AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 1 April 14, 2014 EEE3086F Signals and Systems II 2014 A.J. Wilkinson
[email protected] http:www.ee.uct.ac.za !epart"ent o# $lectrical $n%ineerin% Uni&ersit' o# Cape Town AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 2 April 14, 2014 5.4 Single Sideband Modulation (SSB (.).* SSB concepts (.).+ SSB %eneration &ia side,and #ilterin% (.).- SSB %eneration usin% ./hase Shi#t Method0 (.).) SSB %eneration usin% Wea&er1s "ethod (.).( !e"odulation o# SSB (.).2 SSB34C 5with carrier6 Contents AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 3 April 14, 2014 5.4.1 SSB !on"e#ts A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 4 April 14, 2014 Sin%le Side,and Modulation 5SSB6 ♦DSB-SC/LC requires an RF bandwidth of twice the audio bandwidth. ♦In DSB-SC/LC, there are two ‘sidebands’ on either side of the carrier. ♦Reca !" #B f ( t )cos ω c t ↔ $ # F( ω+ω c )+ $ # F( ω−ω c ) N P N P N = neg components P = pos components DSB-SC c ω c ω − B !" USB LSB N P % &ω F A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 5 April 14, 2014 Sin%le Side,and Modulation 5SSB6 ♦For an' R()L-*aued si+na there e,ists -con.u+ate s'//etr'0 in the Fourier 1ransfor/, i.e. ♦1hus )LL infor/ation is contained in either the 2ositi*e or the ne+ati*e frequenc' co/2onents. ♦3e therefore need on' trans/it a sin+e sideband. % &t f F−ω=F 4 ω sideband Upper c ω or sideband Lower c ω A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 6 April 14, 2014 Spectru" o# !SB3SC si%nal sideband Lower sideband Upper N % &ω F m ω − SC DSB − sideband Lower sideband Upper m ω c ω − c ω P ω ω N P N P ω m =#π B A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 7 April 14, 2014 Spectru" o# SSB si%nal 5upper side,and6 n!" Sideband Upper c ω − c ω ω N P USB % &ω + Φ SSB ω m ω − m ω N P #econstr$cted signa! %he SSB signa! can be demod$!ated b" trans!ation of the spectra! components to the origin& A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 8 April 14, 2014 Spectru" o# SSB si%nal 5lower side,and6 5ote6 1he time domain 7SB and LSB si+nas are rea-*aued since con.u+ate s'//etr' in frequenc' do/ain hods, i.e. N n!" Sideband Lower c ω − ω c ω P ω m ω − m ω LSB ' SSB (−ω)=' SSB 4 ( ω) ⇒ ϕ SSB (t )∈Re N P ' SSB− ω #econstr$cted signa! A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 9 April 14, 2014 SSB Applications ♦SSB sa*es bandwidth. SSB uses haf the bandwidth of DSB-LC )8. 1his aows /ore channes to fit into a radio band. ♦SSB is used for radio broadcasts in the shortwa(e bands &9-9: 8!"% ♦SSB is used for6 Lon+-ran+e co//unications b' shi2s and aircraft. ;oice trans/issions b' a/ateur radio o2erators ♦LSB SSB is +enera' used beow < 8!" and 7SB SSB abo*e < 8!". AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 10 April 14, 2014 5.4.2 SSB gene$ation %ia sideband &ilte$ing A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 11 April 14, 2014 SSB 7eneration 8ia 9ilterin% 5.#ilterin% "ethod06 ♦=enerate DSB-SC Si+na ♦)22' B>F to e,tract desired sideband. : ω % &t DSB φ × cosω c t % & Fiter Sideband ω ) % &t f % &t SSB φ % &ω F A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 12 April 14, 2014 SSB 7eneration 8ia 9ilterin% c ω − ' DSB ω : : )ω c ω c ω − c ω ω ω c ω − ' SSB ω : c ω ω Sideband fi!ter A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 13 April 14, 2014 SSB 7eneration 8ia 9ilterin% ♦5ote6 If f&t% has ow frequenc' co/2onents +oin+ down to DC, then a sideband fiter with a *ar' shar2 ro off is required ♦It is 5?1 so eas' to buid a fiter with a shar2 ro off. ♦1his is 5?1 such a bi+ 2robe/ if does not contain frequenc' co/2onents cose to "ero as de2icted in the 2re*ious and foowin+ iustrations. % &ω F ' SSB ω=' DSB−SC ω⋅) ω Fi!ter Sideband A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 14 April 14, 2014 SSB 7eneration: 9ilter roll o## pro,le" ♦>robe/atic Case ♦Less >robe/atic if no ow freq co/2onents in F&ω% % &ω F : % &ω SC DSB− Φ : % &ω F % &ω SC DSB− Φ %he gap between sidebands a!!ows re!a*ed fi!ter ro!! off& Need +bric, wa!!- fi!ter A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 15 April 14, 2014 SSB 7eneration: 9ilter roll o## pro,le" ♦ 1he ro off 2robe/ worsens if sideband fiterin+ is to be i/2e/ented at hi+h frequencies. 1he required fiter ro off in dB/decade increases as the centre frequenc' of F&ω-ω c % increases. ♦ Fiterin+ 2robe/ can be ae*iated b' usin+ a two-sta+e /i,in+ 2rocess for -u2-con*ersion0 in a trans/itter. ) si/iar a22roach is used in the conte,t of /utista+e down-con*ersion &heterod'nin+%. $ BPF # LSB $ ω $ # ω ω − # ω $ # ω ω + # BPF + SSB φ Desired SSB Si+na # USB % &ω F Note. #adiated SSB signa! is centred on ω # +ω $ +#π B/ # #πB : A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 16 April 14, 2014 Two3sta%e SSB Trans"itter F (ω) #π B : : : : ω # +ω $ ' SSB/ (ω) : : −ω $ ω $ −ω $ ω $ ω # First /i,er ?ut2ut of $st sta+e −ω # ω # −ω $ −(ω # +ω $ ) −(ω # −ω $ ) ω # +ω $ : −(ω # +ω $ ) ?ut2ut of #nd sta+e #nd /i,er B>F$ &accurate' i/2e/ented at a ower frequenc' than the fina RF si+na% B>F# $ #@ ⊛ A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 17 April 14, 2014 Two3sta%e SSB Trans"itter ♦ 1he +a2 between the 7SB and the LSB at the in2ut to the fina B>F is +reater if a two sta+e desi+n is used &i.e. the +a2 between LSB# and 7SB# enterin+ B>F# A see sBetch% . ♦ 1his /uti-sta+e u2-con*ersion technique, athou+h used here to +enerate SSB, is +enera' used to transate &or -heterod'ne0% si+nas to hi+her frequencies &for a /oduation techniques%. ϕ SSB × t $ cosω $ BPF % &t f × t # cosω # BPF 0 s t Sideband fi!ter 1nd Sideband fi!ter A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 18 April 14, 2014 7eneration o# SSB Si%nal 5#ilterin% "ethod6 ♦Fiterin+ 8ethod6 ' SSB ( ω)=' DSB−SC (ω)⋅) ( ω) Fi!ter Sideband t c ω cos ϕ SSB (t )= [ f (t )cos ω c t ] ⊛h(t ) × BPF % &t f ϕ SSB (t ) A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 19 April 14, 2014 9re:uenc' spectru" o# SSB %enerated ,' 9ilterin% ' SSB ( ω) = ' DSB−SC (ω)⋅)( ω) For the USB case 2ass$ming fi!ter passband gain is 03& ' SSB+ (ω)= $ # F − ( ω+ω c )+ $ # F + ( ω−ω c ) For the LSB case& = [ $ # F( ω+ω c )+ $ # F (ω−ω c ) ] ⋅) (ω) ' SSB− ( ω)= $ # F + ( ω+ω c )+ $ # F − ( ω−ω c ) A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 20 April 14, 2014 9re:uenc' spectru" o# SSB %enerated ,' 9ilterin% N m ω − m ω P SSB Sideband Upper c ω − c ω ω P USB % &ω + Φ SSB N ' SSB+ (ω)= $ # F − ( ω+ω c )+ $ # F + ( ω−ω c ) $ # F − (ω+ω c ) $ # F + ( ω−ω c ) % &ω + F % & % & % & ω ω ω + − + = F F F ω % &ω − F AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 21 April 14, 2014 5.4.3 'lte$nati%e met(od &o$ gene$ating SSB using t(e )*(ase S(i&t Met(od+ (,no-n as t(e ).a$tley Modulato$+ A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 22 April 14, 2014 7eneration o# SSB; Si%nal 5phase shi#t "ethod6 ♦Let where re2resents the ne+ati*e frequenc' co/2onents, and re2resents the 2ositi*e frequenc' co/2onents. ♦)n SSBC Fourier s2ectru/ can be constructed fro/46 ♦In*erse transfor/in+ we +et % & % & % & ω ω ω + − + = F F F % & % & % & c c SSB F F ω ω ω ω ω φ − + + = + − + % &ω + F % &ω − F t 4 t 4 SSB c c e t f e t f t ω ω φ % & % & % & + − − + + = % & % & % & % & ω ω + + − − ↔ ↔ F t f F t f 5NB. we ha(e dropped the factor of 60718 present if the SSB signa! is deri(ed b" sideband fi!tering $sing a $nit"-gain BPF& A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 23 April 14, 2014 N m ω − m ω P SSB Sideband Upper c ω − c ω ω P USB % &ω + Φ SSB N % & % & % & c c SSB F F ω ω ω ω ω φ − + + = + − + % & c F ω ω + − % & c F ω ω − + % &ω + F % & % & % & ω ω ω + − + = F F F ω % &ω − F A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 24 April 14, 2014 7eneration o# SSB; Si%nal 5phase shi#t "ethod6 ( ) ( ) t t f t t f t t 4f t 4f t t f t f t t 4f t t f t t 4f t t f e t f e t f t c c c c c c c c t 4 t 4 SSB c c ω ω ω ω ω ω ω ω φ ω ω sin % & D cos % & sin % & % & cos % & % & sin % & cos % & sin % & cos % & % & % & % & − = − + + = + + − = + = − + + − + + − − + − − + and % & % & % & t f t f t f + − + = % & % & % & D t 4f t 4f t f − + + − ≡ where ℱ { ̂ f (t )}=F( ω)=−4F + ( ω)+ 4F − (ω) = { −4F( ω) for ω≥: 4F( ω) for ω<: 9f we transform: we get. A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 25 April 14, 2014 <il,ert Trans#or" )(ω)= { −4 for ω≥: 4 for ω<: )( ω)= { e − 4; /# for ω≥: e 4; / # for ω<: #e-e*pressed as. <e see that this operation is a -=> deg phase shifter: operating o(er a!! fre?$enc" components in F2ω3& -=> deg % &t f % &t f % &ω ) % & D t f ̂ f (t ) %he fre?$enc" domain operations can be e*pressed as a transfer f$nction operation: ,nown as the +)i!bert %ransform- 2the )i!bert %ransform of f2t3 3 # / π − # / π %E & ar+F ω ) A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 26 April 14, 2014 7eneration o# SSB Si%nal t t f t t f t c c SSB ω ω φ sin % & D cos % & % & − = + 1he G indicates that each frequenc' co/2onent in F&H% is dea'ed b' <: : t t f t t f t c c SSB ω ω φ sin % & D cos % & % & + = − @ simi!ar ana!"sis for generating !ower sideband SSB: re(ea!s Upper sideband SSB. A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 27 April 14, 2014 <ardware ="ple"entation o# /hase Shi#t Method 5SSB6 5known as the .<artle' Modulator06 + ± t c ω cos : <: − × % &t f : <: − × t t f c ω sin % & D t c ω sin ∑ t t f c ω cos % & % &t SSB φ >hase shift )LL frequenc' co/2onents in f&t% b' -<: : &i.e. dea' b' <: de+rees% % & D t f Aither add to get SSB- or s$btract to get SSB/ A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 28 April 14, 2014 ♦For the s2ecia case of a sinusoida /oduatin+ si+na, a /ore direct wa' to obtain the e,2ression for SSB is to e,2and usin+ tri+ identities6 t t t t t t c m c m c m SSB ω ω ω ω ω ω φ sin sin cos cos I % cosJ& % & − = + = + t t t t t t c m c m c m SSB ω ω ω ω ω ω φ sin sin cos cos I % cosJ& % & + = + − = − USB LSB %hese e*pressions can easi!" be con(erted to a b!oc, diagram A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 29 April 14, 2014 Co""ent ♦In the 2hase shift /ethod, one is essentia' +eneratin+ a DSB- SC si+na &u22er ar/% and then either addin+ or subtractin+ the si+na fro/ the ower ar/ to cance out either the u22er or the ower sideband. ♦1his /ethod requires a broadband <: de+ree 2hase shifter to obtain . 1his can be tricB' to i/2e/ent 2ractica'. ♦5ote6 1he SSB frequenc' s2ectru/ obtained *ia the 2hase shift /ethod is /athe/atica' equi*aent to that obtained b' 2assin+ the DSB-SC throu+h a sideband fiter )&ω%, which has a 2assband +ain of two. % & D t f AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 30 April 14, 2014 5.4.4 SSB /ene$ation using 0ea%e$1s Met(od (t(is met(od does not $e2ui$e a b$oad3band #(ase s(i&te$ ?ri+ina 2a2er6 K) 1hird 8ethod of =eneration and Detection of Sin+e-Sideband Si+nasK D L 3ea*er, >roc. IR(, Dec. $<MN A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 31 April 14, 2014 SSB <ardware ="ple"entation usin% a .Wea&er Modulator0 + ± × : <: − × ∑ ϕ SSB± (t ) Aither add to get SSB/ or s$btract to get SSB- × : <: − × sin ω $ t L>F L>F 1he L>F cut off frequenc' is B/# !" where B is bandwidth of f&t%. If f&t% ies between DC and B !", then ω $ =#π B/ #=π B f (t ) sin ω # t cos ω $ t cos ω # t A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 32 April 14, 2014 Wea&er1s Method #or %eneratin% SSB F(ω) : ω ω $ : : ω # ω # −ω # : 1ransate to eft b' ω $ . )22' L>F, bandwidth B/#. 1ransate to ri+ht b' ω # . )dd in ne+ati*e frequenc' co/2onents. [ f (t )e −4ω $ t ] LPF *(t )= *(t )+* 4 (t ) ω ω ω B (ω)+B 4 (−ω) B (ω) [ f (t )e − 4ω $ t ] LPF e 4ω # t A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 33 April 14, 2014 !eri&ation o# Wea&er1s Method #or %eneratin% SSB ♦1o create an u22er sideband SSB si+na, we find the ti/e- do/ain equi*aent of the foowin+ frequenc' do/ain o2erations6 1ransate s2ectru/ F&ω% to the eft b' a/ount ω $ >ass throu+h a ow 2ass fiter of bandwidth B/#, re/o*in+ unwanted band. 1ransate to the ri+ht b' a/ount ω # . )dd in ne+ati*e frequenc' co/2onents i.e. add O4&-ω%. ♦Con*ert the abo*e to equi*aent rea ti/e do/ain o2erations6 f (t )e −4ω $ t [ f (t )e −4ω $ t ] LPF *(t )=[ f (t )e −4ω $ t ] LPF e 4ω # t ℱ −$ { B (ω)+ B 4 (−ω)}=*(t )+* 4 (t ) A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 34 April 14, 2014 !eri&ation o# Wea&er1s Method #or %eneratin% SSB ♦ Con*ert the to rea ti/e do/ain o2erations6 ♦ 3ritin+ co/2act' and re-arran+in+6 ♦ )ddin+ the con.u+ate, to +et the rea SSBC si+na6 ♦ Dro2 factor of two, and draw as the bocB dia+ra/. *(t )=[ f (t )(cos ω $ t − 4 sin ω $ t )] LPF (cos ω # t + 4 sin ω # t ) *(t )={[ f C $ ] LPF − 4 [ f S $ ] LPF }(C # + 4 S # ) *(t )={ [ f C $ ] LPF C # +[ f S $ ] LPF S # }+ 4 { [ f C $ ] LPF S # −[ f S $ ] LPF C # } *(t )+* 4 (t )=#{ [ f C $ ] LPF C # +[ f S $ ] LPF S # } *(t )=[ f (t )e −4ω $ t ] LPF e 4ω # t ϕ(t ) =[ f (t )cos ω $ t ] LPF cos ω # t +[ f (t )sin ω $ t ] LPF sin ω # t C n ≡cos ω n t S n ≡sin ω n t *(t )+* 4 (t ) A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 35 April 14, 2014 Wea&er1s Method #or %eneratin% SSB ♦3ea*erPs /ethod does not require a broad-band 2hase shifter &for f&t%% iBe in the !arte' /oduator. 1he quadrature si+nas can be created with a narrow-band 2hase shifter. 1he quadrature si+nas can aso be created without a <: de+ree 2hase shifter A there are ce*er quadrature osciator circuits. ♦3ea*erPs /ethod is the 2referred /ethod for di+ita i/2e/entation. ♦1he out2ut s2ectru/ can be ana'sed b' tracBin+ the 2ath of the in2ut si+na throu+h the /oduator &a +ood tutoria e,ercise%. i.e. sBetch s2ectru/ at each 2oint in the dia+ra/. ♦De2endin+ on whether the si+na fro/ the ower ar/ is added or subtracted fro/ the u22er ar/, either u22er or ower sideband SSB is obtained. )ddition QR u22er sideband. Subtraction QR ower sideband. ♦1he desired sideband is centred on ω # . AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 36 April 14, 2014 5.4.5 4emodulation o& SSB A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 37 April 14, 2014 !e"odulation o# SSB ♦De/oduation of the SSB si+na can be done b' /i,in+ with a cos&ω c t%. &as is done for DSB-SC de/oduation% ♦1his is eas' to see b' +ra2hica con*oution. t t f t t f t c c SSB ω ω φ sin % & D cos % & % & ± = % &t SSB φ t c ω cos × LPF % & : t e A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 38 April 14, 2014 !e"odulation o# SSB; Si%nal % &ω + Φ SSB % &t SSB φ t c ω cos × LPF % & : t e c ω − c ω c ω # − c ω # LPF : : ω ω c ω c ω − : ω π π con(o!(e Upper sideband $ #@ ⊛ A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 39 April 14, 2014 !e"odulation o# SSB3 Si%nal % &t SSB φ t c ω cos × LPF % & : t e c ω # − LPF : ω c ω − % &ω − Φ SSB c ω − : ω : ω π con(o!(e Lower sideband c ω c ω # c ω π $ #@ ⊛ A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 40 April 14, 2014 !e"odulation o# SSB Si%nal = $ # f (t )+ $ # f ( t )cos #H c t − $ # ̂ f (t )sin #H c t ?ut2ut of L>F e : (t )= $ # f ( t ) ϕ SSB+ ( t )cos ω c t = f (t )cosω c # t − ̂ f (t )sin ω c t cos ω c t A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 41 April 14, 2014 !e"odulation o# SSB ♦(ffect of 2hase and frequenc' errors. ♦Let ♦De/oduate with ♦(,2and 2roduct6 Frequenc' (rror cos[ ω c Cωt D ] ϕ SSB+ ( t )=f ( t )cos ω c t − ̂ f (t )sin ω c t >hase (rror [ f t cosω c t − f t sin ω c t ] cos[ ω c Cωt D ] = $ # f t {cos Cωt D cos[ #H c t Cωt D ]} $ # f t {sin Cωt D −sin[ #H c t Cωt D ]} A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 42 April 14, 2014 !e"odulation o# SSB ♦)fter L>F ♦ChecB6 Cω=: case e : t = $ # f t cos Cωt D $ # f t sin Cωt D and D =: } ⇒ e : t = $ # f t 2which is what we e*pect3 %his res$!t re?$ires some interpretation A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 43 April 14, 2014 Case o# /hase $rror onl' 5i.e. , 6 ♦1o see what effect this has on f&t%, consider a sin+e frequenc' co/2onent in f&t%. ♦i.e. consider ♦ 1he 2hasor dia+ra/ shows the reationshi2s. f t Cω=: e : t = $ # f t cos D $ # f t sin D D ≠: ω=ω m f ( t )=e 4ω m t f t e −4D f t ̂ f (t ) ω m D ⇒ ̂ f ( t )=(− 4 )e 4ω m t A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 44 April 14, 2014 Case o# /hase $rror onl' 5i.e. , 6 ♦ 5ote6 (ach frequenc' co/2onent in f&t% wi be 2hase shifted b' the constant θ, i.e. 2hase distortion across band. ♦ 1he hu/an ear is insensiti*e to 2hase dea's, and so s2eech or /usic wi sound fine. : = ∆ω : ≠ θ e : ( t )= $ # e 4ω m t cos D+ $ # (−4 )e 4ω m t sin D = $ # e 4ω m t ( cos D− 4 sin D ) = $ # e 4ω m t e −4D = $ # f ( t )e − 4D A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 45 April 14, 2014 Case o# #re:uenc' $rror 5i.e. , 6 ♦Considerin+ a sin+e frequenc' co/2onent6 e : ( t )= $ # f ( t )cos Cωt + $ # ̂ f (t )sin Cωt f ( t )=e 4ω m t e : ( t )= $ # e 4ω m t cos Cωt + $ # (− 4 )e 4ω m t sin Cωt = $ # e 4ω m t ( cos Cωt −4 sin Cωt ) = $ # e 4ω m t e −4Cωt = $ # e 4 ( ω m −Cω)t fre? shift error Cω : ≠ ∆ω : = θ A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 46 April 14, 2014 Case #re:uenc' $rror ♦1hus an error in the de/oduator osciator frequenc' causes a shift in the s2ectru/ of the reco*ered si+na. ♦S/a frequenc' errors are toerabe in so/e a22ications. ♦3ith *oice, a frequenc' shift can /aBe a s2eaBer sound iBe Donad DucBS ♦ SSB is used for broadcast radio in the so-caed -short wa*e0 bands. A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 47 April 14, 2014 !e"odulation o# SSB > 9re: !o"ain /erspecti&e ♦Frequenc' do/ain 2ers2ecti*e on osciator 2hase and frequenc' errors. Let Let Let ϕ d (t )=cos [( ω c +Cω)t +D ] ϕ d ( ω)=;e −4D E( ω+ω c +Cω)+;e 4D E( ω−ω c −Cω) F ω F( ω)=F + (ω)+F − ( ω) ϕ SSB+ ( ω)=F + ( ω−ω c )+F − ( ω+ω c ) F − ω : ω F ω 2demod$!ator osci!!ator3 A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 48 April 14, 2014 !e"odulation o# SSB % & c F ω ω + − ω ω ∆ − − c ω ω ∆ + c c ω − c ω % &ω + Φ SSB : : ω ω c ω c ω − % & c F ω ω − + ?sciator 3ith >hase and Frequenc' (rror &ne+ freq error% % &ω d Φ $ # F − (ω−Cω) e 4D ∣Cω∣ : ω % & : ω e $ # F + ( ω+Cω)e −4D θ π 4 e − θ π 4 e Con(o!(e. $tp$t A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 49 April 14, 2014 !e"odulation o# SSB ♦?ut2ut6 ♦Concude6 ♦1he frequenc' error resuts in a frequenc' co/2onents bein+ transated b' . 1he 2hase error resuts in a co/2onents bein+ 2hase shifted b' θ. ∣Cω∣ e : ( ω)= { ' SSB + (ω)⊛' d ( ω) $ #@ } ⋅) LPF (ω) e : ( ω)= $ # F + ( ω+Cω)e −4D + $ # F − ( ω−Cω)e 4D AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 50 April 14, 2014 5.4.6 Single Sideband 5a$ge3!a$$ie$ (SSB35! A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 51 April 14, 2014 SSB34C 54ar%e Carrier SSB6 ♦)ows reco*er' of f&t% *ia en*eo2e detection. ♦ 5eeds ar+er carrier than DSB-LC &e*en /ore wastefu of 2ower%. carrier ± SSB t t f t t f t @ t c c c ω ω ω φ sin % & D cos % & cos % & + = en(e!ope % &t r % &t r f (t )+@ % & D t f ω c >hasor re2resentation A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 52 April 14, 2014 SSB34C 54ar%e Carrier SSB6 r (t )= √ [ @+ f (t )] # +[ ̂ f (t )] # ϕ(t )=r ( t )cos[ ω c t +D( t )] ϕ(t )=( @+ f (t )) cos ω c t ̂ f (t )sin ω c t @cos *+Bsin *=C cos( *+D ) where C= √ @ # +B # and D=arctan(−B/ @) A*press SSB-LC as @pp!" trig identit" %h$s: write as where A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 53 April 14, 2014 SSB34C 54ar%e Carrier SSB6 ♦Si+na of For/ where the en*eo2e &i.e. /a+ of resutant 2hasor% is ♦For @RR f (t ) r (t )= √ [ @+ f (t )] # +[ ̂ f (t )] # =[ @ # + f # ( t )+# @f ( t )+ ̂ f # ( t )] $ # =@ [ $+ f # ( t ) @ # + #f ( t ) @ + ̂ f # (t ) @ # ] $ # ϕ(t )=r ( t )cos[ ω c t +D( t )] r (t )≈@ [ $+ #f (t ) @ ] $ # A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II 508 Page 54 April 14, 2014 SSB34C 54ar%e Carrier SSB6 r (t )≈@+ f (t ) %h$s * TT$ ($+*) n =$+n *+ $ #S n(n−$) * # +⋯ r (t )≈@ [ $+ # f (t ) @ ] $ # =@ [ $+ $ # ⋅ # f (t ) @ +⋯ ] ≈@ [ $+ f (t ) @ ] for @RR f (t ) *≡ #f (t ) @ %his shows that f2t3 can be reco(ered from SSB-LC b" en(e!ope detection )22' series e,2ansion6 ($+*) $/ # =$+ $ # *− $ U * # +⋯ 5ote6 If one can o/it hi+her order ter/s.. AM SSB A.J.Wilkinson, UCT EEE3086F Signals and Systems II 508 Page 55 April 14, 2014 EEE3086F Signals and Systems II $nd o# handout