Unfiltered FQPSK: Another Interpretation and Further Enhancements

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▲ Figure 4a. Inphase FQPSK (XPSK) output. ▲ Figure 4b. Quadrature phase FQPSK (XPSK) output. these four transmitted signals in a manner analogous to s 1 (t), s 2 (t), s 9 (t) and s 10 (t), namely, ⎧ πt 2 πt sin + ( 1− A)sin , −Ts / 2≤ t ≤0 ⎪ Ts Ts s5() t = ⎨ ⎪ πt sin , 0 ≤ t ≤Ts / 2 ⎩⎪ Ts s () t =−s () t s 6 13 ⎧ πt sin , −Ts / 2≤ t ≤0 ⎪ Ts () t ⎨ ⎪ πt 2 πt sin −( 1 − A)sin , 0 ≤ t ≤Ts / 2 ⎩⎪ Ts Ts s 14 () t =−s () t 6 5 Note that the signals s 5 (t), s 6 (t), s 13 (t) and s 14 (t) as defined in (9) do not have a slope discontinuity at their midpoint nor, for that matter, anywhere else in the defining interval. Also, the zero slope at their endpoints d In – d I,n–1 d Q,n–1 – d Q,n–2 d Qn – d Q,n–1 s I (t) 2 2 2 0 0 0 d In s 0 (t–nT s ) 0 0 1 d In s 1 (t–nT s ) 0 1 0 d In s 2 (t–nT s ) 0 1 1 d In s 3 (t–nT s ) 1 0 0 d In s 4 (t–nT s ) 1 0 1 d In s 5 (t–nT s ) 1 1 0 d In s 6 (t–nT s ) 1 1 1 d In s 7 (t–nT s ) ▲ Table 1. Mapping for inphase (I)-channel baseband signal y I (t) in in the interval (n–½)T s ≤ t ≤ (n+½)T s . , , (9) has been preserved. Thus, using (9) in place of the corresponding signals of (7b) will result in a modified FQPSK signal that has no slope discontinuity anywhere in time regardless of the value of A. Figure 5 illustrates a comparison of the signal s 6 (t) of (9) with that of (7b) for a value of A=1√2. Figure 6 illustrates the power spectral density of conventional FQPSK and its enhancement obtained by using the waveforms of (9) as replacements for those in (7b). The significant improvement in spectral rolloff rate is clear from a comparison of the two. As it currently stands, the signal set chosen for enhanced FQPSK has a symmetry property for s 0 (t), s 1 (t), s 2 (t) and s 3 (t) that is not present for s 4 (t), s 5 (t), s 6 (t) and s 7 (t). In particular, s 1 (t) and s 2 (t) are each composed of one half of s 0 (t) and one half of s 3 (t), i.e., the portion of s 1 (t) from t = –T s /2 to t = 0 is the same as that of s 0 (t), whereas the portion of s 1 (t) from t = 0 to t = T s /2 is the same as that of s 3 (t), and vice versa for s 2 (t). To achieve the same symmetry property for s 4 (t) – s 7 (t), one would have to reassign s 4 (t) as d Qn – d Q,n–1 d I,n – d I,n–1 d I,n+1 – d I,n s I (t) 2 2 2 0 0 0 d Qn s 0 (t–nT s ) 0 0 1 d Qn s 1 (t–nT s ) 0 1 0 d Qn s 2 (t–nT s ) 0 1 1 d Qn s 3 (t–nT s ) 1 0 0 d Qn s 4 (t–nT s ) 1 0 1 d Qn s 5 (t–nT s ) 1 1 0 d Qn s 6 (t–nT s ) 1 1 1 d Qn s 7 (t–nT s ) ▲ Table 2. Mapping for quadrature (Q)-channel baseband signal y Q (t) in the interval nT s ≤ t ≤ (n+1)T s . 84 · APPLIED MICROWAVE & WIRELESS
▲ Figure 5. Original and new FQPSK shapes. ▲ Figure 6. Power spectra of conventional and enhanced FQPSK: (a) without SSPA; and (b) with SSPA. s 4 ⎧ πt 2 πt sin + ( 1− A)sin , −Ts / 2≤ t ≤0 ⎪ Ts Ts () t = ⎨ , ⎪ πt 2 πt sin −( 1− A)sin , 0 ≤ t ≤Ts / 2 ⎩⎪ Ts Ts () t =−s () t s 12 4 (10) This minor change, which produces a complete symmetry in the waveform set, has an advantage from the standpoint of hardware implementation and produces a negligible change in spectral properties of the transmitted waveform. Nevertheless, for the remainder of the discussion, we shall ignore this minor change and assume the version of enhanced FQPSK first introduced in this section. Interpretation of FQPSK as a trellis-coded modulation The I and Q mappings given in Tables 1 and 2 are alternately described in terms of the (0,1) representation of the I and Q data symbols and their transitions. Specifically, define D In = ∆ (1–d In )/2, D Qn = ∆ (1−d Qn )/2 (11) which both range on the set (0,1). Then, defining the BCD representation of the indices i and j by with 3 i= I3 × 2 + I2 × 2 + I1 × 2 + I0 × 2 3 2 1 j= Q × 2 + Q × 2 + Q × 2 + Q × 2 3 I = DQn ⊕ D Q D D 0 Q n− , = I n ⊕ , 1 0 , + 1 In I = DQn , − ⊕ DQn , − , Q = DIn⊕ DIn , − = I I = 2 D ⊕ In D , I n 1 Q = 2 D ⊕ Qn D = , − Q, n−1 I0 I = D , Q = D 1 1 2 1 1 2 3 In 3 2 2 Qn 1 (12) (13) we have y I (t)=s i (t–nT s ) and y Q (t)=s j (t–(n+½)T s ). That is, in each symbol interval ((n–½)T s ≤ t ≤ (n+½)T s for y I (t) and nT s ≤ t ≤ (n+1)T s for y Q (t)), the I and Q channel baseband signals are each chosen from a set of 16 1 0 0 0 ▲ Figure 7. Alternate implementation of FQPSK baseband signals. ▲ Figure 8. 16-state trellis diagram for FQPSK. 86 · APPLIED MICROWAVE & WIRELESS
Page 1 and 2: Unfiltered FQPSK: Another Interpret
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Unfiltered FQPSK: Another Interpretation and Further Enhancements

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