Coherent phase shift keying In coherent phase shift keying different phase modulation schemes will be covered i.e. binary PSK, quadrature phase shift keying and M-ary PSK Binary PSK will be studied in the next slides 1

Binary Phase shift keying 2

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Signal constellation for binary Phase shift keying 5

If we plot the transmitted symbols for BPSK we may got the following constellation diagram 6

Error probability of BPSK In order to compute the error probability of BPSK we partition the constellation diagram of the BPSK (see slide 28) into two regions If the received symbol falls in region Z 1, the receiver decide in favor of symbol S 1 ( logic 1) was received If the received symbol falls in region Z 2, the receiver decide in favor of symbol S 2 (logic 0) was received 7

Error probability of BPSK- Receiver model 8

Error probability of BPSK 9

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Error probability of BPSK The conditional probability of the receiver deciding in favor of symbol 1, given that symbol zero was transmitted is given by 11

Error probability of BPSK 12

Error probability of error In similar manner we can find probability of error that symbol 1 was sent and the receiver detect 0 mistakenly The average probability as we did in the baseband can be computed as This average probability is equivalent to the bit error rate 13

Generation of BPSK signals 14

Generation of BPSK signals The resulting binary wave and the carrier (basis function) are applied to product multiplier as shown below 15

Detection of BPSK signals 16

Power spectra of binary PSK signals The power spectral density of the binary PSK signal can be found as described for the bipolar NRZ signaling (see problem 3.11 (a) Haykin) This assumption is valid because the BPSK is generated by using bipolar NRZ signaling 17

Power spectra of binary PSK signals The power spectral density can be found as 18

Quadrature phase shift keying QPSK 19

Signal space diagram of QPSK If we expand the QPSK equation using the trigonometric identities we got the following equation Which we can write in vector format as 20

Signal space diagram of QPSK 21

Detailed message points for QPSK 22 iInput DibitPhase of QPSK signaling Coordinate of Message point s i1 s i

Signal space diagram of QPSK 23 22 11 s4s4 s1s1 s3s3 s2s2 (10) (00) (01)(11)

Signal space diagram of QPSK with decision zones The constellation diagram may appear as shown below 24 22 11 s4s4 s1s1 s3s3 s2s2 (10) (00) (10)(11) Z4Z4 Z1Z1 Z3Z3 Z2Z2

Example Sketch the QPSK waveform resulting from the input binary sequence solution 25

Error probability of QPSK 26

Error probability of QPSK 27

Error probability of QPSK 28

Error probability decision rule 29

Error probability of QPSK 30

Error probability of QPSK If symbol is to be received correctly both bits must be received correctly. Hence, the average probability of correct decision is given by Which gives the probability of errors equal to 31

Error probability of QPSK Since one symbol of QPSK consists of two bits, we have E = 2E b The above probability is the error probability per symbol With gray encoding the average probability of error per bit Which is exactly the same as BPSK 32

Error probability of QPSK summery 33

Generation and detection of QPSK 34 Block diagrams of (a) QPSK transmitter and (b) coherent QPSK receiver.

Power spectra of QPSK 35