Definition Recall, from 6.7, that PAM signal waveforms are represented

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1 6.4 Constellations for Digital Modulation Schees PAM Definition Recall, fro 6.7, that PAM signal wavefors are represented as s (t) = A p(t), 1 M where p(t) is a pulse and A A Clearly, PAM signals are one-diensional since all are ultiples of the sae basic signals. We define φ(t) = p(t) Ep as the basis for the PAM signals above. n which case, s (t) = A Ep φ(t), 1 M and the corresponding one-diensional vector representation is s () = A Ep. The corresponding signal space diagras for M = 2, M = 4, and M = 8 are shown in Figure Digital Counications 0 1 FGURE Constellation for PAM signaling. (a)m= (b)m= lll (c)m=8 Figure 29: Constellation for PAM signaling signal aplitudes differ by one binary digit as illustrated in Figure This apping is called Gray coding. t is iportant in the deodulation of the signal because the ost likely errors caused by noise involve the erroneous selection of an adjacent aplitude to the transitted signal aplitude. n such a case, only a single bit error occurs in the k-bit sequence n Aplitude-Shift Keying (ASK), p(t) = g(t) cos(2πf c t) where f c is the carrier frequency. We note that the Euclidean distance between any pair of signal points is 97 (3.2-18) (3.2-19)

6.4.2 Phase-Shift Keying (PSK) Definition 6.48. n digital phase odulation, the M signal wavefors are represented as ( s (t) = g(t) cos 2πf c t + 2π ) ( 1), = 1, 2,.

2 6.4.2 Phase-Shift Keying (PSK) Definition n digital phase odulation, the M signal wavefors are represented as ( s (t) = g(t) cos 2πf c t + 2π ) ( 1), = 1, 2,..., M (38) M where g(t) is the signal pulse shape and θ = 2π M ( 1), = 1, 2,..., M is the M possible phases of the carrier that convey the transitted inforation. Digital phase odulation is usually called phase-shift keying (PSK) The PSK signal wavefors defined in (38) have equal energy: Note that (a) Fro the cos identity we have cos(α ± β) = cos α cos β sin α sin β, s (t) = g(t) cos (θ ) cos (2πf c t) g(t) sin (θ ) sin (2πf c t). (b) g(t) cos (2πf c t) and g(t) sin (2πf c t) are orthogonal. These two observations replace GSOP. (However, can also use GSOP to arrive at the sae conclusion.) 98

3 Suppose we define n which case, φ 1 (t) = 2 g(t) cos (2πf c t), E g (39) φ 2 (t) = 2 g(t) sin (2πf c t). E g (40) Eg s (t) = 2 cos (θ Eg ) φ 1 (t) + 2 sin (θ ) φ 2 (t). Therefore the signal space diensionality is N = 2 and the resulting vector representations are ( ) T s () Eg = 2 cos (θ Eg ), 2 sin (θ ) Signal space diagras for BPSK (binary PSK, M = 2), QPSK (quaternary PSK, M = 4), Chapter and 8-PSK Three: Digital aremodulation shown Schees in Figure FGURE Signal space diagras for BPSK, QPSK, and 8-PSK. M= (BPSK) M=8 (Octal PSK) 10 M=4 (QPSK) Figure 30: Signal and space the iniu diagras distance for corresponding BPSK, QPSK, to l- andnl 8-PSK. = 1 is Note that BPSK corresponds to one-diensional din = J eg ( signals, 1 - cos ~) = J which are identical to binary PAM signals. 2eg sin 2 ~ (3.2-32) Solving Equation for eg and substituting the result in Equation result in 99 (3.2-33) For large values of M, we have sin "fi :::::: - 1, and din can be approxiated by

4 6.4.3 Quadrature Aplitude Modulation (QAM) Definition n Quadrature Aplitude Modulation (QAM), two separate b-bit sybols fro the inforation sequence on two quadrature carriers cos (2πf c t) and sin (2πf c t) are transitted siultaneously. The corresponding signal wavefors ay be expressed as s (t) = A () g (t) cos (2πf c t) A (Q) g (t) sin (2πf c t), = 1, 2,..., M (41) where A () and A (Q) are the inforation-bearing signal aplitudes of the quadrature carriers and g(t) is the signal pulse. Equivalently, {( s (t) = Re A () ) } + ja (Q) g (t) e j2πf ct (42) = Re { r e jθ g (t) e j2πf ct } (43) = r g(t) cos (2πf c t + θ ) (44) where r = and ( ) 2 ( A () + A (Q) θ is the arguent or phase of the coplex nuber A () ) 2 is the agnitude + ja (Q) Fro (44), it is apparent that the QAM signal wavefors ay be viewed as cobined aplitude (r ) and phase (θ ) odulation. n fact, we ay select any cobination of M 1 -level PAM and M 2 -phase PSK to construct an M = M 1 M 2 cobined PAM-PSK signal constellation. f M 1 = 2 b 1 and M 2 = 2 b 2, the cobined PAM-PSK signal constellation results in the siultaneous transission of b 1 + b 2 = log 2 M 1 M 2 binary digits occurring at a sybol rate R/(b 1 + b 2 ). 100

5 6.54. Fro (41), it can be seen that, siilar to the PSK case, φ 1 (t) and φ 2 (t) given in (39) and (40) can be used as an orthonoral basis for QAM signals. The diensionality of the signal space for QAM is N = 2. Using this basis, we have s (t) = A () Eg 2 φ 1 (t) + A (Q) Eg 2 φ 2 (t) which results in vector representations of the for ( ) T s () = A () Eg 2, Eg A(Q). 2 Exaple Exaples of signal space diagras for cobined PAM-PSK are shown in Figure 31, for M = 8 and M = 16. Chapter Three: Digital Modulation Schees 105 M=8 M=16 FGURE Exaples of cobined PAM-PSK constellations. n the special case where the signal aplitudes take the set of discrete values {(2-1-M), = 1, 2,..., M}, the signal space diagra is rectangular, as shown in Figure n this case, the Euclidean distance between adjacent points, i.e., the iniu distance, is (3.2-41) which is the sae result as for PAM. n the special case of a rectangular constellation with M = 2 2 k 1, i.e., M = 4, 16, 64,256,..., and with aplitudes of ±1, ±3,..., ±(v'jij- 1) on both directions, fro Equation we have ---~ ~- 1 ; Figure 31: Exaples of cobined PAM-PSK constellations. Exaple n the special case where the signal aplitudes are taken fro the set of discrete values A = {(2 1 M), = 1, 2,..., M}, the signal space diagra is rectangular, as shown in Figure PAM and PSK can be considered as special cases of QAM. n QAM signaling, both aplitude and phase carry inforation, whereas in PAM and PSK only aplitude or phase carries the inforation ~- 1.,fM.,fM Cavg = M ; LL (A~ +A~) =l n=l cg 2M(M- 1) = - X ---'----..:... 2M 3 M-1 =-3-cg M=64 FGURE ~ ~---, M=32 jj"---~ ~/ ', / '.. / M= 16 ',, (3.2-42) Several signal space diagras for rectangular QAM.

6 ---~ ~- 1 ;..----~- 2M 3 M-1 =-3-cg M=64 FGURE ~ ~---, M=32 jj"---~ ~/ ', / '.. / M= 16 ',, t M=81 1 1M=4: _,. ; : : : : : : 1 + ; *,, , ~---.. : ~ / : // ---~ ~ ~--- t Several signal space diagras for re QAM. Figure 32: Several signal space diagras for rectangular QAM Orthogonal Signaling Definition n orthogonal signaling, the wavefors s (t) are orthogonal and of equal energy E s. n which case, the orthonoral set {φ (t), 1 N} defined by φ (t) = s (t), 1 M Es can be used as an orthonoral basis for representation of {s (t), 1 M}. The resulting vector representation of the signals will be ( ) s (1) = Es, 0, 0,..., 0, ( s (2) = 0, ) E s, 0,..., 0,. =. s (M) = (0, 0, 0,..., ) E s. 102

5 Constellation for Digital Modulation Schemes

5 Constellation for Digital Modulation Schees 5.1 PAM Definition 5.1. Recall, fro 3.6, that PAM signal wavefors are represented as s (t) = A p(t), 1 M where p(t) is a pulse and A A. 5.2. Clearly, PAM signals