represents the amplitude of the signal after modulation and (t) is the phase of the carrier wave.

Transcription

1 1 IQ Sgnals general overvew 2 IQ reevers IQ Sgnals general overvew Rado waves are used to arry a message over a dstane determned by the ln budget The rado wave (alled a arrer wave) s modulated (moded) by the message sgnal mt (); n other words the ampltude and/or phase o the arrer wave s moded so as to nlude the normaton stored wthn the message The general orm o a rado sgnal s as ollows: ( t) s( t) Re{ a( t) e } a( t)os ( t), where a (t) represents the ampltude o the sgnal aer modulaton and (t) s the phase o the arrer wave In the speal ase where there s no modulaton appled to the arrer sgnal (e: no message sent), a we wrte s( t) ( t) osllatng at a requeny o and saled by the arrer ampltude oeent A s the result a( t) (onstant ndependent o tme) A ( t) 2 t (lnear nrease o phase wth tme, 2 radans every 1/ seonds ( t) A os(2 t) A os( t) Ths sgnal s alled the arrer wave (t) and an also be vsualzed as a phasor wth angular requeny 2 and wth perod T 1/ In general the phase o the arrer (t) may have a tme varyng phase omponent (t) that s added to the lnear phase, thus n general: s( t) a( t)os( ( t)) ( t) 2 t ( t) j () 2 s( t) a( t)os(2 t ( t)) Re{ a( t) e t e j },

2 The rado sgnal st () s a osne wave at requeny wth tme-varyng ampltude and phase a( t), ( t) Thus the only way to enode the message mt () on the arrer wave s to vary at () and/or () t n step wth the message m(t) Thus a( t), ( t) are speed as a unton o the message m (t) or some modulaton types and as a onstant or others These untons wll be speed later when we dsuss spe modulaton types Usng the dentty os( A B) os Aos B sn Asn B j () 2 the general rado sgnal s( t) a( t)os(2 ( t)) Re{ a( t) e t e j } may also be wrtten s( t) a( t)os 2 t os ( t) a( t)sn 2 t sn ( t) where and thus I( t) jq( t) a( t) e t Thus we an wrte ( t) j2 s( t) Re{ a( t) e e } I( t)os 2 t Q( t)sn 2 t I( t) at ( )os } () t ( t) Re{ a( t) e, Q( t) a( t) sn ( t) Im{ a( t) e Re{[ I( t) jq( t)][os 2 t jsn2 t]} I( t)os 2 t Q( t) sn2 t a() t os[2 t ( t)] () The general rado sgnal st () must be a real sgnal that we an vew on an osllosope We an wrte st () as the real part o a omplex sgnal st () an be desrbed as ether a osne wave wth ampltude at () 0 and phase () t, or the sum o a osne wave wth ampltude It () and a sne wave wth ampltude Qt (), where I( t), Q( t) an be greater or less than zero In both ases, the message s a two-dmensonal (omplex) sgnal represented usng ether at ( ), () t (polar orm) or I( t), Q( t ) (retangular orm) In the gure, M( t) a( t) () t }

3 I( t), Q( t) are untons o the message sgnal(s), where the exat unton depends on the modulaton type A smple example s to onsder the message sgnal to be a stereo (2- hannel) mus sgnal wrtten as m ( t), m ( t) and hoose m ( t) I( t), m ( t) Q( t) L R I( t) jq( t) s the so-alled omplex baseband sgnal that s a unton o the message For a stereo mus sgnal, the omplex baseband sgnal s ml( t) jmr( t) The rado sgnal st () s the so-alled real passband sgnal that ontans the message modulated onto the arrer wave at requeny In the speal ase where I( t), Q( t) are both onstants IQ,, then st () s a osne wave wth onstant ampltude and phase The gure below shows the rado requeny (RF) sgnal s( t) I os2 Qsn 2 aos2 s a osne wave at requeny wth onstant ampltude and phase a, The n-phase omponent I os2 and the quadrature omponent Qsn 2 are also osne waves wth onstant ampltude and phase L R Rado transmtter (modulator)

4 os2 s(t) sn 2 The gure shows a rado transmtter (or modulator) that produes the rado waveorm s( t) I( t)os2 t Q( t)sn 2 t rom the message sgnals I( t), Q( t ), where n ths example, 107MHz The rado transmtter (modulator) may also be desrbed n omplex orm wth a omplex multplaton o the omplex baseband message I( t) jq( t) wth os 2 t jsn2 t to yeld a omplex sgnal ( t) j2 sˆ ( t) a( t) e e [ I( t) jq( t)][os 2 jsn2 ] whose real part s s( t) I( t)os 2 Q( t) sn2 The omplex message I( t), Q( t ) s multpled by the omplex arrer wave e j2 os2 t jsn2 t The rado sgnal st () s the real part o ths omplex multplaton s( t) I( t)os2 t Q( t)sn 2 t

5 Rado reever (demodulator) The sgnal st () s transmtted over a dstane va some hannel (wred or wreless), attenuated by the path loss L 0 and ped up by a reever n the orm r( t) s( t) / L0 The reever s tas s to reover the message sgnals I( t), Q( t) rom the sgnal rt () Ths an be done usng the reever shown below In ths reever, I( t), Q( t ) are dgtzed by an analog-to-dgtal onverter (A/D or ADC) os2 sn 2 A more omplete drawng o the reever adds some pratal omponents

6 In ths gure, we have added: RX lters needed to lter out undesred sgnals on nearby requenes, a Low Nose Ampler (LNA) to amply rt () that s typally the the mrovolt range to a level n the volt range sutable or ADC, Automat Gan Control (AGC) to adjust the gan to ompensate or varatons n the level rt () and low pass lters (LPF) beore the ADC To see how ths reever wors, we alulate the sgnals x( t), y( t) at the two ADC nputs, and nd that they are equal to I( t), Q( t) Exerse: prove ths Several trgonometr denttes wll be needed: os os [os( ) os( )] / 2 sn os [sn( ) sn( )] / 2 sn sn [os( ) os( )] / 2 Also, the double requeny terms (osne waves at 2 ) are ltered out by the low pass lter (LPF) Alternately, we an prove ths usng the omplex sgnals We wrte the reeved sgnal n omplex orm 1 () t j2 rˆ ( t) L a( t) e e L [ I( t) jqt ( )][os 2 t jsn2 t] The reever demodulates the omplex rado sgnal by multplyng rt ˆ( ) by the omplex j2 loal osllator e os2 t jsn2 t to yeld rˆ( t) e [ L a( t) e e ] e L a( t) e L [ It ( ) jq( t)] j 2 1 j ( t) j 2 j 2 1 ( t)

7 Summary We have reated a ommunaton system wth message sgnals I( t), Q( t ) that are modulated onto a arrer wave at requeny to reate the rado sgnal s( t) I( t)os2 Q( t)sn 2 st () travels over a dstane va a hannel wth path loss L o The reever ps up the sgnal r( t) s( t) / L0 and reovers the messages I( t), Q( t) Messages When we study modulaton, we oen hoose a smple message, m( t) A os(2 t), where m s the modulaton/message requeny and s usually on the order o Hz or Hz, In prate, we wsh to transmt a message wth more than a sngle tone at a sngle requeny A general analog (message would be represented by a sum o osne waves wth derent ampltudes and phases A, at eah requeny m( t) A ( t)os(2 t ( t)) We oen assume that mt () s dvded nto rames o length T n the range 5-20 mllseonds Durng eah rame at tme t T, we assume A ( t) A,, ( t), are onstant, so we an wrte m( t T ) A, os(2 t, ) At the dsrete tmes t T, the message s the sum o osne waves wth requenes ampltudes A, and phases, that are derent or eah rame Frames are normally overlapped An analog message an also represent a dgtal symbol sequene a by wrtng mt ( a p( t T ) ) m m

8 where pt () s a pulse that spans a nte tme perod, a may be bnary symbol 1 (to represent bnary 1 or 0) or multlevel (eg 1, 3 to represent 00, 01, 10, 11) and T s the symbol tme We an have two suh sequenes m () t I a p( t T ) I( t) m () t b p( t T ) Q() t Q Thus we an wrte the omplex baseband sgnal as I( t) jq( t) ( a jb ) p( t T ), where j a jb r e s a omplex data symbol I both a, bare bnary, then represents 2 bts o normaton at eah tme t T I both a, bare or multlevel 1, 3, then represents 4 bts o normaton at eah tme t T

9 2 IQ reevers wth tme- 21 IQ sgnal revew In summary, any sgnal st () an be wrtten as a arrer wave at requeny varyng ampltude and phase, e st () a( t)os[2 t ( t)] ( t) j2 Re{ a( t) e e } Re{[ I( t) jq( t)][os2 t jsn2 t]} I( t)os 2 t Q( t) sn2 t a( t)os[2 ( t)] where () t I( t) at ( )os ( t) Re{ a( t) e }, () t Q( t) a( t) sn ( t) Im{ a( t) e } and j () t s() t I( t) jq( t) a( t) e s alled the omplex envelope o the sgnal The omplex envelope ontans two real waveorms The omplex envelope (or the two real waveorms) ontan the normaton or message The real sgnal st () s obtaned by multplyng the omplex envelope st () wth the j2 omplex arrer wave e os2 jsn2 and tang the real part to yeld j2 st ( ) R e{ s( t) e } 22 USRP dgtal I-Q reever An IQ reever s job s to extrat the omplex envelope j2 rom the sgnal st ( ) R e{ s( t) e } Ths s done by multplyng st () by a omplex arrer wave j2 e os2 t jsn2 t (note the mnus sgn) s() t I( t) jq( t) a( t) j () t e Ths multplaton s alled omplex downonverson It an be done mathematally usng real sgnals or omplex sgnals, as shown below

10 The USRP reever has two stages o o IQ downonverson, one analog IQ downonverter stage n the WBX daughterboard, and a dgtal IQ downonverter (DDC) stage on the motherboard Both IQ downonverters operate by generatng two loal osllator sgnals at LO and mxng (multplyng) t wth a desred rado requeny (RF) sgnal at (ped up by the antenna or ed n by a sgnal generator) to yeld a sgnal at the derene requeny b LO Mathematal proo: In what ollows, we need some trgonometr denttes os os [os( ) os( )] / 2 sn os [sn( ) sn( )] / 2 sn sn [os( ) os( )] / 2 Mathemats o omplex downonverson n real notaton One o the loal osllator sgnals s wrtten os2 t, and the desred RF sgnal that we wsh to reeve s wrtten a( t)os[2 ( t)] We assume a( t) 1, ( t) 0 or the moment, so the desred RF sgnal s smply an unmodulated arrer wave os2 The RF sgnal s a real sgnal that an be seen on a sope LO

11 The IQ reever eetvely has two loal osllators operatng 90 degrees out o phase, os 2 LOt and sn 2 LOtand two mxers Thus there are two outputs that we all It () and Qt () The os mxer unton multples these two sgnals to yeld os 2 os2 LOt 2, 2 we apply a trgonometr dentty wth LOt and wrte os t os 2 t 05os 2 ( ) t 05os 2 ( ) t 2 LO LO LO Thus multplyng two sne (or osne) waves at requenes LO and results n two new sne waves, one at the sum requeny LO and one at the derene requeny LO The sgnal at the derene requeny I( t) 05os 2 ( ) t 05os 2 t LO b LO s wrtten The sn mxer unton multples these two sgnals to yeld os 2 ( sn 2 LOt) 2, 2 we apply a trgonometr dentty wth LOt and wrte sn t os 2 t 05sn 2 ( ) t 05s n 2 ( ) t 2 LO LO LO The sgnal at the derene requeny s wrtten Q( t) 05sn 2 ( ) t 05sn 2 t LO b We an wrte these two sgnals It () and Qt () as one omplex sgnal s() t ) j t () I( t) jq( t) a( t e that has tme varyng ampltude and phase at ( ), () t Mathemats o omplex downonverson n omplex notaton The IQ reever unton an also be desrbed usng omplex sgnals as ollows The omplex RF sgnal () t j2 ˆ ) r( t) a( t e e j2 LOt e os 2 LOt jsn2 LOt s multpled by the omplex loal osllator to yeld j 2 LOt rˆ( t) e j ( t) j [ ( ) 2 j ] 2 LOt a t e e e ( t) j ( ) 2 bt a t e e I( t) jq( t)

12 wth b LO The reeved omplex baseband sgnal s rt ( ) It ( ) jq( t) The dagram below usng omplex sgnals perorms the same unton as the prevous dagram above usng real sgnals Note that the omplex sgnal dagram does not use the low pass lters Fgure apton: IQ reever n omplex notaton, n ths dagram _ = LO For the ase a( t) 1, ( t) 0 where the RF nput sgnal s a smple arrer wave j2 rˆ () t e, and It ( ) os2 b, Qt ( ) sn 2 b, we an wrte j2 bt r( t) I( t) jq( t) os2 bt jsn2 bt e the same result as above, apart rom a ator 05 arsng rom the omplex notaton It () and Qt () as dsplayed on a x-y sope wll show a snusodal wave at the derene requeny, a rle s dsplayed I b 5 Hz or so, then the dot on the sope an be seen trang out the rle Mathemats o omplex downonverson n omplex notaton wth a real arrer wave ( t) j2 Assume sgnal nput, ( ) ( )os[2 ( )] Re{ ( ) t s t a t t a t e e } wth a( t) 1, ( t) 0 j2 LOt Multply s() t os[2 ] by e os 2 LOt jsn2 LOt to obtan omplex downonverter outputs It ( ) os 2 b and Qt ( ) sn 2 b Exerse or the student