Communications II Lecture 7: Performance of digital modulation

Transcription

1 Communicaions II Lecure 7: Performance of digial modulaion Professor Kin K. Leung EEE and Compuing Deparmens Imperial College London Copyrigh reserved

2 Ouline Digial modulaion and demodulaion Error probabiliy of ASK Error probabiliy of FSK Error probabiliy of PSK Noncoheren demodulaion Reference: Lahi, Chap. 13.

3 Bandpass Daa Transmission How do we ransmi symbols 1 and 0? Ampliude Shif Keying ASK Frequency Shif Keying FSK Phase Shif Keying PSK 3

4 ASK, FSK and PSK 4

5 How do we recover he ransmied symbol? Coheren synchronous deecion Use a BPF o rejec ou-of-band noise Muliply he incoming waveform wih a cosine of he carrier frequency Use a LPF Requires carrier regeneraion boh frequency and phase synchronizaion by using a phase-lock loop Noncoheren deecion envelope deecion ec. Makes no eplici effors o esimae he phase 5

6 Ampliude shif keying ASK = on-off keying OOK Or s 0 = 0 s 1 = A cos f c s = A cos f c, A {0, A} 6

7 Coheren Deecion Assume an ideal band-pass filer wih uni gain on [f c W, f c +W ]. For a pracical band-pass filer, W should be inerpreed as he equivalen bandwidh. 7

8 8 Pre-deecion signal: sin ]cos [ sin cos cos f n f n A f n f n f A n s C S C C C S C C C

9 9 Afer muliplicaion wih cos f c : Afer low-pass filering: sin4 cos4 ]1 [ cos sin ]cos [ f n f n A f f n f n A y C S C C C C S C C ~ n A y C 166

10 Reminder: The in-phase noise componen n c has he same variance as he original band-pass noise n The received signal 166 is idenical o he received signal 147 for baseband digial ransmission The sample values of ~ y will have PDFs ha are idenical o hose of he baseband case For ASK he saisics of he receiver signal are idenical o hose of a baseband sysem 10

11 The probabiliy of error for ASK is he same as for he baseband case Assume equiprobable ransmission of 0s and 1s. Then he decision hreshold mus be A/ and he probabiliy of error is given by: P e, ASK A Q 11

12 Phase shif keying PSK s = A cos f c, A { A, A} Use coheren deecion again, o evenually ge he deecion signal: ~ y A n C 1

13 Probabiliy densiy funcions for PSK for equiprobable 0s and 1s in noise: a: symbol 0 ransmied b: symbol 1 ransmied 13

14 Use hreshold 0 for deecion Condiional error probabiliies: n A P e 1 0 ep dn 0 P e 1 n A 0 ep 1 dn 14

15 In he firs se n~ In he second se n A dn dn~ and when n 0, n~ n~ 1 Pe ep dn~ 0 A n~ n n 0, n~ A A n A dn dn~ and when A, and when n, n~ : P e1 1 1 A A n~ ep 1 dn~ n~ ep dn~ 15

16 So: n Pe Pe Pe PSK 1 0 1, ep dn A Change variable of inegraion o z n/σ dn = σdz and when n = A, z = A/σ. Then: 1 z / A P A e dz Q 17 e, PSK Remember ha Q 1 ep / d 16

17 Frequency Shif Keying FSK s 0 = s 1 = A cos f 0, if symbol 0 is ransmied A cos f 1, if symbol 1 is ransmied Symbol recovery: Use wo ses of coheren deecors, one operaing a a frequency f 0 and he oher a f 1. 17

18 Coheren deecor for FSK The wo BPF s are non-overlapping in frequency specrum 18

19 Each branch = an ASK deecor LPF oupu on each branch A noise noise if if symbol presen symbol no presen 19

20 n 0 : he noise oupu of he op branch n 1 : he noise oupu of he boom branch Each of hese noise erms has idenical saisics o n. Oupu if a symbol 1were ransmied y 1 = A + [n 1 n 0 ] Oupu if a symbol 0 were ransmied y0 = -A + [n 1 n 0 ] 0

21 Se deecion hreshold o 0 Difference from PSK: he noise erm is now n 1 n 0. The noises in he wo channels are independen because heir specra are non-overlapping. he variances add he noise variance has doubled! 1

22 Probabiliy of error for FSK: Replace σ in 17 by σ or σ by σ P e, FSK Q A

23 Noise of he sum or difference of wo independen zero mean random variables: 1 : a random variable wih variance σ 1 : a random variables wih variance σ Wha is he variance of y 1 ±? 3

24 4 By definiion, he variance of y 1 ± is For independen variables: E{ 1 } = E{ 1 }E{ } For zero-mean random variables: E{ 1 }=E{ }= 0 E{ 1 }= 0 So } { } { } { } { } { } { } { E E E E E y E y E y 1 1 } { } { E E y

25 Comparison of he hree schemes 5

26 To achieve he same error probabiliy fied P e : PSK can be reduced by 6 db compared wih a baseband or ASK sysem a facor of reducion in ampliude FSK can be reduced by 3 db compared wih a baseband or ASK a facor of reducion in ampliude Cauion: The comparison is based on peak SNR. In erms of average SNR, PSK only has a 3 db improvemen over ASK, and FSK has he same performance as ASK 6

27 7 Properies of he Q-funcion Upper bounds and good approimaions: Becomes igher for large. A beer upper bound for small. 0, 1 / e Q 0, 1 / e Q

28 8

29 Non-coheren deecion Accurae phase synchronizaion may be difficul in a dynamic channel. Phase synchronizaion error is due o frequency drif, insabiliy of he local oscillaor, effecs of srong noise... When he carrier phase is unknown, one mus rely on non-coheren deecion. The phase θ is assumed o be uniformly disribued on [0, ]. Circuiry is simpler, bu analysis is more difficul! 9

30 ASK 30

31 Error analysis When symbol 0 is sen, he envelope noise alone has Rayleigh disribuion r r / f r e, r When symbol 1 is sen, he envelope signal + noise has Rician disribuion f r r A / Ar r e I0, r dominaes he error probabiliy when A/σ >> 1. 31

32 3 Le he hreshold be A/ for simpliciy. The error probabiliy dominaed by symbol 0 is given by Cf. coheren demodulaion /8 / / 1 0, 1 A A r e e r dr e r P /8, 1 A ASK e e A Q P

33 FSK 33

34 Error probabiliy When a symbol 1 is sen, one branch has Rayleigh disribuion, he oher has Rice disribuion. Error occurs if Rice < Rayleigh, and i can be shown ha P e e 1 A /4 Cf. coheren demodulaion P e, FSK Q A 1 e A /4 34

35 DPSK: Differenial PSK I is impossible o demodulae PSK wih an envelop deecor, since PSK signals have he same frequency and ampliude. We can demodulae PSK differenially, where phase reference is provided by a delayed version of he signal in he previous inerval. Differenial encoding is essenial: b n = b n 1 a n, where a n, b n 1. 35

36 Differenial deecion Error probabiliy Cf. coheren demodulaion P e, DPSK e 1 A / P e, PSK A Q 1 A / e 36

37 Conclusions Non-coheren demodulaion reains he hierarchy of performance. Non-coheren demodulaion has error performance slighly worse han coheren demodulaion, bu approaches coheren performance a high SNR. Non-coheren demodulaors are considerably easier o build. 37