1. CHAPTER 6 PASS-BAND DATA TRANSMISSION
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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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Offset QPSK
In the example from the previous lecture we had the following time diagram for QPSK:
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QPSK Equations:
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Figure 6.6 Signal-space diagram of coherent QPSK system.
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…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,
we can make the following conclusions:
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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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:
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…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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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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11. Digital Communication Systems 2012
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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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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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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Figure 6.12
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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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Generation of π/4-shfted DQPSK signals
Based on the symbol pair:
In-phase component
differentially encoded
phase change for symbol k
absolute phase angle
of symbol k-1
Quadrature component
absolute phase angle
of symbol k
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Example 6.2
We have a binary input and a π/4-
shifted 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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Example 6.2
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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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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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Correction rule:
so, after applying the correction rule for the previous example we get:
Δθk = -290o + 360o = 70o
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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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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
carrier frequency
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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:
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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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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:
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Using the above mentioned property and equation we calculate the average probability of symbol error for coherent M-ary PSK as: (M ≥ 4)
Note that M = 4 is the special case discussed before as QPSK.
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Power spectra of M-ary PSK Signals
Symbol duration for M-ary PSK is defined as:
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:
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Figure 6.16 Power spectra of M-ary PSK signals for M  2, 4, 8.
OPSK
QPSK
BPSK
BPSK
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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:
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Also, we have from before
So we can express the bandwidth in terms of bit rate as:
and the bandwidth efficiency as:
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