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Qassim University College of Engineering Electrical Engineering Department Electronics and Communications Course: EE322 Digital Communications Prerequisite:

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1 Qassim University College of Engineering Electrical Engineering Department Electronics and Communications Course: EE322 Digital Communications Prerequisite: EE320 Part B www.wiley.com/go/global/haykin Associate Prof. Dr. Ahmed Abdelwahab

2 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, θ i = 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 f c = n c /T for some fixed integer n c Each s i (t) may be expanded in terms of the same two basis functions ф 1 (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,

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9 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/N o is increased. For M = 4, Equation (6.47) reduces to the same form given in Equation (6.34) for QPSK.

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11 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 R b =1/T b

12 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 E b /N o to compensate for the increase in M.

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14 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.

15 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

16 Let the ith message point s i in the plane be denoted by (a i d min /2, b i d min /2), where d min is the minimum distance between any two message points in the constellation, a i and b i are integers and i = 1, 2,..., M. Let (d min /2) = where E o 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 s k (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.

17 QAM Square Constellations With an even number of bits (n) per symbol, Define & M=2 n, 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.

18 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.

19 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.

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21 Probability of Symbol Error for M-ary QAM To 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 as where is the probability of symbol error for the corresponding L-ary PAM, ( ) which is defined as (Note that L in the M-ary QAM corresponds to M in the M-ary PAM)

22 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

23 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 P e in terms of the average value of the transmitted energy rather than E o. Assuming that the L amplitude levels of the in- phase or quadrature component are equally likely, we have

24 However, the superior performance of M-ary QAM can be realized only if the channel is free of nonlinearities.

25 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 2 n-3 symbols and ignore the corners in the extension. The inner square represents 2 n-1 symbols. The four side extensions add 4* 2 n-3 = 2 n-1 symbols. The total number of symbols in the cross constellation is therefore = 2 n-1 + 2 n-1 =2 n 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.

26 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.

27 BFSK A typical pair of sinusoidal waves is described by where i = 1,2, and E b is the transmitted signal energy per bit. The transmitted frequency f i = (n c +i)/T b for some fixed integer n c. It is a continuous-phase signal in the sense that phase continuity 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)

28 The signals s l (t) and s 2 (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 s ij for i = 1,2, and j = 1,2 is defined by

29 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 s 1 (t) and s 2 (t).

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31 Generation and Detection of Coherent Binary FSK Signals

32 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 f 1 in the upper channel is switched on while the oscillator with frequency f 2 in the lower channel is switched off, with the result that frequency f 1 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 f 2 is transmitted.

33 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.

34 Error Probability of Binary FSK The observation vector x has two elements x 1 and x 2 that are defined by, respectively, Given that symbol 1 was transmitted, x(t) equals s 1 (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 s 2 (t) + w(t). The receiver decides in favor of symbol 1 if The receiver decides in favor of symbol 0 if

35 Define a new Gaussian random variable Y whose sample value y is equal to the difference between x 1 and x 2 ; 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 X 1 and X 2, whose sample values are denoted by x 1 and x 2 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

36 The variance of the random variable Y is independent of which binary symbol was transmitted. Since the random variables X 1 and X 2 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

37 Since the conditional probability of error given that symbol 1 was transmitted, has the same value where p 01 = p 10, 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, E b /N o has to be doubled to maintain the same bit error rate as in a coherent binary PSK system. This result is in perfect accord with the 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

38 Power Spectra of Binary FSK Signals Consider the case of Sunde's FSK, for which the two transmitted frequencies f 1 and f 2 differ by an amount equal to the bit rate l/T b, and their arithmetic mean equals the nominal carrier frequency f c ; 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.

39 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 E b /2T b, and occurring at f = ±1/2T b. 2) The quadrature component is directly related to the input binary wave. During the signaling interval 0 ≤ t ≤ T b, it equals -g(t) for symbol 1, and +g(t) for symbol 0.

40 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

41 The power spectrum of the binary FSK signal contains two discrete frequency components located at (f c + 1/2T b ) = f 1 and (f c - 1/2T b ) = f 2, 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 f 1 and f 2 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.

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44 Consider a continuous-phase frequency-shift keying (CPFSK) signal, which is defined for the interval 0 ≤t ≤ T b 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 f 1 and f 2 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 f c and h. The nominal carrier frequency f c is therefore the arithmetic mean of the frequencies f 1 and f 2. The difference between the frequencies f 1 and f 2, normalized with respect to the bit rate l/T b, defines the dimensionless parameter h, which is referred to as the deviation ratio.

45 Phase Trellis From the above equation, it can be seen that at time t = T b, 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.

46 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 T b, respectively.

47 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 T b, and only the two values 0 and π at even multiples of T b, as in Figure 6.28.

48 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.

49 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 -T b ≤ t ≤ 0, except that the algebraic sign is not necessarily the same in both intervals.

50 Since the phase θ(0) is 0 or π, depending on the past history of the modulation process, we find that, in the interval -T b ≤ t ≤ T b, the polarity of cos[θ(t)] depends only on θ(0), regardless of the sequence of 1s and 0s transmitted before or after t = 0. Thus, for this time interval, the in-phase component s I (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/2T b ) = 0 at t=-T b & t= T b

51 In a similar way, it can be shown that, in the interval 0≤ t ≤ 2T b, the quadrature component s Q (t) consists of a half-cycle sine pulse, whose polarity depends only on θ(T b ) as shown: where the plus sign for θ (T b ) = π/2 and the minus sign for θ (T b ) = - π/2. = ± sin(πt/2T b ) 0≤ t ≤ 2T b Where, sin(πt/2T b ) = 0 at t=0 & t= 2T b

52 Since the phase states θ (0) and θ (T b ) 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 (T b )= π/ 2, corresponding to the transmission of symbol 1. 2.The phase θ (0)= π and θ (T b )= π/ 2, corresponding to the transmission of symbol 0. 3.The phase θ (0)= π and θ (T b )= - π /2 (or, equivalently, 3 π/ 2 modulo 2 π), corresponding to the transmission of symbol 1. 4.The phase θ (0)= 0 and θ (T b )= - 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 (T b ).

53 The orthonormal basis functions ϕ 1 (t) and ϕ 2 (t) for MSK are defined by a pair of sinusoidally modulated quadrature carriers:

54 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 T b with respect to those used to evaluate the coefficients,. The time interval 0 ≤ t ≤ T b, for which the phase states θ (0) and θ (T b ) 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),

55 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, s 1 and s 2, 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 ϕ 1 (t) and ϕ 2 (t), and thereby determine the MSK signal s(t).

56 Generation of MSK Signals Two input sinusoidal waves, one of frequency f c = n c /4T b for some fixed integer n c and the other of frequency 1/4T b, are first applied to a product modulator. This produces two phase-coherent sinusoidal waves at frequencies f 1 and f 2, which are related to the carrier frequency f c and the bit rate l/T b by Equations (6.3 11) and (6.112) for h = 1/2. The resulting filter outputs are next linearly combined to produce the pair of orthonormal basis functions ϕ 1 (t) and ϕ 2 (t). Finally, ϕ 1 (t) and ϕ 2 (t) are multiplied with two binary waves a 1 (t) and a 2 (t), both of which have a bit rate equal to 1/2T b. These two binary waves are extracted from the incoming binary sequence (See Example 6.5). FIGURE 6.31 Block diagrams for (a) MSK transmitter

57 Logic circuit for gener -ating binary waves a 1 & a 2

58 Detection of MSK Signals The received signal x(t) is correlated with locally generated replicas of the coherent reference signals ϕ 1 (t) and ϕ 2 (t). Note that in both cases the integration interval is 2T b seconds, and that the integration in the quadrature channel is delayed by T b seconds with respect to that in the in-phase channel. The resulting in-phase and quadrature channel correlators outputs, x 1 and x 2, are each compared with a threshold of zero, and estimates of the phase θ(0) and θ(T b ) 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.

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60 Error Probability of MSK the received signal is given by where 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 ≤ T b, say, we have to establish a procedure for the use of x(t) to detect the phase states θ(0) and θ(T b ). For the optimum detection of θ(0), we first determine the projection of the received signal x(t) onto the reference signal ϕ 1 (t) over the interval -T b ≤ t ≤ T b, obtaining Where w l 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 x 1 > 0, the receiver chooses the estimate θ(0) =0 and if x 1 < 0, it chooses the estimate θ(0) = π

61 Similarly, for the optimum detection of θ(T b ) we determine the projection of the received signal x(t) onto the second reference signal ϕ 2 (t) over the interval 0 ≤ t ≤ 2T b obtaining Where w 2 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 x 2 > 0, the receiver chooses the estimate θ(T b )= - π /2. If, on the other hand, x 2 < 0, it chooses the estimate θ(T b ) = π /2. To reconstruct the original binary sequence, we interleave the above two sets of phase decisions, as shown in Table (6.5 )

62 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(T b ). Accordingly, using the statistical characterizations of the product-integrator outputs x 1 and x 2, 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 2T b seconds.

63 The two modulation frequencies are: f 1 = 5/4T b and f 2 = 3/4T b. Assuming that, at time t = 0 the phase 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 ϕ 1 (t) and ϕ 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 T b

64 1 1 0 1 0 0 0 0 2T b 4T b 6T b 0 π π 0 a 1 π/2 π/2 π/2 -π/2 a 2

65 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 ψ g (f )/2T b.

66 Depending on the value of the phase state θ(T b ), 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 4E b and the frequency f is normalized with respect to the bit rate l/T b.

67 Figure 6.9 plots S B (f), normalized with respect to 4E b, versus the normalized frequency fT b.

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69 For f >> 1/T b, 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. This is a desirable characteristic of MSK, especially when the digital communication system operates with a bandwidth limitation.

70 GAUSSIAN-FILTERED MSK The desirable properties of the MSK signal may be summarized as follows: –Constant envelope –Relatively narrow bandwidth –Coherent detection performance equivalent to that of QPSK However, 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.

71 When the MSK signal is assigned a transmission bandwidth of l/T b, 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 T b and the two values 0 and π at even multiples of T b as in MSK.

72 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.

73 Error Probability of GMSK The probability of error P e 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 WT b, where W denotes the 3 dB baseband bandwidth of the pulse-shaping filter. we may consider 10log 10 (α/2), expressed in decibels, as a measure of performance degradation of GMSK (with given WT b ) compared to ordinary MSK. This Figure (6.34) shows the machine-computed value of 10log 10 (α/2), versus WT b For GMSK with WT b = 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.

74 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 frequency f c = n c /2T for some fixed integer n c The transmitted symbols are of equal duration T and have equal energy E. Since the individual 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 s i (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.

75 For coherent M-ary FSK, the optimum receiver consists of a bank of M correlators or matched filters, with the ϕ i (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 d min in M-ary FSK is a the average probability of symbol error for M- ary FSK. (assuming equiprobable symbols)

76 Power Spectra of M-ary FSK Signals Bandwidth Effiency of M-ary FSK Signals The channel bandwidth required to transmit M-ary FSK signals is B =M/2T The bandwidth efficiency of M-ary signals is therefore ρ = R b /B = 2 log 2 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.

77 The error rates for all the systems decrease monotonically with increasing values of E b /N o. For any value of E b /N o coherent BPSK, QPSK, and MSK produce a smaller error rate than the other systems. The coherent PSK and the DPSK require an E b /N o that is 3 dB less than the corresponding values for the conventional coherent FSK to realize the same error rate. At high values of E b /N o, 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 again we find that a QPSK system requires a more sophisticated carrier recovery circuit than a BPSK system.

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