Transcript chapter 6 pass-band data transmission

Chapter 6: Pass-band Data Transmission
CHAPTER 6
PASS-BAND DATA
TRANSMISSION
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Chapter 6: Pass-band Data Transmission
Outline
• 6.3 Coherent Phase Shift Keying - QPSK
– Offset QPSK
– π/4 – shifted QPSK
– M-ary PSK
• 6.4 Hybrid Amplitude/Phase Modulation
Schemes
– M-ary Qudarature Amplitude Modulation (QAM)
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Chapter 6: Pass-band Data Transmission
Offset QPSK
• In the example from the previous lecture we had the
following time diagram for QPSK:
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Chapter 6: Pass-band Data Transmission
QPSK Equations:




2
(
t
)
c
o
s
(
2
f
t
)
,0

tT

(
6
.
2
5
)
1
c
T
2
(
t
)
s
i
n
(
2
f
t
)
,0

tT

(
6
.
2
6
)
2
c
T



E
c
o
s
[
(
21
i

)]


4
s
, i
1
,
2
,
3
,
4(
6
.
2
7
)


i

E

s
i
n
[
(
21
i

)]



4

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Chapter 6: Pass-band Data Transmission
Figure 6.6
Signal-space diagram of coherent QPSK system.
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Chapter 6: Pass-band Data Transmission
…translated to a space-signal diagram it looks like this:
Figure 6.10
which shows all the possible paths for switching between the
message points in (a) QPSK and (b) offset QPSK.
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So,
•
Chapter 6: Pass-band Data Transmission
we can make the following conclusions:
1.
2.
3.
The carrier phase changes by ±180o whenever both the in-phase
and the quadrature components of the QPSK signal change sign (01
to 10)
The carrier phase changes by ±90o degrees whenever the in-phase or
quadrature component changes sign (10 to 00 – in-phase changes,
quadrature doesn’t changes)
The carrier phase is unchanged when neither the in-phase nor the
quadrature component change sign. (10 and then 10 again).
Conclusion: Situation 1 is of concern when the QPSK signal is
filtered during transmission because the 180 or also 90
degrees shifts in carrier phase might result in changes in
amplitude (envelope of QPSK), which will cause symbol
errors (for details see chapter 3 and 4 on envelope detection)
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Chapter 6: Pass-band Data Transmission
• To overcome this problem a simple solution is
proposed – delaying the quadrature component with
half a symbol interval (i.e. offset) with respect to
the bit stream responsible for the in-phase
component.
• So the two basis functions are defined as follows:




2
(
t
)
c
o
s
(
2
f
t
)
,0

t

T
(
6
.
4
1
)
1
c
T
2
T3
T
(
t
)

s
i
n
(
2
f
t
)
, 
t
(
6
.
4
2
)
2
c
T
2 2
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Chapter 6: Pass-band Data Transmission
…translated to a space-signal diagram it looks like this:
Figure 6.10
which shows all the possible paths for switching between the
message points in (a) QPSK and (b) offset QPSK.
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Chapter 6: Pass-band Data Transmission
• With this correction the possible phase transitions are
limited to ±90o (see Fig.10b)
• Changes in phase occur with half the intensity in
offset QPSK but twice as often compared to QPSK
• So, the amplitude fluctuations due to filtering in
offset QPSK are smaller than in the case with QPSK
• As for probability of error – it doesn’t change (based
on the statistical independence of the in-phase and
quadrature components)
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Chapter 6: Pass-band Data Transmission
Outline
• 6.3 Coherent Phase Shift Keying - QPSK
– Offset QPSK
– π/4 – shifted QPSK
– M-ary PSK
• 6.4 Hybrid Amplitude/Phase Modulation
Schemes
– M-ary Qudarature Amplitude Modulation (QAM)
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Chapter 6: Pass-band Data Transmission
π/4-Shifted QPSK
• Another variation of the QPSK modulation technique
• In ordinary QPSK the signal may reside in any of the
following constellations:
Figure 6.11
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Chapter 6: Pass-band Data Transmission
π/4-Shifted QPSK – cont’d
• In the so called π/4-shifted QPSK the carrier phase
for the transmission of successive symbols is picked
up alternatively from one of the two QPSK
constellations – so eight possible states.
• Possible transitions are give by dashed lines on the
following figure.
• Relationships between phase transitions and dibits in
π/4-shifted QPSK are given in Table 6.2
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Chapter 6: Pass-band Data Transmission
Figure 6.12
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Chapter 6: Pass-band Data Transmission
Advantages of π/4-shfted QPSK:
• The phase transitions from one symbol to another are
limited to ±π/4 and ±3π/4 radians (compared to ±π/2
and ±π in QPSK) – significantly reduce amplitude
fluctuations due to filtering.
• π/4-shfted QPSK can be noncoherently detected
which simplifies the receiver (offset QPSK cannot)
• in π/4-shfted QPSK signals can be differentially
encoded which creates differential π/4-shfted QPSK
(DQPSK)
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Chapter 6: Pass-band Data Transmission
Generation of π/4-shfted DQPSK signals
• Based on the symbol pair:
In-phase component
I
c
o
s
(




)
k
k

1
k

c
o
s

k
differentially encoded
phase change for symbol k
(
6
.
4
3
)
absolute phase angle
of symbol k-1
Quadrature component
absolute phase angle
of symbol k
Q
s
i
n
(




)
k
k

1
k

s
i
n

k
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6
.
4
4
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Chapter 6: Pass-band Data Transmission
Example 6.2
We have a binary input 01101000 and a π/4shifted DQPSK.
Initial phase shift
is π/4.
Define the symbols
Transmitted according to the convention in Table
6.2 (Formula 6.43 and 6.44)
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Chapter 6: Pass-band Data Transmission
Example 6.2
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Chapter 6: Pass-band Data Transmission
Detection of π/4-shfted DQPSK
•Assume that we have a noise channel (AWGN) and the channel
output is x(t).
•The receiver first computes the projections of x(t) onto the basis
functions φ1(t) and φ2(t).
•Resulting outputs are denoted by I and Q respectively and
applied to a differential detector, which consists of the following
components:
arctangent computing block (extracting phase angle)
phase difference computing block (determining change in
phase)
Modulo-2π correction logic (wrapping errors)
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Chapter 6: Pass-band Data Transmission
Wrapping errors
In this example θk-1 = 350o
θk = 60o (measured counter
clockwise)
Actual Phase change = 70o
but if calculated directly:
60o – 350o = 290o
Correction is required.
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Chapter 6: Pass-band Data Transmission
Correction rule:




I
F



1
8
0
d
e
g
r
e
e
s
T
H
E
N




3
6
0
d
e
g
r
e
e
s
k
k
k
I
F



1
8
0
d
e
g
r
e
e
s
T
H
E
N




3
6
0
d
e
g
r
e
e
s
k
k
k
• so, after applying the correction rule for the previous
example we get:
Δθk = -290o + 360o = 70o
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Chapter 6: Pass-band Data Transmission
Block diagram of the π/4-shfted
DQPSK detector
Figure 6.13
•Relatively simple to implement
•Satisfactory performance in fading Rayleigh channel, static
multipath environment
•Not very good performance for time varying multipath environment
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Chapter 6: Pass-band Data Transmission
M-ary PSK
• More general case than QPSK
• Phase carrier takes one of M possible values,
θi= 2(i-1)π/M, where i = 1,2,…M
• During each signaling interval T one of M possible
signals is sent:


signal energy per symbol
2
E 2
s
(
t
)

c
o
s
(
2
f
t

(
i

1
)
)
,i

1
,
2
,
.
.
.
.
.
,
M
(
6
.
4
6
)
i
c
T
M
carrier frequency
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Chapter 6: Pass-band Data Transmission
• s(t) may be expanded using the same basis functions
defined for binary PSK – φ1(t) and φ2(t).
•The signal constellation is two dimensional.
•The M message points are equally spaced on a circle
of radius
and centered at the origin.
•The Euclidian distance between each two points for
M = 8 can be calculated as:

d
d
2E
s
i
n
( )
1
2
1
8
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Chapter 6: Pass-band Data Transmission
Figure 6.15
(a) Signal-space diagram for
octaphase-shift keying (i.e., M 
8). The decision boundaries are
shown as dashed lines. (b)
Signal-space diagram illustrating
the application of the union
bound for octaphase-shift
keying.
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Chapter 6: Pass-band Data Transmission
Symbol Error
• Note: The signal constellation diagram is circularly
symmetric.
• Chapter 5: The conditional probability of error
Pe(mi) is the same for all I, and is given by:
M
d
1
i
k
P

e
r
f
c
(
f
o
r
a
l
l
i(
5
.
9
2
)

e
2
2
N
k

1
0
0
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Chapter 6: Pass-band Data Transmission
• Using the above mentioned property and equation
we calculate the average probability of symbol
error for coherent M-ary PSK as: (M ≥ 4)

E
P
r
f
c
( s
i
n
()
)(
6
.
4
7
)
e e
N
M
0
• Note that M = 4 is the special case discussed before
as QPSK.
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Chapter 6: Pass-band Data Transmission
Power spectra of M-ary PSK Signals
• Symbol duration for M-ary PSK is defined as:
T

T
l
o
g
M (
6
.
4
8
)
b
2
• Proceeding in a similar manner as with QPSK and
using the results from the introductory part of
chapter 6 we can see that the baseband power
spectral density of M-ary PSK is given by:
2
S
(
f
)

2
E
s
i
n
c
(
T
f
)
B
2

2
E
l
o
g
M
s
i
n
c
(l
T
f
o
g
M
) (
6
.
4
9
)
b2
b
2
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Chapter 6: Pass-band Data Transmission
Figure 6.16
Power spectra of M-ary PSK signals for M  2, 4, 8.
OPSK
QPSK
BPSK
BPSK
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Chapter 6: Pass-band Data Transmission
Bandwidth Efficiency of M-ary PSK
Signals
• From the previous slide of the power spectra of the M-ary
PSK it is visible that we have a well defined main lobe and
spectral nulls.
• Main lobe provides a simple measure for the bandwidth of
the M-ary PSK. (null-to-null bandwidth).
• For the passband basis functions defined with (6.25) and
(6.26) (which are required to pass the M-ary PSK signals)
the channel bandwidth is given by:
2
B

T
(6
.5
0
)
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Chapter 6: Pass-band Data Transmission
Also, we have from before
TT
g2M
blo
Rb 1/ Tb
So we can express the bandwidth in terms of bit rate as:
2
R
B
 b
l
o
g
M
2
and the bandwidth efficiency as:
(
6
.
5
1
)
R
 b or
B
log2M

2
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