Module 5. Carrier Modulation. Version 2 ECE IIT, Kharagpur

Transcription

1 Module 5 Carrier Modulation Version ECE II, Kharagpur

2 Lesson 4 Binary Phase Shift Keying (BPSK) Modulation Version ECE II, Kharagpur

3 After reading this lesson, you will learn aout Binary Phase Shift Keying (BPSK); Power Spetrum for BPSK Modulated Signal; Binary Phase Shift Keying (BPSK) BPSK is a simple ut signifiant arrier modulation sheme. he two time-limited energy signals s (t) and s (t) are defined ased on a single asis funtion ϕ (t) as: E E E s () t =.osπ ft and s() t =.os [ π ft+ π ] =.osπ ft 5.4. he asis funtion, evidently, is, ϕ () t =.osπ ft ; t <. So, BPSK may e desried as a one-dimensional digital arrier modulation sheme. Note that the general form of the asis funtion is, ϕ () t =.os( πf t + φ ), where Φ indiates an aritrary ut fixed initial phase offset. For onveniene, let us set Φ =. As we know, for narrowand transmission, f >>. hat is, there will e multiple yles of the arrier sinusoid within one it duration ( ). For onveniene in desription, let us set, f = n (though this is not a ondition to e satisfied theoretially). Now, we see, s () t = E. ϕ () t and s () t = E. ϕ (), t 5.4. he two assoiated salars are: s () t = s (). t ϕ () t dt = + E and s = s(). t ϕ() t dt = E Fig (a) presents a sketh of the asis funtion ϕ (t) and Fig () shows the BPSK modulated waveform for a inary sequene. Note the arupt phase transitions in the modulated waveform when there is hange in the modulating sequene. On every oasion the phase has hanged y 8. Also note that, in the diagram, we E E have hosen to set =, i.e. = =.5, whih is the power assoiated with an unmodulated arrier sinusoid of unit peak amplitude. Version ECE II, Kharagpur

4 ϕ (t) t Fig. 5.4.: (a) Sketh of the asis funtion ϕ (t) for BPSK modulation Fig. 5.4.: () BPSK modulated waveform for the inary sequene. Note that the amplitude has een normalized to ±, as is a ommon pratie. Fig. 5.4.: () shows the signal onstellation for inary PSK modulation. he two pints are equidistant from the origin, signifying that the two signals arry same energy. Deision Boundary * E * E ϕ Region Z Region Z Fig. 5.4.: () Signal onstellation for inary PSK modulation. he diagram also shows the optimum deision oundary followed y a orrelation reeiver Version ECE II, Kharagpur

5 Fig shows a simple sheme for generating BPSK modulated signal without pulse shaping. A ommonly availale alaned modulator (suh as IC 496) may e used as the produt modulator to atually generate the modulated signal. he asis funtion ϕ (t), shown as the seond input to the produt modulator, an e generated y an osillator. Note that the osillator may work independent of the data lok in general. + E Random input signal BPSK modulated signal Produt E ϕ() t = os + Edt () where d = + or - E Modulator st () =±.os[ wt + θ ] [ wt θ ] E. dt ( )os + [ wt θ ] Fig A simple sheme for generating BPSK modulated signal. No pulse-shaping filter has een used. Fig presents a sheme for oherent demodulation of BPSK modulated signal following the onept of optimum orrelation reeiver. he input signal r(t) to the demodulator is assumed to e entered at an intermediate frequeny (IF). his real narrowand signal onsists of the desired modulated signal s(t) and narrowand Gaussian noise w(t). As is ovious, the orrelation detetor onsists of the produt modulator, shown as an enirled multiplier, and the integrator. he vetor reeiver is a simple inary deision devie, suh as a omparator. For simpliity, we assumed that the asis funtion phase referene is perfetly known at the demodulator and hene the ϕ (t), shown as an input to the produt demodulator, is phase-synhronized to that of the modulator. Now it is straightforward to note that the signal at (A) in Fig is: r A (t) = [ st () + wt (). ].os( wt + θ ) he signal at (B) is: r = E d( t)..os( wt + θ) + wt ( ) os( wt + θ) dt = E. d( t) + w( t).os( wt + θ ) dt Version ECE II, Kharagpur

6 r(t) =s(t) + w(t) φ (t) A B C dt ( ) r o sample at intervals of, if r >, if r Fig A sheme for oherent demodulation of BPSK modulated signal following the onept of optimum orrelation reeiver Note that the first term in the aove expression is the desired term while the seond term represents the effet of additive noise. We have disussed aout similar noise omponent earlier in Module #4 and we know that this term is a Gaussian distriuted random variale with zero mean. Its variane is proportional to the noise power spetral density. It should e easy to follow that, if d(t) = + and the seond term in Eq (i.e. the noise sample voltage) is not less than -., the threshold detetor will properly deide the reeived signal as a logi. Similarly, if d(t) = - and the noise sample voltage is not greater than +., the omparator will properly deide the reeived signal as a logi. hese oservations are ased on positive inary logi. Power Spetrum for BPSK Modulated Signal Continuing with our simplifying assumption of zero initial phase of the arrier and with no pulse shaping filtering, we an express a BPSK modulated signal as: E. st () =. dt ()oswt, where d(t) = ± he aseand equivalent of s(t) is, E ut () = ui () t =. dt () =± gt (), where E gt () = and u Q (t) =. E Now, u I (t) is a random sequene of E + and whih are equi-proale. So, the power spetrum of the ase and signal is: E.sin ( πf ) UB( f) = =. E.sin ( f ) π f ( ) Version ECE II, Kharagpur

7 Now, the power spetrum S(f) of the modulated signal an e expressed in terms of U B (f) as: S( f) = UB ( f f) + UB( f + f) Fig shows the normalized ase and power spetrum of BPSK modulated signal. he spetrum remains the same for aritrary non-zero initial phase of arrier osillator. Fig.5.4.4: Normalized ase and power spetrum of BPSK modulated signal Prolems Q5.4.) Dose BPSK modulated signal have onstant envelope? Q5.4.) Why oherent demodulation is preferred for BPSK modulation? Q5.4.3) Do you think the knowledge of an optimum orrelation reeiver is useful for understanding the demodulation of BPSK signal? Q5.4.4) Sketh the spetrum of the signal at the output of a BPSK modulator when the modulating sequene is,,, Version ECE II, Kharagpur