A First Course in Digital Communications

Transcription

1 A Firs Course in Digial Communicaions Ha H. Nguyen and E. Shwedyk February 9 A Firs Course in Digial Communicaions /58

2 Block Diagram of Binary Communicaion Sysems m { b k } bk = s b = s k m ˆ { bˆ } k r w Bis in wo differen ime slos are saisically independen. a priori probabiliies: P[b k = ] = P, P[b k = ] = P. Signals s ( and s ( have a duraion of seconds and finie energies: E = s (d, E = s (d. Noise w( is saionary Gaussian, zero-mean whie noise wih wo-sided power specral densiy of N / (was/hz: E{w(} =, E{w(w( + τ} = N δ(τ. A Firs Course in Digial Communicaions /58

3 m!" { k } & % $' ( $!*+ '', b = s b k b = s k #$!!%.!*''! m ˆ { bˆ } k -+ % $' (. /, r w Received signal over [(k,k ]: r( = s i ( (k + w(, (k k. Objecive is o design a receiver (or demodulaor such ha he probabiliy of making an error is minimized. Shall reduce he problem from he observaion of a ime waveform o ha of observing a se of numbers (which are random variables. A Firs Course in Digial Communicaions 3/58

4 Geomeric Represenaion of Signals s ( and s ( (I Wish o represen wo arbirary signals s ( and s ( as linear combinaions of wo orhonormal basis funcions φ ( and φ (. Tb Tb φ ( and φ ( are orhonormal if: φ (φ (d = (orhogonaliy, φ (d = Tb The represenaions are φ (d = (normalized o have uni energy. where s ij = s ( = s φ ( + s φ (, s ( = s φ ( + s φ (. Tb s i (φ j (d, i,j {,}, A Firs Course in Digial Communicaions 4/58

5 Geomeric Represenaion of Signals s ( and s ( (II φ ( s s ( s ( = s φ ( + s φ (, s ( = s φ ( + s φ (, s s s ij = Tb s i (φ j (d, i,j {,}, s s φ ( Tb s i (φ j (d is he projecion of s i ( ono φ j (. How o choose orhonormal funcions φ ( and φ ( o represen s ( and s ( exacly? A Firs Course in Digial Communicaions 5/58

6 Gram-Schmid Procedure Le φ ( s ( E. Noe ha s = E and s =. Projec s ( = s ( E ono φ ( o obain he correlaion coefficien: ρ = Tb s ( E φ (d = E E Tb s (s (d. 3 Subrac ρφ ( from s ( o obain φ ( = s ( E ρφ (. 4 Finally, normalize φ ( o obain: φ ( = = Tb ρ φ ( [ = φ (] d φ ( ρ [ s ( ρs ] (. E E A Firs Course in Digial Communicaions 6/58

7 Gram-Schmid Procedure: Summary φ φ s E α E s s d s ρ = cos( α s ρφ φ φ ( = s ( E, φ ( = s = s = d = ρ Tb [ s ( ρs ] (, E E s (φ (d = ρ E, ( ρ E, Tb [s ( s (] d = E ρ E E + E. A Firs Course in Digial Communicaions 7/58

8 Gram-Schmid Procedure for M Waveforms {s i (} M i= φ ( = φ i ( = φ i( = ρ ij = s ( s (d, φ i ( [, φ i (] d s i( i ρ ij φ j (, Ei j= i =,3,...,N, s i ( Ei φ j (d, j =,,...,i. If he waveforms {s i (} M i= form a linearly independen se, hen N = M. Oherwise N < M. A Firs Course in Digial Communicaions 8/58

9 Example s ( s V Tb 3 V φ Tb s ( E 89: s ( E φ (a Signal se. (b Orhonormal funcion. (c Signal space represenaion. A Firs Course in Digial Communicaions 9/58

10 Example s ( s V V φ V T b ; ; Tb <=> φ (? A Firs Course in Digial Communicaions /58

11 Example 3 s ( s V V Tb C α C V φ(, α I G H E, α = F 3E D E ( E, s α = 4 E T α = b increasingα, ρ s ( ( E, ρ = E φ = Tb s (s (d [ V V α V ( α ] = α A Firs Course in Digial Communicaions /58

12 Example 4 s ( V s 3V b J T KLM Tb J φ φ 3 Tb N N OPQ 3 A Firs Course in Digial Communicaions /58

13 φ ρ = E Tb s (s (d = E s ( 3E, E s ( ( E, Tb / φ ( 3 V [ s ( ρ s ] ( = [s ( E E E φ ( = ( s = E, s = E. [ Tb d = [s ( s (] d ] V d = ( = 3 E. 3, 3 s ( ], A Firs Course in Digial Communicaions 3/58

14 Example 5 θ = 3π ρ = φ s ( = s ( = E cos(πf c, E cos(πf c + θ. θ = π ρ = E θ s ( φ locus of s as θ where f c = k, k an ineger. varies from o π. s θ = π ρ = A Firs Course in Digial Communicaions 4/58

15 Represenaion of Noise wih Walsh Funcions 5 x ( 5 5 x ( 5 φ ( φ ( φ 3 ( φ 4 ( Exac represenaion of noise wih 4 Walsh funcions is no possible. A Firs Course in Digial Communicaions 5/58

16 The Firs 6 Walsh Funcions Exac represenaions migh be possible wih many more Walsh funcions. A Firs Course in Digial Communicaions 6/58

17 The Firs 6 Sine and Cosine Funcions Can also use sine and cosine funcions (Fourier represenaion A Firs Course in Digial Communicaions 7/58

18 Represenaion of he Noise I To represen he random noise signal, w(, in he ime inerval [(k,k ], need o use a complee orhonormal se of known deerminisic funcions: w( = w i φ i (, where w i = i= Tb w(φ i (d. The coefficiens w i s are random variables and undersanding heir saisical properies is imperaive in developing he opimum receiver. A Firs Course in Digial Communicaions 8/58

19 Represenaion of he Noise II When w( is zero-mean and whie, hen: { } Tb E{w i } = E w(φ i (d = E{w(}φ i (d =. Tb E{w i w j } = E{ dλw(λφ i (λ } dτw(τφ j (τ = { N, i = j, i j. {w,w,...} are zero-mean and uncorrelaed random variables. If w( is no only zero-mean and whie, bu also Gaussian {w,w,...} are Gaussian and saisically independen!!! The above properies do no depend on how he se {φ i (, i =,,...} is chosen. Shall choose as he firs wo funcions he funcions φ ( and φ ( used o represen he wo signals s ( and s ( exacly. The remaining funcions, i.e., φ 3 (, φ 4 (,..., are simply chosen o complee he se. A Firs Course in Digial Communicaions 9/58

20 Opimum Receiver I Wihou any loss of generaliy, concenrae on he firs bi inerval. The received signal is r( = s i ( + w(, { s ( + w(, if a is ransmied = s ( + w(, if a is ransmied. = [s i φ ( + s i φ (] }{{} s i ( + [w φ ( + w φ ( + w 3 φ 3 ( + w 4 φ 4 ( + ] }{{} w( = (s i + w φ ( + (s i + w φ ( + w 3 φ 3 ( + w 4 φ 4 ( + = r φ ( + r φ ( + r 3 φ 3 ( + r 4 φ 4 ( + A Firs Course in Digial Communicaions /58

21 Opimum Receiver II where r j = Tb r(φ j (d, and r = s i + w r = s i + w r 3 = w 3 r 4 = w 4. Noe ha r j, for j = 3,4,5,..., does no depend on which signal (s ( or s ( was ransmied. The decision can now be based on he observaions r,r,r 3,r 4,... The crierion is o minimize he bi error probabiliy. A Firs Course in Digial Communicaions /58

22 Opimum Receiver III Consider only he firs n erms (n can be very very large, r = {r,r,...,r n } Need o pariion he n-dimensional observaion space ino decision regions. Decide a "" was ransmied if r R falls in his region. R R Observaion space R Decide a "" was ransmied if r R falls in his region. R A Firs Course in Digial Communicaions /58

23 Opimum Receiver IV P[error] = P[( decided and ransmied or ( decided and ransmied]. = P[ D, T ] + P[ D, T ] = P[ D T ]P[ T ] + P[ D T ]P[ T ] = P f( r T d r + P f( r T d r R R = P f( r T d r + P f( r T d r R R R = P f( r T d r + [P f( r T P f( r T ]d r R R = P + [P f( r T P f( r T ] d r. R A Firs Course in Digial Communicaions 3/58

24 Opimum Receiver V The minimum error probabiliy decision rule is { P f( r T P f( r T decide ( D P f( r T P f( r T < decide ( D. Equivalenly, f( r T f( r T D P P. ( D The expression f( r T is called he likelihood raio. f( r T The decision rule in ( was derived wihou specifying any saisical properies of he noise process w(. A Firs Course in Digial Communicaions 4/58

25 Opimum Receiver VI Simplified decision rule when he noise w( is zero-mean, whie and Gaussian: D ( (r s + (r s (r s + (r s + N ln P. P D For he special case of P = P (signals are equally likely: (r s + (r s D D (r s + (r s. minimum-disance receiver! A Firs Course in Digial Communicaions 5/58

26 Minimum-Disance Receiver (r s + (r s D D (r s + (r s. r φ s r ( r, r Choose ( s, s s ( ( s, s s ( Choose s d d φ r A Firs Course in Digial Communicaions 6/58

27 Correlaion Receiver Implemenaion r = s + w i φ d TS b d ( ( = = r r Compue ( r s + ( r s i i N ln( P for i =, and choose he smalles i Decision φ ( r φ ( TV b d TU b d ( ( = = r r Form he do produc a a r s i N E ln( P WXYYZ[ \X[ ]^_`[Z\ Decision φ ( N E ln( P A Firs Course in Digial Communicaions 7/58

28 Receiver Implemenaion using Mached Filers r = Tb h = φ ( T ( b = r bcd efeg h iej dk el Decision h = φ ( Tb r A Firs Course in Digial Communicaions 8/58

29 Example 5.6 I s ( s m.5 m nop φ φ ( q.5 q rs A Firs Course in Digial Communicaions 9/58

30 Example 5.6 II s φ.5 s (.5.5 φ s ( = φ ( + φ (, s ( = φ ( + φ (. A Firs Course in Digial Communicaions 3/58

31 Example 5.6 III s φ r s φ r uvw.5 s xyz.5 s ( Choose s Choose s ( φ Choose s Choose s ( φ.5.5 r.5.5 r φ r { } s Choose s.5 s Choose s ( φ.5.5 r (a P = P =.5, (b P =.5, P =.75. (c P =.75, P =.5. A Firs Course in Digial Communicaions 3/58

32 Example 5.7 I s ( = φ ( + φ (, s ( = φ ( φ (. φ φ ( 3 ~ ~ A Firs Course in Digial Communicaions 3/58

33 Example 5.7 II φ r s N P lnƒ 4 P Choose s ( φ Choose s ( r s ( A Firs Course in Digial Communicaions 33/58

34 Example 5.7 III = TŒ b r( r d φ ( 3 ( ˆ Š Š N = P T ln Ž 4 P r T choose s r < T choose s Tb r( h ( 3 = r š œ œ r T choose s Tb N ž = P T ln Ÿ 4 P r < T choose s A Firs Course in Digial Communicaions 34/58

35 Implemenaion wih One Correlaor/Mached Filer Always possible by a judicious choice of he orhonormal basis. ( φˆ φ s ( φˆ s ˆ = s ˆ s ( ŝ θ φ ŝ ª«[ ˆφ ( ˆφ ( ] [ = cos θ sin θ sin θ cos θ ][ φ ( φ ( ]. A Firs Course in Digial Communicaions 35/58

36 ( φˆ φ s ( φˆ s ˆ = s ˆ s ( ŝ θ φ ŝ f(ˆr, ˆr, ˆr 3,..., T f(ˆr, ˆr, ˆr 3,..., T = f(ŝ + ŵ f(ŝ + ŵ f(ŵ 3... f(ŝ + ŵ f(ŝ + ŵ f(ŵ 3... ˆr D D ŝ + ŝ ( N / + ln( P ŝ ŝ P T. D P P D A Firs Course in Digial Communicaions 36/58

37 = T r( b ˆr d ( ±²³ ³µ rˆ T ¹ rˆ < T ¹ D D ( φˆ ³ Threshold T r( h = ˆ φ( T b = ˆr º»¼½¾ ¾À» rˆ T Ä rˆ < T Ä D D ÁÂà Threshold T s ( s ( ˆφ ( = (E ρ E E + E (, T ŝ + ŝ N / + ln ŝ ŝ ( P P. A Firs Course in Digial Communicaions 37/58

38 Example 5.8 I φ ( ( φˆ s ( E s ˆ = s ˆ ( φˆ θ = π / 4 E s ( φ ( ˆφ ( = [φ ( + φ (], ˆφ ( = [ φ ( + φ (]. A Firs Course in Digial Communicaions 38/58

39 Example 5.8 II = TÌ b rˆ T Ð D r( ˆr ( d ÅÆÇÈÉÊÉËÆÊ rˆ < T Ð D ( φˆ Threshold T Tb Ï ÍÉÎ h( r( = rˆ T Ü D ˆr ÑÒÓÔÕÖÕ ÒÖ Û rˆ < T Ü D ØÙÚ Threshold T A Firs Course in Digial Communicaions 39/58

40 Receiver Performance To deec b k, compare ˆr = ktb ( T = ŝ+ŝ N + (ŝ ŝ ln P P. (k r(ˆφ (d o he hreshold Þßàáâáãä åãæäçèéê ( rˆ f ( r f T ˆ T ŝ ŝ choose Ý choose T T T P[error] = P[( ransmied and decided or ( ransmied and decided] = P[( T, D or ( T, D ]. ˆr A Firs Course in Digial Communicaions 4/58

41 ðñòòñóôõ ðñö øñùúóûöü ýñóþü øñùúóûöü ýñóþü ÿôú öúóõ ú ñöúóõ f r ( rˆ ( f T ˆ T ýñóþü ú ñöúóõ ÿôú öúóõ ýñóþü øñùúóûöü ñõ ðñö øñùúóûöü ðñòòñóôõ úþúòü ÿôú ñõ öûú óú õõ úþúòü ÿôú öûú óú õõ ŝ ŝ ëìíî ï ëìíî choose choose T T T ˆr P[error] = P[ T, D ] + P[ T, D ] = P[ D T ]P[ T ] + P[ D T ]P[ T ] T = P f(ˆr T dˆr +P f(ˆr T dˆr T }{{}}{{} Area B Area A ( [ ( ] T ŝ T ŝ = P Q + P Q. N / N / A Firs Course in Digial Communicaions 4/58

42 Q-funcion " #" $ λ e π % # $ % &'! x λ Area = Q( x Q(x exp ( λ dλ. π x 4 Q(x x A Firs Course in Digial Communicaions 4/58

43 Performance when P = P ( ( ŝ ŝ disance beween he signals P[error] = Q = Q. N / noise RMS value Probabiliy of error decreases as eiher he wo signals become more dissimilar (increasing he disances beween hem or he noise power becomes less. To maximize he disance beween he wo signals one chooses hem so ha hey are placed 8 from each oher s ( = s (, i.e., anipodal signaling. The error probabiliy does no depend on he signal shapes bu only on he disance beween hem. A Firs Course in Digial Communicaions 43/58

44 Relaionship Beween Q(x and erfc(x. The complemenary error funcion is defined as: erfc(x = exp( λ dλ π x = erf(x. erfc-funcion and he Q-funcion are relaed by: Q(x = erfc ( x erfc(x = Q( x. Le Q (x and erfc (x be he inverses of Q(x and erfc(x, respecively. Then Q (x = erfc (x. A Firs Course in Digial Communicaions 44/58

45 Example 5.9 I φ s T φ T s A Firs Course in Digial Communicaions 45/58

46 Example 5.9 II (a Deermine and skech he wo signals s ( and s (. (b The wo signals s ( and s ( are used for he ransmission of equally likely bis and, respecively, over an addiive whie Gaussian noise (AWGN channel. Clearly draw he decision boundary and he decision regions of he opimum receiver. Wrie he expression for he opimum decision rule. (c Find and skech he wo orhonormal basis funcions ˆφ ( and ˆφ ( such ha he opimum receiver can be implemened using only he projecion ˆr of he received signal r( ono he basis funcion ˆφ (. Draw he block diagram of such a receiver ha uses a mached filer. A Firs Course in Digial Communicaions 46/58

47 Example 5.9 III (d Consider now he following argumen pu forh by your classmae. She reasons ha since he componen of he signals along ˆφ ( is no useful a he receiver in deermining which bi was ransmied, one should no even ransmi his componen of he signal. Thus she modifies he ransmied signal as follows: ( s (M ( = s ( s (M ( = s ( componen of s ( along ˆφ ( ( componen of s ( along ˆφ ( Clearly idenify he locaions of s (M ( and s (M ( in he signal space diagram. Wha is he average energy of his signal se? Compare i o he average energy of he original se. Commen. A Firs Course in Digial Communicaions 47/58

48 Example 5.9 IV s s 3 3 A Firs Course in Digial Communicaions 48/58

49 Example 5.9 V (*+,+-. /-.34 D φ M s ( ˆ φ D M s ( π θ = 4 s T φ T s ˆ φ A Firs Course in Digial Communicaions 49/58

50 Example 5.9 VI [ ˆφ ( ˆφ ( ] = = [ cos( π/4 sin( π/4 sin( π/4 cos( π/4 [ ] [ ] φ (. φ ( ][ φ ( φ ( ] ˆφ ( = [φ ( φ (], ˆφ ( = [φ ( + φ (]. A Firs Course in Digial Communicaions 5/58

51 Example 5.9 VII ˆ ( φ φ ˆ ( / / r( h = ˆ φ ( / = rˆ 5 rˆ < 5 D D A Firs Course in Digial Communicaions 5/58

52 PSD of Digial Ampliude Modulaion I p Informaion bis b k {,} 67889:; <= >?7@ :ABC?>D c k Ampliude modulaion s c k is drawn from a finie se of real numbers wih a probabiliy ha is known. Examples: c k {,+} (anipodal signaling, {,} (on-off keying, {,, +} (pseudoernary line coding or {±, ±3,, ±(M } (M-ary ampliude-shif keying. p( is a pulse waveform of duraion. A Firs Course in Digial Communicaions 5/58

53 PSD of Digial Ampliude Modulaion II The ransmied signal is s( = k= c k p( k. To find PSD, runcae he random process o a ime inerval of T = N o T = N : s T ( = N k= N c k p( k. Take he Fourier ransform of he runcaed process: S T (f = k= c k F{p( k } = P(f k= c k e jπfk. A Firs Course in Digial Communicaions 53/58

54 PSD of Digial Ampliude Modulaion III Apply he basic definiion of PSD: E { S T (f } S(f = lim T T P(f = lim E N (N + = P(f m= N k= N R c (me jπmf. c k e jπfk. where R c (m = E {c k c k m } is he (discree auocorrelaion of {c k }, wih R c (m = R c ( m. A Firs Course in Digial Communicaions 54/58

55 PSD of Digial Ampliude Modulaion IV The oupu PSD is he inpu PSD muliplied by P(f, a ransfer funcion. E S f R m ( c k k in ( = F j π mftb ( e T c b m= LTI Sysem h = p P( f Transmied signal s S( f = S ( f P( f in S(f = P(f m= R c (me jπmf. A Firs Course in Digial Communicaions 55/58

56 PSD Derivaion of Arbirary Binary Modulaion I Applicable o any binary modulaion wih arbirary a priori probabiliies, bu resriced o saisically independen bis. T b st s s ( 3 Tb T b 3T b 4T b s T ( = k= g k (, g k ( = { s ( k, wih probabiliy P s ( k, wih probabiliy P. A Firs Course in Digial Communicaions 56/58

57 PSD Derivaion of Arbirary Binary Modulaion II Decompose s T ( ino a sum of a DC and an AC componen: s T ( = E{s T (} +s T ( E{s T (} = v( + q( }{{}}{{} DC AC v( = E{s T (} = [P s ( k + P s ( k ] S v (f = n= D n δ k= (f ntb, D n = [ P S ( n + P S ( n ], where S (f and S (f are he FTs of s ( and s (. S v (f = n= ( P S n + P S ( n δ (f ntb. A Firs Course in Digial Communicaions 57/58

58 PSD Derivaion of Arbirary Binary Modulaion III To calculae S q (f, apply he basic definiion of PSD: E{ G T (f } S q (f = lim = = P P S (f S (f. T T Finally, S st (f = P P S (f S (f + ( ( P S n + P S n n= δ (f ntb. A Firs Course in Digial Communicaions 58/58