5 Constellation for Digital Modulation Schemes

Size: px

Start display at page:

Download "5 Constellation for Digital Modulation Schemes"

April Tate
5 years ago
Views:

3 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 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 The apping or assignent of b inforation bits to the M = 2 b possible signal aplitudes ay be done in a nuber of ways. The preferred assignent is one in which the adjacent signal aplitudes differ by one binary digit. 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 b-bit sequence For carrier-odulated bandpass PAM signals, we have p(t) = g(t) cos(2πf c t). 19

4 100 Digital Co 0 1 FGURE Constellation for PAM signali (a)m= (b)m= lll Note that E p = E g (c)m=8 Figure 4: Constellation for PAM signaling signal aplitudes differ by one binary digit as illustrated in Figure This is called Gray coding. t is iportant in the deodulation of the signal because likely errors caused by noise involve the erroneous selection of an adjacent a to the transitted signal aplitude. n such a case, only a single bit error occ k-bit sequence. 2. We note that the Euclidean distance between any pair of signal points is where the last relation corresponds to a bandpass PAM. For adjacent sign A - An = 2, and hence the iniu distance of the constellation is give The bandpass digital PAM is also called aplitude-shift keying (ASK). We can express the iniu distance of an M-ary PAM syste in ters o by solving Equations and for [ P and eg, respectively, and substi result in Equation The resulting expression is drn = 12log 2 M & M2-1 bavg 20 The carrier-odulated PAM signal represented by Equation is a sideband (DSB) signal and requires twice the channel bandwidth of the e lowpass signal for transission. Alternatively, we ay use single-sideband (SS which has the

5 5.2 Phase-Shift Keying (PSK) Definition 5.5. n digital phase odulation, the M signal wavefors are represented as ( s (t) = g(t) cos 2πf c t + 2π ) ( 1), = 1, 2,..., M (2) 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 (2) have equal energy: 5.7. Note that (a) Fro the cos identity cos(α ± β) = cos α cos β sin α sin β, we have 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. 21

6 Therefore, we define n which case, φ 1 (t) = 2 g(t) cos (2πf c t), E g (3) φ 2 (t) = 2 g(t) sin (2πf c t). E g (4) s (t) = 2 cos (θ ) φ 1 (t) + 2 sin (θ ) φ 2 (t). Therefore the signal space diensionality is N = 2 and the resulting vector representations are ( ) s () = 2 cos (θ ), 2 sin (θ ) Signal space diagras for BPSK (binary PSK, M = 2), QPSK (quaternary PSK, M = 4), and Chapter 8-PSK Three: Digital are Modulation shown in Schees Figure 5. 0 FGURE Signal space diagras for BPSK, QPSK, and 8-PSK. M= (BPSK) M=8 (Octal PSK) 10 M=4 (QPSK) Figure 5: Signal and space the iniu diagras distance for BPSK, corresponding QPSK, to l- and 8-PSK. nl = 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 22 (3.2-33) For large values of M, we have sin "fi :::::: - 1, and din can be approxiated by

7 5.3 Quadrature Aplitude Modulation (QAM) Definition 5.9. 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 (5) 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 (6) = Re { r e jθ g (t) e j2πf ct } (7) = r cos (2πf c t + θ ) (8) 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 (8), 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 ). 23

8 5.11. Fro (5), it can be seen that, siilar to the PSK case, φ 1 (t) and φ 2 (t) given in (3) and (4) 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 () 2 φ 1 (t) + A (Q) which results in vector representations of the for ( ) s () = A () 2, A(Q) 2 2 φ 2 (t) Exaples of signal space diagras for cobined PAM-PSK are shown in Figure 6, for M = 8 and M = 16. Chapter Three: Digital Modulation Schees 105 M=8 M=16 FGURE Figure 6: Exaples of cobined PAM-PSK constellations. Exaples of cobined PAM-PSK constellations. n the special case where the signal aplitudes take the set of discrete values 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 7. {(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) 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. 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.,fM.,fM Cavg = M ; LL (A~ +A~) =l n=l cg 2M(M- 1) = - X ---'----..:... 2M 3 M-1 =-3-cg (3.2-42) ---~ ~- 1 ;..----~- M=64 FGURE ~ ~---, M=32 jj"---~ ~/ ', / '.. / M= 16 ',, Several signal space diagras for rectangular QAM.

9 ---~ ~- 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 7: Several signal space diagras for rectangular QAM. 5.4 Orthogonal Signaling Definition n orthogonal signaling, the wavefors s (t) are orthogonal and of equal energy E. n which case, the orthonoral set {φ (t), 1 N} defined by φ (t) = s (t), 1 M E 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) = E, 0, 0,..., 0, ( s (2) = 0, ) E, 0,..., 0,. =. ( s (M) = 0, 0, 0,..., ) E. 25

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

Definition Recall, from 6.7, that PAM signal waveforms are represented 6.4 Constellations for Digital Modulation Schees 6.4.1 PAM Definition 6.45. 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. 6.46. Clearly,