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

In a coherent PSK system the pair of signals and are used to represent binary logics 1 and 0 respectively

2

Binary Phase shift keying

Where , and is the transmitted signal energy per bit

The carrier frequency is selected such that so that each bit contains an integral number of cycles

From the pair of symbols and we can see only one basis function (carrier) is needed to represent both and

3

Binary Phase shift keying

The basis function is given by

Now we can rewrite and on the interval

4

Signal constellation for binary Phase shift keying

In order to draw the constellation diagram we need to find the projection of each transmitted symbol on the basis function

The projection of the logic (1); ; is given by

The projection of the second symbol on the basis function is given by

5

Signal constellation for binary Phase shift keying

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 Z1, the receiver decide in favor of symbol S1 ( logic 1) was received

If the received symbol falls in region Z2, the receiver decide in favor of symbol S2 (logic 0) was received

7

Error probability of BPSK-Receiver model

The receiver in the pass band can be modeled as shown

The received signal vector

8

Error probability of BPSK

The observable element (symbol zero was sent and the detected sample was read in zone 1) is given by

9

Error probability of BPSK

To calculate the probability of error that symbol 0 was sent and the receiver detect 1 mistakenly in the presence of AWGN with , we need to find the conditional probability density of the random variable , given that symbol 0, was transmitted as shown below

10

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

By letting

the above integral for can be rewritten as

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

To generate a binary PSK signal we need to present the binary sequence in polar form

The amplitude of logic 1 is whereas the amplitude of logic 0 is

This signal transmission encoding is performed by using polar NRZ encoder

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

To detect the original binary sequence we apply the received noisy PSK signal to a correlator followed by a decision device as shown below

The correlator works a matched filter

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

In quadrature phase shift keying 4 symbols are sent as indicated by the equation

Where ; is the transmitted signal energy per symbol, and is the symbol duration

The carrier frequency is for some fixed integer

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

TttiEtiE

tfT

iEtfT

iEts cci

0;4

12sin4

12cos

2sin2

412sin2cos

2

412cos

21

412sin

412cos

iE

iEis

Signal space diagram of QPSK

There are four message points defined by

According to this equation , A QPSK has a two-dimensional signal constellation (i.e. or two basis functions)

21

Detailed message points for QPSK

22

i Input Dibit Phase of QPSK

signaling

Coordinate of Message pointsi1 si2

1 10

2 00

3 01

4 11

4/

4/3

4/5

4/7

2/E

2/E

2/E

2/E

2/E

2/E2/E

2/E

Signal space diagram of QPSK

23

2

1

s4

s1

s3

s2

(10)(00)

(01) (11)

Signal space diagram of QPSK with decision zones

The constellation diagram may appear as shown below

24

2

1

s4

s1

s3

s2

(10)(00)

(10) (11)

Z4

Z1

Z3

Z2

2/E

2/E

2/E

2/E

Example

Sketch the QPSK waveform resulting from the input binary sequence 01101000

solution

25

Error probability of QPSK

In coherent QPSK, the received signal is defined by

Where is the sample function of AWGN with zero mean and power spectral density of

26

Error probability of QPSK

The observation vector has two elements, and defined by

27

Error probability of QPSK

28

Error probability decision rule

If the received signal point associated with the observation vector falls inside region , the receiver decide that was transmitted

Similarly the receiver decides that was transmitted if falls in region

The same rule is followed for and

29

Error probability of QPSK

We can treat QPSK as the combination of 2 independent BPSK over the interval

since the first bit is transmitted by and the second bit is transmitted by

Probability of error for each channel is given by

30

00

12

2erf

2

1

2erfc

2

1

N

Ec

N

dP

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

21 PPc

0

0

2

0

2erfc

2erfc

4

1

2erfc1

N

E

N

E

N

EPP Ce

Error probability of QPSK

Since one symbol of QPSK consists of two bits, we have E = 2Eb

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

0erfcsymbolper

N

EPe b

0erfc

2

1symbolper

2

1bitper

N

EPePe b

Error probability of QPSK summery

We can state that a coherent QPSK system achieves the same average probability of bit error as a coherent PSK system for the same bit rate and the same but uses only half the channel bandwidth

33

Generation and detection of QPSK

34

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

Power spectra of QPSK

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Power spectrum density of BPSK vs. QPSK

Normalized frequency,fTb

Nor

mal

ized

PS

D,S

f/2E

bBPSKQPSK