Refresher on Digital Communications Channel, Modulation, and Demodulation Philippe Ciblat Université Paris-Saclay & Télécom ParisTech
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
Section 1: Digital Communication scheme Philippe Ciblat DC: Channel, Modulation, and Demodulation 3 / 36
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
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
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
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
A few systems System D b B P e Spectral efficiency DVB 10Mbits/s 8MHz 10 11 1,25 bits/s/hz 2G 13kbits/s 25kHz 10 2 0,5 bits/s/hz ADSL 500kbits/s 1MHz 10 7 0,5 bits/s/hz Philippe Ciblat DC: Channel, Modulation, and Demodulation 7 / 36
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
Section 2: A toy example Philippe Ciblat DC: Channel, Modulation, and Demodulation 9 / 36
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 0 00000000000000000000000000000000000000 11111111111111111111111111111111111111 t 1 0 0 1 1 0 1 0 but Light has a color ( wavelength) 1 08 06 04 02 0 x c (t) = x(t) cos(2πf 0 t) 02 04 06 08 1 0 1 2 3 4 5 6 7 8 Philippe Ciblat DC: Channel, Modulation, and Demodulation 9 / 36
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
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
Section 3: Baseband/carrier signals Philippe Ciblat DC: Channel, Modulation, and Demodulation 12 / 36
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
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
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 00000000 00000000 00000000 00000000 11111111 11111111 11111111 00000000 00000000 00000000 11111111 11111111 11111111 00000000 00000000 11111111 11111111 11111111 00000000 00000000 00000000 11111111 11111111 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 1100000000 111111110000 I/Q mod 00000000 111111110000 Channel 0000 1111 I/Q demod 00000000 111111110000 111100000000 1111111100 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 hilippe Ciblat DC: Channel, Modulation, and Demodulation 14 / 36
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
Section 4: Propagation channel Philippe Ciblat DC: Channel, Modulation, and Demodulation 16 / 36
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
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
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
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
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
Section 5: Transmitter (Modulation) Philippe Ciblat DC: Channel, Modulation, and Demodulation 21 / 36
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
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 0 1 1 0 0 1 0 1 0 0 t Philippe Ciblat DC: Channel, Modulation, and Demodulation 22 / 36
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 0 1 1 0 0 1 0 1 0 0 t 3A Philippe Ciblat DC: Channel, Modulation, and Demodulation 23 / 36
Constellations Constellation = set of possible symbols Pulse Amplitude Modulation (PAM) 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 000 11 1 00 11 00 11 00 11 Phase Shift Keying (PSK) 00 11 00 11 00 11 00 11 00 11 00 11 1 00 0 11 00 11 00 11 00 11 Quadrature Amplitude Modulation (QAM) 00 11 00 11 00 11 00 11 1100 1100 1100 0 00 11 1 0 00 11 1 0 00 11 1 1100 1100 1100 Philippe Ciblat DC: Channel, Modulation, and Demodulation 24 / 36
Section 6: Receiver (Demodulation) Philippe Ciblat DC: Channel, Modulation, and Demodulation 25 / 36
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
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
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
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
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
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
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
Nyquist filter 12 08 06 04 02 1 0 1 05 0 05 1 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 08 06 h(t) 04 H(f) 02 0-02 -10-5 0 5 10 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
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
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
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
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