# Contents. A Error probability for PAM Signals in AWGN 17. B Error probability for PSK Signals in AWGN 18

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1 Contents 5 Signal Constellations Pulse-amplitude Modulation (PAM) Performance of PAM in Additive White Gaussian Noise Phase-shift Keying Performance of PSK in Additive Gaussian Noise Quadrature Amplitude-modulation (QAM) Performance of QAM in Additive Gaussian Noise Frequency-shift keying Performance in additive Gaussian noise Continuous-phase modulation The Power Spectrum of Linearly Modulated Signals A Error probability for PAM Signals in AWGN 17 B Error probability for PSK Signals in AWGN 18 C Error probability for PSK Signals in AWGN -Alternative Form 19 1

2 List of Figures 5.1 The signal space representation of binary PAM, 4-PAM and 8-PAM constellations Symbol error probability for 2, 4 and 8-PAM as a function of SNR per bit Signal space representation of various PSK constellations The function f(θ) Symbol error probability for BPSK, QPSK, 8-PSK and 16-PSK as a function of the signal-to-noise ratio per bit Signal-space representation of various QAM constellations Symbol error probability as a function of SNR per bit for 4, 16, and 64-QAM Error probability comparison between coherent and incoherent FSK. 13 2

3 Chapter 5 Signal Constellations In this chapter we study a number of signal constellations often used in practice. 5.1 Pulse-amplitude Modulation (PAM) As the name implies, pulse-amplitude modulation (PAM) conveys information by assigning k = log 2 (M) bits to a set of M discrete amplitudes of a transmitted signal (baseband or passband). M-ary PAM is a one-dimensional signaling scheme described mathematically by s i (t) = a i φ(t), i = 1, 2,, M, t T, (5.1) where a i = (2i 1 M)A, i = 1, 2,, M, (5.2) φ(t) is a unit-energy signal, and A is half the spacing between adjacent signals in the constellation (i.e. d min = 2A where d min is the minimum Euclidean distance of the contellation). Figure 5.1 shows the signal-space representation of PAM signals assuming A = 1. Clearly not every signal in the constellation has the same energy. 8-PAM 4-PAM 2-PAM Figure 5.1: The signal space representation of binary PAM, 4-PAM and 8-PAM constellations. 3

4 CHAPTER 5. SIGNAL CONSTELLATIONS 4 The average energy of the constellation is E av = 1 M and the peak-energy is M i=1 a 2 i = A2 M ( M M (2i 1 M) 2 2 ) 1 = A 2, (5.3) 3 i=1 E p = (M 1) 2 A 2. (5.4) Everything else being the same, it is desirable to have constellations whose peakenergy is not much different than their average energy. This is desirable in practice, since the transmitter must be built to accommodate the peak power required and if the latter is much larger than the average (which determines performance), then the higher cost of hardware does not result in a proportional performance gain. A figure of merit for signal constellations is the peak-to-average ratio, which for PAM constellations is E p = 3 6 E av M + 1, (5.5) which tends to 3 as M. This is a large peak-to-average energy ratio, and it is one of the reasons multi-level PAM is rarely used in communication systems Performance of PAM in Additive White Gaussian Noise Based on the data r(t) received (as given in (??)), the maximum-likelihood receiver for PAM signaling chooses as the most likely signal transmitted the signal that maximizes l i = a i r a2 i 2, or equivalently the signal that minimizes (r a i ) 2, where r = T r(t)φ(t)dt. In signal space, the decision boundaries for this receiver are midway between constellation points, and a decision is made accordingly, based on where r falls on the real line. The error probability for M-ary PAM signals is derived in Appendix?? and is given by P PAM (e) = ( (M 1) M erfc 3 M 2 1 ) E av. (5.6) N The error probability for various PAM constellations is shown in Figure 5.2 as a function of SNR per bit. Cleraly, even for the same SNR per bit, as M increases, performance degrades. The gain, of course, is in increased bit-rate for the same bandwidth. 1 One notable exception is in PCM modems, where the system makes use of the 256 PCM levels in the digital part of the telephone network

5 CHAPTER 5. SIGNAL CONSTELLATIONS Error-Probability PAM 8-PAM PAM SNR, db Figure 5.2: Symbol error probability for 2, 4 and 8-PAM as a function of SNR per bit. 5.2 Phase-shift Keying Under phase-shift keying (PSK), the information bits determine the phase of a carrier, which takes values from a discrete set in accordance with the information bits. The general form of M-ary PSK signals is given by s i (t) = 2E T cos(2πf ct + θ i ), i = 1, 2,, M, t T, (5.7) where and θ i = 2π(i 1), i = 1, 2,, M M E = T s 2 i (t)dt

6 CHAPTER 5. SIGNAL CONSTELLATIONS 6 is the signal energy (the same for all signals). Equation (5.7) can be re-written in a slightly different form as s i (t) = [ ] 2 2 E cos(θ i ) T cos(2πf ct) sin(θ i ) T sin(2πf ct) = E [cos(θ i )φ 1 (t) sin(θ i )φ 2 (t)] where 2 φ 1 (t) = T cos(2πf ct) (5.8) 2 φ 2 (t) = T sin(2πf ct) (5.9) are easily seen to be orthonormal. Thus, PSK signals are points in a two-dimensional space spanned by φ 1 (t) and φ 2 (t). Figure 5.3 illustrates various PSK signal constellations, including binary PSK (BPSK), and 4-ary PSK, also known as quadrature PSK (QPSK). The figure also illustrates the mapping of information bits to each ϕ 2 () t ϕ 2 1 () t E ϕ 1 () t 11 E ϕ 1 () t BPSK:d 111 min = () t ϕ 2 11 E 11 2 E 1 ϕ 1 1 () t π 8-PSK: dmin = 2 Esin( ) 8 QPSK: d 16-PSK: d min = min ϕ 2 () t E 2E ϕ 1 () t π = 2 Esin( ) 16 Figure 5.3: Signal space representation of various PSK constellations. signal in the constellation for 4-PSK and 8-PSK. The illustrated mapping, known as Gray coding, has the property that adjacent signals are assigned binary sequences that differ in only one bit. This is desirable in practice, because, when a detection error is made, it is more likely to be to a signal adjacent to the transmitted signal. Then Gray coding results in a single bit error for the most likely signal errors.

7 CHAPTER 5. SIGNAL CONSTELLATIONS Performance of PSK in Additive Gaussian Noise For PSK signals, the optimum receiver decides which of the M possible PSK signals was transmitted by finding the signal that maximizes where l i = T r c = r(t)s i (t)dt = r c a i r s b i, (5.1) T T r(t)φ 1 (t)dt r s = r(t)φ 2 (t)dt a i = [ ] 2π(i 1) E cos M b i = [ 2π(i 1) E sin M For binary PSK, which corresponds to antipodal signaling, the probability that the optimal receiver makes a decision error is ( P BPSK (e) = 1 ) 2 erfc E. (5.11) The performance of QPSK is also derived easily and is given by N ]. P QPSK (e) = P BPSK (e)[2 P BPSK (e)] where P BPSK (e) is as given in Eq. (5.11). An exact expression for the error probability of larger PSK constellations also exists and is derived in Appendix B. It is given by π/m P MPSK (e) = 1 f(θ)dθ, (5.12) π/m where f(θ) = 1 [ 2π e SNR 1 + ( π SNR cos(θ)e SNR cos2 (θ) erfc )] SNR cos(θ), (5.13) where, as defined earlier, SNR = E/N. Figure 5.4 plots the function f(θ) for various SNR values. Figure 5.5 shows the error probability of various PSK constellations as a function of the SNR per information bit. Another interesting expression for the error-probability of two-dimensional signal constellations, and in particular M-PSK, was obtained by Craig [1]: P MPSK (e) = 1 π (M 1)π M [ ( )] exp SNR sin2 π M sin 2 dθ, (5.14) (θ)

8 CHAPTER 5. SIGNAL CONSTELLATIONS dB f( θ).8.6 3dB.4.2 db θ, Radians Figure 5.4: The function f(θ). The derivation is given in Appendix C. From (5.14) it is easy to derive an upperbound to the probability of error for arbitrary PSK signals by noting the the integrand is maximized at θ = π/2. Thus: P MPSK (e) ( )] (M 1) π [ SNR M exp sin 2 M An interesting implication of the expression in (5.14) is that 1 2 erfc(x) = 1 π 2 π e x 2 sin 2 (θ) dθ. 5.3 Quadrature Amplitude-modulation (QAM) (5.15) Quadrature amplitude modulation (QAM) is a popular scheme for high-rate, high bandwidth efficiency systems. QAM is a combination of both amplitude and phase modulation. Mathematically, M-ary QAM is described by s i (t) = Ep(t) [a i cos(2πf c t) + b i sin(2πf c t)], t T, i = 1, 2,, M (5.16)

9 CHAPTER 5. SIGNAL CONSTELLATIONS 9 Symbol Error Probability PSK 4-PSK 8-PSK 16-PSK SNR per bit, db Figure 5.5: Symbol error probability for BPSK, QPSK, 8-PSK and 16-PSK as a function of the signal-to-noise ratio per bit. where a i and b i take values from the set {±1, ±3, ±5, } and E and p(t) are as defined earlier. The signal-space representation of QAM signals is shown in Figure 5.6 for various values of M which are powers of 2, that is, M = 2 k, k = 2, 3,. For even values of k, the constellations are square, whereas for odd values of k the constellations have a cross shape and are thus called cross constellations. For square constellations, QAM corresponds to the independent amplitude modulation of an in-phase carrier (i.e., the cosine carrier) and a quadrature carrier (i.e., the sine carrier). 5.4 Performance of QAM in Additive Gaussian Noise The optimum receiver for QAM signals chooses the signal that maximizes E ( ) l i = a i r c + b i r s a 2 i + b 2 i 4

10 CHAPTER 5. SIGNAL CONSTELLATIONS 1 64-QAM 32-QAM 16-QAM 8-QAM 4-QAM Figure 5.6: Signal-space representation of various QAM constellations. where and r c = r s = T T r(t)p(t) cos(2πf c t)dt r(t)p(t) sin(2πf c t)dt For square constellations which correspond to independent PAM of each carrier, an exact error probability is derived easily and is given by [ ( P QAM (e) = ) ( )] 2 3 erfc M 2(M 1) Eav. N For cross constellations, tight upper-bounds and good approximations are available. Figure 5.7 plots the symbol-error probability of various square QAM constellations as a function of SNR per bit. 5.5 Frequency-shift keying As the name implies, frequency-shift keying (FSK) modulates the frequency of a carrier to convey information. FSK is one of the oldest digital modulation techniques

11 CHAPTER 5. SIGNAL CONSTELLATIONS QAM Error-Probability QAM QAM SNR per bit, db Figure 5.7: Symbol error probability as a function of SNR per bit for 4, 16, and 64-QAM. and was the modulation of choice for the first, low-rate modems. Its main attribute that makes it of interest in some applications is that it is detected incoherently (and coherently), which reduces the cost of the receiver. Mathematically, the modulated M-ary FSK signal is described by where s i (t) = 2E T cos[2π(f c + f i )t], t T, i = 1, 2,, M ( 2i 1 M f i = 2 ) f. f is the minimum frequency separation between modulation tones. For orthogonal signaling (i.e. when the correlation between all pairs of distinct signals is zero), the minimum tone spacing is 1/2T. This is often a condition imposed in practice. Orthogonal signaling performs well as a function of energy per bit, but it is also rather bandwidth-inefficient, making it impractical for high-speed, band-limited applications.

12 CHAPTER 5. SIGNAL CONSTELLATIONS Performance in additive Gaussian noise FSK is detected coherently or incoherently. Coherent detection requires a carrier phase synchronization subsystem at the receiver which generates locally a carrier which is phase-locked to the received carrier. The optimum receiver for coherent detection makes decisions by maximizing the following (implementation assumes phase-coherence) l i = T r(t)s i (t)dt. For binary (orthogonal) signaling, the error probability is given simply by P FSK (e) = 1 2 erfc ( E 2N ), (coherent FSK) which is 3dB worse than BPSK. For M-ary signaling, an exact expression exists in integral form and is found, for example, in [?]. Incoherent detection does not assume phase coherence and does not attempt to phase-lock the locally generated carrier to the received signal. In this case, it is easy to argue that the phase difference between the LO carrier and the received carrier is completely randomized. An optimum receiver is also derived in this case and it is one that maximizes over the set of frequency tones where and r 2 ci = r 2 si = T T l i = r 2 ci + r 2 si r(t) cos[2π(f c + f i )t]dt r(t) sin[2π(f c + f i )t]dt The exact error-probability performance of this incoherent receiver is available in analytical form, but it is rather complicated to compute for the general M-ary case (see, for example, [?]). For the binary case, the error probability has a simple form given by P FSK (e) = 1 E 2 e 2N, (incoherent FSK) Figure 5.8 compares the performance of coherent and Incoherent binary FSK. At an error probability of about 1 6, incoherent detection is inferior only slightly more than half a db compared with coherent detection. However, this small loss is well compensated for by the fact that no carrier phase synchronization is needed for the former.

13 CHAPTER 5. SIGNAL CONSTELLATIONS Incoherent Detection Error-Probability Coherent Detection SNR per bit, db Figure 5.8: Error probability comparison between coherent and incoherent FSK. 5.7 Continuous-phase modulation All of the modulation schemes described so far are memoryless, in the sense that the signal transmitted in a certain symbol interval does not depend on any past (or future) symbols. In many cases, for example, when there is a need to shape the transmitted signal spectrum to match that of the channel, it is necessary to constrain the transmitted signals in some form. Invariably, the imposed constraints introduce memory into the transmitted signals. One important class of modulation signals with memory are continuous phase modulation (CPM) signals. These signals constrain the phase of the transmitted carrier to be continuous, thereby reducing the spectral sidelobes of the transmitted signals. Mathematically, the modulation signals for CPM are described by the expression u(t) = A cos [2πf c t + φ(t; d)] where n φ(t; d) = 2π d k h k q(t kt ), k= nt t (n + 1)T.

14 CHAPTER 5. SIGNAL CONSTELLATIONS 14 The d k are the modulation symbols and h k are the modulation indices, which may vary from symbol to symbol. For binary modulation, the modulation symbols are either 1 or -1. Finally, q(t)is the integral of some baseband pulse p(t) containing no impulses (thus guaranteeing that q(t) is continuous) q(t) = t p(τ)dτ. When p(t) is zero for t T, we have what is called full response CPM, otherwise we have partial-response CPM. In general, partial-response CPM achieves better spectral sidelobe reduction than full-response CPM. A special case of CPM in which the modulation indices are all equal and p(t) is a rectangular pulse of duration T seconds is called continuous-phase FSK (CPFSK). If, further, h = 1/2, we have what is called minimum-shift keying (MSK). A variation of MSK, in which the rectangular baseband pulse is first passed through a filter with a Gaussian-shape impulse response for further reduction in the spectral sidelobes, is called Gaussian MSK (GMSK). Various simple ways for detecting GMSK are available, which, combined with its spectral efficiency, has made it a popular modulation scheme. In particular, it is the modulation scheme used for the European digital cellular radio standard, known as GSM. For more information on CPM signaling, including spectral characteristics and performance in noise, refer to [?]. 5.8 The Power Spectrum of Linearly Modulated Signals Since the discrete modulation symbols which are mapped into transmitted signals are random, the resulting transmitted signal is a stochastic process. It is of interest to determine the frequency content of a transmitted stochastic signal, which is given by its power spectrum, for a number of reasons. One important reason is to determine its bandwidth and thus determine the channel bandwidth required for its undistorted transmission. Another use of the power spectrum of a modulated signal is in timesynchronization systems. A class of time synchronization systems extract their timing information by tracking a discrete frequency component, which is at some multiple of the signaling rate, in the spectrum of the modulated signal. A general, linearly modulated signal can be expressed as x(t) = k d k p(t kt ), (5.17) where {d k } is a sequence of complex symbols, 1/T is the signaling rate, and p(t) is the transmitted pulse. We will assume that the sequence {d k } is wide-sense stationary having mean µ and autocorrelation function r dd (k) = 1 2 E[d jd j+k ]. (5.18)

15 CHAPTER 5. SIGNAL CONSTELLATIONS 15 In computing the power spectrum of x(t), we first derive an expression for the autocorrelation function of x(t) R xx (t, t + τ) = 1 2 E[x (t)x(t + τ)] (5.19) = 1 E[d 2 kd j ]p (t kt )p(t + τ jt ) k j = r dd (j k)p (t kt )p(t + τ jt ) k j = r dd (m)p (t kt )p (t + τ (k + m)t ) k m = m r dd (m) k p (t kt )p (t + τ (k + m)t ) (5.2) Clearly, R xx (t, t + τ) is periodic in t with period T. In addition, the mean of x(t) E[x(t)] = µ k p(t kt ) is periodic in t with period T. Thus, x(t) is what is referred to as a cyclostationary process. For cyclostationary processes, the dependence on t is removed (and thus the resulting autocorrelation becomes only a function of a single variable τ) by averaging R xx (t, t + τ) over a period: Letting R xx (τ) = 1 T = m T/2 T/2 r dd (m) k = r dd (m) m k = 1 r dd (m) T m R xx (t, t + τ)dt 1 T 1 T R pp (τ) = T/2 T/2 T/2 kt T/2 kt p (t kt )p (t + τ (k + m)t ) dt p (t)p (t + τ mt ) dt p (t)p (t + τ mt ) dt (5.21) be the autocorrelation function of p(t) we have R xx (τ) = 1 T Taking Fourier transforms on both sides above we obtain p (t)p (t + τ) dt (5.22) r dd (m)r pp (τ mt ) (5.23) m S X (f) = 1 T P (f) 2 S d (f) (5.24)

16 CHAPTER 5. SIGNAL CONSTELLATIONS 16 where S d (f) = k r dd (k)e j2πfkt (5.25) is the discrete Fourier transform of {r dd (k)}.

17 Appendix A Error probability for PAM Signals in AWGN 17

18 Appendix B Error probability for PSK Signals in AWGN 18

19 Appendix C Error probability for PSK Signals in AWGN -Alternative Form 19

20 Bibliography [1] J.W. Craig, A New, Simple and Exact Result for Calculating the Probability of Error for Two-Dimensional Signal Constellations, Proceedings IEEE MILCOM 91, Boston, MA, pp

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