Multiphase Shift Keying (MPSK) Lecture 8. Constellation. Decision Regions. s i. 2 T cos 2π f c t iφ 0 t As iφ 1 t. t As. A c i.

Transcription

1 π fc uliphase Shif Keying (PSK) Goals Lecure 8 Be able o analyze PSK modualion s i Ac i Ac Pcos π f c cos π f c iφ As iφ π i p p As i sin π f c p Be able o analyze QA modualion Be able o quanify he radeoff beween daa rae and energy. for i φ sin, where φ π f p. cos p and A c i A s i Ecos πi Esin πi VIII- VIII- Consellaion φ Decision Regions φ s s s s 4 s φ φ s 5 s 7 s 6 VIII- VIII-4

2 ψ ψ φ Noes on PSK For his modulaion scheme we should use Gray coding o map bis ino signals. BPSK 4 QPSK his ype of modulaion has he properies ha all signals have he same power hus he use of nonlinear amplifiers (class C amplifiers) affecs each signal in he same manner. Furhermore if we are resriced o wo dimensions and every signal mus have he same power hen his signal se minimizes he error probabiliy of all such signal ses. φ QPSK and BPSK are special cases of his modulaion. VIII-5 VIII-6 Symbol Error Probabiliy for PSK he error probabiliy for PSK can be deermined as follows. Consider a signal ransmied where wih he consellaion shown above. he probabiliy of error given signal ransmied is he probabiliy ha he noise brings he received signal ouside he region R where he decision is ha signal was ransmied. his is given as π ψ π ψ π ψ π π R r R u π exp r σ exp R ψ σ exp σ dψ r σ drdψ u dudψ P e Rc Rc Rc π ψ f Z H z πσ exp E R r H f Z H σ πσ exp z z σ r π σ exp E H dz dz z z z r σ dz dz dz dz drdψ where R c is he complemen of R and R is he disance from he signal poin s o he line wih slope π R. sin π E sin E sin sin ψ ψ π π π π VIII-7 VIII-8

3 π π hus R σ E sin N sin ψ π In he derivaion of he error probabiliy he las line follows from a change of variables where he poin z axis wih reference poin s and a magniude r from he poin s. he symbol error probabiliy is hus φ z is mapped o an angle φ from he horizonal φ P e s π ψ π exp E sin N sin ψ π dψ E b E log VIII-9 VIII- φ Consellaion (E b N db).5.5 Quadraure Phase.5.5 π ψ R E φ VIII- VIII-

4 Consellaion (E b N db) Consellaion (E b N 4dB) Quadraure Phase.5.5 Quadraure Phase VIII- VIII-4 Consellaion (E b N 6dB) Consellaion (E b N 8dB) Quadraure Phase.5.5 Quadraure Phase VIII-5 VIII-6

5 Consellaion (E b N db) Consellaion (E b N db) Quadraure Phase.5.5 Quadraure Phase VIII-7 VIII-8 Consellaion (E b N 4dB) Consellaion (E b N 6dB) Quadraure Phase.5.5 Quadraure Phase VIII-9 VIII-

6 Sym bol Error Probabiliy Consellaion (E b N db) Consellaion (E b N db) Quadraure Phase.5.5 Quadraure Phase VIII- VIII- Consellaion (E b N 5dB) Symbol Error Probabiliy.5 Perform ance of PSK odulaion Quadraure Phase = = 4 = 8 = 6 = E b /N (db) Figure 9: Symbol Error Probabiliy for PSK Signaling VIII- VIII-4

7 Bi Error Probabiliy A Bi Error Probabiliy -ary Pulse Ampliude odulaion (PA) Bi Error Rae for PSK s i Ai φ - where A i i Ai - - s s E i Ai φ s s -4 =,4 = 8 = 6 = A A A φ Eb/N (db) Figure 4: Bi Error Probabiliy for PSK Signaling (wih Gray coding) E E i i A i i A VIII-5 VIII-6 Error Probabiliy he error probabiliy (for 4-ary) is P e P e Pe Pe Q Q A N A N he average error probabiliy (for 4-ary PA) is In general he error probabiliy is P e s Q Q 6E 6E b log N N P e 4 P e 4 P e 4 P e 4 P e Q A N Q Ē 5N VIII-7 VIII-8

8 Symbol Error Probabiliy P e,s 4 5 = =4 =8 =6 = E b /N (db) P e,b =4 =8 = E b /N (db) Figure 4: Symbol Error Probabiliy for PA Signaling Figure 4: Bi Error Probabiliy for PA Signaling VIII-9 VIII- Quadraure Ampliude odulaion Consellaion φ For i s i s i Ai cosπ fc Bi sinπ fc Ai φ Bi φ φ VIII- VIII-

9 Decision Regions Error Probabiliies for QA φ Since his is wo PA sysems in quadraure. P e wih signals P e for PA φ he applicaion of QA is o bandwidh consrained channels. We can consider as a baseline a wo dimensional modulaion sysem ransmiing a 4 symbols per second. If each symbol represens 4 bis of informaion hen he daa rae is 96 bis per second. So we would like o have more signals per dimension in order o increase he daa rae. However, we mus ry o keep he signals as far apar from each oher as possible (in oder o keep he error rae low). So an increase of he size of he signal consellaion for fixed minimum disance would likely increase he oal signal energy ransmied. VIII- VIII-4 -ary signal ses Consider a -ary QA signal se shown below. he average energy is. he minimum disance is and he rae is 5 bis/dimension ary signal ses Consider a 64-ary QA signal se shown below. he average energy is VIII-5 VIII-6

10 odified QA (used in Paradyne 4.4kbi modem). his has average energy of VIII-7 he following hexagonal consellaion has energy 5.5 bu each inerior poin now has 6 neighbors compared o he four neighbors for he recangular srucures VIII-8 Figure 4: Consellaion used in v4 8.8kbps modems (96 poins) VIII-9 Capaciy E b N (db) R W BPSK QPSK 8PSK 6PSK PSK VIII-4

11 Capaciy Eb N (db) BPSK R W 8PSK QPSK PSK 6PSK VIII-4 VIII-4