Analog Communication (10EC53) Unit 3 Quadrature Carrier Multiplexing

Transcription

1 Analog Couniation (0EC53) Unit 3 Quadrature Carrier Multiplexing A Quadrature Carrier Multiplexing (QCM) or Quadrature Aplitude Modulation (QAM) ethod enables two DSBSC odulated waves, resulting ro two dierent essage signals to oupy the sae transission band width and two essage signals an be separated at the reeiver. The transitter and reeiver or QCM are as shown in igure 3.. Figure 3.: QCM transitter and reeiver The transitter involves the use o two separate produt odulators that are supplied with two arrier waves o the sae requeny but diering in phase by -90 o. The ultiplexed signal s( onsists o the su o the two produt odulator outputs given by the equation 3.. s( = A ( os(π + A ( sin(π (3.) where ( and ( are two dierent essage signals applied to the produt odulators. Thus, the ultiplexed signal s( oupies a transission band width o W, entered at the arrier requeny where W is the band width o essage signal ( or (, whihever is larger. Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore

2 Analog Couniation (0EC53) At the reeiver, the ultiplexed signal s( is applied siultaneously to two separate oherent detetors that are supplied with two loal arriers o the sae requeny but diering in phase by -90 o. The output o the top detetor is ( A. botto detetor is ( A and that o the For the QCM syste to operate satisatorily, it is iportant to aintain orret phase and requeny relationships between the loal osillators used in the transitter and reeiver parts o the syste. Hilbert transor The Fourier transor is useul or evaluating the requeny ontent o an energy signal, or in a liiting ase that o a power signal. It provides atheatial basis or analyzing and designing the requeny seletive ilters or the separation o signals on the basis o their requeny ontent. Another ethod o separating the signals is based on phase seletivity, whih uses phase shits between the appropriate signals (oponents) to ahieve the desired separation. In ase o a sinusoidal signal, the siplest phase shit o 80 o is obtained by Ideal transorer (polarity reversal). When the phase angles o all the oponents o a given signal are shited by 90 o, the resulting untion o tie is alled the Hilbert transor o the signal. Consider an LTI syste with transer untion deined by equation 3.. H j, > 0 = 0, = 0 j, < (3.) and the Signu untion given by sgn, > 0 = 0, = 0, < 0 The untion H() an be expressed using Signu untion as given by 3.3. H = j sgn (3.3) Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore

3 Analog Couniation (0EC53) We know that π, and. e j = j e j π ± θ = j e j = os( θ ) ± j sin( θ ) Thereore, Thus the agnitude H H =, e = e jπ jπ,, > 0 < 0 or all, and angle H π, = + π, > 0 < 0 The devie whih possesses suh a property is alled Hilbert transorer. When ever a signal is applied to the Hilbert transorer, the aplitudes o all requeny oponents o the input signal reain unaeted. It produes a phase shit o -90 o or all positive requenies, while a phase shit o 90 o or all negative requenies o the signal. I x( is an input signal, then its Hilbert transorer is denoted by xˆ ( and shown in the ollowing diagra. x ( xˆ ( Hilbert Transorer To ind ipulse response h( o Hilbert transorer with transer untion H(). Consider the relation between Signu untion and the unit step untion. sgn Dierentiating both sides with respet to t, ( t ) = u( = x(, Apply Fourier transor on both sides, jω Applying duality property o Fourier transor, d dt { x( } = δ ( sgn( sgn( Sgn jπt jπ Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 3

4 Analog Couniation (0EC53) We have H = j sgn equation 3.4, H πt Thereore the ipulse response h( o an Hilbert transorer is given by the h ( = πt (3.4) Now onsider any input x( to the Hilbert transorer, whih is an LTI syste. Let the ipulse response o the Hilbert transorer is obtained by onvolving the input x( and ipulse response h( o the syste. ( = x( h( The equation 3.5 gives the Hilbert transor o x(. xˆ xˆ xˆ ( ( = x( + πt x( τ ) ( t τ ) = dτ π (3.5) The inverse Hilbert transor x( is given by We have xˆ x ( ( = x( h( + xˆ ( τ ) ( t τ ) = dτ π (3.6) The Fourier transor Xˆ o ( xˆ is given by X H X ˆ = Xˆ = j sgn X (3.7) Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 4

5 Analog Couniation (0EC53) Appliations o Hilbert transor. It is used to realize phase seletivity in the generation o speial kind o odulation alled Single Side Band odulation.. It provides atheatial basis or the representation o band pass signals. Note: Hilbert transor applies to any signal that is Fourier transorable. Exaple: Find the Hilbert transor o x t = os( π t. This represents Fourier transor o the sine untion. Thereore the Hilbert transor o osine untion is sin untion given by Pre-envelope Consider a real valued signal x(. The pre-envelope x + ( or positive requenies o the signal x( is deined as the oplex valued untion given by equation 3.8. Apply Fourier transor on both the sides, X + x ( ) ) X + = [ δ ( ) + δ ] Xˆ ( = x( + jxˆ ( (3.8) = X + j[ j sgn X ] = j sgn X X ˆ + = j sgn [ δ ( ) + δ ] X ˆ + j = [ δ ( ) δ ] X ˆ + j = [ δ ( ) δ ] xˆ ( = sin(π Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 5

6 Analog Couniation (0EC53), ( 0 ), X > 0 X + = X = (3.9) 0, < 0 The pre-envelope x ( or negative requenies o the signal is given by x ( = x( jxˆ ( The two pre-envelopes x + ( and ( that is x ( x ( * + = x are oplex onjugate o eah other, The spetru o the pre-envelope x + ( is nonzero only or positive requenies as ephasized in equation 3.9. Hene plus sign is used as a subsript. In ontrast, the spetru o the other pre-envelope x ( is nonzero only or negative requenies. That is Thus the pre-envelopes x + ( and ( oplex valued signals. 0, > 0 X = X ( 0 ), = (3.0) X, < 0 x onstitute a opleentary pair o X X(0) W Figure 3.: Spetru o the low pass signal x( Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 6

7 Analog Couniation (0EC53) X + X(0) W Figure 3.3: Spetru o pre-envelope ( x + Properties o Hilbert transor. A signal x( and its Hilbert transor xˆ ( have the sae aplitude spetru. The agnitude o jsgn() is equal to or all requenies. Thereore x( and xˆ ( have the sae aplitude spetru. ˆ or all That is X = X. I xˆ ( is the Hilbert transor o x(, then the Hilbert transor o xˆ (, is x(. To obtain its Hilbert transor o x(, x( is passed through a LTI syste with a transer untion equal to jsgn(). A double Hilbert transoration is equivalent to passing x( through a asade o two suh devies. The overall transer untion o suh a asade is equal to [ sgn] = j or all The resulting output is x(. That is the Hilbert transor o xˆ ( is equal to x(. Canonial representation or band pass signal The Fourier transor o band-pass signal ontains a band o requenies o total extent W. The pre-envelope o a narrow band signal x( is given by x t ( ) ( ) e j π t x t = ~ +, (3.) where x~ ( is oplex envelope o the signal x(. Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 7

8 Analog Couniation (0EC53) X X ( ) - - +w Figure 3.4: Spetru o the band pass signal x( X + X ( ) - -w Figure 3.5: Spetru o the pre-envelope ( x + - +w X ~ X ( ) Figure 3.6: spetru o the oplex envelope ( Figure 3.4 shows the aplitude spetru o band pass signal x(. Figure 3.5 shows aplitude spetru o pre envelope x + (. Figure 3.6 shows aplitude spetru o oplex envelope x~ (. Equation 3. is the basis o deinition or oplex envelope x~ ( in ters o preenvelope x + (. The spetru o ( 0 w x + is liited to the requeny band x~ Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 8

9 Analog Couniation (0EC53) w < < w as shown in igure 3.5. Thereore, applying the requeny-shit + property o Fourier transor to equation 3., we ind that the spetru o the oplex envelope ( x~ is liited to the band w < < w and entered at the origin as shown in igure 3.6. That is, the oplex envelope x~ ( o a band pass signal x( is a low-pass signal. Given signal x( is the real part o the pre-envelope x + (. So, we express the original band pass signal x( in ters o the oplex envelope x~ (, as ollows But x~ ( is a oplex quantity. x I ( ) ( ) (3.) (3.3) where t and t are both real valued low pass untions. This low pass property is x Q inherited ro the oplex envelope x~ (. Thereore original band pass signal x( is expressed in anonial or by using equations 3. and 3.3 as ollows. x (3.4) where t is in-phase oponent and t is quadrature oponent o the band pass signal x(. This noenlature reognizes that sin ( π is in phase-quadrature Both t and t are low-pass signals liited to the band w < < w. Hene, exept or saling ators, they ay be derived ro the band pass signal x( using the blok diagra shown 3.7. x x ( = Re x( ~ x [ ~ e ] j π t ( = x ( jx ( I + [ e ] j π t j π t j t [ x t e jx t e ] ( = Re ~ x( ( = Re ( ) + ( ) π x I Q [ ] ( = Re x ( [ os( π + j sin( π ] + jx ( [ os( π + j sin( π ] I x ( = x ( os( π x sin( π x I ( ) ( ) with respet to os ( π. x I ( ) ( ) x Q I x Q Q Q Q Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 9

10 Analog Couniation (0EC53) * Low Pass Filter x I ( x( os( π -90 o Phase Shiter Osillator * sin( π Low Pass Filter x Q ( Figure 3.7: Blok diagra to produe in-phase and quadrature oponents Single Side Band Suppressed Carrier odulation Standard AM and DSBSC require transission bandwidth equal to twie the essage bandwidth. In both the ases spetru ontains two side bands o width W Hz, eah. But the upper and lower sides are uniquely related to eah other by the virtue o their syetry about the arrier requeny. That is, given the aplitude and phase spetra o either side band, the other an be uniquely deterined. Thus i only one side band is transitted, and i both the arrier and the other side band are suppressed at the transitter, no inoration is lost. This kind o odulation is alled SSBSC and spetral oparison between DSBSC and SSBSC is shown in the igures 3.8 and 3.9. Figure 3.8: Spetru o the DSBSC wave Figure 3.9: Spetru o the SSBSC wave Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 0

11 Analog Couniation (0EC53) Frequeny-doain desription: - Consider a essage signal ( with a spetru M() band liited to the interval w < < w as shown in igure 3.0, the DSBSC wave obtained by ultiplexing ( by the arrier wave ( = A ) os( π t and is also shown, in igure 3.. The upper side band is represented in dupliate by the requenies above and those below -, and when only upper M() -w 0 Figure 3.0: Spetru o essage wave S DSB () w - -w - - +w 0 -w Figure 3.: Spetru o DSBSC wave +w S SSB () - - +w -w Figure 3.: Spetru o SSBSC-LSB wave S SSB () - -w - Figure 3.3: Spetru o SSBSC-USB wave +w Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore

12 Analog Couniation (0EC53) side band is transitted; the resulting SSB odulated wave has the spetru shown in igure 3.3. Siilarly, the lower side band is represented in dupliate by the requenies below and those above - and when only the lower side band is transitted, the spetru o the orresponding SSB odulated wave shown in igure 3.. Thus the essential untion o the SSB odulation is to translate the spetru o the odulating wave, either with or without inversion, to a new loation in the requeny doain. The advantage o SSB odulation is redued bandwidth and the eliination o high power arrier wave. The ain disadvantage is the ost and oplexity o its ipleentation. Frequeny Disriination Method or generating an SSBSC odulated wave Consider the generation o SSB odulated signal ontaining the upper side band only. Fro a pratial point o view, the ost severe requireent o SSB generation arises ro the unwanted sideband, the nearest oponent o whih is separated ro the desired side band by twie the lowest requeny oponent o the essage signal. It iplies that, or the generation o an SSB wave to be possible, the essage spetru ust have an energy gap entered at the origin as shown in igure 3.4. This requireent is naturally satisied by voie signals, whose energy gap is about 600Hz wide. M() 0 Figure 3.4: Message spetru with energy gap at the origin The requeny disriination or ilter ethod o SSB generation onsists o a produt odulator, whih produes DSBSC signal and a band-pass ilter to extrat the desired side band and rejet the other and is shown in the igure 3.5. Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore

13 Analog Couniation (0EC53) Figure 3.5: Frequeny disriinator to generate SSBSC wave Appliation o this ethod requires that the essage signal satisies two onditions:. The essage signal ( has no low-requeny ontent. Exaple: - speeh, audio, usi.. The highest requeny oponent W o the essage signal ( is uh less than the arrier requeny. Then, under these onditions, the desired side band will appear in a non-overlapping interval in the spetru in suh a way that it ay be seleted by an appropriate ilter. In designing the band pass ilter, the ollowing requireents should be satisied: ) The pass band o the ilter oupies the sae requeny range as the spetru o the desired SSB odulated wave.. The width o the guard band o the ilter, separating the pass band ro the stop band, where the unwanted sideband o the ilter input lies, is twie the lowest requeny oponent o the essage signal. When it is neessary to generate an SSB odulated wave oupying a requeny band that is uh higher than that o the essage signal, it beoes very diiult to design an appropriate ilter that will pass the desired side band and rejet the other. In suh a situation it is neessary to resort to a ultiple-odulation proess so as to ease the iltering requireent. This approah is illustrated in the ollowing igure 3.6 involving two stages o odulation. Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 3

14 Analog Couniation (0EC53) ( Produt Modulator Figure 3.6: Two stage requeny disriinator The SSB odulated wave at the irst ilter output is used as the odulating wave or the seond produt odulator, whih produes a DSBSC odulated wave with a spetru that is syetrially spaed about the seond arrier requeny. The requeny separation between the side bands o this DSBSC odulated wave is eetively twie the irst arrier requeny, there by peritting the seond ilter to reove the unwanted side band. Tie-doain desription The tie doain desription o an SSB wave s( in the anonial or is given by the equation (3.5) where S I ( is the in-phase oponent o the SSB wave and S Q ( is its quadrature oponent. The in-phase oponent S I ( exept or a saling ator, ay be derived ro S( by irst ultiplying S( by os ( π and then passing the produt through a low-pass ilter. Siilarly, the quadrature oponent S Q (, exept or a saling ator, ay be derived ro s( by irst ultiplying s( by sin ( π and then passing the produt through an idential ilter. The Fourier transoration o S I ( and S Q ( are related to that o SSB wave as ollows, respetively. ( A os π s Band Pass Filter s ( Produt Modulator ( A os π ( = s ( os( π s ( sin( π I Q Band Pass Filter s( SSB wave S S Q I ( ) + S( + ) S = 0, elsewhere [ S( ) S( + )] j = 0, elsewhere, w, w w w (3.6) (3.7) where w < < w deines the requeny band oupied by the essage signal (. Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 4

15 Analog Couniation (0EC53) Consider the SSB wave that is obtained by transitting only the upper side band, shown in igure 3.. Two requeny shited spetra S( ) and S ( + ) are shown in igure 3. and igure 3.3 respetively. Thereore, ro equations 3.6 and 3.7, it ollows that the orresponding spetra o the in- phase oponent S I ( and the quadrature oponent S Q ( are as shown in igure 3.4 and 3.5 respetively. S 0 + w Figure 3.: Spetru o SSBSC-USB ( ) S ( ) 0 Figure 3.: Spetru o SSBSC-USB shited right by S + 0 Figure 3.3: Spetru o SSBSC-USB shited let by s I A M ( 0) w 0 Figure 3.4: Spetru o in-phase oponent o SSBSC-USB w A M ( 0) S j Q w 0 w Figure 3.5: Spetru o quadrature oponent o SSBSC-USB Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 5

16 Analog Couniation (0EC53) Fro the igure 3.4, it is ound that where M() is the Fourier transor o the essage signal (. Aordingly in-phase oponent S I ( is deined by equation 3.8. Now on the basis o igure 3.5, it is ound that where sgn But ro the disussions on Hilbert transors, it is shown that (3.8) (3.9) (3.0) where Mˆ is the Fourier transor o the Hilbert transor o (. Hene the substituting equation (3.0) in (3.9), we get SQ = A Mˆ (3.) s Q t is deined by equation 3.. Thereore quadrature oponent ( ) (3.) Thereore substituting equations (3.8) and (3.) in equation in (3.5), we ind that anonial representation o an SSB wave s( obtained by transitting only the upper side band is given by the equation 3.3. s U S = A M (3.3) Following the sae proedure, we an ind the anonial representation or an SSB wave s( obtained by transitting only the lower side band is given by Q S s I I ( = A ( j A M = 0, = 0 j A M j SQ = A sgn is the Signu untion. M Mˆ j sgn = s Q ( = A ˆ (,, > 0 < 0 M ( = A ( os( π A ˆ ( sin ( π Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 6

17 Analog Couniation (0EC53) s L ( = A ( os( π + A ˆ ( sin( π (3.4) Phase disriination ethod o SSB generation Tie doain desription o SSB odulation leads to another ethod o SSB generation using the equations (3.3) or (3.4). The blok diagra o phase disriinator is as shown in igure 3.6. ( Figure 3.6: Blok diagra o phase disriinator The phase disriinator onsists o two produt odulators I and Q, supplied with arrier waves in-phase quadrature to eah other. The inoing base band signal ( is applied to produt odulator I, produing a DSBSC odulated wave that ontains reerene phase sidebands syetrially spaed about arrier requeny. The Hilbert transor ˆ ( o ( is applied to produt odulator Q, produing a DSBSC odulated that ontains side bands having idential aplitude spetra to those o odulator I, but with phase spetra suh that vetor addition or subtration o the two odulator outputs results in anellation o one set o side bands and reinoreent o the other set. The use o a plus sign at the suing juntion yields an SSB wave with only the lower side band, whereas the use o a inus sign yields an SSB wave with only the upper side band. This odulator iruit is alled Hartley odulator. Single tone SSB-LSB odulation transor -90 o Phase Shiter ˆ ( Produt Modulator I -90 o Phase Shiter Produt Modulator Q ( π Consider a single-tone essage signal ( A sin( π ˆ = A A os sin ( π In-phase path Osillator Quadrature path + ( = A os( π. Substituting these in equation (3.4), we get ± s( t ) and its Hilbert Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 7

18 Analog Couniation (0EC53) s L ( = A A os( π os( π + A A sin( π sin( π s L (3.5) Thereore the single tone SSBSC wave is a sinusoidal wave o requeny equal to su/ dierene o arrier and essage requenies or USB/LSB. Deodulation (oherent detetion) o SSBSC wave Deodulation o SSBSC wave using oherent detetion is as shown in 3.7. The SSB wave s( together with a loally generated arrier t is applied to a produt odulator and then low-pass iltering o the odulator output yields the essage signal. ( = A A os( π ( ) ( ) = A os(π t + φ) Figure 3.7: Blok diagra o oherent detetor or SSBSC The produt odulator output v( is given by v v v = A t 4 ( = A os( π t + φ) s(, Put φ = (3.6) The irst ter in the above equation 3.6 is desired essage signal. The other ter represents an SSB wave with a arrier requeny o as suh; it is an unwanted oponent, whih is reoved by low-pass ilter. ( = A os( π [ ( os( π ± ˆ ( sin( π ] 4 ( ( ) + A [ ( os( 4π ± ˆ ( sin( 4π ] Prashanth CR, Asst. Proessor, ECE, Veana Institute o Tehnology, B lore 8