Advanced I/Q Signal Processing for Communication Systems

Transcription

1 Advancd I/ Signal Procssing or Communication Systms Mikko Valkama and Markku Rnors Tampr Univrsity o Tchnology Institut o Communications Enginring P.O.Box 553, FIN-33 Tampr, FINLAND Tl: , Fax: valkama@cs.tut.i and mr@cs.tut.i SDR-3 Tchnical Conrnc, Nov. 3, Orlando, FL CONTENTS. Complx Signals and Systms Basic concpts and dinitions Analytic signals and Hilbrt transorms Frquncy translations and mixing Complx signals and sampling. Sampling and Multirat DSP with Bandpass and I/ signals Sampling o bandpass signals Multirat procssing o bandpass and I/ signals Eicint polyphas structurs 3. I/ Mismatch Problms in Analog I/ Signal Procssing I/ signal procssing in rcivrs Signal modls or I/ imbalanc Imbalanc cts in rcivr ront-nds Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd SDR-3 Tchnical Conrnc, Nov. 3, Orlando, FL

2 4. Adaptiv DSP or I/ Imbalanc Compnsation Blind signal stimation basd imbalanc compnsation Simulation xampls 5. Scond-Ordr Sampling and its Enhancmnts Basic scond-ordr sampling schm Imbalanc problm and imag rjction Enhancing th imag rjction using DSP 6. I/ Signal Procssing in Frquncy Synthsizrs Digitally synthsizd complx ton and analog I/ mixing I/ imbalanc problm rvisitd Summary Rrncs SDR-3 Tchnical Conrnc, Nov. 3, Orlando, FL Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd

3 Background and Motivation. COMPLEX SIGNALS AND SYSTEMS All physical signals and wavorms ar ral-valud so why bothr to considr complx-valud signals and systms? Th original complx signal concpts can b tracd back to th introduction o lowpass quivalnt notation, i.., analysis o bandpass signals and systms using thir lowpass/basband quivalnts in gnral, a ral-valud bandpass signal/systm has a complx-valud lowpass quivalnt or xampl, linar I/ modulation and dmodulation principls ar basd on ths idas also all advancd rquncy translation tchniqus and thus th rlatd rcivr architcturs low-if, dirct-convrsion, tc. utiliz complx signals sampling and icint multirat procssing o bandpass signals is anothr good xampl 3 Basic Concpts and Dinitions By dinition, th tim domain wavorm or squnc xt o a complx signal is complx-valud, i.., xt = x I t + jx t. In practic, this is nothing mor than a pair o two ral-valud signals x I t and x t carrying th ral and imaginary parts. Similarly, a complx systm is dind as a systm with complx-valud impuls rspons. In th rquncy domain, ral-valud signals hav always symmtric amplitud spctrum complx signals don t nd to hav any symmtry proprtis in gnral th spctral support rgion o non-zro amplitud spctrum can basically b anything On basic opration rlatd to complx quantitis is complx-conjugation i th spctrum o xt is dnotd by X, thn th spctrum o x * t is X * this simpl-looking rsult is surprisingly usul whn intrprting som proprtis o complx signals in th continuation Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 3

4 an immdiat consqunc is that i you considr th ral part o xt, i.., yt = R[xt] = xt+x * t/, its spctrum is Y = X + X * / i X and X * ar not ovrlapping, yt = R[xt] contains all th inormation about xt this rsult will ind good us,.g., in undrstanding rquncy translations Anothr ky opration rlatd to linar systms in gnral is convolution. In th gnral complx cas, this can b writtn as xt ht = xi t + jx t hi t + jh t = x t h t x t h t + j x t h t + x t h t I I I I In othr words, 4 ral convolutions ar ndd in gnral. Obvious simpliications occur i ithr th iltr input or th iltr itsl is ral valud in ths cass, only two ral convolutions nd to b calculatd 3 3 Analytic Signals and Hilbrt Transorms Hilbrt transormr is gnrally dind as an allpass linar iltr which shits th phas o its input signal by 9 dgrs. Th anticausal impuls and rquncy rsponss can b ormulatd as H continuous-tim hht t = πt j, = + j, < HT h H HT HT discrt-tim, n vn n = / π n, n odd jω j, ω < π = + j, π ω < In practic this bhaviour can b wll approximatd ovr a init bandwidth. On ascinating proprty rlatd to Hilbrt iltrs/transormrs is that thy can b usd to construct signals with only positiv rquncy contnt. This kind o signals ar gnrally trmd analytic and thy ar always complx. Th simplst xampl is to tak a cosin wav Acosω t whos Hilbrt transorm is Asinω t just a 9 dgr phas shit! Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 4 3

5 ths togthr whn intrprtd as I and componnts o a complx signal rsult in Acosω t + jasinω t = Axpjω t whos spctrum has an impuls at ω but nothing on th othr sid o th spctrum Th limination o th ngativ rquncis can mor gnrally b ormulatd as ollows. Starting rom an arbitrary signal xt w orm a complx signal xt + jx HT t whr x HT t dnots th Hilbrt transorm o xt. Thn th spctrum o th complx signal is X[ + jh HT ] whr continuous-tim + j j,, + jh HT = = + j j, <, < which shows th limination o th original ngativ rquncy contnt. Similar concpts carry on to discrt-tim world and w can writ discrt-tim jω + j j, ω < π, ω < π + jh HT = = + j j, π ω <, π ω < 5 3 This ida o using Hilbrt transorm to gnrat analytic signals is urthr illustratd graphically in th ollowing igur. In practic th Hilbrt iltring causs a dlay and a corrsponding dlay nds to b includd also in th uppr I branch. input spctrum input HT I output output spctrum Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 6 3

6 Dsign Exampl: Hilbrt transormr o ordr 5, dsign bandwidth.π.9π π dnots hal th sampling rquncy, Rmz dsign this rsults in about 87 db attnuation or th ngativ rquncis wrt. corrsponding positiv band Hilbrt Transormr HT n Hilbrt Transormr HT Amplitud Frquncy ω / π Phas wrt. π/ Frquncy ω / π Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 8 3

7 .5 Complx +j H HT Rspons Amplitud Frquncy ω / π Amplitud [db] Frquncy ω / π 9 3 Frquncy Translations and Mixing On ky opration in communications signal procssing is th shiting o a signal spctrum rom on cntr-rquncy to anothr convrsions btwn basband and bandpass rprsntations ar spcial cass o this Th basis o all th rquncy translations lis in multiplying a signal with a complx xponntial, gnrally rrrd to as complx or I/ mixing. This will indd caus a pur rquncy shit, i.., jωlot yt = xt Y = X LO which orms th basis or all th linar modulations. This is illustratd in rquncy domain blow. input spctrum output spctrum Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights C Rsrvd 3

8 In gnral, our ral mixrs ar ndd to implmnt a complx mixr as jωlot x t = xi t + jx tcos ωlot + jsin ωlot = x tcos ω t x tsin ω t + j x tcos ω t + x tsin ω t I LO LO LO I LO in th spcial cas o a ral input, only two mixrs ndd Ral mixing is obviously a spcial cas o th prvious complx on and rsults in two rquncy translations: yt = xt cos ωlot jω LOt jωlot = x t + Y = X LO + X + LO Hr, th original spctral componnt appar twic in th mixr output, th two rplicas bing sparatd by LO in rquncy. In rcivrs, this rsults in th so calld imag signal problm sinc th signals rom both C + LO and C LO will appar at C atr th ral mixing stag i ral mixing is usd, th imag signal nds to b attnuatd bor th actual mixr stag w ll talk about this in mor dtail in th rcivr architctur sction 3 input spctrum C LO C C + LO ral mixing output spctrum C LO C LO C C + LO C + LO Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 3

9 Linar I/ modulation mthods ar basically just a spcial cas o complx mixing. Givn a complx mssag signal xt = x I t + jx t, it is irst modulatd as xtxpjω C t, atr which only th ral part is actually transmittd: j Ct j Ct * j Ct yt R[ xt ω ] xicos t Ct xsin t Ct xt ω = = ω ω = + x t ω intrprtation #: x I t and x t ar modulatd onto two orthogonal cosin and sin carrirs; nic rom th implmntation point o viw intrprtation #: xt and x * t ar modulatd onto two complx xponntials xpjω C t and xpjω C t; ky in building gnral undrstanding and rcovring xt back rom yt Notic that both trms/spctral componnts at + C and C contain all th original inormation i.., xt. This procss, also trmd lowpass-to-bandpass transormation, is picturd in th igur blow. 3 3 input spctrum output spctrum C C I/ dmodulation: In th rcivr, th goal is to rcovr th original mssag xt rom th modulatd signal yt. Basd on th prvious discussion, it s asy to undrstand that ithr o th signal componnts at + C or C can b usd or that purpos, whil th othr on should b rjctd. Sinc yt = xt + x t = xt + x t j ω t j ω t * j ω t j ω t * j ω t C C C C C th mssag can b ully rcovrd by simply lowpass iltring th complx rcivr mixr output. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 4 3

10 Formal block-diagrams or th modulator and dmodulator in trms o complx signals ar prsntd blow. MODULATOR: j Ct ω input R[ ] output DEMODULATOR: j Ct ω input LPF output 5 3 Complx Signals and Sampling In priodic sampling sampl rat S th rsulting discrt-tim signal has a priodic spctrum whr th original continuous-tim spctrum is rplicatd around th intgr multipls o th sampling rquncy. Intrstingly, any o ths spctral rplicas or imags can b considrd as th usul part and thus b usd or urthr procssing. Consquntly, sampling and multirat oprations in gnral can also b usd, in addition to mixing tchniqus, in prorming rquncy translations. Lt B dnot th doubl-sidd bandwidth o a complx-valud basband signal i.., th spctrum is nonzro only or Bng Bpos, B = B ng + B pos to avoid aliasing, th sampling rquncy S should simply b high nough such that th spctral imags don't ovrlap, i.., S Bng Bpos S Bng + Bpos S B this is th traditional Nyquist sampling thorm naturally, sinc th signal to b sampld is complx-valud, thr xist two ral-valud sampl strams I and both at rat S Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 6 3

11 i th signal to b sampld consists o multipl rquncy channls, sampling rats blow S = B ar possibl i only som o th channls ar o intrst th sampling rquncy should simply b slctd in such a mannr that aliasing is avoidd on top o thos intrsting rquncy bands Exampl spctra which both hav th sam lowr limit S rquncy ar dpictd blow. = B or th sampling original B sampld B S S S S B B S S S S 7 3 Conclusion: It dosn t mattr whthr th signal is ral or complx or whthr it is locatd symmtrically with rspct to origin th minimum sampling rat is always S = B So in gnral th traditional statmnt signal should b sampld at last at rat two tims its highst rquncy componnt can b concludd inaccurat what rally mattrs is th doubl-sidd bandwidth a good xampl is th sampling o a ral-valud lowpass signal, say xt, with spctral support W W whn sampld dirctly, th minimum sampling rat is S = W as an altrnativ, you can orm an analytic signal xt + jx HT t, whr x HT t dnots th Hilbrt transorm o xt, or which th minimum sampling rat is only S = W vn though th highst rquncy componnt prsnt in both signals is W Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 8 3

12 . SAMPLING AND MULTIRATE DSP WITH SAMPLING OF BANDPASS SIGNALS BANDPASS AND I/ SIGNALS Starting point is th traditional Nyquist sampling thorm: Any signal occupying th band B ng B pos [Hz] is compltly charactrizd by its discrt-tim sampls givn that th sampling rat is at last B ng + B pos two-sidd bandwidth. Popl commonly intrprt this that i th highst rquncy componnt in a signal is MAX, you nd to tak at last MAX sampls pr scond strictly spaking, this is inaccurat as concludd bor i.., sampling at or abov rat MAX is clarly always suicint but.g. in cas o bandpass signals w can also us usually much lowr sampl rat mor spciically, sampling at rat blow MAX will indd rsult in aliasing but as long as all th inormation about th original signal is prsnt in th sampls, w ar doing good kp in mind also that th Nyquist accssibl band or any sampl rat S is S / S /, so with blow MAX sampling rats it is rally on o th imags that appar on this band!!! 9 3 ths kind o tchniqus ar gnrally rrrd to as subsampling Th on and only principl to rmmbr in sampling is that th rsulting signal has a priodic spctrum and any part o that spctrum can b slctd/usd or urthr procssing. Mor spciically, in communications rcivrs, aliasing du to sub-sampling can b takn advantag o to bring th signal closr to basband. W considr two cass; starting rom a ral-valud bandpass signal, th rsulting sampl stram is ithr ral-valud or complx-valud. Ral Subsampling Basic stup: ral-valud bandpass signal, bandwidth B, cntr-rquncy C, uppr band-dg U = C + B/ and lowr band-dg L = C B/. original B U C L L C U Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 3

13 Now sampling at som rat S rsults in a signal whr th prvious spctrum is rplicatd at intgr multipls o th sampling rat th basic ct o sampling. With S < U, aliasing will tak plac but as long as th aliasing componnts don t all on top o ach othr, vrything is OK!! So an xampl spctrum o th sampld signal could look lik in th igur blow, whn thr is no harmul aliasing and yt S < U. sampld B U C L L C U L +n S U +n+ S Basd on th abov igur, it is asy to ormulat th rgions o allowabl sampling rats. Ths ar in gnral o th orm + C B C B S n+ n whr C B n loor B 3 Commnts: as can b sn, th possibl valus o th sampling rat dpnd on both th bandwidth B and th cntr-rquncy C or n = w gt S C + B = C + B/ = U which is th traditional Nyquist sampling thorm th uppr limit bcoms ininity or n > w ar rally sampling at lowr rquncy than givn by th traditional Nyquist thorm or n > aliasing dos occur but with givn valus o S, not on top o th dsird signal band no harmull aliasing th lowst possibl sampling rat is in gnral givn by C + B C + B C + B C + B S = = = n max + C B C B C + B loor + loor + loor B B B th ultimat sampling rat S =B is utilizabl i C + B is an intgr thn B C + B C + B and only thn loor = B B Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 3

14 Numrical xampl: C = khz and B = khz, so 4 n loor = loor.5 = and th possibl valus or S ar n = : 5 khz S n = : 5 khz S 3 khz try.g. with S = 7 khz and you s that no harmull aliasing occurs Complx subsampling Instad o sampling dirctly th ral-valud signal, th ida is to sampl th corrsponding analytic signal!!! So th sampling structur looks lik HT dnots Hilbrt transormr input S I HT S 3 3 Now sinc th analytic signal is r rom ngativ rquncy componnts, sampling rquncy o S = B or any rat abov is always indpndntly o th cntr-rquncy C! suicint to avoid harmull aliasing!!! Som xampl spctral igurs with th sam input signal as in th prvious subsction: S = B: sampld B L C U S > B: sampld B L C U Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 4 3

15 Notic: I th cntr-rquncy C is an intgr multipl o th sampl rat S i.., S = C / k, th cntr-rquncy o th k-th spctral rplica will coincid with zro rquncy and a dirct bandpass-to-lowpass transormation is obtaind!!! this is asy to undrstand basd on spctral intrprtations but can also b sn as ollows: th ral bandpass input, say rt, can b writtn in trms o its basband quivalnt zt as rt = zt = zt + z t thn th corrsponding analytic signal is o th orm j * R[ ω Ct j Ct j Ct ] ω ω rt jr t zt z t j j zt j z t jωct = zt jωct * jωct jωct * jωct + HT = thus sampling at S = C / k with k intgr rsults in r nt jr nt z nt z nt z nt z nt j ωc nt S j π C nt S j π nk S + HT S = S = S = S = S which ar indd just th sampls o th basband quivalnt 5 3 MULTIRATE PROCESSING OF BANDPASS AND I/ SIGNALS Th two undamntal multirat oprations dcimation/down-sampling and intrpolation/up-sampling nabl th sampl rat to b altrd digitally. In addition to this, thy also or an intrsting altrnativ to mixing in prorming rquncy translations. Dcimation or Down-Sampling with Complx Signals Th basic block-diagram to rduc th sampl rat by an intgr actor L is prsntd in th igur blow, with S dnoting th original sampl rat. In th down-sampling L, vry L-th sampl o th input squnc is pickd up to orm th output squnc and th nw sampl rat bcoms S /L. As a consqunc, all th rquncy bands locatd at th intgr multipls o S /L within S / S / o cours ar aliasd down to basband. Ths ar also illustratd in th ollowing igur. Thus, i th gnrally complx input signal is o lowpass typ passband B ng B pos, th dcimation iltr Hz should attnuat all th rquncy bands o width B=B ng +B pos locatd corrspondingly at th prvious critical rquncis. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 6 3

16 Naturally, th bandwidth B=B ng +B pos nds to b smallr than S /L but thr s no rstriction such that,.g., B pos should b smallr than S /L/ as long as S /L B, th down-sampld signal is r rom harmul aliasing On th othr hand, i th dsird signal is originally a bandpass signal with a gnrally complx-valud basband quivalnt, th inhrnt aliasing can b xploitd to chang th signal cntr-rquncy. Now, th dcimation iltr Hz is a bandpass iltr slcting th dsird rquncy band, and aliasing can b usd to bring th signal closr to basband as a spcial cas o this, i th signal is cntrd at any multipl o th output sampl rquncy S /L, an analytic bandpass iltr and dcimation will rsult in a dirct bandpass-to-lowpass transormation this basically rprsnts a digital quivalnt o th complx I/ subsampling schm o th prvious sction Two xampl cass ollow in th igurs blow. 7 3 S Hz S L S /L bor down-sampling: B S / S /L S /L S /L S /L S / atr down-sampling: S /L S /L B S /L S /L Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 8 3

17 S Hz S L S /L bor down-sampling: B S / S /L S /L S /L S /L S / atr down-sampling: S /L S /L B S /L S /L 9 3 Intrpolation or Up-Sampling with Complx Signals Th basic block-diagram to incras th sampl rat by an intgr actor L is prsntd in th igur blow, with S again dnoting th original sampl rat. Th output squnc o th up-samplr L is ormd by adding L zros btwn ach original input sampl. As a consqunc, all th spctral imags within L S / L S / o th input spctrum appar now at th output signal at th multipls o th input sampl rat S. Ths ar illustratd in th ollowing igur. Traditionally, th intrpolation iltr Hz attnuats ths xtra imags rtaining only th original spctral componnt. This is, howvr, not th only possibility. Mor prcisly, any o th spctral imags within L S / L S / can b considrd to b th dsird on. As an xampl, th original lowpass signal can b transormd into a bandpass signal by simply using a propr bandpass iltr as th intrpolation iltr Hz. This iltr now rtains th spctral imag at th dsird cntr-rquncy and attnuats Procding o th SDR othrs. 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 3 3

18 S L L S Hz L S B L S / S S S S L S / B L S / S S S S L S / 3 3 EFFICIENT POLYPHASE STRUCTURES Polyphas iltring rprsnts on intrsting approach to implmnt dcimation or intrpolation in a lxibl yt computationally icint way. In th traditional approachs, th dcimation/intrpolation iltrs oprat at th highr sampling rat, i.., ithr bor down-sampling or atr up-sampling. Givn that th down-/up-sampling ratio is L, th ida in th polyphas structurs is to split th rlatd iltring into L paralll stags oprating at th lowr sampling rat in applications rquiring vry high opration spds, this can b a crucial bnit Furthrmor, th polyphas structurs ar quit lxibl i usd,.g., in channlization applications. Ths aspcts will b xplaind in mor dtail in th ollowing. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 3 3

19 Polyphas Dcomposition o FIR iltrs Th so calld polyphas dcomposition o a init lngth iltr Hz is usually ormulatd as L i L H z = z Hi z To put it in words, th output o any FIR iltr in gnral can b constructd as a sum o th outputs o th iltrs H z L, H z L,, H L z L whos input signals ar dlayd by z, z,, z L. Givn that th impuls rspons o Hz is hn, th impuls rsponss o H z, H z,, H L z ar simply L i= h n = { h, h L,...} h n = { h, h L+,...}! h n = { h L, hl,...} 33 3 Exampl: Two-phas dcomposition o Hz=h+hz + +h4z 4 is H z = h + h z + h4 z + h6 z + h8 z + h z + h z + h4 z z h + h3 z + h5 z + h7 z + h9 z + h z + h3 z = H z + z H z whr h n={h, h, h4, h6, h8, h, h, h4} h n={h, h3, h5, h7, h9, h, h3} = H z + z H z Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 34 3

20 Basic Polyphas Dcimators and Intrpolators In dcimation applications, th down-samplr which traditionally oprats on th iltr output can now b transrrd to th ront o th branch iltrs givn that th L tim dlays in th iltrs H z L,, H L z L ar rplacd by ordinary unit dlays this opration is intuitivly clar and thortically justiid by th amous nobl idntity o multirat signal procssing So in th i-th branch, th input signal is dlayd by z i, down-sampld by L, and iltrd using H i z. Finally, th outputs o th L branchs ar summd to orm th inal dcimatd output signal. Notic that all th branch iltrs oprat at th lowr sampling rat! This is illustratd in th ollowing using a simpl xampl Exampl cont d: Down-sampling by L=. H z z H z H z z H z Sinc in gnral th dlay z i is dirnt in vry branch and th down-samplrs oprat synchronously, th low-rat data squncs ntring th branch iltrs ar all disjoint. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 36 3

21 Thus, th ront-nd dlays and down-samplrs can actually b discardd by simply ding vry L-th with a propr tim-shit sampl to th polyphas branchs using a commutativ switch. This inal structur is dpictd in th igur blow. Figur. a Th basic dcimation down-sampling schm. b Th corrsponding polyphas implmntation Similar rasoning can b usd also in th intrpolation cas. Th traditional approach basd on zro-padding and iltring maps into a polyphas structur whr th input signal to b intrpolatd is d dirctly into th branch iltrs H z, H z,, H L z. Th outputs o ths iltrs ar thn up-sampld and dlayd, th dlay bing z i in th i-th branch, and inally summd to orm th intrpolator output. Du to zro-packing and dirnt branch dlays, only on branch iltr output actually contributs to th inal output with othr outputs bing zro at any givn tim instant. Thus, as in th dcimation cas, th inal structur can b implmntd by simply multiplxing th branch iltr outputs using a commutator. This approach is picturd blow. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 38 3

22 Figur. a Th basic intrpolation up-sampling schm. b Th corrsponding polyphas implmntation On intrsting intrprtation o th prvious polyphas structurs is rlatd to aliasing and th amplitud and phas charactristics o th polyphas iltrs. As an xampl, considr th dcimation cas. Sinc th input signal is dirctly down-sampld in ach polyphas branch, all th bands locatd at th intgr multipls o th inal output sampling rat alias to basband. Howvr, du to th rlativ dlays o th dirnt branchs as wll as dirnt phas charactristics o th polyphas iltrs, only th inormation within th passband o th prototyp iltr Hz will appar in th inal dcimator output. To b mor spciic, all th polyphas branch iltrs ar actually allpass iltrs whos amplitud rspons is idally constant. Furthrmor, th phas dlays o th dirnt iltrs dir by /L with H z having th largst dlay. This is asy to s vn intuitivly whn considring how th polyphas iltrs ar obtaind rom th prototyp Hz. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 4 3

23 Bandpass Polyphas Structurs Th prvious discussion is basically valid or both lowpass and bandpass dcimator/intrpolator structurs. Th only dirnc is, o cours, rlatd to th charactristics and dsign o th iltr Hz. A straight-orward way to handl th lowpass and bandpass cass is to considr thm sparatly. Howvr, th ull lxibility o th polyphas structurs can only b capitalizd by trating thm togthr. To illustrat th basic ida, assum th iltr Hz is dsignd or a spciic lowpass dcimation scnario passband B ng B pos in gnral. Now, suppos w wish to chang th structur to procss a bandpass signal locatd around a cntr-rquncy C which is an intgr multipl o th output rat S /L. In trms o th normalizd rquncy variabl ω =π/ S, this mans that ω C is an intgr multipl o π/l. 4 3 In gnral, an analytic iltr Gz to xtract th intrsting band can b obtaind simply by rquncy translating th corrsponding lowpass prototyp iltr Hz as gn=hnxp{jω C n}. Th impuls rsponss o th corrsponding polyphas implmntation o gn ar thn obtaind as prsntd bor, i.., g n={g, gl, }, g n={g, gl+, }, tc. Howvr, atr som manipulations, th polyphas impuls rsponss or th analytic iltr Gz can simply b writtn as g g g L n = h n = h n = h g n = h! n n xp{ jkπ / L} n xp{ jkπ / L} L n xp{ jkπ L / L} whr th intgr k is th channl indx, i.., ω C = kπ/l. => th sam polyphas rprsntation as in th lowpass cas xcpt or th constant complx multiplirs xpjkπi/l!!! Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 4 3

24 Exampl: Bandpass down-sampling by L=4 with ω C = π/ i.., C = S /4, gn=hnxpjπ/n g g g 3 n = { h g n = { h n = { h n = { h3 π j π j π j π j 3, h4, h5, h6, h7 π j 4 π j 5 π j 6 π j 7, h8, h9 π j 8 π j 9, h, h,...} = { h, h5, h9,...} π j π j,...} = { h, h4, h8,...} π j,...} = { h, h6, h,...},...} = { h3, h7, h,...} So as statd bor, th modulating xponntials in th polyphas iltrs rduc to constant multiplirs w i,k =xp{jkπi/l}, i =,,, L. This ct, in turn, can b implmntd by simply multiplying th outputs o th lowpass polyphas iltrs by th constants w i,k!!! As a consqunc, th sam polyphas ront-nd th sam iltrs!!! can b usd to xtract any channl locatd at th multipl o th output rat. This gnral polyphas dcimator structur is illustratd in th igur blow. jπ 3π j 43 3 Figur. Gnral polyphas dcimator. Similar typ rasoning can again b usd also in th intrpolation up-sampling cas. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 44 3

25 Matlab Dmo Th prvious idas o bandpass polyphas iltring and dcimation ar illustratd using Matlab. As an xampl cas, w hav thr channls #: ω C =.5π, 8PSK modulatd signal, 6 sampls pr symbol #: ω C =.5π, 4PSK modulatd signal, 6 sampls pr symbol #3: ω C3 =.75π, 6AM modulatd signal, 6 sampls pr symbol In all th cass, raisd-cosin puls-shap with 35% roll-o is usd. Sinc th channls ar cntrd at intgr multipls o π/4 i.., S /8, w us a polyphas structur with L=8 atr downsampling and polyphas iltring, a urthr dcimation by is includd to gt symbol rat sampls in ordr to plot th output constllations Th simulation stup and th rsults ar illustratd in th ollowing igurs H z π xp j k 8 H z π xp j k 8 H z k=: k=: π xp j k 7 8 H 7 z k=3: Figur. Dmonstration st-up, L =8. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 46 3

26 Ral bandpass signal and th analytic channl iltr, k =.8 Amplitud Frquncy ω / π h n h n h n h 3 n h 4 n h 5 n h 6 n h 7 n. 5 n. 5 n Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 48 3

27 H [db] H [db] ω / π ω / π H [db] H 3 [db] ω / π ω / π H 4 [db] H 5 [db] ω / π ω / π H 6 [db] H 7 [db] ω / π ω / π 49 3 Phas dlays o th polyphas iltrs argh i / ω ω / π Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 5 3

28 .5 Symbol rat output sampls, channl #.5 IM RE 5 3 Ral bandpass signal and th analytic channl iltr, k =.8 Amplitud Frquncy ω / π Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 5 3

29 .8 Symbol rat output sampls, channl #.6.4. IM RE 53 3 Ral bandpass signal and th analytic channl iltr, k = 3.8 Amplitud Frquncy ω / π Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 54 3

30 4 Symbol rat output sampls, channl #3 3 IM RE 55 3 Notic that vn though analytic channl iltrs ar shown in th thr igurs, th actual implmntation is rally basd on th polyphas concpts. In othr words, thr is no singl iltr with th givn rspons but th polyphas dcimator ctivly implmnts that kind o unction. Rmmbr also that th branch iltrs or ach channl k =,, 3 ar idntical, only th scaling coicints w i,k atr th iltrs dpnd on th channl indx. Notic also that th structur works similarly or complx input signals as wll in th prvious xampl, th input is ral-valud. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 56 3

31 3. I/ MISMATCH PROBLEMS IN ANALOG I/ Signal Procssing in Rcivrs I/ SIGNAL PROCESSING In communication rcivrs, on o th ky ront-nd unctionalitis is to downconvrt th dsird channl signal rom RF closr to basband, in th prsnc o othr channls/signals. In this contxt, th undamntal problm o imag signal attnuation is a major concrn. Traditionally, in suprhtrodyn and its variants rcivrs, th imag band is attnuatd using RF iltring bor th down-convrsion stag th basic imag signal problm is illustratd in th ollowing igur whr th targt is to translat th dsird channl signal to an intrmdiat rquncy IF IF th dsird channl is illustratd in gry and it s imag in dark, sparatd in rquncy by IF 57 3 IF IF LO LO IF IF IF IF Th lowr part illustrats th signal atr down-convrsion and lowpass iltring without lt and with right RF imag rjction iltring. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 58 3

32 In practic a trado is ndd in th slction o th IF: th highr th IF, th asir it is to implmnt th RF imag rjction iltr sinc th sparation o th dsird and imag bands is IF on th othr hand, th lowr th IF, th asir it is to implmnt th channl slctivity iltring In ordr to rduc th ndd RF imag rjction iltring, complx I/ down-convrsion can b usd instad o ral mixing thortically a pur rquncy translation and imag problms ar avoidd during th rquncy shit this ida can b usd in th rcivr ront-nd simply or down-convrsion purposs, indpndntly o th dsird channl modulation A gnralizd rcivr basd on this ida appars in igur blow. Th 9 phas shit can basically b introducd ithr btwn th local oscillator LO signals point A or btwn th input signal branchs point B point A: cos. and sin. LO signals point B: a widband Hilbrt transormr and two in-phas LOs 59 3 AGC LPF A/D I RF LNA LO DSP A AGC B LPF A/D In thory: th rsulting I and channls should hav qual amplituds and a phas dirnc o 9 ininit attnuation or th imag signal band In practic: ral-world analog componnts, such as th mixrs, LPFs, tc., can nvr b prctly matchd som imbalanc will always xist th imag attnuation is init ormal proo will b givn latr Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 6 3

33 - and -% imbalanc valus ar ralistic, rsulting in -4 db imag attnuation. To b mor prcis, lt s start to look at th ollowing situation in mor dtail: widband ront-nd, no imag rjction iltring idalizd cas multichannl s th ollowing igur bandpass signal bandwidth B cntrd at LO targt: produc a widband basband quivalnt o th rcivd signal widband downconvrsion how: using I/ signal procssing Obtain a ormal charactrization o th imbalanc cts du to th mismatchs o practical analog lctronics irst som basic rsults ar givn in th cas whn th amplituds and/or phass o th two componnts I and o a complx signal bcom mismatchd ths rsults ar thn applid to th widband rcivr cas 6 3 R B LO LO Z Z'? IF IF IF IF Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 6 3

34 63 3 Narrowband Imbalanc Modl Considr an idal singl rquncy signal sin cos t j t t z t j ω ω ω + = =. Hr, w assum that th mismatchd I and componnts hav rlativ amplituds g and g rlativ phass φ and φ In othr words, w writ th imbalancd signal t z as sin cos φ ω φ ω = t jg t g t z Rcap: Eulr s ormulas cos jx jx x + = and j x jx jx sin =. Thn, using th Eulr s ormulas, t z can b writtn as t j j j t j j j g g g g t z ω φ φ ω φ φ + + = 64 3 So, in addition to th original componnt t j ω, t z consists also o th mirror rquncy componnt t j ω. Considring th rlativ strngths o th two rquncy componnts, t z can also b writtn as φ ω φ φ φ ω φ φ ω φ φ φ ω φ φ φ = + + = t j j t j j t j j j t j j j g g g g g g g g g g g g t z So w notic that th rlativ strngths o th two rquncy componnts dpnd only on th rlativ amplitud and phas imbalancs / g g and φ φ Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd

35 65 3 Thror, to simpliy th notations, w assum = g, g g = and = φ, φ φ = Thn, th modl can b writtn in a mor simpl orm as t j j t j j g g t z ω φ ω φ + + = Widband Imbalanc Modl Thorm: Considr a complx-valud signal t jz t z t z I + = and its complx conjugat * t jz t z t z I = whos Fourir transorms ar givn by jz Z Z t z I + = * * * * jz Z jz Z Z t z I I = = Thn, th Fourir transorm o Z jh Z H Z I I + = can b writtn as 66 3 * Z H H Z H H Z I I + + = Proo: Dirct substitution nxt pag will yild q..d. * Z jh Z H Z jh Z H Z jh Z H Z jh Z H Z jh Z H jz Z H H jz Z H H Z H H Z H H Z I I I I I I I I I I I I I I I I + = = = + + = Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd

36 As can b obsrvd, th mismatchs caus a signal componnt rlativ to z * t to appar in addition to th original signal componnt z t! Again, only th dirnc btwn H I and H contribut to th rlativ * strngth o th mirror componnt Z i.., z * t. Clarly, a gnralization o th narrowband cas. Imbalanc Ects in Rcivr Front-Ends In gnral, all th analog componnts such as th quadratur mixing stag branch iltrs A/D convrtrs acting th I and branch signals contribut to th ctiv amplitud and phas mismatchs. Motivatd by this, th modl o th ollowing igur is usd hratr x LO,I t = cosπ LO t x I t H NOM H I z I ' t rt x t H NOM H z ' t x LO, t = gsinπ LO t + φ Th ct o quadratur dmodulator: th local oscillator signal x LO t o an imbalancd quadratur dmodulator is hr modlld as x t = cos π t jgsin π t+ φ LO LO LO jπ LOt jπ LOt = K + K whr g and φ rprsnt th dmodulator amplitud and phas imbalancs, rspctivly. Th mismatch coicints K and K ar givn by Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 68 3

37 idally j K = [ + g φ ]/ =, idally j K = [ g φ ]/ =. For mor dtails, s th narrowband cas in th bginning o th matrial us g g. Th ct o branch componnts: th branch componnt mismatchs can b asily modlld as imbalancd lowpass iltrs LPF as givn by H H, I H NOM H I LPF =, H NOM H LPF = H NOM is th nominal LPF rspons rjcting th high-rquncy componnts. H I and H rprsnt th actual mismatch cts du to branch iltrs, AGCs, A/Ds, tc. with prct matching idal cas, H I = H 69 3 Now, to xplicitly charactriz th imbalanc cts on th individual channl signals, w writ th multichannl rcivd signal rt as rt = R[ zt ] = zt + z t j π t j π t * j π t LO LO LO As a modl, th rcivd signal rt is down-convrtd to basband by mixing it with x LO t. Assuming that H NOM = or B/ and H NOM = or > B/, th downconvrtd signal xt can b asily writtn as x t K z t K z t * = + To analyz th ct o branch mismatchs, th signal xt can irst b writtn as xt = x I t + jx t, whr x t = z t I x t = gcos φ z t gsin φ z t I Thn, in trms o Fourir transorms, th signal z't = z I 't + jz 't atr branch mismatchs is givn by I Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 7 3

38 Z = Z I + jz = HI XI + jh X = HI ZI + jh [ gcos φ Z gsin φ ZI ] = [ HI jh gsin φ] ZI + j[ H gcos φ] Z. = A Z + ja Z I I Atr som manipulation, th abov rsult can b writtn in a mor convnint orm as s th gnral thorm in th widband imbalanc modl sction whr * Z = G Z + G Z G [ ]/ = AI + A j = [ H + H g φ ]/ I G [ ]/ = AI A j = [ H H g φ ]/ I 7 3 In th abov modl, th trm rlativ to Z * is causd by th imbalancs and rprsnts th imag aliasing ct. with prct matching, I/ procssing allows us to considr ngativ and positiv rquncis sparatly this sparabililty o ngativ and positiv rquncis th idal cas is lost du to mismatchs Now, th imag attnuation o th analog ront-nd can b dind as L = G / G With practical analog lctronics as statd arlir, this attnuation is usually in th ordr o 4 db. Th important qustion is whthr this 4 db attnuation is suicint dpnds on th architctur dirct-convrsion zro-if rcivr: suicint, spcially with low-ordr modulations low-if rcivr: insuicint, vn though th systm spcs hlp to som xtnt gnral widband rcivr: clarly insuicint Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 7 3

39 R LO Z Z * LO IF IF IF IF Z' = G Z + G Z * IF IF 73 3 Possibl solutions to nhanc th imag attnuation: analog RF imag rjct iltring ral or complx diicult to implmnt, complicats th analog ront-nd limits th intgrability limits th lxibility advancd DSP at basband rrrd to as imbalanc compnsation stimat th amplitud and phas mismatchs and corrct thm training signals ndd usually diicultis with rquncy-slctiv mismatchs novl statistical signal procssing basd mthods to rmov th imag intrrnc signal stimation vs. mismatch stimation basic tools: adaptiv intrrnc cancllation or blind signal sparation blind, i.., no training signals ndd can asily cop with rquncy- and tim-dpndnt imbalancs to b introducd nxt Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 74 3

40 4. ADVANCED DSP FOR I/ IMBALANCE COMPENSATION Basband Signal Modl or Digital Imbalanc Compnsation Hr, in our ormulation, th task o imbalanc compnsation is to nhanc th init imag attnuation L o th analog procssing. Mor prcisly, th targt is to obtain an imag-r obsrvation o a spciic channl rrrd to as th dsird channl signal locatd at non-zro intrmdiat rquncy IF atr th initial widband downconvrsion this signal stimation basd approach is dirnt rom traditional imbalanc compnsation tchniqus traditional approach is to try to stimat th imbalanc paramtrs and us thm in som kind o a corrction ntwork hr w dirctly stimat th inal quantity o intrst ; th dsird signal Som notations dsird channl gry basband quivalnt signal s t imag channl dark basband quivalnt signal i t P X = E xt in gnral 75 3 As in th idal prct matching cas, th imbalancd multichannl signal Z' contains a dsird signal componnt around + IF an imag componnt around IF Du to imbalancs s th prvious igur, Z' has also a dstructiv imag signal componnt around + IF also a dsird signal componnt around IF Motivatd by this, w gnrat two basband obsrvations, dt and vt dt is th basband obsrvation o th combind signal around + IF vt is th mirrord complx-conjugatd basband obsrvation o th combind signal around IF this obsrvation gnration is illustratd in th ollowing igur th dsird signal st is thn stimatd as s ˆ t = L{ d t, d t, d t,..., v t, v t, v t,...} what kind o procssing L{.} is usd, is not dind yt Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 76 3

41 Z' D V B C IF IF In n j j IFn ω j IF n + ω H LP z H LP z. * dn vn ALGORITHM COMPENSATION sˆ n 77 3 Atr som manipulation, th rquncy domain xprssion or ths obsrvations can b writtn using matrix ormulation as X = A S whr X = [D V] T, S = [S I * ] T, and st and it dnot th basband quivalnts o th dsird and imag signals, rspctivly. Th matrix A is givn by G + IF G + A = * * G G IF IF IF P, B whr B C dnots th individual channl bandwidth and P,B C = diag[π,b C, Π,B C ] with Π,B C = or B C / and zro othrwis. I H I = H, th gnral modl rducs to an instantanous mixtur modl d t K PS K PI s t = * * v t K PS K PI s t whr s t = st/sqrtp S, s t = i * t/sqrtp I, and P X = E xt in gnral. This Procding is a valid o th SDR modl 3 Tchnical i th Conrnc I/ dmodulator and Product Exposition. is Copyright th main 3 sourc SDR Forum. o All imbalanc. Rights Rsrvd C 78 3

42 Notic that compnsation is actually ndd only i th imag signal is mor powrul than th dsird signal, i.., s ˆ t = d t is a good stimator i P S >> P I Evn though th basband modl was hr drivd in continuous-tim domain, th obsrvations can in practic b gnratd digitally atr A/D xcss imbalanc cts ar avoidd Consquntly, discrt-tim notations dn, vn, sn, and in ar usd hratr. In gnral, th obsrvations x n = dn and x n = vn appar as convolutiv mixturs o th ctiv sourc squncs s n = sn and s n = i * n th instantanous mixtur modl is a spcial cas o this 79 3 Adaptiv Intrrnc Cancllation IC Basd Compnsation I th dsird channl signal is originally mor powrul than th imag signal P S >> P I, th imag attnuation o analog procssing is suicint th dsird channl obsrvation dn can b usd dirctly as an stimat o sn On th othr hand, in th diicult cas o a strong imag signal P I >> P S, th attnuation o analog procssing is insuicint vn is highly corrlatd with th intrring signal componnt s th prvious igur but only wakly corrlatd with th dsird signal componnt o dn Motivatd by this, adaptiv intrrnc cancllr can b usd to stimat sn as N k IC sˆ n = d n = w n v n k IC Th iltr coicints w k n, k = {,, N IC }, can b adaptd with any practical algorithm, such as th wll-known last-man-squar LMS or rcursiv lastsquars RLS algorithms. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd k 8 3

43 Multichannl Blind Dconvolution MBD Basd Compnsation Assumption: Signals in dirnt rquncy channls ar statistically indpndnt. Obsrvation: With practical imbalanc valus, th gnral mixtur modl is invrtibl i.., A is non-singular. blind signal sparation or multichannl blind dconvolution in gnral algorithms can b usd to stimat th sourc vctor sn = [sn i * n] T as NMBD = k = Wk n x n k s ˆ n whr xn = [dn vn] T. th actual dsird channl signal sn is thn stimatd as sˆ MBD T i n = sˆ n, i = or, whr = [ ] T or = [ ] T is usd, dpnding on th possibl sourc prmutation i.., MBD producs also an stimat o th imag signal. In gnral, thr xists a wid varity o dirnt approachs to masur th indpndnc o th sparatd output signals, and thus, to adapt th dmixing matrics W k n, k = {,, N MBD }. 8 3 Comparisons In gnral, th IC basd compnsator is only utilizabl i th imag signal is mor powrul than th dsird channl signal. Consquntly, som kind o powr stimation o th dirnt channl signals is ndd to dcid whn to switch th IC structur on and o. This problm, usually rrrd to as signal lakag, can in thory b avoidd using th MBD basd compnsator, though, no compnsation is actually ndd i th imag signal is wak. On th othr hand, th prormanc o th IC basd solution is likly to b mor insnsitiv to th cts o additiv nois and symbol timing rrors, and, spcially, to dirnt intrrr typs. Naturally, thr is also th issu o computational complxity. Both tchniqus ar blind, i.., no training signals ndd!! ar abl to handl rquncy-slctiv mismatchs ar also abl to cop with tim-variant mismatch cts du to inhrnt adaptiv natur Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 8 3

44 Simulation Exampl Front-End Paramtrs: Th rcivd signal consists o th dsird and imag channls o bandwidth.π locatd originally around.7π and.3π, rspctivly. Th dsird and imag signals ar PSK- and 8PSK-modulatd, rspctivly, with raisd-cosin puls-shaps roll-o.35. Th rlativ powr dirnc is 4 db. In translating th dsird channl signal to an IF o.π, imbalanc valus o g =. and φ = ar usd or th quadratur dmodulator. Atr that, th branch mismatchs ar modlld as H I z =. + z +.z and H z =. + z +.z. Finally, th symbol rat basband obsrvations dn and vn ar gnratd by propr rquncy translations o ±.π, lowpass iltring, and dcimation IC Simulation: Th standard RLS algorithm with a orgtting actor o.999 is usd to adapt th IC iltr o lngth 5 N IC = 4. Th total numbr o sampls is, to guarant stady-stat opration. MBD Simulation: Th natural gradint basd algorithm o Amari t al. is usd with a stp-siz o. to adapt th dmixing iltrs o lngth 5 N MBD = 4. Th total numbr o sampls is, to quarant stady-stat opration. To stabiliz th adaptation, th sourc sparator input signals x n = dn and x n = vn ar normalizd as ad-hoc x n x n/sqrtp, x n x n/sqrtp, whr th powr stimats P n and P n ar obtaind rcursivly as P i n =.995P i n +.5 x i n, i = and. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 84 3

45 Commnts: Front-nd imbalanc proprtis usd in th simulations ar illustratd in Figur and Figur. Singl ralizations o th absolut valu o th cntr taps w n and W,ij n = [W n] ij ar prsntd in Figur 3 and Figur 4 to illustrat th convrgnc proprtis. Clarly, th IC algorithm convrgs much astr than th MBD algorithm. Howvr, as vriid by Figur 5, thr is no dirnc in th stady-stat opration btwn th two mthods only th dsird signal stimat is shown or th MBD mthod, it also producs an stimat o th original imag signal H / H I argh argh I Frquncy ω / π Figur. Rlativ amplitud and phas mismatchs o th branch componnts. Positiv gry and ngativ dark IF bands ar illustratd in dirnt colours. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 86 3

46 .. G G Frquncy ω / π Figur. Gnral imbalanc coicints G and G. Positiv gry and ngativ dark IF bands ar illustratd in dirnt colours Absolut valu W, 5 W, Absolut valu 5 W, W, Itration numbr n Figur 3. Cntr tap valus W,ij n = [W n] ij o th dmixing iltrs using th natural gradint algorithm stp-siz.. Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 88 3

47 . Absolut valu.5..5 w Itration numbr n Figur 4. Cntr tap valu w n o th RLS orgtting actor.999 basd intrrnc cancllr. ŝ IC ŝ MBD Figur 5. Compnsatd output sampls atr convrgnc o both th IC lt and MBD right basd compnsators. Also shown ar th idal PSK symbol locations whit astrisks Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd

48 5. SECOND-ORDER SAMPLING AND ENHANCEMENTS Starting point: Complx I/ sampling o bandpass signals. input S I HT S Ida: Approximat th ndd 9 dgr phas shit Hilbrt transormr using a simpl tim dlay o on quartr o th carrir cycl. input S I DELAY S 9 3 Th tim dlay o T =/4 C corrsponds to a rquncy-dpndnt phas shit o π π T = C thus th idal Hilbrt iltring is achivd only at th carrir-rquncy ± C as a rsult, th suprssion o th ngativ rquncy contnt imag componnts is gnrally imprct input spctrum input spctrum output spctrum output spctrum Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd? 9 3

49 Filtring Ect o th Dlay Procssing W writ th input bandpass signal r t in trms o its basband quivalnt signal z t = z t jz t as scaling by is irrlvant and could b ignord I + r t = R[ z t = z I jπ Ct tcosπ C ] = z t t z jπ Ct + z * tsinπ t C t jπ Ct Thn th complx signal, say xt, ntring th samplrs is gnrally o th orm x t = r t + jr t T whr T = /4. C Th spctrum o this signal can b writtn as whr X = [ + jh ] R H D D j π T =. This ct is illustratd in rquncy domain in th ollowing igur. 9 3 WITH HILBERT TRANSFORMER +jh HT C C WITH DELAY +jh D C C Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 93 3

50 Now suppos you us a subharmonic sampling rquncy S = C / r, whr th intgr r dnots th subsampling ratio. th signal is aliasd dirctly to basband in cas o dlay procssing, th componnt rom ngativ rquncis alls dirctly on top o th dsird componnt kind o sl-intrrnc actually dpnds on th structur o th signal o in cas o a singl rquncy channl as shown blow, this intrrnc is not rally problmatic, spcially with low-ordr modulations o but how about in th widband/multichannl cas? WITH HILBERT TRANSFORMER WITH DELAY S S S S 94 3 Whn applid to multichannl signals, th imag band signal ban b much mor powrul than th dsird channl obvious problm quantitativ masurs givn shortly +jh HT C C IF IF +jh D C C IF IF Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 95 3

51 Imag Attnuation Analysis Atr som manipulations, th sampld complx signal with S = C / r, r intgr can b writtn as x nts = rnt S + jrnt S T =... = zi nts + j z nts T = I n + j n Thus it is intrsting to not that th ront-nd dlay maps dirctly into a corrsponding dlay o th basband obsrvation. For analysis purposs, w can ormally think that th discrt-tim signal is obtaind by sampling th corrsponding continuous tim basband signal z t = z t + jz t T This can b writtn in a mor inormativ orm as I z t = z t + z t T + z t z t T * * 96 3 th part including zt and zt T corrsponds to th signal componnt originating rom positiv rquncis th part including z * t and z * t T corrsponds to th signal componnt originating rom ngativ rquncis To gt th xact imag attnuation, w writ th Fourir Transorm o z't as jπ T jπ T Z = + Z + Z * Thn, th imag attnuation L providd by th scond-ordr sampling is givn by L jπ T + + cos π T = =... = jπ T sin π T To illustrat, i th cntr rquncy C = MHz and th bandwidth B=5 MHz, th imag attnuation L at th band dg is only around db in cas o multichannl downconvrsion, th powr dirnc o th individual channl signals can b vn 5 db thn, it is clar that th imag attnuation L and thus, th basic scondordr sampling schm, ar not suicint as such or multichannl rcivrs!! Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 97 3

52 Scond-Ordr Sampling and Enhancd Imag Rjction In narrowband singl channl rcivrs, th imag signal is inhrntly a "slimag" and th attnuation o th basic scond-ordr sampling schm as such can b adquat. In multichannl rcivrs, as discussd bor, th imag band signal can b up to 5- db strongr than th dsird channl signal, and th imag attnuation o th basic scond-ordr schm alon is clarly insuicint. Two altrnativ mthods to nhanc this imag attnuation ar prsntd nxt. For simplicity, th ollowing compnsation mthods ar analyzd in continuoustim domain Intrrnc Cancllation Th basic ida: To nhanc th obtainabl imag attnuation, and thus, to rproduc an accurat basband obsrvation o th multichannl signal z t, th intrrnc cancllr typ o compnsation structur is proposd. I'n=rnT S rt S z nts + zˆ nt S DELAY S 'n=rnt S T j. * C jω Procding o th SDR 3 Tchnical Conrnc and Product Exposition. Copyright 3 SDR Forum. All Rights Rsrvd 99 3