Passband Digital Transmission• The transmitter consists of a signal transmission encoder followed by a

modulator. With the vector si as input, the modulator constructs a distinct signal si(t) of duration T seconds as the representation of the symbol mi generated by the message source.

• With a sinusoidal carrier, the feature that is used by the modulator to distinguish one signal from another is a step change in the amplitude, frequency, or phase of the carrier. (sometimes, a hybrid form of modulation

that combines changes in both amplitude and phase or amplitude and frequency is used.)

• The bandpass communication channel, coupling the transmitter to the receiver, is assumed to have two characteristics:

1. The channel is linear, with a bandwidth that is wide enough to accommodate the transmission of the modulated signal si(t) with negligible or no distortion.

2. The channel noise w(t) is the sample function of a white Gaussian noise process of zero mean and power spectral density No/2.

• The receiver, which consists of a detector followed by a signal transmission decoder, performs two functions:

1. It reverses the operations performed in the transmitter. 2. It minimizes the effect of channel noise on the estimate output computed

symbol for the transmitted symbol mi.

The following points are noteworthy from Figure 6.1:Although in continuous-wave modulation it is usually difficult to distinguish between phase-modulated and frequency-modulated signals by merely looking at their waveforms, this is not true for PSK and FSK signals. Unlike ASK signals both PSK and FSK signals have a constant envelope.

Coherent Phase-shift Keying (PSK)There are binary PSK (BPSK), Quadriphase-shift keying (QPSK) and M-ary PSK

In a coherent binary PSK system, the pair of signals s1(t) and s2(t) used to represent binary symbols 1 and 0, respectively, is defined by

where 0 ≤ t ≤ Tb, and Eb is the transmitted signal energy per bit. To ensure that each transmitted bit contains an integral number of cycles of the carrier wave, the carrier frequency fc is chosen equal to nc/Tb for some fixed integer nc

in the case of binary PSK, there is only

one basis function of unit energy, namely,

Then we may express the transmitted signals s1(t) and s2(t) in terms of Φ1(t) as follows

A coherent binary PSK system is therefore characterized by having a signal space that is one-dimensional (i.e., N = 1), with a signal constellation consisting of two message points (i.e., M = 2). The coordinates of the message points are

Error Probability of Binary PSK

The signal space is divided into two regions:The set of points closest to message point 1 at The set of points closest to message point 2 at This is accomplished by constructing the midpoint of the line joining these two message points, and then marking off the appropriate decision regions. These decision regions are marked Z1 and Z2 according to the message point around which they are constructed.

The decision rule is now simply to decide that signal sl(t) (i.e., binary symbol 1) was transmitted if the received signal point falls in region Z1, and decide that signal s2(t) (i.e., binary symbol 0) was transmitted if the received signal point falls in region Z2. Noise is assumed to be AWGN with zero mean and variance No/2.

Note that the signal space of BPSK is symmetric with respect to the origin. It follows therefore that p01, the conditional probability of the receiver deciding in favor of symbol 0, given that symbol 1 was transmitted, has the same value as p10 above. Thus, averaging the conditional error probabilities p10 and p01, we find that the average probability of symbol error or, equivalently, the bit error rate for coherent BPSK is (assuming equiprobable symbols)

Note that increasing the transmitted signal energy per bit, Eb, for a specified noise spectral density No, the message points corresponding to symbols 1 and 0 move further apart, and the average probability of error Pe is correspondingly reduced.

Generation and Detection of Coherent Binary PSK Signals

a product modulator

The carrier and the timing pulses used to generate the binary wave are usually extracted from a common master clock where

Power Spectra of Binary PSK Signals

the power spectral density of a random binary wave (symbols 1 and 0 equally likely and the symbols transmitted during the different time slots being statistically independent.) is equal to the energy spectral density of the symbol shaping function g(t) divided by the symbol duration. The energy spectral density of a Fourier transformable signal g(t) is defined as the squared magnitude of the signal's Fourier transform. Hence, the baseband power spectral density of a binary PSK signal

The power spectral density, Ss(f), of the original band-pass signal s(t) is a frequency-shifted version of SB(f), except for a scaling factor, as shown by

It is therefore sufficient to evaluate the baseband power spectral density SB(f), where the calculation of SB(f) should be simpler than the calculation of Ss(f ).

DIFFERENTIAL PHASE-SHIFT KEYING (DPSK)

In the coherent PSK receiver, it was assumed that the receiver was perfectly synchronized in frequency and had exact knowledge of the transmitted carrier phase. In practice, however, the receiver does not have exact knowledge of this carrier phase, although it may be able to establish a phase reference to the exact phase. Provided that it remains essentially constant over a period of two-bit intervals, this phase ambiguity may be resolved by using differential encoding. A signaling technique that combines differential encoding with binary phase-shift keying is known as differential phase-shift keying (DPSK). Thus, by using DPSK, digital information is encoded, not by the absolute identification of zero carrier phase with symbol 1 and 180 degrees phase with symbol 0, but rather in terms of the phase change between successive pulses in the given binary data stream. For example, symbol 1 is represented by zero phase change from the previous pulse of the binary sequence, whereas symbol 0 is represented by a phase change of 180 degrees, as illustrated in Figure 9.13, where zero phase is arbitrarily chosen to represent the reference bit.

1

For the differentially coherent detection of a DPSK signal, we may use the receiver shown in Figure 9.14. At any particular instant of time, we have the received DPSK signal as one input into the multiplier and a delayed version of the received signal by a bit duration Tb, as the other input. The integrator output is proportional to cos ϕ, where ϕ is the difference between the carrier phase angles in the received DPSK signal and its delayed version, measured in the same bit interval. Therefore, when ϕ = 0 (corresponding to symbol 1), the integrator output is positive; on the other hand, when ϕ = π (corresponding to symbol 0), the integrator output is negative. Thus, by comparing the integrator output with a decision level of zero volts, the above receiver can reconstruct the binary sequence, which, in the absence of noise, is exactly the same as the original binary data input.

The major difference between a DPSK system as described above and a coherent PSK system is not in the differential encoding, which can be used in any case, but rather it lies in the way in which the reference signal is derived for the phase detection of the received signal. Specifically, in a DPSK receiver the reference is contaminated by additive noise to the same extent as the information pulse; that is, both have the same signal-to noise ratio. This makes the determination of the overall probability of error using differentially coherent detection of differentially encoded PSK signals somewhat complicated. Therefore, it will not be given here. The result is, however,

Quadriphase-shift keying (QPSK)A very low probability of error is one important goal in the design of a digital communication system. Another important goal is the efficient utilization of channel bandwidth. The coherent quadriphase-shift keying, which is an example of quadrature-carrier multiplexing is a bandwidth-conserving modulation scheme in which the phase of the carrier takes on one of four equally spaced values, such as π/4, 3π/4, 5π/4 and 7π/4. Each possible value of the phase corresponds to a unique Gray-encoded dibit. For this set of values we may define the transmitted signal aswhere i = 1, 2, 3, 4; E is the transmitted

signal energy per symbol, and T is the symbol

Duration and fc=nc/T.

Signal-Space Diagram of QPSK

where i = 1,2,3,4. Based on this representation, we can make the following observations:

There are two orthonormal basis functions, φ1(t) and φ2(t) , contained in the expansion

of si(t). Specifically, φ1(t) and φ2(t) are defined by a pair of quadrature carriers

&

fc= nc/T.

Generation and Detection of Coherent QPSK Signals

From Equation (6.24), 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; this is merely a statement of the quadrature-carrier multiplexing property of coherent QPSK. The in-phase channel output x1 and the quadrature channel output x2 (i.e., the two elements of the observation vector x) may be viewed as the individual outputs of the two coherent binary PSK systems.

Symbols 1 and 0 are represented by respectively. This binary wave is next divided by means of a demultiplexer into two separate binary waves consisting of the odd- and even- numbered input bits. These two binary waves are denoted by a1(t) and a2(t) which are used to modulate a pair of quadrature carriers (orthonormal basis functions φ1(t) and φ2(t))

In any signaling interval, the amplitudes of al(t) and a2(t) equal to si1 and si2, respectively,depending on the particular dibit that is being transmitted.

Figure 6.7 illustrates the sequences and waveforms involved in the generation of a QPSK signal for the input binary sequence 01101000. The two waveforms si1 φ1(t) and si2 φ2(t) are alsoshown in Figures 6.7b and 6.7c, respectively, which may individually be viewed as examples of a binary PSK signal. Adding them, we get the QPSK waveform shown in Figure 6.7d.

Error Probability of QPSK

In a coherent QPSK system, the received signal x(t) is defined by

where w(t) is the sample function of a white Gaussian noise process of zero mean and power

spectral density No/2. Correspondingly, the observation vector x has two elements, x1 and x2 defined by

Thus the observable elements xl and x2 are sample values of independent Gaussian random variables with mean values equal to respectively, and with a common variance equal to No/2.

The decision rule is now simply to decide that s1(t) was transmitted if the received signal point associated with the observation vector x falls inside region Z1, decide that s2(t) was transmitted if the received signal point falls inside region Z2, and so on.

An erroneous decision will be made if, for example, signal s4(t) is transmitted but the noise w(t) is such that the received signal point falls outside region Z4.

To calculate the average probability of symbol error, we note from Equation (6.24) that a coherent QPSK system is in fact equivalent to two coherent binary PSK systems working in parallel. These two binary PSK systems may be characterized as follows: The signal energy per bit is E/2 and The noise spectral density is No/2. Hence, using Equation (6.20) for the average probability of bit error of a coherent binary PSK system, we may now state that the average probability of bit error in each channel of the coherent QPSK system is

Another important point to note is that the bit errors in the in-phase and quadrature channels of the coherent QPSK system are statistically independent. Accordingly, the average probability of a correct decision resulting from the combined action of the two channels working together is

The average probability of symbol error for coherent QPSK is therefore

In a QPSK system, we note that since there are two bits per symbol, the transmitted signal energy per symbol is twice the signal energy per bit,

With Gray encoding used for the incoming symbols, the bit error rate of QPSK is exactly

We may therefore state that •A coherent QPSK system achieves the same average probability of bit error as a coherent binary PSK system for the same bit rate and the same Eb/No but uses only half the channel bandwidth. •Stated in a different way, for the same Eb/No and therefore the same average probability of bit error, a coherent QPSK system transmits information at twice the bit rate of a coherent binary PSK system for the same channel bandwidth. •For a prescribed performance, QPSK uses channel bandwidth better than binary PSK, which explains the preferred use of QPSK over binary PSK in practice.

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, the in-phase and quadrature components have a common power spectral density, namely, E sinc2(Tf ).

The in-phase and quadrature components are statistically independent. Accordingly, the baseband power spectral density of the QPSK signal equals the sum of the individual

power spectral densities of the in-phase and quadrature components, so we may write

Figure 6.9 plots SB(f), normalized with respect to 4Eb, versus the normalized frequency fTb.

M-ARY PSK

QPSK is a special case of M-ary PSK, where the phase of the carrier takes on one of M possible values, namely, θi = 2(i - l ) π / M where i = 1,2,. . . , M. Accordingly, during each signaling interval of duration T, one of the M possible signals is sent, where E is the signal energy per symbol. The carrier frequency fc = nc/T for some fixed integer nc

Each si(t) may be expanded in terms of the same two basis functions ф1(t) and ф 2(t)) defined in Equations (6.25) and (6.26), respectively. The signal constellation of M-ary PSK is therefore two-dimensional. The M message points are equally spaced on a circle of radius and center at the origin,

Note that since the signal-space diagram is circularly symmetric, an approximate formula for the average probability of symbol error for coherent M-ary PSK can be calculated as

where it is assumed that M ≥ 4. The approximation becomes extremely tight, for fixed M, as E/No is increased.

For M = 4, Equation (6.47) reduces to the same form given in Equation (6.34) for QPSK.

The power spectra of M-ary PSK signals possess a main lobe bounded by well-defined spectral nulls (i.e., frequencies at which the power spectral density is zero). Accordingly, the spectral width of the main lobe provides a simple and popular measure for the bandwidth of M-ary PSK signals. This definition is referred to as the null-to-null bandwidth which contains most of the signal power. the channel

bandwidth required to pass the main spectral lobe of

M-ary signals is given by B = 2/T =

Where the bit rate Rb=1/Tb

Bandwidth Efficiency ρ

the bandwidth efficiency of M-ary PSK signals is given by

Note that as the number of states, M, is increased, the bandwidth efficiency is improved at the expense of error performance. To ensure that there is no degradation in error performance, we have to increase Eb/No to compensate for the increase in M.

In an M-ary PSK system, the in-phase and quadrature components of the modulated signal are interrelated in such a way that the envelope is constrained to remain constant which can be seen from the circular constellation of the message points. However, if this constraint is removed, and the in-phase and quadrature components are thereby permitted to be independent, we get a new modulation scheme called M-ary quadrature amplitude modulation (QAM) which is hybrid in nature in that the carrier experiences amplitude as well as phase modulation.

In Chapters 4, we studied M-ary pulse amplitude modulation (PAM), which is one dimensional. M-ary QAM is a two-dimensional generalization of M-ary PAM in that its formulation involves two orthogonal passband basis functions, as shown by

Let the ith message point si in the plane be denoted by (aidmin/2, bidmin/2), where dmin is the minimum distance between any two message points in the constellation, ai and bi are integers and i = 1, 2,. . . , M. Let (dmin/2) = where Eo is the energy of the signal with the lowest amplitude. The transmitted M-ary QAM signal for symbol k, say, is then 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. Depending on the number of possible symbols M, we may distinguish two distinct QAM constellations: square constellations for which the number of bits per symbol is even, and cross constellations for which the number of bits per symbol is odd.

QAM Square ConstellationsWith an even number of bits (n) per symbol, Define & M=2n, where n is an even number and L is a positive integer. Under this condition, an M-ary QAM square constellation can always be viewed as the Cartesian product of a one-dimensional L-ary PAM constellation with itself.

By definition, the Cartesian product of two sets of coordinates

(representing a pair of one-dimensional constellations) is made up of the set of all possible ordered pairs of coordinates with the first coordinate in each such pair taken from the first set involved in the product and the second coordinate taken from the second set in product.

Two of the four bits, namely, the left-most two bits, specify the quadrant in the plane in which a message point lies. Thus, starting from the first quadrant and proceeding counterclockwise, the four quadrants are represented 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 plane. Note that the encoding of the four quadrants and also the encoding of the symbols in each quadrant follow the Gray coding rule.

Thus the square constellation of Figure 6.17a is the Cartesian product of the 4-PAM constellation shown in Figure 6.17b with itself.

In the case of a QAM square constellation, the ordered pairs of coordinates naturally form a square matrix, as shown by

For the previous example, we have L = 4.

Probability of Symbol Error for M-ary QAMTo calculate the probability of symbol error for M-ary QAM, we exploit the property that a QAM square constellation can be factored into the product of the corresponding PAM constellation with itself. We may thus proceed as follows:The probability of correct detection for M-ary QAM may be written aswhere is the probability of symbol error for the corresponding L-ary PAMwhich is defined as(Note that in the M-ary QAM corresponds to M in the M-ary PAM)

The probability of symbol error for M-ary QAM is given by

the probability of symbol error for M-ary QAM is approximately given by

The transmitted energy in M-ary QAM is variable in that its instantaneous value depends on the particular symbol transmitted. It is therefore more logical to express Pe in terms of the average value of the transmitted energy rather than Eo.

Assuming that the L amplitude levels of the in-phase or quadrature component are equally likely, we have

To generate an M-ary QAM signal with an odd number of bits per symbol, a cross constellation is used as following:

• Start with a QAM square constellation with (n-1) bits per symbol.

•Extend each side of the QAM square constellation by adding 2n-3 symbols and ignore the corners in the extension.

The inner square represents 2n-1symbols. The four side extensions add 4* 2n-3 = 2n-1 symbols. The total number of symbols in the cross constellation is therefore = 2n-1+ 2n-1 =2n and therefore represents n bits per symbol as desired. Unlike QAM square constellation, it is not possible to express a QAM cross constellation as the product of a PAM constellation with itself. Note also that it is not possible to perfectly Gray code a QAM cross constellation.

M-ary PSK and M-ary QAM share a common property: Both are examples of linear modulation.

In this section we study a nonlinear method of passband data transmission, namely, coherent frequency-shift keying (FSK). Considering the simple case of binary FSK where symbols 1 and 0 are distinguished from each other by transmitting one of two sinusoidal waves that differ in frequency by a fixed amount.

BFSKA typical pair of sinusoidal waves is described by

where i = 1,2, and Eb is the transmitted signal energy per bit. The transmitted frequency fi = (nc+i)/Tb for some fixed integer nc. It is a continuous-phase signal in the sense that phasecontinuity is always maintained, including the inter-bit switching times (Sunde's FSK). This form of digital modulation is an example of continuous-phase frequency-shift keying (CPFSK)

The signals sl(t) and s2(t) are orthogonal, but not normalized to have unit energy. The most useful form for the set of orthonormal basis functions is deduced as

where i = 1,2. Correspondingly, the coefficient sij for i = 1,2, and j = 1,2 is defined by

Thus, unlike coherent binary PSK, a coherent binary FSK system is characterized by having a signal space that is two-dimensional (i.e., N = 2) with two message points (i.e., M = 2), as shown in Figure 6.25. The two message points are defined by

The Euclidean distance between

them equal to

Figure 6.25 also includes a couple of inserts, which show waveforms representative of signals s1(t) and s2(t).

Generation and Detection of Coherent Binary FSK Signals

The output of the on-off level encoder, symbol 1 is represented by a constant amplitude of volts and symbol 0 is represented by zero volts.

The use of an inverter in the lower channel in Figure 6.26a, makes sure that when symbol 1 is at the input, the oscillator with frequency f1 in the upper channel is switched on while the oscillator with frequency f2 in the lower channel is switched off, with the result that frequency f1 is transmitted. Conversely, when symbol 0 is at the input, the oscillator in the upper channel is switched off and the oscillator in the lower channel is switched on, with the result that frequency f2 is transmitted.

The two oscillators are synchronized, so that their outputs satisfy the requirements of the two orthonormal basis functions:

Alternatively, we may use a single keyed (voltage controlled) oscillator. In either case, the frequency of the modulated wave is shifted with a continuous phase, in accordance with the input binary wave.

To detect the original binary sequence from the noisy received signal x(t), the receiver consists of two correlators with a common input which are supplied with locally generated coherent reference signals . The correlators outputs are then subtracted, one from the other, and the resulting difference is compared with a threshold of zero volts. If y > 0, the receiver decides in favor of 1. On the other hand, if y < 0, it decides in favor of 0. if y is exactly zero, the receiver makes a random guess in favor of 1 or 0.

Error Probability of Binary FSKThe observation vector x has two elements x1 and x2 that are defined by, respectively,

Given that symbol 1 was transmitted,

x(t) equals s1(t) + w(t), where w(t) is the

sample function of a white Gaussian noise process of zero mean and power spectral density No/2.

If, on the other hand, symbol 0 was transmitted,

x(t) equals s2(t) + w(t).

The receiver decides in favor of symbol 1 if

The receiver decides in favor of symbol 0 if

Define a new Gaussian random variable Y whose sample value y is equal to the difference between x1 and x2; that is,

The mean value of the random variable Y depends on which binary symbol was transmitted. Given that symbol 1 (0) was transmitted, the Gaussian random variables X1 and X2, whose sample values are denoted by x1 and x2 have mean values equal to and zero

(zero and ), respectively. Correspondingly, the conditional mean of the random variable Y, given that symbol 1(0) was transmitted, respectively, is

The variance of the random variable Y is independent of which binary symbol was transmitted. Since the random variables X1 and X2 are statistically independent, each with a variance equal to N0/2, it follows that

= No.

Given symbol 0 was transmitted, The conditional probability density function of the random variable Y is then given by

Therefore, the conditional probability of error, given that symbol 0 was transmitted, is

Since the conditional probability of error given that symbol 1 was transmitted, has the same value where p01 = p10, the average probability of bit error or, equivalently, the bit error for coherent binary FSK is (assuming equiprobable symbols)

Note that in a coherent binary FSK system, the bit energy-to-noise density ratio, Eb/No has to be doubled to maintain the same bit error rate as in a coherent binary PSK system. This result is in perfect accord withthe signal-space diagrams of Figures 6.3 and 6.25, where it can be seen that in a binary PSK system the Euclidean distance between the two message points is equal to whereas in a binary FSK system the corresponding distance is

Power Spectra of Binary FSK SignalsConsider the case of Sunde's FSK, for which the two transmitted frequencies f1 and f2 differ by an amount equal to the bit rate l/Tb, and their arithmetic mean equals the nominal carrier frequency fc; phase continuity is always maintained, including inter-bit switching times. We may express this special binary FSK signal as follows:

The plus sign corresponds to transmitting symbol 0, and the minus sign corresponds to transmitting symbol 1.

As before, we assume that the symbols 1 and 0 in the random binary wave at the modulator input are equally likely, and that the symbols transmitted in adjacent time slots are statistically independent.1) The in-phase component is completely independent of the input binary wave equals for all values of t. The power spectral density of this component therefore consists of two delta functions, weighted by the factor Eb/2Tb, and occurring at f = ±1/2Tb.2) The quadrature component is directly related to the input binary wave. During the signaling interval 0 ≤ t ≤ Tb, it equals -g(t) for symbol 1, and +g(t) for symbol 0.

The symbol shaping function g(t) is defined by

The power spectral density of the quadrature component equals the The baseband power spectral density of Sunde's FSK signal equals the sum of the power spectral densities of these two components, as shown in Figure 6.5

The power spectrum of the binary FSK signal contains two discrete frequency components located at (fc+ 1/2Tb) = f1 and (fc - 1/2Tb) = f2, with their average powers adding up to one-half the total power of the binary FSK signal. The presence of these two discrete frequency components provides a means of synchronizing the receiver with the transmitter.

Note also that the baseband power spectral density of a binary FSK signal with continuous phase ultimately falls off as the inverse fourth power of frequency. This is readily established by taking the limit in Equation (6.107) as f approaches infinity If, however, the FSK signal exhibits phase discontinuity at the inter-bit switching instants (this arises when the two oscillators applying frequencies f1 and f2 operate independently of each other), the power spectral density ultimately falls off as the inverse square of frequency. Accordingly, an FSK signal with continuous phase does not produce as much interference outside the signal band of interest as an FSK signal with discontinuous phase.

M-ARY FSKConsider the M-ary version of FSK, for which the transmitted signals are defined by

where i = 1, 2, . . . , M, and the carrier frequency fc = nc/2T for some fixed integer nc

The transmitted symbols are of equal duration T and have equal energy E. Since theindividual signal frequencies are separated by 1/2T Hz, the signals in Equation (6.137) are orthogonal. This property of M-ary FSK suggests that we may use the transmitted signals si(t) themselves, except for energy normalization, as a complete orthonormal set of basis functions, as shown by

Accordingly, the M-ary FSK is described by an M-dimensional signal-space diagram.

For coherent M-ary FSK, the optimum receiver consists of a bank of M correlators or matched filters, with the ϕi(t) of the previous Equation providing the pertinent reference signals. At the sampling times t = kT, the receiver makes decisions based on the largest matched filter output in accordance with the maximum likelihood decoding rule.

Noting that the minimum distance dmin in M-ary FSK is a the average probability of symbol error for M-ary FSK. (assuming equiprobable symbols)

Power Spectra of M-ary FSK Signals

Bandwidth Effiency of M-ary FSK Signals

The channel bandwidth required totransmit M-ary FSK signals is

B =M/2T

The bandwidth efficiency of M-ary signals is therefore ρ = Rb/B = 2 log2 M/M

Comparing Tables 6.4 and 6.6, we see that increasing the number of levels M tends to increase the bandwidth efficiency of M-ary PSK signals, but it also tends to decrease the bandwidth efficiency of M-ary FSK signals. In other words, M-ary PSK signals are spectrally efficient, whereas M-ary FSK signals are spectrally inefficient.

Consider a continuous-phase frequency-shift keying (CPFSK) signal, which is defined for the interval 0 ≤t ≤ Tb as follows:The phase θ(0) denoting the value of the phase at time t = 0, sums up the past history of the modulation process up to time t = 0. The frequencies f1 and f2 are sent in response to binary symbols 1 and 0 appearing at the modulator input, respectively. s(t) can be expressed in the conventional form of an angle-modulated signal as shown where the plus sign corresponds to sending symbol 1, and the minus sign corresponds to sending symbol 0.Solve for fc and h. The nominal carrier frequency fc is therefore the arithmetic mean of the frequencies f1 andf2. The difference between the frequencies f1 and f2, normalized with respect to the bit rate l/Tb, defines the dimensionless parameter h, which is referred to as the deviation ratio.

Phase Trellis

From the above equation, it can be seen that at time t = Tb,

That is to say, the sending of symbol 1 increases the phase of a CPFSK signal s(t) by πh radians, whereas the sending of symbol 0 reduces it by an equal amount. The variation of phase θ(t) with time t follows a path consisting of a sequence of straight lines, the slopes of which represent frequency changes.

The tree makes clear the transitions of phase across interval boundaries of the incoming sequence of data bits. Moreover, it is evident from Figure 6.27 that the phase of a CPFSK signal is an odd or even multiple of πh radians at odd or even multiples of the bit duration Tb, respectively.

The deviation ratio h is exactly unity in the case of Sunde's FSK, which is a CPFSK scheme as previously described. Hence, according to Figure 6.27 the phase change over one bit interval is ± π radians. But, a change of +π radians is exactly the same as a change of - π radians, modulo 2π. It follows therefore that in the case of Sunde's FSK there is no memory; that is, knowing which particular change occurred in the previous bit interval provides no help in the current bit interval.

In contrast, the situation is completely different when the deviation ratio h is equal the special value of 1/2. Now in this case, the phase can take on only the two values ± π /2 at odd multiples of Tb, and only the two values 0 and π at even multiples of Tb, as in Figure 6.28.

Note that a "trellis" is a treelike structure with remerging branches. Each path from left to right through the trellis of Figure 6.28 corresponds to a specific binary sequence input.

Signal-Space Diagram of MSK

The CPFSK signal s(t) in terms of its in-phase and quadrature components can be written as follows:

Consider first the in-phase component

With the deviation ratio h = 1/2, where

Where the plus sign corresponds to symbol 1 and the minus sign corresponds to symbol 0.

A similar result holds for θ(t) in the interval -Tb ≤ t ≤ 0, except that the algebraic sign is not necessarily the same in both intervals. Since the phase θ(0) is 0 or π, depending on the past history of the modulation process, we find that, in the interval -Tb ≤ t ≤ Tb, the polarity of cos[θ(t)] depends only on θ(0), regardless of the sequence of 1s and 0stransmitted before or after t = 0. Thus, for this time interval, the in-phase component

sI(t) consists of a half-cycle cosine pulse defined as follows:where the plus sign corresponds to θ(0) = 0 and the minus sign corresponds to θ (0) = π.

Where, cos(πt/2Tb) = 0 at t=-Tb & t= Tb

In a similar way, it can be shown that, in the interval 0≤ t ≤ 2Tb, the quadrature component sQ(t) consists of a half-cycle sine pulse, whose polarity depends only on θ(Tb) as shown:

where the plus sign for

θ(Tb) = π/2 and the

minus sign for

θ(Tb) = - π/2.

= ± sin(πt/2Tb) 0≤ t ≤ 2Tb

Where, sin(πt/2Tb) = 0 at t=0 & t= 2Tb

Since the phase states θ(0) and θ(Tb) each assume one of two possible values, any one of four possibilities can arise, as described here:

1. The phase θ(0)= 0 and θ(Tb)= π/2, corresponding to the transmission of symbol 1.

2. The phase θ(0)= π and θ(Tb)= π/2, corresponding to the transmission of symbol 0.

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

4. The phase θ(0)= 0 and θ(Tb)= - π/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 θ(0) and θ(Tb).

The orthonormal basis functions ϕ1(t) and ϕ2(t) for MSK are defined by a pair of sinusoidally modulated quadrature carriers:

Both integrals are evaluated for a time interval equal to twice the bit duration.Both the lower and upper limits of the product integration used to evaluate the coefficients, are shifted by the bit duration Tb with respect to those used to evaluate the coefficients,.The time interval 0 ≤ t ≤ Tb, for which the phase states θ(0) and θ(Tb) are defined, is common to both integrals. The signal constellation for an MSK signal is two-dimensional (i.e., N = 2), with four possible message points (i.e., M = 4),

The signal-space diagram of MSK is thus similar to that of QPSK in that both of them have four message points. 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 θ(0).

Note that the coordinates of the message points, s1 and s2, have opposite signs when symbol 1 is sent in this interval, but the same sign when symbol 0 is sent. Accordingly, for a given input data sequence, we may use the entries of Table 6.5 to derive, on a bit-by-bit basis, the two sequences of coefficients required to scale ϕ1(t) and ϕ2(t) , and thereby determine the MSK signal s(t).

Generation of MSK Signals

Two input sinusoidal waves, one of frequency fc = nc/4Tb for some fixed 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 f1 and f2,which are related to the carrier frequency fc and the bit rate l/Tb by Equations (6.3 11) and (6.112) for h = 1/2. The resulting filter outputsare next linearly combined to produce the pair of orthonormal basis functions ϕ1(t) and ϕ2(t). Finally, ϕ1(t) and ϕ2(t) are multiplied with two binary waves a1(t) and a2(t), both of which have a bit rate equal to 1/2Tb. These two binary waves are extracted from the incoming binary sequence (See Example 6.5).

FIGURE 6.31 Block diagrams for (a) MSK transmitter

Logic circuit for gener-ating binary waves a1 & a2

Detection of MSK SignalsThe received signal x(t) is correlated with locally generated replicas of the coherent reference signals ϕ1(t) and ϕ2(t). Note that in both cases the integration interval is 2Tb seconds, and that the integration in the quadrature channel is delayed by Tb seconds with respect to that in the in-phase channel. The resulting in-phase and quadrature channel correlators outputs, x1 and x2, are each compared with a threshold of zero, and estimates of the phase θ(0) and θ(Tb) are derived. Finally, these phase decisions are interleaved so as to reconstruct the original input binary sequence (According to Table 7.2) with the minimum average probability of symbol error in an AWGN channel.

Error Probability of MSKthe received signal is given bywhere s(t) is the transmitted MSK signal, and w(t) is the sample function of a white Gaussian noise process of zero mean and power spectral density No/2. To decide whether symbol 1 or symbol 0 was transmitted in the interval 0 ≤ t ≤ Tb, say, we have to establish a procedure for the use of x(t) to detect the phase states θ(0) and θ(Tb). For the optimumdetection of θ(0), we first determine the projection of the received signal x(t) onto the reference signal ϕ1(t) over the interval -Tb ≤ t ≤ Tb, obtaining

Where wl is the sample value of a Gaussian Random variable of zero mean and variance No/2. From the signal-space diagram of Figure 6.29,we observe that if x1 > 0, the receiver chooses the estimate θ(0) =0 and if x1 < 0, it chooses the estimate θ(0) = π

Similarly, for the optimum detection of θ(Tb) we determine the projection of the received signal x(t) onto the second reference signal ϕ2(t) over the interval 0 ≤ t ≤ 2Tb obtaining

Where w2 is the sample value of another independent Gaussian random variable of zero mean and variance No/2. Referring again to the signal space diagram of Figure 6.29, we observe that if x2 > 0, the receiver chooses the estimate θ(Tb)= - π /2. If, on the other hand, x2 < 0, it chooses the estimate θ(Tb) = π /2.

To reconstruct the original binary sequence, we interleave the above two sets of phase decisions, as shown in Table (6.5 )

The signal from other bits does not interfere with the receiver's decision for a given bit in either channel. The receiver makes an error when the I-channel assigns the wrong value to θ(0) or the Q-channel assigns the wrong value to θ(Tb). Accordingly, using the statistical characterizations of the product-integrator outputs x1 and x2, of these two channels, defined before, we readily find that the bit error rate for coherent MSK is given by

which is exactly the same as that for binary PSK and QPSK and it is 3 dB more energy-efficient than the Sunde’s BFSK. It is important to note, however, that this good performance (better bit error rate) is the result of the detection of the MSK signal being performed in the receiver on the basis of observations over 2Tb seconds.

The two modulation frequencies are: f1 = 5/4Tb and f2 = 3/4Tb. Assuming that, at time t = 0 thephase θ(0) is zero, the sequence of phase states is as shown in Figure 6.30, modulo 2 π. The polarities of the two sequences of factors used to scale the time functions ϕ1(t) and ϕ2(t) are shown in the top lines of Figures 6.30b and 6.30c. Note that these two sequences are offsetrelative to each other by an interval equal to the bit duration Tb

1 1 0 1 0 0 0

0 2Tb 4Tb 6Tb

0 π π 0

a1

π/2 π/2 π/2 -π/2

a2

Power Spectra of MSK Signals

Assuming the input binary wave is random with symbols 1 and 0 equally likely, and the symbols transmitted during different time slots being statistically independent.

Depending on the value of phase state θ(0), the in-phase component equals +g(t) o r -g(t), where

Hence, the power spectral density of the in-phase component equals ψg(f )/2Tb.

Depending on the value of the phase state θ(Tb), the quadrature component equals +g(t) or -g(t), where we now have

The energy spectral density of this second symbol-shaping function is the same as the first one. Thus, the in-phase and quadrature components have the same power spectral density. The in-phase and quadrature components of the MSK signal are also statistically independent. Hence, the baseband power spectral density of the MSK signal is given by

This baseband power spectrum is plotted in Figure 6.9 which also includes the corresponding plot of the QPSK signal, where the power spectrum is normalized with respect to 4Eb and the frequency f is normalized with respect to the bit rate l/Tb.

Figure 6.9 plots SB(f), normalized with respect to 4Eb, versus the normalized frequency fTb.

For f >> 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 interestas QPSK. This is a desirable characteristic of MSK, especially when the digital communicationsystem operates with a bandwidth limitation.

GAUSSIAN-FILTERED MSKThe desirable properties of the MSK signal may be summarized as follows:– Constant envelope– Relatively narrow bandwidth– Coherent detection performance equivalent to that of

QPSKHowever, the out-of-band spectral characteristics of MSK signals, as good as they are, still do not satisfy the severe requirements of certain applications such as wireless communications.

When the MSK signal is assigned a transmission bandwidth of l/Tb, the adjacent channel interference of a wireless communication system using MSK is not low enough to satisfy the practical requirements of such a multiuser communications environment. These requirements can be achieved through the use of a premodulation low-pass filter, which can be called a baseband pulse-shaping filter. Desirably, the pulse-shaping filter should satisfy the following properties:

1. Frequency response with narrow bandwidth and sharp cutoff characteristics.2. Impulse response with relatively low overshoot.3. Evolution of a phase trellis where the carrier phase of the

modulated signal assumes the two values at odd multiples of Tb and the two values 0 and π at even multiples of Tb as in MSK.

These desirable properties can be achieved by passing a nonreturn-to-zero (NRZ) binary data stream through a baseband pulse-shaping filter whose impulse response (and likewise its frequency response) is defined by a Gaussian function. The resulting method of binary frequency modulation is naturally referred to as Gaussian-filtered MSK or just GMSK.

Error Probability of GMSKThe probability of error Pe of GMSK using coherent detection in the presence of additive white Gaussian noise:

The factor α is a constant whose value depends on the time-bandwidth product WTb, where W denotes the 3 dB baseband bandwidth of the pulse-shaping filter. we may consider 10log10 (α/2), expressed in decibels, as a measure of performance degradation of GMSK (with given WTb) compared to ordinary MSK.

This Figure (6.34) shows the machine-computed value of 10log10 (α/2), versus WTb For GMSK with WTb = 0.3, there is a degradation in performance of about 0.46 dB, which corresponds to (a/2) = 0.9, This degradation in performance is a small price to pay for the highly desirable spectral compactness of the GMSK signal.

The error rates for all the systems decrease monotonically with increasing values of Eb/No.For any value of Eb/No coherent BPSK, QPSK, and MSK produce a smaller error rate than the other systems.The coherent PSK and the DPSK require an Eb/No that is 3 dB less than the corresponding values for the conventionalcoherent FSK to realize the same error rate.At high values of Eb/No, the DPSK performs almost as well (to within about 1 dB) as the coherent PSK for the same bit rate and signal energy per bit.The QPSK system transmits, in a given bandwidth, twice as many bits of information as a conventional coherent BPSK system with the same error rate performance. Here againwe find that a QPSK system requires a more sophisticated carrier recovery circuit than a BPSK system.