1. Band-pass modulations. 2. 2D signal set. 3. Basis signals p(t)cos(2πf 0 t) e p(t)sin(2πf 0 t) 4. Costellation = m signals, equidistant on a circle

Transcription

1 TUTORIAL ON DIGITAL MODULATIONS Part 14: m-psk [last modified: ] Roerto Garello, Politecnico di Torino Free download at: (personal use only) 1

2 m-psk modulations 1. Band-pass modulations 2. 2D signal set Characteristics 3. Basis signals p(t)cos(2πf 0 t) e p(t)sin(2πf 0 t) 4. Costellation = m signals, equidistant on a circle 5. Information associated to the carrier phase R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 2

3 m-psk: : constellation SIGNAL SET M = { s ( t) = Ap( t)cos(2 π f t ϕ ) } m i 0 i i= 1 2π ϕi = Φ + ( i 1) m Information associated to the carrier phase R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 3

4 m-psk: : constellation s ( t) = Ap( t)cos(2 π f t ϕ ) i 0 i ϕ = Φ + i ( i 1) 2π m We can write s ( t) = ( Acos ϕ ) p( t)cos(2 π f t) + ( Asin ϕ ) p( t)sin(2 π f t) i i 0 i 0 Clearly, we have two versors ( t) = p( t)cos(2 π f t) 1 0 ( t) = p( t)sin(2 π f t) 2 0 R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 4

5 m-psk: : constellation SIGNAL SET M = { s ( t) = Ap( t)cos(2 π f t ϕ ) } ϕ = Φ + ( i 1) m i 0 i i= 1 i 2π m VERSORS 1 ( t) = p( t)cos(2 π f0t) ( t) = p( t)sin(2 π f t) 2 0 VECTOR SET M = { s = ( α, β ) } R α = i β = i m i i i i= 1 Acosϕ Asinϕ 2π ϕi = Φ + ( i 1) m i i 2 R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 5

6 Example 8-PSK Φ = 0 M = { s = ( A,0), s = ( A/ 2, A/ 2), s = (0, A), s = ( A/ 2, A/ 2), s = ( A,0), s = ( A/ 2, A/ 2), s = (0, A), s = ( A/ 2, A/ 2)} R R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 6

7 Example 16-PSK M = { s = ( A,0), s = (0.924 A,0.383 A), s = ( A/ 2, A/ 2), s = (0.383 A,0.924 A), s = (0, A), s = ( A,0.924 A,), s = ( A/ 2, A/ 2), s = ( A,0.383 A), s = ( A,0), s = ( A, A), s = ( A/ 2, A/ 2), s = ( A, A), s = (0, A), s = (0.383 A, A,), s = ( A/ 2, A/ 2), s = (0.924 A, A)} R Φ = 0 2 R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 7

8 m-psk: : inary laeling e : Hk M It is always possile to uild Gray laelings R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 8

9 m-psk: : transmitted waveform k = log 2 m T kt R k = Each symol has duration T Each symol component (α and β) lasts for T second R = Transmitted waveform s( t) = α[ n] p( t nt ) cos(2 π f0t) + β[ n] p( t nt ) sin(2 π f0t) n n i( t ) q( t) I component (in phase) Q component (in quadrature) R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 9

10 example: 8-PSK transmitted waveform p( t) = P T ( t) T T A = 2 f = R t/t R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 10

11 example: 8-PSK I and Q components I COMPONENT i( t) = α[ n] p( t nt ) q( t) = β[ n] p( t nt ) n Q COMPONENT n u T T 2T 3T 4T 5T 6T 7T 8T 1 9T i( t) T 2T 3T q( t) T 2T 3T R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 11

12 m-psk: : andwidth and spectral efficiency Transmitted waveform s( t) = α[ n] p( t nt ) cos(2 π f0t) + β[ n] p( t nt ) sin(2 π f0t) n n 2 2 Gs ( f ) = z P( f f0) + P( f + f0) z R Each symol α[n] and β[n] has time duration T = kt R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 12

13 m-psk: : andwidth and spectral efficiency Case 1: p(t) = ideal low pass filter f 0 f 0 R R 1/T 1/T Total andwidth (ideal case) B id = R = R k Spectral efficiency (ideal case) R η id = = k ps / Hz B id R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 13

14 m-psk: : andwidth and spectral efficiency Case 2: p(t) = RRC filter with roll off α f 0 f 0 Total andwidth B R = R(1 + α ) = (1 + α) k 1 1 T 1 1 T R (1 + α ) R (1 + α ) ( + α ) ( +α ) Spectral efficiency η R B = = k (1 + α) ps / Hz R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 14

15 Exercize Given a andpass channel with andwidth B = 4000 Hz, centred around f 0 =2 GHz, compute the maximum it rate R we can transmit over it with an 8-PSK constellation or a 16-PSK constellation in the two cases: Ideal low pass filter RRC filter with α=0.25 R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 15

16 m-psk: : modulator u [ ] T i vt [ n] k e ( ) α[ n] β[ n] cos 2 p( t) ( π f t) 0 i ( t) 90 cos sin s( t) p( t) q( t) FOR m > 4 NOT CARTESIAN PRODUCT R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 16

17 m-psk: : demodulator p( t) t0 + nt ρ [ n] 1 90 cos 2 sin 2 ( π f t) 0 ( π f t) 0 SYMBOL SYNCHRONIZATION R ML CRITERION s [ ] R n v [ n ] u [ i] e -1 R k R CARRIER SYNCHRONIZATION cos 2 ( π f t) 0 p( t) t0 + nt ρ2[ n] FOR m > 4 NOT CARTESIAN PRODUCT Voronoi regions = plane sectors R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 17

18 m-psk: : eye diagram 8-PSK constellation with RRC filter (α=0.5) [ α and β components = 0.924,0.383,-0.383,-0.924] Channel I Channel Q R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 18

19 m-psk: : eye diagram 16-PSK constellation with RRC filter (α=0.5) [ α and β components = 0.981,0.832,0.556,0.195,-0.195,-0.556,-0.832,-0.981] Channel I Channel Q R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 19

20 m-psk constellation: : error proaility By applying the asymptotic approximation we can otain 1 E π 2 P ( e) erfc k sin k N0 m BER performance decreases for increasing m (minimum distance decreases) R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 20

21 m-psk constellation: : error proaility 4-PSK: P ( e) 1 E erfc 2 N 0 1 E 8-PSK: P ( e) erfc N db with respect to 4-PSK 0 1 E 16-PSK: P ( e) erfc N db with respect to 8-PSK No one uses m-psk for m > 16: very poor BER performance R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 21

22 m-psk constellation: : error proaility E-3 1E-4 1E-5 1E-6 2-PSK,4-PSK 8-PSK 16-PSK BE ER 1E-7 1E-8 1E-9 1E-10 1E-11 1E-12 1E-13 1E E/N0 [db] The BER performance decreases for increasing m R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 22

23 m-psk constellation: performance/spectral efficiency trade-off Given a andpass channel with andwidth B Given an m-psk constellation, y increasing the numer of signals m=2 k 1. the spectral efficiency increases: R η id = = B id k ps / Hz we can transmit a higher it rate R. 2. the BER performance decreases: fixed a target BER value, the signal-to-noise ratio E /N 0 (proportional to the received power S) necessary to achieve it increases with m. practical applications: m 8 R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 23

24 m-psk: analytic signal s( t) = α[ n] p( t nt ) cos(2 π f0t) + β[ n] p( t nt ) sin(2 π f0t) n n i( t ) q( t) [ ] 2 0 s( t) = Re s& ( t) = Re s% ( t) e j π f t s% ( t) = i( t) jq( t) = γ[ n] p( t nt) n γ[ n] = α[ n] jβ[ n] R. Garello. Tutorial on digital modulations - Part 14 m-psk [ ] 24