ch06-part 4

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Signal-Space Diagram of MSK

Using a well-known trigonometric identity in the equation of

CPFSK signal, we may express the CPFSK signal s(t) in terms

of its in-phase and quadrature components as follows:

Consider first the in-phase component (2Eb/Tb) cos[(t)].

With the deviation ratio h= 1/2, we have

where the plus sign corresponds to symbol 1 and the minus sign

corresponds to symbol 0.

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Since the phase states (0) and (Tb) can each assume one of two

possible values, any one of four possibilities can arise, as described

here:

The phase (0) = 0 and (Tb) = /2, corresponding to the transmissionof symbol 1

The phase (0) = and (Tb) = /2, corresponding to the transmissionof symbol 0

The phase (0) = and (Tb) = -/2,(or, equivalently, 3/2 modulo2), corresponding to the transmission of symbol 1.

The phase (0) = 0 and (Tb) = -/2, corresponding to the transmission

of symbol 0.

This, in turn, means that the MSK signal itself may assume any one of

four possible forms, depending on the values of (0) and (Tb) .

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Accordingly, the signal constellation for an MSK signal is two-

dimensional (i.e., N = 2), with four possible message points (i.e.,

M = 4), as illustrated in Figure 6.29.

The coordinates of the message points are as follows in a

counterclockwise direction: (+Eb, +Eb), (-Eb, +Eb), (-Eb, -Eb), and (+Eb, -Eb).

The possible values of (0) and (Tb), corresponding to these

four message points, are also included in Figure 6.29.

The signal-space diagram of MSK is thus similar to that ofQPSK in that both of them have four message points.

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Figure

6.29

Signal-spacediagram for

MSK system.

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However, they differ in a subtle way that should be carefully

noted: In QPSK the transmitted symbol is represented by any

one of the four message points, whereas in MSK one of two

message points is used to represent the transmitted symbol at

any one time, depending on the value of (0). The next table presents a summary of the values of (0) and

(Tb), as well as the corresponding values of sl and s2 that are

calculated for the time intervals -Tb tTb and 0 t 2Tb,respectively.

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The first column of this table indicates whether symbol 1 or symbol 0was sent in the interval 0 tTb.

Note that the coordinates of the message points, sl and s2, haveopposite signs when symbol 1 is sent in this interval, but the same signwhen symbol 0 is sent.

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Example

Figure 6.30 shows the sequences and waveforms involved in the

generation of an MSK signal for the binary sequence 1101000. The

input binary sequence is shown in Figure 6.30a. The two modulation

frequencies are: fl= 5/4Tband f2= 3/4Tb. Assuming that, at time t= 0

the phase (0) is zero, the sequence of phase states is as shown in Figure

6.30, modulo 2. The polarities of the two sequences of factors used toscale the time functions l(t) and 2(t) are shown in the top lines of

Figures 6.30b and 6.30c. Note that these two sequences are offset

relative to each other by an interval equal to the bit duration Tb. The

waveforms of the resulting two components of s(t), namely,sll (t) and

s22(t), are also shown in Figures 6.30b and 6.30c. Adding these twomodulated waveforms, we get the desired MSK signal s(t) shown in

Figure 6.30d.

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Figure 6.30

(a) Input binarysequence.

(b) Waveform of

scaled time function

s1f

1(t). (c) Waveform

of scaled time

function s2f2(t). (d)

Waveform of the

MSK signal s(t)

obtained by addings1f1(t) and s2f2(t) on

a bit-by-bit basis.

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Error Probability of MSK

Referring to the signal-space diagram of Figure 6.29, we see

that the decision made by the receiver is between the message

points ml and m3 for symbol 0, or between the message points

m2and m4for symbol 1.

The corresponding decisions whether (0) = 0 or and whether(Tb) is -/2 or +/2 (i.e., the bit decisions) are made alternatelyin the I- and Q-channels of the receiver, with each channel

looking at the input signal for 2Tbseconds.

The signal from other bits does not interfere with the receiversdecision for a given bit in either channel.

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Generation and Detection of MSK Signals

Figure 6.31a shows the block diagram of a typical MSK transmitter.

The advantage of this method of generating MSK signals is that the

signal coherence and deviation ratio are largely unaffected by

variations in the input data rate.

Two input sinusoidal waves, one of frequency fc = nc/4Tb for somefixed integer nc, and the other of frequency 1/4Tb, are first applied to a

product modulator.

This produces two phase-coherent sinusoidal waves at frequencies fl

andf2, which are related to the carrier frequencyfcand the bit rate 1/Tb

for h= 1/2. These two sinusoidal waves are separated from each other by two

narrowband filters, one centered atfland the other atf2.

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The resulting filter outputs are next linearly combined toproduce the pair of quadrature carriers or orthonormal basisfunctions l(t) and 2(t).

Finally, l(t) and 2(t) are multiplied with two binary waves a1(t)and a

2(t), both of which have a bit rate equal to 1/2T

b.

Figure 6.31b shows the block diagram of a typical MSKreceiver.

The received signal x(t) is correlated with locally generatedreplicas of the coherent reference signals l(t) and 2(t).

Note that in both cases the integration interval is 2Tb seconds,and that the integration in the quadrature channel is delayed byTbseconds with respect to that in in-phase channel.

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The resulting in-phase and quadrature channel correlatoroutputs, x1and x2, are each compared with a threshold of zero,and estimates of the phase (0) and (Tb) are derived in themanner described below.

If we have the estimates (0) = 0 and (Tb) = -/2, or

alternatively if we have the estimates (0) = and (Tb) = /2,the receiver makes a decision in favor of symbol 0.

If we have the estimates (0) = and (Tb) = -/2, oralternatively if we have the estimates (0) = 0 and (Tb) = /2,the receiver makes a decision in favor of symbol 1.

Finally, these phase decisions are interleaved so as to reconstructthe original input binary sequence with the minimum average

probability of symbol error in an AWGN channel.

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Power Spectra of MSK Signals

As with the binary FSK signal, we assume that the input binary

wave is random with symbols 1 and 0 equally likely, and the

symbols transmitted during different time slots being

statistically independent.

The baseband power spectral density of the MSK signal is givenby

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Power Spectra of MSK Signals

The baseband power spectrum is plotted in Figure 6.9, where

the power spectrum is normalized with respect to 4Eb and the

frequencyfis normalized with respect to the bit rate 1/Tb.

Forf>> 1/Tb, the baseband power spectral density of the MSK

signal falls off as the inverse fourth power of frequency,whereas in the case of the QPSK signal it falls off as the inverse

square of frequency.

Accordingly, MSK does not produce as much interference

outside the signal band of interest as QPSK.

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M-ary FSK

Consider the M-ary version of FSK, for which the transmitted

signals are defined by

where i= 1, 2, . . . , M, and the carrier frequencyfc= nc/2Tfor

some fixed integer nc.

Since the individual signal frequencies are separated by 1/2T

Hz, the signals in the above equation are orthogonal; that is

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M-ary FSK

Specifically, noting that the minimum distance dmin in M-ary

FSK is 2E:

For fixed M, this bound becomes increasingly tight as E/N0 is

increased.

Indeed, it becomes a good approximation toPefor values of Pe

10-3.

Moreover, forM= 2 (i.e., binary FSK), the bound of the aboveequation becomes an equality.

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Power Spectra of M-ary FSK Signals

The spectral analysis of M-ary FSK signals is much more

complicated than that of M-ary PSK signals.

A case of particular interest occurs when the frequencies

assigned to the multilevels make the frequency spacing uniform

and the frequency deviation k= 0.5. That is, theMsignal frequencies are separated by 1/2T, where T

is the symbol duration.

For k= 0.5, the baseband power spectral density of M-ary FSK

signals is plotted in Figure 6.36 forM= 2, 4, 8.

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Figure 6.36Power spectra of M-ary FSK signals forM2, 4, 8.

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Detection of Signals with Unknown Phase

Up to this point in our discussion, we have assumed that the receiver isperfectly synchronized to the transmitter, and the only channelimpairment is noise.

In practice, however, it is often found that in addition to the uncertaintydue to channel noise, there is also uncertainty due to the randomness of

certain signal parameters. The usual cause of this uncertainty is distortion in the transmission

medium.

Synchronization with the phase of the transmitted carrier may then betoo costly, and the designer may choose to disregard the phaseinformation in the received signal at the expense of some degradation

in noise performance. A digital communication receiver with no provision made for carrier

phase recovery is said to be noncoherent.

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Optimum Quadratic Receiver

Consider a binary digital communication system in which the

transmitted signal is

where E is the signal energy, T is the duration of the signalinginterval, and the carrier frequency i for symbol i is an integral

multiple of 1/2T.

The system is assumed to be noncoherent, in which case the

received signal for an AWGN channel may be written in the

form

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where is the unknown carrier phase, and w(t) is the sample

function of a white Gaussian noise process of zero mean and

power spectral densityN0/2.

In a real-life situation it is realistic to assume complete lack of

prior information about and to treat it as a random variablewith uniform distribution:

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We may formulate the conditional likelihood function of symbol

si, given the carrier phase , as

We may express the likelihood function for the signal detectionproblem described herein in the compact form

The binary hypothesis test (i.e., the hypothesis that signal s1(t)or signals2(t) was transmitted) can now be written as

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where hypothesisH1andH2correspond to signalss1(t) ands2(t),

respectively.

For convenience of implementation, however, the hypothesis

test is carried out in terms of li2 instead of li, as shown by

A receiver based on the above equation is known as the

quadratic receiver.

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Two Equivalent Forms of Quadratic Receiver

We next derive two equivalent forms of the quadrature receivershown in Figure 6.37a.

The first form is obtained easily by replacing each correlator inFigure 6.37a with a corresponding equivalent matched filter.

We thus obtain the alternative form of quadrature receivershown in Figure 6.37b.

In one branch of this receiver, we have a filter matched to thesignal cos(2fit), and in the other branch we have a filtermatched to sin(2fit), both of which are defined for the timeinterval 0 tT.

The filter outputs are sampled at time t= T, squared, and thenadded together.

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Figure 6.37Noncoherentreceivers.

(a) Quadrature

receiver usingcorrelators.

(b) Quadrature

receiver using

matched filters.(c) Noncoherent

matched filter.

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To obtain the second equivalent form of the quadrature receiver,

suppose we have a filter that is matched to s(t) = cos(2fit+ )

for 0 tT.

The envelope of the matched filter output is obviously

unaffected by the value of phase . The output of such a filter in response to the received signalx(t)

is given by

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The envelope of the matched filter output is proportional to the

square root of the sum of the squares of the integrals in the

above equation.

The output (at time T) of a filter matched to the signal cos(2fit

+ ), of arbitrary phase , followed by an envelope detector isthe same as the corresponding output of the quadrature receiver

of Figure 6.37a.

This form of receiver is shown in Figure 6.37c.

The combination of matched filter and envelope detector shown

in Figure 6.37c is called a noncoherent matched filter.

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Noncoherent Orthogonal Modulation

Consider a binary signaling scheme that involves the use of twoorthogonal signalssl(t) ands2(t), which have equal energy.

During the interval 0 tT, one of these two signals is sentover an imperfect channel that shifts the carrier phase by anunknown amount.

Let gl(t) and g2(t) denote the phase-shifted versions of sl(t) ands2(t), respectively.

lt is assumed that the signals gl(t) and g2(t) remain orthogonaland have the same energyE, regardless of the unknown carrier

phase.

We refer to such a signaling scheme as noncoherent orthogonalmodulation.

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The channel also introduces an additive white Gaussian noisew(t) of zero mean and power spectral densityN0/2.

We may thus express the received signalx(t) as

The requirement is to use x(t) to discriminate between s1(t) and

s2(t), regardless of the carrier phase.

For this purpose, we employ the receiver shown in Figure 6.39a.

The receiver consists of a pair of filters matched to the

transmitted signalss1(t) ands2(t).

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Because the carrier phase is unknown, the receiver relies onamplitude as the only possible discriminant.

If the upper path in Figure 6.39a has an output amplitude llgreater than the output amplitude l2 of the lower path, thereceiver makes a decision in favor ofs1(t).

If the converse is true, it decides in favor ofs2(t).

When they are equal, the decision may be made by flipping afair coin.

In any event, a decision error occurs when the matched filterthat rejects the signal component of the received signalx(t) has alarger output amplitude (due to noise alone) than the matchedfilter that passes it.

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We note that a noncoherent matched filter (constituting theupper or lower path in the receiver of Figure 6.39a) may be

viewed as being equivalent to a quadrature receiver.

The quadrature receiver itself has two channels.

One version of the quadrature receiver is shown in Figure 6.39b.

In the upper channel, called the in-phase channel, the received

signalx(t) is correlated with the function i(t) which represents

a scaled version of the transmitted signals1(t) ors2(t) with zero

carrier phase.

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In the lower channel, called the quadrature channel, on the otherhand, x(t) is correlated with another function i(t), which

represents the version of i(t) that results from shifting the

carrier phase by -90 degrees.

The average probability of error for the noncoherent receiver ofFigure 6 .39a is given by the simple formula

where E is the signal energy per symbol, and N0

/2 is the noise

spectral density.

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