ch06-part 2

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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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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 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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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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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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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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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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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

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