] g(kt s `T s ) r[k] 5 Interference Free Transmission Example: lowpass filter with impulse function g(t) = T s ³ sinc π t T s g(t) symbols do not interfere at t = T s t t = k T s - T s equally spaced zeros, interval f N = T s t no intersymbol interference (ISI)! 6

9 st Nyquist Condition Sampling rate: f s = T s =f N delay t = k T s Time Domain: Impulse response has equally spaced zeros at t = T s, = ±, ±, st Nyquist Condition in time domain ( for k = k T s g[k] =T s g(k T s )= for k 6= k Frequency Domain: Spectrum shifted by /T s sums up to a constant value symmetry condition st Nyquist Condition in frequency domain X µ G jω jπ ` =const. T s `= 7 st Nyquist Condition: Time Domain Ideal lowpass has (theoretically) infinite duration not realizable Multiply with shaping function f(t) to limit the length of impulse response Shaping function f(t) =f( t) F (jω) R f(t) = for t t Example: Limitation of length ( t t ) by multiplication with rectangle f(t) f(t) shaping function g(t) g(t) f(t) t g(t) f(t) G(jω) F (jω) with t t F (jω) =t sinc(ωt ) infinite bandwidth 8

10 Nyquist Systems The impulse response of Nyquist systems has equally spaced zeros at T = f N If the system is linear phase and band-limited with f f N, than the spectrum is symmetric with respect to f N (Nyquist frequency) G (jω),5 a b Δf Δf a All impulse responses achieved by windowing of the ideal lowpass have equally spaced zeros b f N = T s Due to symmetry the magnitude of sum of spectra is constant X µ G jω jπ ` =const. T s `= f G(jω) e jωk ot s f s =f N 4f N Σ= f 9 Raised Cosine Filter (RC) Shaping function with constant spectrum up to frequency (-r)(t s ) and cosine slope down to zero g RC (t) = sin( πt T s ) cos(r πt T πt s ) T s T s (rt/t s ) h ³ ³ i for ω ω N r G RC (jω) = π +cos ω r ω N ( r) for r ω ω N +r for ω ω N +r Rolloff factor r < dominates shape of slope r = : ideal lowpass r = : common cosine slope Shaping function f (t) with smooth spectrum F(jω)

11 Raised Cosine Filter (RC) Impulse response and spectrum for varying rolloff factor r =,.,.4,.6,.8, g RC (t) t/t s r.6 G RC (jω) f/f N r Eye Pattern Data sequence d[k] =[......] d[k] g(t kt s ).5.5 r(t) t/t s t/t s

12 nd Nyquist Condition Discrete sequence r[k] is achieved by sampling of r(t) r[k] continuous receive signal r(t) at arbitrary sampling time Optimum sampling time of Rx corresponds to sampling time of Tx r[k]= ± Example: RC filter with rolloff factor r =. Δτ r(t) t/t s Horizontal eye opening determines the required accuracy for synchronization τ opt k T s t/t s 3 nd Nyquist condition ν = g(ν T s )= ν = ± else Impulse response contains additional zeros at twice of the sampling rate nd Nyquist Condition Example: raised cosine filter with r =, g(t) If nd Nyquist condition is fulfilled, the eye pattern shows the maximum horizontal opening Raised Cosine with r=. t/t s 4

13 Raised Cosine Filter: Eye pattern nd Nyquist st Nyquist: nd Nyquist: st Nyquist: nd Nyquist: st Nyquist st Nyquist: nd Nyquist: st Nyquist: nd Nyquist: 5 Bandwidth Efficiency Bandwidth efficiency indicates required bandwidth to transmit one bit (BPSK) η = data rate bandwidth = /T s ( + r)/t s = +r bit/s Hz Influence of rolloff factor: bit/s Hz +r < bit/s Hz r = : ideal lowpass B = f N bit/s/hz perfect synchronization required r = : cosine slope B = 4 f N bit/s/hz max. horizontal eye opening Partial Response Transmission: Advanced strategies for spectral shaping of transmit signal bit/s/hz realizable 6

14 Impact of the Channel s[k] r[k] Transmitter Channel Receiver Channel limits performance of the system two dominating factors: Noise Signal attenuation (fading) Chapter 4 Noise: Unwanted signals that disturb the transmission We have no control over noise stochastic process Independent of transmitted signal s of noise Atmospheric noise, Shot noise arises in devices like diodes/ transistors due to discrete nature of current flow Thermal noise arises due to random motion of electrons in electronic devices, e.g. filter, amplifier, resistor, 7 Additive White Gaussian Noise (AWGN) Idealized noise model for combining the effects of different noise sources Additive: noise influences transmit signal additively s[k] n[k] r[k] White: Power spectral density is independent of frequency samples are independent S nn (jω) = N Gaussian: noise samples are Gaussian distributed with probability density function (pdf) and variance σn = N / p N (n) = e σ n N πσn N S nn (jω) p N (n) ω 8 n

15 Matched Filter d[k] g Tx (t) s(t) n a (t) g Rx (t) k T s r[k] =r [k]+n[k] Task: Design g Rx (t) to maximize signal-to-noise-ratio (SNR), i.e. power ratio of usable signal to disturbance collect usable signal but avoid noise G Rx (jω) S ss (jω) = σ d T s G Tx (jω) S nn (jω) = N G Rx (jω) = G Tx (jω) leads to maximum SNR (matched filter). Nyquist condition is satisfied for Root Raised Cosine filter G Rx (jω) = G Tx (jω) = p G RC (jω) ω symbol energy S N = σ d σ N E s = σ d R = E s N / gtx (t) dt 9 Bit Error Probability What is the probability that the receiver estimates the data incorrectly? Assumptions: Binary transmission with d[k] {d,d } and. Nyquist is fulfilled Binary transmission induces two conditional pdfs for receive signal p (x) = p N (x d ) p (x) = p N (x d ) Area corresponds to probability of wrong decision d φ x Threshold φ φ d x No closed form solution for area integral define erfc(x) = π Z x e t dt 3

16 Error function complement and BER Gaussian error function erf(x) = π Z x Error function complement erfc(x) = erf(x) Z = π x e t dt e t dt erf(x),erfc(x) erf(x) erfc(x) x Bit error probability (BER) for binary modulation scheme {d,d } µ r P b = erfc d d P ±d b = σ erfc Es N N 3 BER Bit Error Probability antipodal unipolar E s /N in db BER-Plot Tool for comparing properties of transmission systems Dependency of error rate from transmission quality, i.e. SNR X-axis: SNR in logarithmic scale; here E s /N : SNR = log( E s N ) Y-axis: average number of bit errors Line: BER for a given SNR Reliability depends on: SNR and signal constellation Antipodal {±} outperforms unipolar {,} scheme by 3 db twice power 3

17 Outline Chapter 3 (continued) Linear Digital Modulation Methods Definition of PSK, QAM Gray Coding Transmitter Structures Receiver Structures Coherent Receivers Noncoherent Receivers Error Probability Simplified System Model Coherent and noncoherent (D)PSK QAM 33 Linear Digital Modulation Methods Up to now: Real-valued data d I [k] - d I [k] + g Tx (t) X s I (t) =T s d I [k] g Tx (t kt s ) k= Bandpass transmission allows the application of complex-valued baseband signals s(t). Complex-valued data d[k] = d I [k] + j d Q [k] X Complex signal s(t) = T s d[k] g Tx (t kt s ) = T s k= X (d I [k] + j d Q [k]) g Tx (t kt s ) k= 34

18 Transmitter Configuration Carrier modulation for bandpass transmission d I [k] cos(ω t) d Q [k] + g Tx (t) g Tx (t) sin(ω t) + s BP (t) Compact description: s BP (t) = T s Re (d I [k] + j d Q [k]) g Tx (t) e jωtª QPSK, bits/symbol 4 different symbols d[k] = d I [k] + j d Q [k] g Tx (t) s(t) s(t) = d[k] g Tx (t) e jω t Re{} x BP (t) 35 6QAM, 4 bits/symbol d[k] M-ary modulation, 6QAM Quadrature Amplitude Modulation (QAM) e jω t g Tx (t) Re{} x BP (t) d I [k] cos(ω t) g Tx (t) + x BP (t) d Q [k] g Tx (t) sin(ω t) 36

19 Examples for Signal Space Constellations 4 ASK (M=4) d''( i) QPSK, λ=π/4 (M=4) d''( i) QPSK, λ= (M=4) d''( i) d'( i) d'( i) d'( i) 8 PSK (M=8) d''( i) 6 QAM (M=6) d''( i) 6 PSK/ASK (M=6) d''( i) d'( i) d'( i) d'( i) Phase Shift Keying (PSK) M: Number of signal points Every signal point represents log (M) bits. 37 Transmitter Configuration b[i] {, } cos(ω t) S/P M ld(m) signal mapper (ROM) d I [k] d Q [k] Impulse generator Impulse generator g Tx (t) g Tx (t) + x BP (t) - sin(ω t) 38

20 Bit-to-Symbol-Mapping: Gray-Coding 4-QAM, 4-PSK (QPSK) Im Re 6-PSK 8-PSK Im Re Gray Mapping: binary representation of neighboring symbols differ in only one bit achieves minimum BER for uncoded transmission 6-QAM Im Re 39 Linear Modulation with Nyquist Impulse Shaping QPSK diagram under bandwidth-limited conditions: If system (tx and rx filter) meets st Nyquist: 4 sharp signal points (right diagram) 4

21 Data Rate Symbol time T s : Physical time devoted to each symbol Defines organization and timing at transmitter and receiver Baud rate: numbers of symbols transmitted per second /T s Bit time T b : Time devoted to each bit For M-ary modulation each symbol carries log (M) = ld(m) bit Bit rate: numbers of bits per second /T b T s =log (M )T b =ld(m )T b Examples: BPSK with T s = /4: 4 baud and 4 bit/s 4-QAM with T s =/4: 4 baud but 48 bit/s Drawback of higher modulation schemes: Increases error probability due to smaller distance between symbols Forward Error Correction 4 Outline Chapter 3 (continued) Linear Digital Modulation Methods Definition of PSK, QAM Gray Coding Transmitter Structures Receiver Structures Coherent Receivers Noncoherent Receivers Error Probability Simplified System Model Coherent and noncoherent (D)PSK QAM 4

22 Basic Structures of Coherent Receivers Lowpass filter structure yt () ( ) e j ω t + ψ g Rx (t) g Rx (t) r'( t) r''( t) it r'( it) r''( it) Carrier sync Data decision di ˆ( ) Drawback: Delay in synchronisation loop caused by group delay of lowpass filters Quadrature filter structure yt () h BP (t) hˆ () t BP it yit ( ) j yit ˆ( ) e j( ωt+ ψ ) rit ( ) Carrier sync Data decision di ˆ( ) Advantage: Fast synchronisation loop 43 Coherent QPSK Demodulator 44

23 Noncoherent Demodulation Definition of a noncoherent structure Transfering received signal into baseband without carrier synchronisation Typical: Use of nonlinear systems for demodulation Example: FM demodulation Problem: Linear channel distortion leads to nonlinear distortion after demodulation. Difficult equalization 45 Noncoherent Receiver for Differential Detection e j( ω Δω) t i.t z- * y () DPSK t h () TP t r% DPSK ( it) rδ ( it) d ˆ ( it) Δ Demap. P/S ~ ω = ω + Δω Differentially encoded receive signal r% () i = e e DPSK j ϕ () i j( Ψ ΔωTi) Phase error due to carrier frequency offset Differential encoding of phase ϕ( i) = ϕ( i ) + Δϕ ( i) μ =, K, M μ { Information 46

24 Noncoherent DPSK Receiver (cont.) Removal of differential encoding rδ ( it) = r% DPSK ( i) r% DPSK ( i ) [ ] rδ ( i) = exp j( ϕ( i) ϕ( i ) Δ ωt) = exp[ j( Δϕμ ( i) ΔωT)] Correct decision, if decision threshold is not exceeded M-ary PSK: Δ ω T < π / M (otherwise: decision errors) Drawbacks of noncoherent DPSK demodulation:. Nyquist criterion is not fulfilled Influence of noise (multiplication of noisy signal) 47 Outline Chapter 3 (continued) Linear Digital Modulation Methods Definition of PSK, QAM Gray Coding Transmitter Structures Receiver Structures Coherent Receivers Noncoherent Receivers Error Probability Simplified System Model Coherent and noncoherent (D)PSK QAM 48

25 Simplified System Model in Baseband Equivalent discrete channel Digital b[`] Bit/Symbol Mapping d[k] g Tx (t) s(t) Upconverter Channel Digital Sink ˆd[k] ˆb[`] Decision r[k] k T s r(t) g Rx (t) r a (t) Downconverter To simplify the further analysis the equivalent discrete system model is considered Transmitter Digital b[k] Bit/Symbol Mapping s[k] AWGN Channell r[k] ˆb[k] Receiver Digital Sink 49 Bit Error Probabilities Approximation for PSK signals (Gray coded) P E π erfc log ( M ) sin M N M b bm PSK log ( ) E b = = Es log (M) Energy per bit QAM signals (Gray coded) 3 log ( M) E b bm QAM = erfc log ( M) M ( M ) N P Noncoherent binary DPSK E exp b Pb = N Coherent MSK with precoding P bm PSK E erfc b = N 5

26 Error Probability for M-PSK bit error probability symbol error probability 5 Bit Error Probability for M-DPSK: Examples (AWGN) coherent noncoherent Between coherent and noncoherent reception there is a.3db loss in E b /N. 5

27 Error Probability for M-QAM: Examples (AWGN) bit error probability symbol error probability 53

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