Signal detection, Fouriertransformation, phase correction and quadrature detection

Transcription

1 2/35 Signal deecion, Fourierransformaion, phase correcion and quadraure deecion Peer Schmieder Schmilka 2004 Wha is his seminar abou? Signaldeecion Wha kind of signal do we deec in NMR Fourierransformaion How is he ime dependen signal ransformed ino a specrum Phase correcion Wh is a phase correcion necessar Quadraur deecion Wh do we need quadraure deecion 3/35 4/35 90 pulse FID The simples eperimen in -NMR consiss of one pulse and deecion of an FID Afer he pulse magneizaion sars o precess around he z-ais, i.e. he main magneic field This precession induces a curren in he deecion coil resuling in he signal ha is recorded 5/35 6/35 An oscillaing signal viewed from and ais resuls in a cosine and a sine sin Ω Ω Boh signals are deeced o ield a comple NMR signal dependenc cos Ω M = M + i M M = ep (iω) ep (-/T 2 ) M M M = cos Ω ep (-/T 2 ) M = sin Ω ep (-/T 2 ) The above relaion resuls from he famous Euler equaions ep(iα) = cosα + i sinα ep(-iα) = cosα -isinα

2 7/35 8/35 M M Berechnung von Pulssequenzen We jus saw he saring poin of he precession a M = 1 und M = 0, bu ha is no a necessi M = cos Ω 0 ep (-/T 2 ) M = sin Ω 0 ep (-/T 2 ) M (0) = 1 M (0) = 0 M (0) = 0 M (0) = 1 M = M + i M = [cos Ω 0 + i sin Ω 0 ] ep (-/T 2 ) M = ep (iω 0 ) ep (-/T 2 ) 9/35 10/35 M M Berechnung von Pulssequenzen M = - sin Ω 0 ep (-/T 2 ) M = cos Ω 0 ep (-/T 2 ) M = - sin Ω 0 ep (-/T 2 ) = cos (Ω 0 + π/2) ep (-/T 2 ) M = cos Ω 0 ep (-/T 2 ) = sin (Ω 0 + π/2) ep (-/T 2 ) The saring posiion of he signal and he posiion of he deecion ais are variable, he signal has a phase φ cos φ sin φ M = M + i M M = ep (iω 0 + π/2) ep (-/T 2 ) M = ep (π/2) ep (iω 0 ) ep (-/T 2 ) M = ep (iφ) ep (iω 0 ) ep (-/T 2 ) 11/35 12/35 The phase ells us a which poin of he wave he signal deecion has begun The phase can be influenced b he choice of he pulse phase bu does also depend on he elecronics (i.e. lengh of he cables) M z B 1 aus z B 1 aus

3 13/35 14/35 To process he daa on a compuer using he D (discree fourier ransform) he signal needs o b digiized. The analog-o-digial-converer (ADC) does he job To make he digiizaion echnicall feasible he range of frequencies has o be as small as possible Therefore he original signal (sen as a pulse) is subraced from he received signal o obain onl he modulaion In addiion, he carrier is placed in he cener of he specrum allowing posiive and negaive frequencies (clockwise and counerclockwise roaion) 15/35 16/35 The deecion uni Puing he carrier in he cener of he specrum creaes he need o disinguish posiive and negaive frequencies: quadraure deecion 17/35 18/35 We wan o know he frequenc of he oscillaion The ransforms a miure of imedependen signals (oscillaions) ino a specrum, which shows a frequenc dependence depiced below

4 19/35 20/35 We sar ou wih guessing a frequenc Then we mulipl boh and sum up all values of he resuling oscillaion (i.e. we inegrae) If he frequencies do no mach we ge an equal amoun of posiive and negaive values and he resul is 0 21/35 22/35 If he frequencies do mach, mos poins will be posiive and he resul of he summaion will be large If we do his ssemaicall for all possible frequencies, we obain a specrum of all frequencies conained in he oscillaion 23/35 24/35 Pu in equaions he easies wa o eplain he is o use a comple funcion s() = ep (iφ) ep (iω 0 ) ep (-/T 2 ) phase (ime independen!) oscillaion (he frequenc) deca (relaaion) We ignore he phase facor a firs and do he muliplicaion and summaion (inegraion) S(Ω) = m s() ep (-iω) d 0 S(Ω) = m ep (iω 0 ) ep (-/T 2 ) ep (-iω) d 0 1 S(Ω) = lorenian line (1/T 2 ) i(ω Ω 0 ) A comple funcion consiss of real and imaginar par S (Ω) = R(Ω) + i I(Ω)

5 25/35 26/35 In he simples (and bes) case he real par of an lorenian is absorbive, he imaginar par dispersive S (Ω) = A(Ω) + i D(Ω) ha s wha we wan A(Ω)= (1/T 2 ) (1/T 2 ) 2 + (Ω Ω 0 ) 2 Bu we remember ha he signal had a phase S(Ω) = [A(Ω) + i D(Ω) ] ep(iφ) S(Ω) = R(Ω) + i I(Ω) This makes real and imaginar par a miure of he (desired) absorbive and he (unwaned) dispersive par D(Ω)= (Ω Ω 0 ) 2 (1/T 2 ) 2 + (Ω Ω 0 ) 2 R(Ω) = A(Ω) cos φ -D(Ω) sin φ I(Ω) = D(Ω) cosφ + A(Ω) sinφ 27/35 28/35 And ha s wha we correc b doing a phase correcion A(Ω) = R(Ω) cos φ + I(Ω) sin φ D(Ω) = I(Ω) cosφ -R(Ω) sinφ phase correcion R I...which works as long as all signals have similar phase 29/35 30/35 Oherwise we calculae a magniude specrum S = (R) 2 + (I) 2 or a power specrum S = (R) 2 + (I) 2 s() = ep (iφ) ep (iω 0 ) ep (-/T 2 ) phase (ime independen!) oscillaion (frequenc) deca (relaaion)

6 31/35 32/35 Recording a comple signal is essenial no onl for phase correcion bu also for quadraure deecion The resul are wo lines, he can hen no disinguish beween posiive and negaive frequencies Cosinus-Signal Sinus-Signal Wha if we would onl record cosine and sine specra cosα = ep(iα) + ep(-iα) sinα = ep(iα) - ep(-iα) 33/35 34/35 Which is wh we have o combine boh. o be able o pu he carrier in he cener of he specrum Real-Teil 3 Signale: 800 Hz 100 Hz -300 Hz nur cos Imaginär-Teil cos und sin 35/35 Tha s i