EE456 Digital Communications

Transcription

1 EE456 Digial Communicaions Professor Ha Nguyen Sepember 6 EE456 Digial Communicaions

2 Inroducion o Basic Digial Passband Modulaion Baseband ransmission is conduced a low frequencies. Passband ransmission happens in a frequency band oward he high end of he specrum. Saellie communicaion is in he 6 8 GHz band, while mobile phones sysems are in he 8 MHz. GHz band. Bis are encoded as a variaion of he ampliude, phase or frequency, or some combinaion of hese parameers of a sinusoidal carrier. The carrier frequency is much higher han he highes frequency of he modulaing signals (or messages. Shall consider binary ampliude-shif keying (BASK, binary phase-shif keying (BPSK and binary frequency-shif keying (BFSK: Error performance, opimum receivers, specra. Exensions o quadraure phase-shif keying (QPSK, offse QPSK (OQPSK and minimum shif keying (MSK. EE456 Digial Communicaions

3 Examples of Binary Passband Modulaed Signals (a Binary daa (b Modulaing signal m Tb Tb 3Tb 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb V (c BASK signal V V (d BPSK signal V V (e BFSK signal V EE456 Digial Communicaions 3

4 Binary Ampliude-Shif Keying (BASK { s( =, T s ( = V cos(πf c, T, < T b, f c = n/t b s ( s ( E BASK φ r T b ( d = T b r Comparaor r T h r < T h D D φ ( (b Choose Choose s ( s ( T T T E BASK N P ln = E + P h BASK T b ( φ r = E r d BASK ( ( E BASK E b P[error] BASK = Q = Q, N N where E b = E BASK = E BASK is he energy per bi. EE456 Digial Communicaions 4

5 PSD of BASK S BASK (f = V 6 [ ] δ(f f c+δ(f +f c+ sin [πt b (f +f c] π T b (f +f c + sin [πt b (f f c] π T b (f f c. S ( f BASK V 6 fc T b f c fc + T b f Approximaely 95% of he oal ransmied power lies in a band of 3/T b (Hz, cenered a f c. The carrier componen could be helpful for frequency and phase synchronizaion a he receiver. EE456 Digial Communicaions 5

6 Binary Phase-Shif Keying (BPSK { s ( = V cos(πf c, if T s ( = +V cos(πf c, if T, < T b, s ( s ( EBPSK EBPSK φ = cos( π fc T b P[error] BPSK = Q ( E BPSK N = Q ( E b N where E b =.5 E BPSK +.5 E BPSK = E BPSK is he energy per bi. S BPSK (f = V 4 [ sin ] [π(f f ct b ] π (f f c + sin [π(f +f ct b ] T b π (f +f c. T b Similar o ha of BASK, bu no impulse funcions a ±f c., EE456 Digial Communicaions 6

7 Binary Frequency-Shif Keying (BFSK s( { s( = V cos(πf + θ, if T s ( = V cos(πf + θ, if T, < T b. (i Minimum frequency separaion for coheren orhogonaliy (θ = θ : ( f [coheren] min = T b. (ii Minimum frequency separaion for noncoheren orhogonaliy (θ θ : ( f [noncoheren] min = T b. EE456 Digial Communicaions 7

8 φ ( = s ( EBFSK, φ ( = s ( EBFSK. [ φ φ ( ] ( Choose T r s φ( (, E BFSK ( E, BFSK s ( Decision boundary when P = P Choose r φ( T P[error] BFSK = Q ( E BFSK N ( E b = Q, N where E b =.5 E BFSK +.5 E BFSK = E BFSK is he energy per bi. EE456 Digial Communicaions 8

9 PSD of BFSK S BFSK (f = + V 6 V 6 [ δ(f f + δ(f + f + sin [πt b (f + f ] π T b (f + f + sin [πtb (f f ] ] π T b (f f [ δ(f f + δ(f + f + sin [πt b (f + f ] π T b (f + f + sin [πtb (f f ] π T b (f f ]. Bandwidh W = ( f f + 3/ Tb V 6 V 6.5 f f f + f f Tb Tb Tb f.5 + T b EE456 Digial Communicaions 9

10 Performance Comparison of BASK, BPSK and BFSK BASK and BFSK P[error] 3 4 BPSK E /N (db ( b ( E b E b P[error] BPSK = Q, P[error] BASK = P[error] BFSK = Q. N Quesion: You have designed a BPSK sysem ha achieves P[error] = 6. Wha needs o be changed if you wan o double he ransmission bi rae and sill mee P[error] = 6? N EE456 Digial Communicaions

11 Comparison Summary of BASK, BPSK and BFSK The BER performance curves of BASK, BPSK and BFSK shown in he previous slide can only be realized wih coheren receivers, i.e., when perfec carrier frequency and phase synchronizaion can be esablished a he receivers. A receiver ha does no require phase synchronizaion is called a non-coheren receiver, which is simpler (hence less expensive han a coheren receiver. Using he peaks of he firs side lobes on boh sides of he carrier frequency (or frequencies in BFSK, he ransmission bandwidhs for BASK, BPSK and BFSK are approximaed as: 3 W BASK = W BPSK = 3r b T b W BFSK f f + 3 { 3.5rb, coherenly orhogonal = T b 4r b, non-coherenly orhogonal Taking ino accoun boh bandwidh and power, BPSK is he bes scheme if a coheren receiver can be afforded! This is followed by BASK and BFSK. The quesion is when would one use BASK or BFSK? The answers are as follows: If he carrier frequency can be synchronized, bu no he phase, hen BPSK does no work! On he conrary one can sill use BASK bu he receiver has o be redesigned o be a non-coheren receiver. See Assignmen 7 (Problem 7.7, and he nex few slides. Similarly, BFSK can be deeced non-coherenly wihou phase synchronizaion. EE456 Digial Communicaions

12 Non-Coheren Deecion of BASK { sr (+w( = +w( : T r( = s R (+w( = E cos(πf c+θ+w( : T T b Wihou he noise, he receiver sees one of he wo signals { s R ( = : T s R ( = E T b cos(πfc+θ : T Wrie s R ( as follows: s R ( = ( Ecosθ cos(πf c T b } {{ } φ ( ( + Esinθ sin(πf c T b } {{ } φ ( Thus, for an arbirary θ wo basis funcions are required o represen s R ( and sr (. The signal s R ( always lies a he origin, while sr ( could be anywhere on he circle of radius E, depending on θ.. EE456 Digial Communicaions

13 φ φ locus of sr ( s R s R E s θ E φ decision boundary assuming θ = s D θ decision boundary ino accoun φ aking phase uncerainy (a (b (a If he receiver assumes θ =, he opimum decision boundary is a line perpendicular o φ ( and a disance E/ away from s R (. The error is: ( ( E E P[error] =.5Q +.5Q [cosθ.5] N N (b Wihou phase synchronizaion, he decision boundary is a circle cenered a s R ( and wih some diameer D. I can be shown ha he opimum value of D is: D = ( E + 4N E E (for high SNR The error probabiliy is well approximaed as P[error] e E 4N. EE456 Digial Communicaions 3

14 = T b r φ T b ( d T b ( d = T b r r Compue r + r and compare o D Decision φ ( Noncoheren Pr[error] 3 4 θ= θ=π/6 θ=π/3 θ=π/ 5 5 E /N (db b EE456 Digial Communicaions 4

15 Quadraure Phase Shif Keying (QPSK Basic idea behind QPSK: cos(πf c and sin(πf c are orhogonal over [,T b ] when f c = k/t b, k ineger Can ransmi wo differen bis over he same frequency band a he same ime. The symbol signaling rae (i.e., he baud rae is r s = /T s = /(T b = r b / (symbols/sec, i.e., halved. Thus, he required bandwidh is halved compared o BPSK. An alernaive view is ha, for he same bandwidh as in BPSK, he bi rae can be doubled. Bi Paern Message Signal Transmied m s ( = V cos(πf c, T s = T b m s ( = V sin(πf c, T s = T b m 3 s 3( = V cos(πf c, T s = T b m 4 s 4( = V sin(πf c, T s = T b V m = m 3 = m = m 4 = V T b 4Tb 6Tb 8Tb EE456 Digial Communicaions 5

16 Comparison of BPSK and QPSK Waveforms BPSK /Tb.5 QPSK Approach : Same rb wih BPSK, same power o achieve he same Eb, hence he same P[bi error], half he bandwidh since he baud rae is halved (Ts = Tb /Tb.5 QPSK Approach : Twice bi rae compared o BPSK, wice he power o achieve he same Eb, hence he same P[bi error], same bandwidh since he baud rae is unchanged (Ts = Tb /Tb EE456 Digial Communicaions 6

17 Signal Space Represenaion of QPSK Ts s i (d = V Ts = V T b = E s, φ ( = s( Es, φ ( = s( Es. φ s s ( 3 E s s ( φ s 4 EE456 Digial Communicaions 7

18 Opimum Receiver for QPSK m-dimensional observaion space r = ( r, r,, r m R Choose s or m R Choose s or m R Choose s or m R Choose s or m P[correc] = + P f( r s (d r + P f( r s (d r R R P 3f( r s 3(d r + P 4f( r s 4(d r. R 3 R 4 Choose s i( if P if( r s i( > P jf( r s j(, j =,,3,4; j i. EE456 Digial Communicaions 8

19 Minimum-Disance Receiver When P i =.5, i =,...,4, he decision rule is he minimum-disance rule: Choose s i ( if (r s i +(r s i is he smalles! "#$%& φ, r s -./ **+, ( s ( 3 π / 4 s ( φ, r s 4! "'$&% EE456 Digial Communicaions 9

20 Simplified Decision Rule and Receiver Implemenaion Choose s i ( if N lnp i +r s i +r s i > N lnp j +r s j +r s j j =,,3,4; j i. = T = T s b r φ ( T s ( : d T s ( : d r = T = T s b r Compue N r s j + r s j + ln( Pj for j =,,3, 4 and choose he larges Decision φ EE456 Digial Communicaions

21 Symbol (Message Error Probabiliy of QPSK r ˆr s / ˆr LMN F G HIJK s ( 3 E s π / 4 / s ( r T U VW O PQRS E s s 4 P[error] = 4 P[error s i(]p[s i(] = 4 P[error s i(] = P[error s i(] = P[correc s i(] 4 i= i= P[correc s (] = P[(r,r shaded quadran s (] = P[(ˆr,ˆr shaded quadran s (] = P[(ˆr AND (ˆr s (] = P[ˆr s (] P[ˆr s (] [ ] [ ] [ ] = Q ( E s/n Q ( E s/n = Q ( E s/n [ ( ] P[error] = Q E s/n = Q ( E s/n Q ( E s/n Q ( E s/n. EE456 Digial Communicaions

22 Illusraion of Symbol Error Probabiliy Analysis for QPSK Join pdf f(ˆr, ˆr Observaion ˆr 4 4 Observaion ˆr 4 Observaion ˆr Observaion ˆr EE456 Digial Communicaions

23 Bi Error Probabiliy of QPSK XYZZ[\ ]^_`a r ˆr s / ˆr ijk c de fgh s ( 3 s 4 XYZZ[\ ]b_a` E s π / 4 / E s s ( r q r s l mnop P[m m ] = Q Es Q Es, N N P[m 3 m ] = Q Es, N P[m 4 m ] = Q Es Q Es. N N P[bi error] =.5P[m m ] +.5P[m 4 m ] +.P[m 3 m ] ( ( E s E b = Q = Q, because E s = E b. N N Gray mapping: Neares neighbors are mapped o he bi pairs ha differ in only one bi. EE456 Digial Communicaions 3

24 An Alernaive Represenaion of QPSK V cos( πf c a I ( a uvwxyz{ yv}v~ s( a Q ( Bi sequence a V sin( πf c a a a a3 a4 a5 a6 a7 a8 Tb Tb 3T b 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb a a a a 4 6 a8 a I ( Tb Tb 3T b 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb a a3 a5 a7 a Q ( Tb Tb 3T b 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb EE456 Digial Communicaions 4

25 V a I (Vcos(πf c V Tb 4Tb 6Tb 8Tb Tb a Q (Vsin(πf c V V Tb 4Tb 6Tb 8Tb Tb s QPSK (=a I (Vcos(πf c +a Q (Vsin(πf c V V Tb 4Tb 6Tb 8Tb Tb s( = a I(V cos(πf c + a Q(V sin(πf c ( aq( = a (πf I ( + aq (V cos c an = V cos[πf c θ(]. a I( π/4, if a I = +,a Q = + (bis are π/4, if a θ( = I = +,a Q = (bis are. 3π/4, if a I =,a Q = + (bis are 3π/4, if a I =,a Q = (bis are EE456 Digial Communicaions 5

26 Signal Space Represenaion of QPSK φ ( = V cos(πfc = cos(πf V T T b s c φ ( = V sin(πfc = sin(πf V T T b s c φ = sin( π fc T s, < < T s = T b, a =, a = θ = 3 π / 4 I Q a =, a = θ = π / 4 I Q V T s π / 4 φ = cos( π fc T s a =, a = θ = 3 π / 4 I Q a =, a = θ = π / 4 I Q EE456 Digial Communicaions 6

27 Receiver Implemenaion of QPSK φ = sin( π fc T s a =, a = θ = 3 π / 4 I Q a =, a = ƒ θ = π / 4 I Q V T s π / 4 φ = cos( π fc T s a =, a = θ = 3 π / 4 I Q a =, a = θ = π / 4 I Q = T s T s ( d r Threshold = ( a I r φ T s ( d = T s r Threshold = Muliplexer a Q ( a φ ( V P[bi error] = Q T s E b = = Q. N N EE456 Digial Communicaions 7