ch06-part 2
TRANSCRIPT
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Quadriphase Shift Keying (QPSK)
In quadriphase-shift keying (QPSK), as with binary PSK,information carried by the transmitted signal is containedin the phase.
In particular, the phase of the carrier takes on one of four
equally spaced values, such as /4, 3/4, 5/4, and 7/4. For this set of values we may define the transmitted signal
as
where i= 1, 2, 3, 4; Eis the transmitted signal energy persymbol, and Tis the symbol duration.
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Signal-Space Diagram of QPSK
Using a well-known trigonometric identity, we may use the
previous equation to redefine the transmitted signal si(t) for
the interval 0 tTin the equivalent form:
where i= 1, 2, 3, 4.
There are two orthonormal basis functions, 1(t) and 2(t),
contained in the expansion of si(t).
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Signal-Space Diagram of QPSK
There are 4 message points and the associated signal
vectors are defined as
The elements of the signal vectors, si1 and si2, have their
values summarised in the next table.
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Signal-Space Diagram of QPSK
Signal-space characterization of QPSK
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Figure 6.6Signal-space diagram of coherent QPSK
system.
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Error Probability of QPSK
A coherent QPSK system is in fact equivalent to two
coherent binary PSK systems working in parallel and using
two carriers that are in phase quadrature.
The average probability of bit error in each channel of the
coherent QPSK system is
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Error Probability of QPSK
The in-phase and quadrature channels of the coherent
QPSK system are statistically independent.
The in-phase channel makes a decision on one of the two
bits constituting a symbol (dibit) of the QPSK signal, and
the quadrature channel takes care of the other bit.
Accordingly, the average probability of a correct decision
resulting from the combined action of the two channels
working together is
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Error Probability of QPSK
The average probability of symbol error for coherent
QPSK is therefore
In the region where (E/2No) >> 1, we may ignore the
quadratic term on the right hand side of the equation, so we
approximate the formula for the average probability ofsymbol error for coherent QPSK as
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Error Probability of QPSK
In a QPSK system, since there are two bits per symbol, the
transmitted signal energy per symbol is twice the signal
energy per bit, as shown by
Thus expressing the average probability of symbol error in
terms of the ratio Eb/N0, we may write
With Gray encoding used for the incoming symbols, the bit
error rate of QPSK is exactly
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Generation and Detection of Coherent
QPSK Signals Figure 6.8a shows a block diagram of a typical QPSKtransmitter.
The incoming binary data sequence is first transformed
into polar form by a nonreturn-to-zero level encoder.
Thus, symbols 1 and 0 are represented by +Eband - Eb,respectively.
This binary wave is next divided by means of a
demultiplexer into two separate binary waves, denoted
denoted by a1(t) and a2(t), consisting of the odd- and even-
numbered input bits.
The two binary waves a1(t) and a2(t) are used to modulate
a pair of quadrature carriers or orthonormal basis
functions.
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Generation and Detection of Coherent
QPSK Signals The result is a pair of binary PSK signals, which may bedetected independently due to the orthogonality of 1(t)
and 2(t) .
Finally, the two binary PSK signals are added to produce
the desired QPSK signal.
The QPSK receiver consists of a pair of correlators with a
common input and supplied with a locally generated pair
of coherent reference signals 1(t) and 2(t), as in Figure
6.8b.
The correlator outputs xl and x2, produced in response to
the received signal x(t), are each compared with a
threshold of zero.
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Figure
6.8Block
diagrams of
(a) QPSK
transmitterand (b)
coherent
QPSKreceiver.
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Generation and Detection of Coherent
QPSK Signals If xl> 0, a decision is made in favor of symbol 1 for the in-
phase channel output, but if xl< 0, a decision is made in
favor of symbol 0.
Similarly, if x2> 0, a decision is made in favor of symbol 1
for the quadrature channel output, but if x2< 0, a decision
is made in favor of symbol 0.
Finally, these two binary sequences at the in-phase and
quadrature channel outputs are combined in a multiplexer
to reproduce the original binary sequence at the transmitterinput with the minimum probability of symbol error in an
AWGN channel.
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Power Spectra of QPSK Signals
Assume that the binary wave at the modulator input is random,
with symbols 1 and 0 being equally likely, and with the symbols
transmitted during adjacent time slots being statistically
independent.
We make the following observations pertaining to the in-phaseand quadrature components of a QPSK signal:
1. The in-phase and quadrature components have a comnon power
spectral density, namely, Esinc2(Tf).
2. The in-phase and quadrature components are statistically
independent, so we may write
Figure 6.9 plots SB(f), normalized with respect to 4Eb, versus
the normalized frequency fTb.
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Figure 6.9Power spectra of QPSK and MSK signals.
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Offset QPSK
The signal space diagram of Figure 6.10a embodies all thepossible phase transitions that can arise in the generation of a
QPSK signal.
The extent of amplitude fluctuations exhibited by QPSK signals
may be reduced by using offset QPSK.
In this variant of QPSK, the bit stream responsible for
generating the quadrature component is delayed (i.e., offset) by
half a symbol interval with respect to the bit stream responsible
for generating the in-phase component.
Specifically, the two basis functions of offset QPSK are defined
by
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Figure 6.10
Possible paths for switching between themessage points in (a) QPSK and (b)
offset QPSK.
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Offset QPSK
Accordingly, unlike QPSK, the phase transitions likely to occurin offset QPSK are confined to 90 degrees, as indicated in the
signal space diagram of Figure 6.10b.
However, 90 degree phase transitions in offset QPSK occur
twice as frequently but with half the intensity encountered in
QPSK.
Despite the delay T/2 applied to the basis function 2(t), the
offset QPSK has exactly the same probability of symbol error in
an AWGN channel as QPSK.
We may therefore say that the error probability in the in-phaseor quadrature channel of a coherent offset QPSK receiver is still
equal to (1/2) erfc( (E/2N0)).
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M-ary PSK
QPSK is a special case of M-ary PSK, where the phase of thecarrier takes on one of Mpossible values, namely, i= 2(i - 1)
/M, where i= 1, 2, . . . , M.
Accordingly, during each signaling interval of duration T, one of
the Mpossible signals
is sent, where Eis the signal energy per symbol.
Each si(t) may be expanded in terms of the same two basis
functions 1(t) and 2(t).
The signal constellation of M-ary PSK is therefore two-
dimensional.
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M-ary PSK
The M message points are equally spaced on a circle of radiusEand center at the origin, as illustrated in Figure 6.15a, for thecase of octaphase-shift-keying (i.e., M = 8).
From Figure 6.15a we note that the signal-space diagram is
circularly symmetric.
The average probability of symbol error for coherent M-ary
PSK is given as
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Figure 6.15
(a) Signal-space diagramfor octaphase-shift keying
(i.e., M8). 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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Power Spectra of M-ary PSK Signals
The symbol duration of M-ary PSK is defined by
where Tbis the bit duration.
Proceeding in a manner similar to that described for a QPSK
signal, we may show that the baseband power spectral density ofan M-ary PSK signal is given by
In Figure 6.16, we show the normalized power spectral density
SB(f)/2Eb plotted versus the normalized frequency fTb for three
different values of M, namely, M= 2, 4, 8.
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Figure 6.16
Power spectra ofM
-ary PSK signals for M2, 4, 8.
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Hybrid Amplitude/Phase Modulation Schemes
In an M-ary PSK system, the in-phase and quadraturecomponents of the modulated signal are interrelated in
such a way that the envelope is constrained to remain
constant.
If this constraint is removed, and the in-phase andquadrature components are thereby permitted to be
independent, we get a new modulation scheme called M-
ary quadrature amplitude modulation.
This latter modulation scheme is hybrid in nature in that
the carrier experiences amplitude as well as phase
modulation.
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M-ary Quadrature Amplitude Modulation
M-ary QAM is a two-dimensional generalization of M-ary PAMin that its formulation involves two orthogonal passband basis
functions, as shown by
Let the ith message point si in the (1, 2) plane be denoted by
(aidmin/2, bidmin/2), where dminis the minimum distance between
any two message points in the constellation, ai and bi are
integers, and i= 1, 2, . . . , M.
Let (dmin/2) = E0, Where E0is the energy of the signal with thelowest amplitude.
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M-ary Quadrature Amplitude Modulation
The transmitted M-ary QAM signal for symbol k, say, is defined
by
The signal sk(t) consists of two phase-quadrature carriers with
each one being modulated by a set of discrete amplitudes, hence
the name quadrature amplitude modulation.
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QAM Square Constellations
With an even number of bits per symbol, we may write
where Lis a positive integer.
In the case of a QAM square constellation, the ordered pairs of
coordinates naturally form a square matrix, as shown by
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Example
Consider a 16-QAM whose signal constellation is depicted in Figure6.17a. The encoding of the message points shown in this figure is as
follows:
*Two of the four bits, namely, the left-most two bits, specify the quadrant
in the (1, 2) plane in which a message point lies. Thus, starting from the
first quadrant and proceeding counterclockwise, the four quadrants arerepresented by the dibits 11, 10, 00, and 01.
* The remaining two bits are used to represent one of the four possible
symbols lying within each quadrant of the (1, 2)-plane.
Note that the encoding of the four quadrants and also the encoding of the
symbols in each quadrant follow the Gray coding rule.
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Example
For the example at hand, we have L= 4. Thus the square constellation ofFigure 6.17a is the Cartesian product of the 4-PAM constellation shown
in Figure 6.17b with itself. Moreover, the matrix of the previous equation
has the value
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Figure 6.17(a) Signal-space diagram of M-ary QAM for M 16; the
message points in each quadrant are identified with Gray-encoded quadbits. (b) Signal-space diagram of the
corresponding 4-PAM signal.
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QAM Square Constellations
The probability of symbol error for M-ary QAM isapproximately given by
It is more logical to express Pein terms of the average value ofthe transmitted energy rather than E0.
Assuming that the L amplitude levels of the in-phase or
quadrature component are equally likely, we have
Accordingly, we may rewrite the previous equation in terms of
Eavas