coherent phase shift keying in coherent phase shift keying different phase modulation schemes will...
TRANSCRIPT
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