BPSK so that we have a discrete set of RF signals. t)cos(

Transcription

1 BPSK. BPSK Introdution Reall that the most general modulation has the form s( t) a( t)os[ t ( t)]. We remared earlier that phase modulation was not an effetive way to implement analog ommuniation, one reason eing eause it requires a phase referene. For digital systems this prolem an e overome. Phase modulation is an effetive digital ommuniation strategy, and we refer to it as phase shift eying or PSK. In PSK we limit ourselves to disrete values of the phase, so that we have a disrete set of RF signals s ( t) A os( t ) (.) We use the different phases to represent different logi states. We an rewrite (.) in the form s ( t) A os os t sin A a os t sin t sin t whih shows that phase shift eying signals are linear ominations of os t and sin t. (.) For inary phase shift eying, or BPSK, we use to represent a logi and to represent logi. In this ase we have simply s ( t) A s ( t) A os( t) os( t ) A os( t) (.3) Sine a phase shift of just hanges the sign of the arrier, we an write s( t) A m( t)os( t) where m ( t), and represents logi, and represents logi. hus a BPSK signal has the form of an AM signal. his is the most useful point of view, partiularly when filtering is applied to redue the andwidth. An example BPSK signal is shown in Fig... EE43: RF Engineering for eleommuniations Sott Hudson, Washington State University 5/3/7

2 BPSK Figure.: BPSK signals. he modulation is m ( t) and the phase transitions are sharp. Constellation Diagram As shown aove, phase shift eying signals are linear ominations of os t and sin t. We an represent one of these signals as a point in Cartesian oordinates where the horizontal oordinate gives the oeffiient a and the vertial oordinate. he result is alled a onstellation diagram. It is simply a phase diagram where amplitude and phase are given y A and. he onstellation diagram for BPSK is illustrated in Fig... sin t os t -A A Figure.: Constellation diagram for BPSK. he la dots denote the phasors of the ideal signals that represent logi states and. EE43: RF Engineering for eleommuniations Sott Hudson, Washington State University 5/3/7

3 BPSK.3 BPSK Demodulation Previously we noted that phase modulation suffered from the fat that phase is a relative onept, so to perform demodulation a reeiver would need to now the phase of the arrier used in the transmitter. With BPSK the arrier an e otained y a proess alled arrier reovery. A BPSK signal is proportional to times os. Sine t t, if we square a BPSK signal the modulation disappears and we get os os t/. his is a pure sinusoid at twie the arrier frequeny (plus some DC). Using phase lo loop iruits and a frequeny divider, the arrier os t an e reovered from this. Assuming we have reovered the arrier, we an demodulate a BPSK signal y alulating r A m A m X A mos t n( t) os t dt n( t)os t dt (.4) where m is either and X is a Gaussian RV with zero mean and variane n N /. If m, i.e., logi. r we assume m, i.e., that a logi was sent. Otherwise we tae Say m. hen we mae an error if X A eause that will ause r to e greater than zero. he proaility of this is P e A Q n Q n E N A e x n dx (.5) Compare this to (.) for FSK. Here we get an additional fator of. his means that with half the power BPSK an give the same BER as FSK. herefore, BPSK is more power effiient than FSK. In fat we an say it is a fator of, or 3 db, more effiient. EE43: RF Engineering for eleommuniations Sott Hudson, Washington State University 5/3/7

4 BPSK.4 BPSK Spetrum Realling that our BPSK signal is s( t) A m( t)os( t), we an use the onvolution theorem to alulate its spetrum. he spetrum of the produt of two signals is the onvolution of their individual spetra. he spetrum of os( t) is f f ) ( f f ) /, so S ( A A f ) M ( f f) M ( f f ) (.6) ( where M ( f ) is the Fourier transform of m (t). o alulate this you d need to now the speifi sequene of logi states. If you assume the its are random, then you an show that the power spetrum of m (t) is idential to the power spetrum of a single it. Hene we have S( f ) sin ( f f ) (.7) where we ve limited ourselves to positive frequenies and normalized the power spetrum to have unit energy. his is illustrated in.3 and ompared to the MSK spetrum Figure.3: BPSK spetrum (solid red urve) ompared to MSK spetrum (dashed lue urve). Both urves have unit energy (on linear sale). Horizontal axis is normalized frequeny f relative to the arrier, vertial axis is amplitude in db. he wider main loe of BPSK as ompared to MSK, and the muh larger side loes show that BPSK is not as andwidth effiient as MSK. We an quantify andwidth effiieny as follows EE43: RF Engineering for eleommuniations Sott Hudson, Washington State University 5/3/7

5 BPSK.5 R B RF its s Hz / (.8) where R is the it rate and BRF is the RF andwidth. his, of ourse, depends on how we define the andwidth. And, in pratie, these modulation tehniques are usually aompanied y filtering to redue the side loes. If we tae the frequeny spread etween the nulls of the main loe as an effetive andwidth, then we find that the spetral effiieny of BPSK is. 5 while that of MSK is. 67. So, we get 34% more its per seond through an MSK hannel than through a BPSK hannel with the same andwidth. he side loes of the sin funtion (Fig..3) are very large and is left unfiltered they ould ause interferene with adjaent RF hannels. herefore filtering is typially applied. his smoothes the retangular pulses that mae up m (t). he result for a single pulse is shown in Fig B=infinity B=/ B=4/ Figure.4: Retangular pulse and filtered versions. Horizontal axis is time in units of. Solid (red) urve orresponds to elimination of all sideloes of the BPSK spetrum. Dashed (lue) urve orresponds to eeping only the first sideloes. his has the effet of smoothing the it transitions as shown in Fig..5. EE43: RF Engineering for eleommuniations Sott Hudson, Washington State University 5/3/7

6 BPSK Figure.5: Filtered BPSK. he it transitions are smoothed, ut this introdues amplitude flutuations. he filtering, while reduing side loes and hene total andwidth, introdues amplitude flutuations. his an ause a prolem if the signal passes through a non-linear RF power amplifier, and all real amplifiers are to some extent non-linear. he resulting distortion an lead to spetral regrowth that reates unwanted signals outside the desired andwidth. his is a relative disadvantage of BPSK as opposed to FSK. With FSK the signal always has onstant amplitude, even when the filtering is modulated. Referenes. Anderson, J. B., Digital ransmission Engineering, IEEE Press, 999, ISBN Proais, J. G. and M. Salehi, Communiation Systems Engineering, nd Ed., Prentie Hall,, ISBN EE43: RF Engineering for eleommuniations Sott Hudson, Washington State University 5/3/7

7 BPSK.7 EE43: RF Engineering for eleommuniations Sott Hudson, Washington State University 5/3/7