Principles of Digital Communications Part 1

Transcription

1 Principles of Digital Communications Part 1 Philippe Ciblat Université Paris-Saclay & Télécom ParisTech

2 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Outline Section 1: What are Digital Communications? Section 2: A toy example Section 3: Baseband and carrier signals Section 4: Continuous-time propagation channel Section 5: Modulation (TX): from discrete-time to continuous-time Section 6: Demodulation (RX): from continuous-time to discrete-time Matched filter + sampler Equivalent discrete-time channel model Section 7: Detection Theory Optimal detector General performances Section 8: Application to Gaussian channel Section 9: Application to Frequency-Selective channel Viterbi Algorithm Linear equalization OFDM Philippe Ciblat Digital Communications 2 / 86

3 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Section 1: What are Digital Communications? Philippe Ciblat Digital Communications 3 / 86

4 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Introduction Except audio broadcasting (radio), current communication systems are digital 2G, 3G, 4G, 5G, DVBT, Wifi, Free Space Optical Communications ADSL, Optical fiber communications MP3, DVD Channels: copper twisted pair, powerline, wireless, optical fiber, Sources: analog (voice) or digital (data) Philippe Ciblat Digital Communications 3 / 86

5 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 4 / 86

6 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI What is digital? Analog system: d(t) analog source + Pros: low complexity transmit signal : x(t) = f (d(t)) Cons: data transmission, multiple access, performance, limited information processing Digital system: d n digital source (composed by 0 and 1) transmit signal : x(t) = f (d n ) Philippe Ciblat Digital Communications 5 / 86

7 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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) practical constraints (propagation, complexity) Practical case: depends on Quality of Service (QoS) 2G/3G/4G (voice): target L with fixed D b and variable P e ADSL (data): max D b with target P e and fixed B and P Philippe Ciblat Digital Communications 6 / 86

8 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI A few systems System D b B P e Spectral efficiency TV 10Mbits/s 8MHz ,25 bits/s/hz voice 13kbits/s 25kHz ,5 bits/s/hz Data (ADSL) 500kbits/s 1MHz ,5 bits/s/hz Philippe Ciblat Digital Communications 7 / 86

9 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 8 / 86

10 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Section 2: A toy example Philippe Ciblat Digital Communications 9 / 86

11 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 9 / 86

12 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 10 / 86

13 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 11 / 86

14 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Section 3: Baseband/carrier signals Philippe Ciblat Digital Communications 12 / 86

15 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 12 / 86

16 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 13 / 86

17 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 14 / 86

18 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 15 / 86

19 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Section 4: Propagation channel Philippe Ciblat Digital Communications 16 / 86

20 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 16 / 86

21 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 17 / 86

22 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 18 / 86

23 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 19 / 86

24 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 20 / 86

25 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Section 5: Modulation (TX): from discrete-time to continuous-time Philippe Ciblat Digital Communications 21 / 86

26 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 21 / 86

27 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 22 / 86

28 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 23 / 86

29 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Constellations Constellation = set of possible symbols Pulse Amplitude Modulation (PAM) Phase Shift Keying (PSK) Quadrature Amplitude Modulation (QAM) Philippe Ciblat Digital Communications 24 / 86

30 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Section 6: Demodulation (RX): from continuous-time to discrete-time Philippe Ciblat Digital Communications 25 / 86

31 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 25 / 86

32 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 26 / 86

33 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 27 / 86

34 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 28 / 86

35 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 29 / 86

36 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 30 / 86

37 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 31 / 86

38 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 32 / 86

39 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 33 / 86

40 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI 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 Digital Communications 34 / 86

41 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI General model Any previous case can be written as follows with y = [y(0),, y(n)] T y = Hs + w s = [s L,, s 1, s 0,, s N ] T }{{} known Goal: decode the data s from the observations y Matrix model valid for no-isi case, ISI case, but also wireless MIMO, multiple access, multi-core/multi-mode fibers (with different matrix structures) Philippe Ciblat Digital Communications 35 / 86

42 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Section 7: Detection Theory Philippe Ciblat Digital Communications 36 / 86

43 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Main results How to choose optimally s given y? y = f (s, w, ), with s {s m } m C By optimality, we mean minimal error probability min P e, with P e = Prob(ŝ s) where ŝ is the output of the detector Optimal detectors If data are not equilikely, then the optimal receiver (minimizing the error probability) is the Max A Posteriori (MAP), ŝ = arg max p(s y) s If data are equilikely, then the optimal receiver (minimizing the error probability) is the Maximum Likelihood (ML), ŝ ML = arg max p(y s) s Philippe Ciblat Digital Communications 36 / 86

44 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Decision regions If y is seen as a point in a vector space, we can define a region decision for any s m as follows R m = {y p(s m y) p(s m y), m m} R m1 s m1 R m2 s m2 Philippe Ciblat Digital Communications 37 / 86

45 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Application to Gaussian additive noise s model We consider y = d + w with d = f (s) with s the transmit information vector d m = f (s m ), and d m D (received alphabet) Example: f (s) = Hs ML = Least Square (LS) ˆd ML = arg min d y d 2 ŝ ML = arg min s y f (s) 2 Philippe Ciblat Digital Communications 38 / 86

46 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Application to Gaussian additive noise s model (cont d) Decision regions are described through the medians of the received alphabet (if s, or equivalently d, are equilikely) R m1 d m1 R m2 d m2 Philippe Ciblat Digital Communications 39 / 86

47 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Performances Only available for Gaussian additive noise s model d min d m1 d m2 d min = min m m d m d m (= min m m f (s m ) f (s m ) ) N min = 1 M M m=1 N min,m with N min,m number of points at distance d min from d m ( ) dmin P e N min Q 2N0 Philippe Ciblat Digital Communications 40 / 86

48 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Section 8: Application to Gaussian Channel Philippe Ciblat Digital Communications 41 / 86

49 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Signal model with w Gaussian white noise y = s + w ŝ = arg min s y s 2 with the Euclidian distance Main result If y = [y(1),, y(n)] T and s = [s(1),, s(n)] T, then ŝ = arg min s N y(n) s(n) 2 n=1 ŝ(n) = arg min y(n) s(n) 2 s(n) Separable function Component per component optimization is still optimal Philippe Ciblat Digital Communications 41 / 86

50 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Application to linear modulations y = s + w (scalar) with s belonging to PAM, or PSK, or QAM We look for the nearest constellation point from y (threshold detector, denoted by : ŝ = (y)) Philippe Ciblat Digital Communications 42 / 86

51 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Application to linear modulations (cont d) Constellation ( ) Performance Q γ E b N 0 M-PAM γ = 6 log 2 (M)/(M 2 1) M-PSK γ = log 2 (M)(1 cos( 2π M )) M-QAM γ = 3 log 2 (M)/(M 1) with E b the energy used per transmit information bit Remark QAM > PSK > PAM Philippe Ciblat Digital Communications 43 / 86

52 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Numerical illustrations Taux Erreur Symbole en fonction du Eb/No Taux Erreur Symbole en fonction du Eb/No 1 MDP-2 MDP-4 MDP-8 MDP-16 1 MDP-2 MAQ-16 MDP-16 MDA-16 Taux Erreur Symbole 0 00 Taux Erreur Symbole e e Eb/No (en db) P e for different M-PSK Eb/No (en db) P e for different constellations (with fixed M) Philippe Ciblat Digital Communications 44 / 86

53 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Section 9: Application to Frequency-Selective Channel Philippe Ciblat Digital Communications 45 / 86

54 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Outline y = Hs + w with H a N H N H (non-diagonal) Toeplitz-band matrix Section 91: Optimal receiver Some special cases Viterbi algorithm Section 92: Suboptimal receivers ZF (with or without CSIT) MMSE DFE Section 93: OFDM Philippe Ciblat Digital Communications 45 / 86

55 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Section 91 : Optimal receiver Philippe Ciblat Digital Communications 46 / 86

56 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Optimal detector ŝ ML = arg min s y Hs 2 Simple optimization without constraint on s Optimization problem consists in finding out the minimum of a (positive) quadratic form under (nonconvex) conditions Exhaustive search complexity in O(M N H ) with M-QAM ex: 4 4 MIMO with 64-QAM leads to 16M s Special simple cases: N H = 1 or unitary matrix Tree approach suitable for wireless MIMO with small N H Sphere decoding Using the structure of H suitable for ISI since H Toeplitz Viterbi algorithm Suboptimal detectors Remove constraint and threshold (ŝ = (H 1 y)) ZF MMSE, DFE Philippe Ciblat Digital Communications 46 / 86

57 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Optimal detector ŝ ML = arg min y Hs 2 s C N H Hard optimization due to constraint on s Optimization problem consists in finding out the minimum of a (positive) quadratic form under (nonconvex) conditions Exhaustive search complexity in O(M N H ) with M-QAM ex: 4 4 MIMO with 64-QAM leads to 16M s Special simple cases: N H = 1 or unitary matrix Tree approach suitable for wireless MIMO with small N H Sphere decoding Using the structure of H suitable for ISI since H Toeplitz Viterbi algorithm Suboptimal detectors Remove constraint and threshold (ŝ = (H 1 y)) ZF MMSE, DFE Philippe Ciblat Digital Communications 46 / 86

58 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Special case: N H = N H = 1 We consider scalar signal We have y = hs + w ŝ ML = arg min y hs s C = arg min s C h h 1 y s = arg min s C h 1 y s = (h 1 y) y h 1 ŝ ML ZF=ML Easy to implement Philippe Ciblat Digital Communications 47 / 86

59 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Special case: SIMO We consider a multivariate signal (typically N H antennas) y(l) = h l s + w(l) with l = 1,, N H y = hs + w with h a N H 1 column-vector We have ŝ ML = arg min y hs 2 s C = arg min s C y 2 + h 2 s 2 y H hs sh H y y h H Normalization ŝ ML 1/ h 2 ML=MRC=ZF (with pseudo-inverse h # = (h H h) 1 h H = h H / h 2 ) Easy to implement Philippe Ciblat Digital Communications 48 / 86

60 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Special case: SIMO We consider a multivariate signal (typically N H antennas) y(l) = h l s + w(l) with l = 1,, N H y = hs + w with h a N H 1 column-vector We have ŝ ML = arg min y hs 2 s C = arg min s C h 2 h H 2 ( ) y h 2 + h 2 s 2 yh h h 2 s s hh y h 2 y h H Normalization ŝ ML 1/ h 2 ML=MRC=ZF (with pseudo-inverse h # = (h H h) 1 h H = h H / h 2 ) Easy to implement Philippe Ciblat Digital Communications 48 / 86

61 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Special case: SIMO We consider a multivariate signal (typically N H antennas) y(l) = h l s + w(l) with l = 1,, N H y = hs + w with h a N H 1 column-vector We have ŝ ML = arg min y hs 2 s C = arg min h H y s C h 2 s 2 y h H Normalization ŝ ML 1/ h 2 ML=MRC=ZF (with pseudo-inverse h # = (h H h) 1 h H = h H / h 2 ) Easy to implement Philippe Ciblat Digital Communications 48 / 86

62 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Special case: unitary matrices We consider HH H = I NH, or equivalently, Hx = x for any x We have ŝ ML = arg min y Hs s C N H = arg min H ( H H y s ) s C N H = arg min H 1 y s s C N H y ŝ ML H 1 = H H ZF=ML Easy to implement Not true in general since usually AB A B Philippe Ciblat Digital Communications 49 / 86

63 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Go back to ISI problem Idea Due to filtering, H is Toeplitz Such a structure enables a low-complex optimal detector Dynamic programming Viterbi s algorithm [Viterbi1973] We remind that y Hs 2 = N L 2 y(n) h l s n l n=0 Fundamental property l=0 J N (s (N) ) = J N 1 (s (N 1) ) + J(s N, E N ) = def = J N ([s 0,, s N ]) }{{} s (N) N J(s n, E n ) with J(s n, E n ) = y(n) L l=0 h ls n l 2 and E n = [s n 1,, s n L ] T n=0 Philippe Ciblat Digital Communications 50 / 86

64 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Definition of Trellis enables to view s (N) through states {E n } n corresponds to the possible transitions between states Example : M = 2, L = 1 two states E (0) = [ 1] et E (1) = [1] E (0) = 1 received 1 path (one possible s ) E (1) = 1 received 1 time 0 branch node Branch: only characterized by s n and E n (at time n) J : Branch metric (cost to go through this branch) Goal: find optimal path (with the lowest sum of branch metrics) Philippe Ciblat Digital Communications 51 / 86

65 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Algorithm principle For sake of simplification two states only E (0) = [ 1] et E (1) = [1] inspecting between time n 1 and n E (0) = 1 s (n 1) E (0) s n = 1 E (1) = 1 s (n 1) E (1) s n = 1 time n 1 time n Amongst paths arriving at E (0), only one minimizes J n 1 () s (n 1) st J opt E (0) n 1 (s (n 1) ) J opt E (0) n 1 (s (n 1) ) E (0) s (n 1) st J opt E (1) n 1 (s (n 1) ) J opt E (1) n 1 (s (n 1) ) (similar result for E (1) ) E (1) Philippe Ciblat Digital Communications 52 / 86

66 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Algorithm principle (cont d) What s happened at time n in state E (0)? In other words, what do we know about s (n) E (0) and s (n) opt E (0) J n 1 (s (n 1) opt E (0) ) J n 1 (s (n 1) E (0) ) Philippe Ciblat Digital Communications 53 / 86

67 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Algorithm principle (cont d) What s happened at time n in state E (0)? In other words, what do we know about s (n) E (0) and s (n) opt E (0) J n 1 (s (n 1) opt E (0) ) + J( 1, E (0) ) J n 1 (s (n 1) E (0) ) + J( 1, E (0) ) Philippe Ciblat Digital Communications 53 / 86

68 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Algorithm principle (cont d) What s happened at time n in state E (0)? In other words, what do we know about s (n) E (0) and s (n) opt E (0) J n ([s (n 1) opt E (0), 1]) J n ([s (n 1) E (0), 1]) Any suboptimal path at time n 1 remains suboptimal Philippe Ciblat Digital Communications 53 / 86

69 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Algorithm principle (cont d) What s happened at time n in state E (0)? In other words, what do we know about s (n) E (0) and s (n) opt E (0) J n ([s (n 1) opt E (0), 1]) J n ([s (n 1) E (0), 1]) Any suboptimal path at time n 1 remains suboptimal Finally, s (n) = [s (n 1), 1] or [s (n 1), 1] opt E (0) opt E (0) opt E (1) according to respective values of J n Philippe Ciblat Digital Communications 53 / 86

70 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Complexity analysis Complexity in O((N + 1)M L+1 ) linear in N polynomial in M exponential in L So M and L have to be small enough Refresher: max k τ k = LT s, D = log 2 (M)/T s and B = 1/T s Increasing data rate leads to either increasing M bad news for Viterbi s algorithm or increasing B and thus L bad news for Viterbi s algorithm Philippe Ciblat Digital Communications 54 / 86

71 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Numerical example Filter: h 0 = 1 and h 1 = 025 (L = 1) Noiseless case Opening: 1 Data: 1, 1, 1 Closing: 1 Sought vector: [ 1, s 0?, s 1?, s 2?, 1] Received vector: [y( 1)?, 075, 075, 125, 125, y(4)?] Branch metric (at time n) J = y(n) s n 025s n 1 2 Philippe Ciblat Digital Communications 55 / 86

72 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Time Philippe Ciblat Digital Communications 56 / 86

73 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Time Philippe Ciblat Digital Communications 57 / 86

74 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Time Philippe Ciblat Digital Communications 58 / 86

75 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Time Philippe Ciblat Digital Communications 59 / 86

76 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Section 93 : Suboptimal receivers Philippe Ciblat Digital Communications 60 / 86

77 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Principle Goal Viterbi s algorithm not applicable due to high data rate Design a simple receiver (but suboptimal) Idea: linear compensation for linear interference linear receivers w s H y Equalizer P z ŝ ISI input ISI output Question: how choosing matrix P? Philippe Ciblat Digital Communications 60 / 86

78 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM ZF equalizer Principle So Forcing interference to be zero Zero-Forcing (ZF) Mathematically, PH = I P ZF = H 1 z = s + w with w = H 1 w Drawback : noise enhancement (we so derive the SNR per component) SNR input = E ( s trace ) HH H 2N 0 N H Thanks to convexity of x 1/x, one can prove SNR output = E s 2N 0 1 trace(h 1 H H ) N H SNR output SNR input with equality when HH H = I, ie, unitary matrix (cf slide 49) hilippe Ciblat Digital Communications 61 / 86

79 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Precompensation: ZF at the transmitter side! If H known at the transmitter side, one can precompensate it! How? by sending x instead of s with x = Ps where P corresponds to a linear precoder How choosing P? eg ZF principle Drawback: No noise enhancement anymore, but P = ah 1 with a = N H /trace ( H 1 H H) (for energy purpose), then SNR w/o comp = E s ( trace ) HH H 2N 0 N H SNR comp = E s 2N 0 1 trace(h 1 H H ) N H Same SNR issue (just not located at the same place!) Solution suitable if CSIT and unitary transformation Philippe Ciblat Digital Communications 62 / 86

80 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM MMSE equalizer Principle Choosing P st Py close to s P MMSE = arg min P E s,w [ Py s 2 ] After algebraic manipulations, P MMSE = E s ( Es H H H + 2N 0 I ) 1 H H Remarks : At high SNR: P MMSE P ZF At low SNR: P MMSE H H Philippe Ciblat Digital Communications 63 / 86

81 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM DFE equalizer Principle Using previous decision on s for removing interference Decision Feedback Equalizer (DFE) Problem: causality principle in MIMO Solution: the so-called QR decomposition H = QR with a unitary matrix Q and a upper triangular matrix R z = Q H y = Rs + w where w is still white Gaussian noise s H w y DFE equalizer Q H z ŝ R ZF-DFE: decode first ŝ NH, then ŝ NH 1 based on (z NH 1 r NH 1,N H ŝ NH ), and so on Philippe Ciblat Digital Communications 64 / 86

82 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Implementation issue When N H is high, implementation issue for matrix inversion When H corresponds to ISI, equivalence between Toeplitz matrices and filtering Implementation through filtering Example: ZF P ZF H = I p ZF h = δ so p ZF (e 2iπf ) = 1 h(e 2iπf ) Philippe Ciblat Digital Communications 65 / 86

83 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Numerical illustrations We assume L = 1 with h 0 = ρ 2 and h 1 = ρ 1 + ρ 2 Matrix implementation of introduced equalizers The larger ρ is, the stronger ISI is Philippe Ciblat Digital Communications 66 / 86

84 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Viterbi s algorithm performance 10 0 ρ=0 ρ=02 ρ=04 ρ= ρ=08 ρ=1 BER E b /N 0 (db) ISI can not be totally removed Philippe Ciblat Digital Communications 67 / 86

85 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Performance of various equalizers for weak ISI ρ = Performance with rho= Viterbi No equalizer ZF MMSE ZF-DFE BER E b /N 0 (db) Close performance between any solution Philippe Ciblat Digital Communications 68 / 86

86 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Performance of various equalizers for strong ISI ρ = Performance with rho= BER Viterbi No equalizer ZF MMSE ZF-DFE E b /N 0 (db) DFE outperforms linear equalizers Philippe Ciblat Digital Communications 69 / 86

87 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Section 93 : OFDM Philippe Ciblat Digital Communications 70 / 86

88 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Refresher: Input/Ouput model Let x(n) be the transmitted signal (may be different from symbols s n ) y(n) be the received signal {h l } l=0,,l be the filter associated with propagation channel L y(n) = h l x(n l) l=0 Let us consider one block of size N y = [y(n 1),, y(0)] T x = [x(n 1),, x(0)] T x = [x( 1),, x( L)] T Philippe Ciblat Digital Communications 70 / 86

89 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Matrix model for Input/Output y = T 1 x + T 2 x T 1 : N N Toeplitz matrix whose the k-th row is given by [0 k 1, h 0, h 1,, h L, 0 N L k ] (if k N L) [0 k 1, h 0, h 1,, h N k 1 ] (if k > N L) T 2 : N L Toeplitz matrix whose the k-th row is given by 0 L (if k N L) [h L, h L 1,, h N k+1, 0 N k ] (if k > N L) In noisy case, we find again y = Hx + w when null guard interval between blocks: H = T 1 and x = 0 when guard interval x is not empty and linearly depends on current bloc, ie, x = T 3 x: H = T 1 + T 2 T 3 Question: how obtaining an interference-free system at the receiver side (by modifying the transmitter)? Philippe Ciblat Digital Communications 71 / 86

90 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Lemma 1 Assumption: Perfect CSIT (H known at the transmitter side) Actually, extension of ZF principle described in slide 23 Let H = U H ΛV be the singular value decomposition (svd) of H with U, V two unitary matrices, and Λ = diag(λ 0,, λ N 1 ) Instead of sending x = s, we send x = V H s (no energy purpose) Instead of detecting on y, we detect on z = Uy z = Λs + w with w = Uw still white Gaussian noise Remarks Information located in eigenvectors (not interfer in-between!) Eigenvectors usually depend on H Issue: perfect CSIT assumption unrealistic in wireless Philippe Ciblat Digital Communications 72 / 86

91 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Lemma 2 Let C be a N N circulant matrix associated with {h l } l=0,,l h 0 h 1 h L 0 0 C = 0 h 0 h 1 h L 0 0 h 1 h L 0 0 h 0 Properties of C Eigenvectors: Fourier vectors (independent of the channel!) Eigenvalues: filter responses at Fourier frequencies with F FFT matrix (so, F H = F 1 ) C = F H ΛF λ n = H(e 2iπn/N ) = L l=0 h ln 2iπ N le Philippe Ciblat Digital Communications 73 / 86

92 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM OFDM principle If x = [x(n 1),, x(n L)] T, then y = T 1 x + T 2 x y = Cx Thus we have z = Λs with z = Fy and x = F 1 s Finally, for the k-th block and the n-th subcarrier, we get z (k) n = H(e 2iπn/N )s (k) n n, k Remarks cyclic prefix transforms Toeplitz into Circulant (who diagonalizes within a basis independent of the channel, so no required CSIT!) OFDM: Orthogonal Frequency Division Multiplexing Philippe Ciblat Digital Communications 74 / 86

93 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Another way for introducing OFDM The historical way C(f) y(t) = c(t) x(t) f Bcoh B c : coherence bandwidth If B < B c, almost no ISI with c(t) propagation channel If B > B c, ISI B: signal bandwidth C(f) C(f) X(f) X(f) f0 f f Y (f) C(f0)X(f) y(t) C(f0)x(t) Y (f) = C(f)X(f) y(t) = c(t) x(t) How being always in the configuration B < B c? Philippe Ciblat Digital Communications 75 / 86

94 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Another way naive idea Idea: as B = 1/T s (symbol period), splitting symbols sequence into N subsequence (with period T = NT s ) st C(f) X0(f) Xn(f) XN 1(f) 1 T < B c Then each subsequence n transmitted to different subcarriers f n f0 f fn fn 1 On each subcarrier, no ISI f Let s (k) n = s kn+n be a subsequence, the transmitted signal is N 1 x(t) = s (k) n g(t kt )e 2iπfnt n=0 k Z N: subcarriers number N 1 n=0 s(k) n e 2iπfnt : OFDM symbol with period T = NT s Philippe Ciblat Digital Communications 76 / 86

95 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Another way orthogonality principle x(t) = N 1 s (k) n Φ n,k (t) with Φ n,k (t) = g(t kt )e 2iπfnt k Z n=0 Orthogonal subcarriers Φ n,k (t)φ n,k (t)dt = δ n,n δ k,k R We force {Φ n,k (t)} n,k to be orthonormal basis (i) (ii) Without overlap f With overlap f Case (ii) : OFDM (Orthogonal Frequency Division Multiplexing) f n equally spaced f n = n f g(t) rectangular function of support [0, T ] orthogonality iff f = 1/T = 1/NT s Philippe Ciblat Digital Communications 77 / 86

96 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Another way transmitter description Bandwidth: almost same spectral efficiency than single carrier B tot = NB sc N 1 T = N 1 NT s = 1 T s x (k) (m) = x(kt + mt s ) s (k) 0 x (k) (0) = N 1 1 s (k) n e 2iπ nm N N n=0 } {{ } inverse FFT sn S / P s (k) N 1 IFFT x (k) (N 1) Rate 1/Ts Rate 1/T Rate 1/T ADC x(t) Remarks x in time domain k OFDM block number and m time index (within block) s in frequency domain n frequency index hilippe Ciblat Digital Communications 78 / 86

97 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Another way receiver description z (k) (n) = = R Ts y(t)φ n,k (t)dt N 1 y (k) (m)e 2iπ nm N N m=0 with y (k) (m) = y(kt + mt s ) When channel present, y(t) Ts y (k) (0) y (k) (N 1) FFT z (k) (0) z (k) (N 1) P / S z n (k) ŝ (k) n y(t) = c(t) x(t) = N 1 s (k) n Ψ n,k (t) with Ψ n,k (t) = c(t) Φ n,k (t) k Z n=0 Problem: Ψ n,k (t) is not orthonormal anymore (actually, almost) Solution: go back to the previously-developed approach Cyclic prefix hilippe Ciblat Digital Communications 79 / 86

98 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM TX/RX Scheme TX RX s FFT 1 x Add CP Channel h Remove CP y FFT z Freq EQ (typ ZF) s Convolution Toeplitz matrix Circular convolution / Circulant matrix Cyclic prefix is crucial! Philippe Ciblat Digital Communications 80 / 86

99 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Detection: OFDM-SISO Formally, we have z = Hs + w with H = diag(h(1),, H(e 2iπn/N ),, H(e 2iπ(N 1)/N )) }{{} H(n) As H is diagonal (no interference occurs), one can work subcarrier per subcarrier, ie, z(n) = H(n)s n + w(n) with z(n) received signal after FFT s n transmitted symbol (before IFFT) H(n) filter response at the n-th subcarrier Optimal detector (application of Slide 47) Threshold detector on H(n) 1 z(n) Philippe Ciblat Digital Communications 81 / 86

100 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM History end-50: multicarriers concept (Case (i)) end-60: orthogonal multicarriers (Case (ii)) OFDM beginning-70: FFT mid-80: European project Eurêka for DAB cyclic prefix coding and OFDM relationship in wireless context beginning-90: First standard based on OFDM (DAB) end-90: Very popular standards (ADSL, DVBT, Wifi, LTE, ) Philippe Ciblat Digital Communications 82 / 86

101 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM OFDM design: two rules N should be large enough L N LT s NT s T d NT s N should be small enough N B/B c f B c (L + N)T s T c NT s T c N B/B d f B d but also Mis-synchronization of VCO (a few ppm) FFT complexity (O(N log(n))) Latency Philippe Ciblat Digital Communications 83 / 86

102 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM OFDM design: four examples Wifi ADSL DVBT Optic 100G Carrier freq 52GHz 06MHz 700MHz 1550nm Bandwidth 20MHz 11MHz 915MHz 5GHz Sampling period 50ns 09µs 1µs 02ns Filter length 800ns 135µs 224µs 069ns Filter degree Cyclic prefix Spect eff loss 20% 125 % 20% 3125% # Subcarrier OFDM duration 4µs 256µs 896µs 528ns Subcarrier spac 3125kHz 431kHz 111kHz 1953 MHz Coherence band 125MHz 74kHz 447kHz 144GHz Doppler band 52Hz (3m/s) 0 52Hz (3m/s) 0 Remark: in ADSL, Time Equalizer added for channel shortening Philippe Ciblat Digital Communications 84 / 86

103 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM Perspectives MIMO : how to code, how to decode? Multi-user interference: how to manage resource allocation Artificial Intelligence tools : decoding and resource optimization Philippe Ciblat Digital Communications 85 / 86

104 DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w ISI Optimal receiver Suboptimal receivers OFDM 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 [Viterbi1967] A Viterbi, Error bounds for convolutional codes and an asymptotic optimum decoding algorithm, IEEE Trans on Information Theory, 1967 [VanTrees23] H VanTrees, K Bell, Detection Estimation and Modulation Theory Part I, Wiley, 23 Philippe Ciblat Digital Communications 86 / 86