Refresher on Digital Communications Channel, Modulation, and Demodulation

Transcription

1 Refresher on Digital Communications Channel, Modulation, and Demodulation Philippe Ciblat Université Paris-Saclay & Télécom ParisTech

2 Outline Section 1: Digital Communication scheme Section 2: A toy example Section 3: Baseband and carrier signals Section 4: Propagation channel Section 5: Transmitter (Modulation) Section 6: Receiver (Demodulation) Matched filter + sampler Nyquist filter Philippe Ciblat DC: Channel, Modulation, and Demodulation 2 / 36

3 Section 1: Digital Communication scheme Philippe Ciblat DC: Channel, Modulation, and Demodulation 3 / 36

4 Introduction Except audio broadcasting (radio), current communication systems are digital 2G, 3G, DVBT, Wifi ADSL MP3, DVD Channels: copper twisted pair, powerline, wireless, optical fiber, Sources: analog (voice) or digital (data) Philippe Ciblat DC: Channel, Modulation, and Demodulation 3 / 36

5 If analog source, Sampling (no information loss) Nyquist-Shannon Theorem Let t x(t) be a continuous-time signal of bandwidth B x(t) is perfectly characterized by the sequence {x(nt )} n where T is the sampling period satisfying 1/T B Quantization (information loss) Example Let us consider voice signal Quality Bandwidth Sampling Quantization 2G [300Hz, 3400 Hz] 8kHz 8 bits Hifi [20Hz, 20kHz] 44kHz 16 bits hilippe Ciblat DC: Channel, Modulation, and Demodulation 4 / 36

6 What is digital? Analog system: s(t) analog source + Pros: low complexity transmit signal : x(t) = f (s(t)) Cons: data transmission, multiple access, performance, limited information processing Digital system: s n digital source (composed by 0 and 1) transmit signal : x(t) = f (s n ) Philippe Ciblat DC: Channel, Modulation, and Demodulation 5 / 36

7 Design parameters Goal but Data rate D b bits/s Bandwidth B Hz Error probability P e Transmit power (SNR) P mw or dbm Latency L max D b with min B, P e, P, L theoretical limits (information theory) physical constraints (propagation, complexity) Practical case: depends on Quality of Service (QoS) 2G/3G: target L with fixed D b and variable P e ADSL: max D b with target P e and fixed B and P Philippe Ciblat DC: Channel, Modulation, and Demodulation 6 / 36

8 A few systems System D b B P e Spectral efficiency DVB 10Mbits/s 8MHz ,25 bits/s/hz 2G 13kbits/s 25kHz ,5 bits/s/hz ADSL 500kbits/s 1MHz ,5 bits/s/hz Philippe Ciblat DC: Channel, Modulation, and Demodulation 7 / 36

9 Transceiver/Receiver structure Source d n Channel coding a n Modulation x(t) propagation channel Destination ˆd n Channel decoding â n Demodulation y(t) Question? How to design Modulation/demodulation boxes Coding/decoding boxes depending on propagation channel Philippe Ciblat DC: Channel, Modulation, and Demodulation 8 / 36

10 Section 2: A toy example Philippe Ciblat DC: Channel, Modulation, and Demodulation 9 / 36

11 The old optical fiber t/t b Goal: Sending a bit stream a n {0, 1} at data rate D b bits/s Data a n will be sent at time nt b with T b = 1/D b s How? x(t) = 0 if a n = 0 within [nt b, (n + 1)T b ) No light x(t) = A if a n = 1 within [nt b, (n + 1)T b ) Light Tb A t but Light has a color ( wavelength) x c (t) = x(t) cos(2πf 0 t) Philippe Ciblat DC: Channel, Modulation, and Demodulation 9 / 36

12 Mathematical framework Each data has a shape Here, the rectangular function Each shape is multiplied by an amplitude Here, either A or 0 Each data is shifted at the right time x(t) = n s n g(t nt s ) with g(t) shaping filter Here, g(t) rectangular function s n symbol sequence Here s n = Aa n T s symbol period Here, T s = T b Finally x c (t) = x(t) cos(2πf 0 t) Philippe Ciblat DC: Channel, Modulation, and Demodulation 10 / 36

13 Degrees of freedom carrier frequency f 0 impact on propagation condition impact on data rate (see later) shaping filter g(t) impact on bandwidth S x(f ) G(f ) 2 with G(f ) Fourier Transform of g(t) impact on receiver complexity and performance (see later) symbol s n impact on data rate: multi-level impact on performance (see later) symbol period T s impact on data rate impact on bandwidth (through the choice of g(t)) Philippe Ciblat DC: Channel, Modulation, and Demodulation 11 / 36

14 Section 3: Baseband/carrier signals Philippe Ciblat DC: Channel, Modulation, and Demodulation 12 / 36

15 Questions x c (t) = x(t) cos(2πf 0 t) with x c (t): carrier signal x(t): baseband signal (complex) envelope Q1: Is there another way to translate the signal? x(t) x c (t) YES I/Q modulator Complex-valued signal Q2: How retrieving x(t) from x c (t)? I/Q demodulator Philippe Ciblat DC: Channel, Modulation, and Demodulation 12 / 36

16 Mathematical framework Instead of using only cos, we can use simultaneously cos and sin with x c (t) = x p (t) cos(2πf 0 t) x q (t) sin(2πf 0 t) = R ((x ) p (t) + ix q (t))e 2iπf 0t x p (t) a baseband real-valued signal of bandwidth B: In-phase x q (t) another real-valued signal of bandwidth B: Quadrature We may have two streams in baseband for one carrier signal! Complex envelope The baseband signal can be represented by the so-called complex envelope x(t) = 1 2 (x p (t) + ix q (t)) Philippe Ciblat DC: Channel, Modulation, and Demodulation 13 / 36

17 Mathematical framework (cont d) Assuming B/2 < f 0, we have I/Q modulator I/Q demodulator x p(t) x x x p(t) π/2 f 0 + x c(t) x c(t) π/2 f 0 x q(t) x x x q(t) In practice, we work with complex envelope smaller bandwidth B instead of 2f 0 + B no cos and sin disturbing terms TX Supra-channel RX a n x(t) x c (t) y c (t) y(t) â n I/Q mod Channel I/Q demod hilippe Ciblat DC: Channel, Modulation, and Demodulation 14 / 36

18 A few wireless systems When f 0 increases propagation degrades (1/f 2 ) antenna size decreases (1/f ) bandwidth B may increase System f 0 B Antenna size Intercont 10MHz (HF) 100kHz 100m DVBT 600MHz (UHF) 1 MHz 1m 2G 900MHz 1 MHz 10cm Wifi 54 GHz 10MHz 1cm Satellite 11GHz 100MHz Personal Network 60GHz Philippe Ciblat DC: Channel, Modulation, and Demodulation 15 / 36

19 Section 4: Propagation channel Philippe Ciblat DC: Channel, Modulation, and Demodulation 16 / 36

20 Multipath channel typical wireless channel valid also for ADSL and optical fiber (low SNR) (ρ 1, τ 1) (ρ 2, τ 2) (ρ 0, τ 0) y(t) = ρ k x(t τ k ) + w(t) k = c(t) x(t) + w(t) with noise w(t) Dispersion time: T d = max k τ k Coherence bandwidth: B c = min f arg max δ { C(f ) C(f + δ) < ε} B c = O(1/T d ) Philippe Ciblat DC: Channel, Modulation, and Demodulation 16 / 36

21 Noise property Let w c (t) be the (random) noise at carrier level w c (t) is zero-mean (real-valued) Gaussian variable w c (t) is stationary (E[w c (t) 2 ] independent of t) w c (t) is almost white N0/2 f0 P = S w (f )df = N 0 B B f What s happened for complex envelope w(t)? Philippe Ciblat DC: Channel, Modulation, and Demodulation 17 / 36

22 Noise property (cont d) with w(t) = 1 2 (w p (t) + iw q (t)) 1 w p (t) and w q (t) zero-mean (real-valued) stationary Gaussian variable with the same spectrum N0 2 w p (t) and w q (t) are independent B f Philippe Ciblat DC: Channel, Modulation, and Demodulation 18 / 36

23 Model: Gaussian channel Short multipaths (T d ) compared to symbol period (T s ) Holds for Hertzian beams Holds for Satellite Holds also for very low data rate transmission y(t) = x(t) + w(t) Philippe Ciblat DC: Channel, Modulation, and Demodulation 19 / 36

24 Model: Frequency-Selective channel Holds for cellular systems (2G with T d = 4T s ) Holds for Local Area Network (Wifi with T d = 16T s ) Holds for ADSL (T d = 100T s ) Holds also for Optical fiber (the so-called chromatic dispersion) y(t) = c(t) x(t) + w(t) InterSymbol Interference (ISI) Remark Channel type (ISI?) is modified according to data rate The higher the rate is, the stronger the ISI is (T d T s ) Philippe Ciblat DC: Channel, Modulation, and Demodulation 20 / 36

25 Section 5: Transmitter (Modulation) Philippe Ciblat DC: Channel, Modulation, and Demodulation 21 / 36

26 Question a n Modulation x(t) I/Q modulator x c (t) "Modulation" How associating bits a n with analog (baseband) signal x(t)? Philippe Ciblat DC: Channel, Modulation, and Demodulation 21 / 36

27 Binary modulation Waveform: x 0 (t) if bit 0 and x 1 (t) if bit 1 Binary linear modulation x 0 (t) = Ag(t) and x 1 (t) = Ag(t) with symbols A and A, and the shaping filter g(t) If the symbol period is T s, then x(t) = k s k g(t kt s ) with s k { A, A} Example (g(t) rectangular function) Ts A A t Philippe Ciblat DC: Channel, Modulation, and Demodulation 22 / 36

28 Multi-level modulation Bandwidth of x(t) (B) identical of that of g(t): - If B 1/T s, InterSymbol Interference (see rectangular case) - If B 1/T s, bandwidth is wasted (signal oscillates at 1/T s) B = O(1/T s) Spectral efficiency is 1bit/s/Hz in binary modulation Multi-level modulation: one symbol contains more than one bit Exemple (M = 4) 00 A A 10 3A 11 3A 3A A Ts A t 3A Philippe Ciblat DC: Channel, Modulation, and Demodulation 23 / 36

29 Constellations Constellation = set of possible symbols Pulse Amplitude Modulation (PAM) Phase Shift Keying (PSK) Quadrature Amplitude Modulation (QAM) Philippe Ciblat DC: Channel, Modulation, and Demodulation 24 / 36

30 Section 6: Receiver (Demodulation) Philippe Ciblat DC: Channel, Modulation, and Demodulation 25 / 36

31 Question y c (t) y(t) z(n) â n I/Q modulator Demodulation Detector Two main boxes: How coming back to discrete-time signal: demodulation How detecting optimally the transmit bits (from z(n)): detector Goal Describing and justifying the demodulation Philippe Ciblat DC: Channel, Modulation, and Demodulation 25 / 36

32 A mathematical tool: signal space Let L 2 be the space of energy-bounded function { } L 2 = f st f (t) 2 dt < + Properties L 2 is an infinite-dimensional vectorial space L 2 has an inner product < f 1 (t) f 2 (t) >= f 1 (t)f 2 (t)dt leads to orthogonality principle: < f 1 (t) f 2 (t) >= 0 leads to a norm: f (t) = < f (t) f (t) > L 2 has an infinite-dimensional orthonormal (otn) basis: {Ψ m (t)} m f L 2, {β m } m, f (t) = m β m Ψ m (t) with β m =< f (t) Ψ m (t) > Any function is described by complex-valued coefficients Philippe Ciblat DC: Channel, Modulation, and Demodulation 26 / 36

33 A signal subspace Let E be a subspace of L 2 generated by the functions {f m (t)} m=1,,m { M } E = span({f m (t)} m=1,,m ) = α m f m (t) for any complex α m Property m=1 This subspace has a finite dimension and a finite otn basis D = dim C E and E = span{φ l (t)} l {1,,D} For instance, let f (t) be a function in E f (t) = D s (l) Φ l (t) with s (l) =< f (t) Φ l > C l=1 s = [s (1),, s (D) ] T corresponds to the analog signal f (t) Usually, we prefer to work with s (which will carry information) Philippe Ciblat DC: Channel, Modulation, and Demodulation 27 / 36

34 Exhaustive demodulator y(t) = s k h(t kt s ) + w(t) k with any symbol s k and any filter h(t) Question How sampling without information loss? Nyquist-Shannon Theorem: sampling at f e > B Then y(n/f e ) contains all the information on y(t) Actually information ({s k }) is only a part of y(t) Exhaustive demodulator based on subspace principle Information {s k } belongs to the subspace E E = span({h(t kt s )} k ) Noise w(t) belongs to E and E (orthogonal of E) w(t) = w E (t) + w E (t) (w E (t) and w E (t) independent) Consequently, projection on E contains any information on {s k } in y(t) Philippe Ciblat DC: Channel, Modulation, and Demodulation 28 / 36

35 Exhaustive demodulator (cont d) Projection on E z(n) = < y(t) h(t nt s ) > = y(τ)h(τ nt s )dτ = h( t) y(t) t=nts y(t) h( t) nt s z(n) Projection = Matched filter + Sampling Remark: Sampling at T s and not at T e Philippe Ciblat DC: Channel, Modulation, and Demodulation 29 / 36

36 Input/output discrete-time model s n h(t) w(t) y(t) h( t) h(t) z(n) z(n) = l h(lt s )s n l + w(n) with h(t) = h( t) h(t) w(n) = h( t) w(t) t=nts zero-mean complex-valued stationary Gaussian with spectrum S w (e 2iπf ) = N 0 h(e 2iπf ) = N 0 h(e 2iπf ) 2 Philippe Ciblat DC: Channel, Modulation, and Demodulation 30 / 36

37 Orthogonal basis case What s happened when {h(t kt s )} k is an otn basis No ISI z(n) = s n + w(n) Equivalent proposition {h(t kt s )} k otn basis h(t) Nyquist filter h(lt s ) = δ l,0 k ) H (f kts = T s h(t) square-root Nyquist h(t) = h( t) h(t) H(f ) = H(f ) In practice, h(t) square-root Nyquist iff Gaussian channel no ISI provided by propagation channel g(t) square-root Nyquist no ISI provided by shaping filter Philippe Ciblat DC: Channel, Modulation, and Demodulation 31 / 36

38 Nyquist filter Main property If h(t) square-root Nyquist, then B > 1 T s Examples: h(t) rectangular h(t) triangular h(t) square-root raised cosine (srrc) h(t) raised cosine 1 Raised cosine with rho=05 Ts = 1 et ρ = 05 B h(t) 04 H(f) t with roll-off ρ (ρ = 022 in 3G, ρ = 005 in DVB-S2, ρ = 0 in WDM-Nyquist) 1+ρ 2Ts 1 ρ 2Ts f 1 ρ 2Ts 1+ρ 2Ts Philippe Ciblat DC: Channel, Modulation, and Demodulation 32 / 36

39 Consequence on noise If h(t) square-root Nyquist, then w(n) white noise w(n) = w R (n) + iw I (n) w R (n) and w I (n) independent E[w R (n) 2 ] = E[w I (n) 2 ] = N 0 /2 E[ w(n) 2 ] = N 0 and E[w(n) 2 ] = 0 Probability density function (pdf) p w (x) = p wr,w I (x R, x I ) = p wr (x R )p wi (x I ) = = 1 e x 2 R N 0 1 e x 2 I N 0 = 1 e x R 2 +x2 I N 0 πn0 πn0 πn 0 1 e x 2 N 0 πn 0 Philippe Ciblat DC: Channel, Modulation, and Demodulation 33 / 36

40 Non-orthogonal basis case What s happened when {h(t kt s )} k is a non-otn basis ISI Colored noise Equivalent model By using whitening filter f, we have y(n) = f z(n) = with w(n) white Gaussian noise L h(l)s n l + w(n) l=0 Philippe Ciblat DC: Channel, Modulation, and Demodulation 34 / 36

41 Conclusion Lecture stops here but it remains to do Detector recovering s n from z(n) in no-isi case recovering s n from z(n) in ISI case: see Digital Information Processing course Performances System design Channel coding/decoding Philippe Ciblat DC: Channel, Modulation, and Demodulation 35 / 36

42 References [Tse2005] D Tse and P Viswanath, Fundamentals of wireless communications, 2005 [Goldsmith2005] A Goldsmith, Wireless Communications, 2005 [Proakis2000] J Proakis, Digital Communications, 2000 [Benedetto1999] S Benedetto and E Biglieri, Principles of digital transmission with wireless applications, 1999 [Viterbi1979] A Viterbi and J Omura, Principles of digital communications and coding, 1979 [Gallager2008] R Gallager, Principles of digital communications, 2008 [Wozencraft1965] J Wozencraft and I Jacob, Principles of communications engineering, 1965 [Barry2004] J Barry, D Messerschmitt and E Lee, Digital communications, 2004 [Sklar20] B Sklar, Digital communications : fundamentals ans applications, 20 [Ziemer20] R Ziemer and R Peterson, Introduction to digital communication, 20 Philippe Ciblat DC: Channel, Modulation, and Demodulation 36 / 36