Cross correlated trellis coded quatrature modulation transmitter and system

[0001] This application is a divisional of U.S. application Ser. No. 09/496,135 filed Feb. 1, 2000, which is a continuation of U.S. application Ser. No. 09/412,348, filed Oct. 5, 1999, which claims priority to U.S. provisional application Ser. No. 60/103,227, filed Oct. 5, 1998.

[0002] Information can be sent over a channel using modulation techniques. Better bandwidth efficiency allows this same channel to hold and carry more information. A number of different systems for efficiently transmitting over channels are known. Examples include Gaussian minimum shift keying, staggered quadrature overlapped raised cosine modulation, and Feher's patented quadrature phase shift keying.

[0003] Many of these systems provide a transmitted signal with a constant or pseudo-constant envelope. This is desirable when the transmitter has a nonlinear amplifier that operates in or near saturation.

[0004] Many of these phase shift keying signals systems can operate using limited groups of the information at any one time.

[0005] Trellis coded modulation techniques are well known. Trellis coded techniques operate using multi-level modulation techniques, and hence can be more efficient than symbol-by-symbol transmission techniques.

[0006] The present application teaches a special cross correlated trellis coded quadrature modulation technique that can be used with a variety of different transmission schemes. Unlike conventional systems that use constant envelopes for the modulating waveforms, the present system enables mapping onto an arbitrarily chosen waveform that is selected based on bandwidth efficiency for the particular channel.

[0007] The system uses a special cross correlator that carries out the mapping in a special way.

[0008] This system can be used with offset quadrature phase shift keying along with conventional encoders, matched filters, decoders and the like. The system uses a special form of trellis coding in the modulation process that shapes the power spectrum of the transmitted signal over and above bandwidth efficiency that is normally achieved by an M-ary (as opposed to binary) modulation.

[0009] These and other aspects of the invention will be described in detail with reference to the accompanying drawings, wherein:

[0010] FIG. 1 shows a basic block diagram of a preferred transmitter of the present application;

[0011] FIG. 2 shows a specific cross correlation mapper;

[0012] FIG. 3 shows a specific embodiment that is optimized for XPSK;

[0013] FIG. 4 shows waveforms for FQPSK;

[0014] FIG. 5 shows a block diagram of the system for FQPSK;

[0015] FIGS. 6a and 6b respectively show the waveforms for in phase and out of phase FQPSK outputs;

[0016] FIG. 7 shows a trellis diagram for FQPSK;

[0017] FIG. 8 shows an FQPSK shaper;

[0018] FIG. 9 shows waveforms for full symbols of OQPSK;

[0019] FIG. 10 shows a trellis coded OQPSK;

[0020] FIG. 11 shows a 2 state trellis diagram;

[0021] FIG. 12 shows an uncoded OQPSK transmitter; and

[0022] FIG. 13 shows paths.

[0023] The present application describes a system with a transmitter that can operate using trellis coding techniques, which improve the operation as compared with the prior art techniques.

[0024] The present application focuses on the spectral occupancy of the transmitted signal. A special envelope property is described that improves the power efficiency of the demodulation and decoding operation. The disclosed structure is generic, and can be applied to different kinds of modulation including XPSK, FQPSK, SQORC, MSK and OP or OQPSK.

[0025] FIG. 1 shows a block diagram of a cross correlated quadrature modulation (XTCQM) transmitter 100.

[0026] An input binary (1) datastream 105 is an independent, identically distributed information sequence d_n at a bit rate R_b1/T_b. A quadrature converter 110 separates this sequence into an inphase (I) sequence 102 and a quadriphase (Q) sequence 104 d_in and d_Qn. As conventional, every second bit becomes part of the different phase. Hence, the phases can be formed by the even and odd bits of the information bit sequence d_n. The bits hence occur on the I and Q channels at a rate R1/TT_h; where T_h is the bit rate, and Tis the symbol rate.

[0027] For this explanation, it is assumed that the I and Q sequences d and d_Qn are time synchronous. Hence, each bit d_m (or d_Qn) occurs during the interval (n)T,t(n)Twhere n represents a count of adjacent symbol time periods T.

[0028] Rather than analyzing these levels as extending from 1 to 1, it may be more convenient to work with the (0,1) equivalents of the I and Q data sequences. This can be defined as $\begin{array}{cc}{D}_{In}\stackrel{}{=}\frac{1-d_{In}}{2},D_{Qn}\stackrel{}{=}\frac{1-d_{Qn}}{2}& \left(1\right)\end{array}$

[0029] which both range within the set (0,1). The sequences D_In and D_Qn are separately and respectively applied to rate r1/N convolutional encoders 120, 125. The two encoders are in general different, i.e., they have different tap connections and different modulo 2 summers but are assumed to have the same code rate.

[0030] We can define $\left\{E_{\mathrm{Ik}}|\begin{array}{c}N\\ k=1\end{array}\right\},\left\{E_{Qk}|\begin{array}{c}N\\ k=1\end{array}\right\}$

[0031] respectively as the sets of N(0,1) output symbols 122, 127 respectively, of the I and Q convolutional encoders 120, 125 corresponding to a single bit input to each of the encoders.

[0032] These sets of output symbols 122, 127 will be used to determine a pair of baseband waveforms s_t(t).s_Q(t) which ultimately modulate I and Q carriers for transmission over the channel. The signal s_Q(t) is delayed by delay element 130 for T/2T_h seconds prior to modulation on the quadrature carrier. This delay offsets the signal s_Q(t) relative to the s_t(t) signal, and thereby provides an offset modulation. Delaying the waveform by one half of a symbol at the output of the mapping allows synchronous demodulation and facilitates computation of the path metric at the receiver. This is different than the approach used for conventional FQPSK.

[0033] The present application teaches mapping of the symbol sets $\left\{E_{\mathrm{Ik}}|\begin{array}{c}N\\ k=1\end{array}\right\}\mathrm{and}\left\{E_{Qk}|\begin{array}{c}N\\ k=1\end{array}\right\}$

[0034] into s_t(t) and s_Q(t) using a waveform with a desired size and content (waveshape).

[0035] The mapping of the sets $\{E_{\mathrm{lk}}{}_{k=1}^N\}\mathrm{and}\{E_{\mathrm{Qk}}{}_{k=1}^N\}$

[0036] into s_t(t) and s_Q(t) uses a crosscorrelation mapper 140. Details of the mapping is shown in FIG. 2. Each of these sets of N (0,1) output symbols is partitioned into one of three groups as follows.

[0037] The I and Q signals are separately processed. For the I signals, the first group uses $I_{l_1},I_{l_2},,I_{N_1}$

[0038] as a subset of N₁ elements of $\{E_{\mathrm{lk}}{}_{k=1}^N\}$

[0039] which will be used only in the selection of s_t(t). The second group uses $Q_{l_1},Q_{l_2},,Q_{N_2}$

[0040] as a subset N₂ elements of $\{E_{\mathrm{lk}}{}_{k=1}^N\}$

[0041] which will be used only in the selection of s_Q(t). The third group uses $I_{l_{N_1+1}},I_{l_{N_1+2}},,I_{l_{N_1+N_3}}=Q_{l_{N_2+1}},Q_{l_{N_2+2}},,Q_{l_{N_2+N_3}}$

[0042] as a subset of N₃ elements of $\{E_{\mathrm{lk}}{}_{k=1}^N\}$

[0043] which will be used both for the selection of s_t(t) and s_Q(t). The term crosscorrelation in this context refers to the way in which the groups are formed.

[0044] All of the output symbols of the I encoder are used either to select s_t(t)_ts_Q(t) or both. Therefore, N₁N₂N₃N.

[0045] A similar three part grouping of the Q encoder output symbols $\{E_{\mathrm{Qk}}{}_{k=1}^N\}$

[0046] occurs. That is, for the first group let $Q_{m_1},Q_{m_2},,Q_{m_{i_1}}$

[0047] be a subset L₁ elements of $\{E_{\mathrm{Qk}}{}_{k=1}^N\}$

[0048] which will be used only in the selection of s_Q(t). For the second group, let $I_{m_1},I_{m_2},,I_{m_{i_2}}$

[0049] be a subset of L₂ elements of $\{E_{\mathrm{Qk}}{}_{k=1}^N\}$

[0050] which will be used only in the selection of s_t(t). Finally, for the third group let $Q_{m_{t_1}1},Q_{m_{t_{1}2}},,Q_{m_{t_{1}1}},=I_{m_{t_{2}1}},I_{m_{t_1}2},,I_{m_{t_{2}1}},$

[0051] be a subset of L₃ elements of $\{E_{\mathrm{Qk}}{}_{k=1}^N\}$

[0052] which will be used both for the selection of s_t(t) and s_Q(t). Once again, since all of the output symbols of the Q encoder are used either to select s_E(t), s_Q(t) or both, then L₁L₂L₃N.

[0053] A preferred mode exploits symmetry properties associated with the resulting modulation by choosing L₁N₁, L₂N₂ and L₃N₃. However, the present invention is not restricted to this particular symmetry.

[0054] In summary, based on the above, the signal S_E(t) is determined from symbols $I_{t_1},I_{t_2},,I_{l_{s_1{s}_3}}$

[0055] from the output of the I encoder and symbols $I_{l_1},I_{l_2},,I_{l_{L_2{L}_3}}$

[0056] from the output of the Q encoder. Thus, the size of the signaling alphabet used to select s_E(t) is 2^{N₁N₃L₂L₃}2^N₁. Similarly, the signal s_Q(t) is determined from symbols $Q_{l_1},Q_{l_2},,Q_{l_1{l}_2}$

[0057] from the output of the Q encoder and symbols $Q_1{Q}_{l_2},,Q_{l_{s_2{s}_1}}$

[0058] from the output of the I encoder. Thus, the size of the signaling alphabet used to select S_Q(t) is 2^{N₁30 N₃N₂N₃}2^N_Q.

[0059] An interesting embodiment results when the size of the signaling alphabets for selecting s_t(t) and s_Q(t) are equal. In that case, N_tN_Q or equivalently L₁N₂N₁L₂. This condition is clearly satisfied if the condition L₁N₁, L₂N₂ is met; however, the former condition is less restrictive and does not require the latter to be true.

[0060] FIG. 3 shows an example of the above mapping corresponding to N₁N₂N₃1 and L₁L₂L₃1, i.e., r1/N encoders for FQPSK, which is one particular embodiment of the XTCQM invention. The specific symbol assignments for the three partitions of the I encoder output are I₃ (group 1), Q₀ (group 2), I₂Q₁ (group 3). Similarly, the specific symbol assignments for the three partitions of the Q encoder output are: Q₃ (group 1), I₁ (group 2), I₀Q₂ (group 3). Since N₁N_Q4, the size of the signaling alphabet from which both s_E(t) and s_Q(t) are to be selected has 2⁴16 signals.

[0061] After assigning the encoder output symbols to either s_E(t), s_Q(t) or both, appropriate binary coded decimal (BCD) numbers are formed from these symbols. These numbers are used as indices i and j for selecting s_i(t) s_j(t) and s_Q(t)s₁(t) where $\left\{s_i\left(t\right)|\begin{array}{c}{N}_I\\ i=1\end{array}\right\}\mathrm{and}\left\{s_j\left(t\right)|\begin{array}{c}{N}_Q\\ j=1\end{array}\right\}$

[0062] are the signal waveform sets assigned for transmission of the I and Q channel signals.

[0063] I₀,I₁, . . . , I_N1 are defined as the specific set of symbols taken from both I and Q encoder outputs used to select s_t(t) and s_Q(t). Then the BCD indices needed above are iI_N112^N₁1 . . . I₁2¹ . . . I₀2⁰ and jQ_NQ12^N_Q1 . . . Q₁2¹ . . . Q₀2⁰. The FIG. 2 embodiment uses iI₃2³I₂2²I₁2¹ . . . I₀2⁰ and jQ₃2³Q₂2²Q₁2¹ . . . Q₀2⁰. This is shown in FIG. 3.

[0064] Numerically speaking, in a particular transmission interval of Tseconds, the contents of the I and Q encoders in FIG. 3 can be D_1.n11,D_1n0,D_1.n10 and D_Q.n1,D_Q.n10,D_Q.n21, then the encoder output symbols $\left\{E_{\mathrm{Ik}}|\begin{array}{c}3\\ k=1\end{array}\right\}\mathrm{and}\left\{E_{\mathrm{Qk}}|\begin{array}{c}3\\ k=1\end{array}\right\}$

[0065] would respectively partition as I_i0 (group 1), Q₀1 (group 2), I₂Q₁0 (group 3) and Q_i1 (group 1), I₁1 (group 2), I₀Q₂1 (group 3). Thus, based on the above, i3 and j13 and hence the selection for s_t(t) and S_Q(t) would be s₁(t)s₃(t) and s_Q(t)s₁₃(t)

[0066] An important function of the present application is that any set of N₁ waveforms of duration T, seconds (defined on the interval (T/2tT,/2) can be used for selecting the I channel transmitted signal. Likewise, any set of N_Q waveforms of duration Tseconds, also defined on the interval (T/2tT/2) can be used for selecting the Q channel transmitted signal s_Q(t). However, certain properties can be invoked on these waveforms to make them more power and spectrally efficient.

[0067] This discussion assumes the special case of N₁N_QN^*, although other embodiments are contemplated. Maximum distance in the waveform set can improve power efficiency. The distance can be increased by dividing the signal set $\left\{s_i\left(t\right)|\begin{array}{c}{N}^{*}\\ i=1\end{array}\right\}$

[0068] into two equal parts; with the signals in the second part being antipodal to (the negatives of) those in the first part. Mathematically, the signal set has the composition s₀(t).s₁(t) . . . s_{N2 1}(t),s₀(t),s₁(t), . . . ,s_N21(t). To achieve good spectral efficiency, one should choose the waveforms to be as smooth, i.e., as many continuous derivatives, as possible, since a smoother waveform gives better power spectrum roll off. Furthermore, to prevent discontinuities at the symbol transition time instants, the waveforms should have a zero first derivative (slope) at their endpoints tT/2.

[0069] An example of a signal set that satisfies the first requirement and part of the second requirement is still illustrated in FIG. 4. This shows the specific FQPSK embodiment.

[0070] Generic FQPSK is described in U.S. Pat. Nos. 4,567,602; 4,339,724; 4,644,565 and 5,491,457. This is conceptually similar to the cross-correlated phase-shift-keying (XPSK) modulation technique introduced in 1983 by Kato and Feher. This technique was in turn a modification of the previously-introduced (by Feher et al) interference and jitter free QPSK (IJF-QPSK) with the purpose of reducing the 3 dB envelope fluctuation characteristic of IJF-QPSK to 0 dB. This made the modulation appear as a constant envelope, which was beneficial in nonlinear radio systems. It is further noted that using a constant waveshape for the even pulse and a sinusoidal waveshape for the odd pulse, IJF-QPSK becomes identical to the staggered quadrature overlapped raised cosine (SQORC) scheme introduced by Austin and Chang. Kato and Feher achieved their 3 dB envelope reduction by using an intentional but controlled amount of crosscorrelation between the inphase (I) and quadrature (Q) channels. This crosscorrelation operation was applied to the IJF-QPSK (SQORC) baseband signal prior to its modulation onto the I and Q carriers.

[0071] FIG. 5 shows a conceptual block diagram of FPQSK. Specifically, this operation has been described by mapping, in each half symbol, the 16 possible combinations of I and Q 20 channel waveforms present in the SQORC signal. The mapping moves the signals into a new set of 16 waveform combinations chosen in such a way that the crosscorrelator output is time continuous and has a unit (normalized) envelope at all I and Q uniform sampling instants.

[0072] The present embodiment describes restructuring the crosscorrelation mapping into one mapping, based on a full symbol representation of the I and Q signals. The FPQSK signal can be described directly in terms of the data transitions on the I and Q channels. As such, the representation becomes a specific embodiment of XTCQM.

[0073] Appropriate mapping of the transitions in the I and Q data sequences into the signals s_t(t) and s_Q(t) is described by Tables 1 and 2.

TABLE 1
Mapping for Inphase (I)-Channel Baseband Signal
sI(t) in the Interval (n )TS t < (n )TS
d In - d I n - 1 2	d Qn - 1 - d Qn - 2 2	d Qn - d Qn - 1 2	sI(t)
0	0	0	dIns0(t nT1)
0	0	1	dIns1(t nT1)
0	1	0	dIns2(t nT1)
0	1	1	dIns3(t nT1)
1	0	0	dIns4(t nT1)
1	0	1	dIns5(t nT1)
1	1	0	dIns6(t nT1)
1	1	1	dIns7(t nT1)

[0074]

TABLE 2
Mapping for Quadrature (Q)-Channel Baseband Signal
sQ(t) in the Interval (n )T1 t (n )T1
d Qn - d Qn - 1 2	d Q n - d Q n - 1 2	d Q n + 1 - d Q n 2	sQ(t)
0	0	0	dQns0(t nT1)
0	0	1	dQns1(t nT1)
0	1	0	dQns2(t nT1)
0	1	1	dQns3(t nT1)
1	0	0	dQns4(t nT1)
1	0	1	dQns5(t nT1)
1	1	0	dQns6(t nT1)
1	1	1	dQns7(t nT1)

[0075] Note that the subscript i of the transmitted signal s₁(t) or s_Q(t) as appropriate is the binary coded decimal (BCD) equivalent of the three transitions. Since d_tn and d_Qn take on values 1, Tables 1 and 2 specify the mapping of I and Q symbol transitions 16 different waveforms, namely, $s_i\left(t\right)|\begin{array}{c}15\\ i=0\end{array}$

[0076] where s₁(t)s_1-8(t).i8.9, . . . , 15.

[0077] The specifics are as follows: $\begin{array}{cc}{s}_0\left(t\right)=A,-T_s/2t{T}_s/2,s_8\left(t\right)=-s_0\left(t\right)s_1\left(t\right)=\{\begin{array}{cc}A,-T_s/2t0& \\ 1-\left(1-A\right){\cos }^2\frac{t}{T_s},& 0t{T}_s/2\end{array}{s}_9\left(t\right)=-s_1\left(t\right)s_2\left(t\right)=\{\begin{array}{cc}1-\left(1-A\right){\cos }^2\frac{t}{T_s},& -T_s/2t0\\ A,0t{T}_s/2& \end{array}{s}_{10}\left(t\right)=-s_2\left(t\right)s_3\left(t\right)=1-\left(1-A\right){\cos }^2\frac{t}{T_s},-T_s/2t{T}_s/2s_{11}\left(t\right)=-s_3\left(t\right)\mathrm{and}{s}_4\left(t\right)=A\sin \frac{t}{T_s},-T_s/2t{T}_s/2,s_{12}\left(t\right)=-s_4\left(t\right)s_5\left(t\right)=\{\begin{array}{cc}A\sin \frac{t}{T_s},& -T_s/2t0\\ \sin \frac{t}{T_s},& 0t{T}_s/2\end{array},s_{13}\left(t\right)=-s_5\left(t\right)s_6\left(t\right)=\{\begin{array}{cc}\sin \frac{t}{T_s},& -T_s/2t0\\ A\sin \frac{t}{T_s},& 0t{T}_s/2\end{array},s_{14}\left(t\right)=s_6\left(t\right)s_7\left(t\right)=\sin \frac{t}{T_s},-T_s/2t{T}_s/2,s_{15}\left(t\right)=s_7\left(t\right)& \left(\text{2a}\right)\end{array}$

[0078] Applying the mappings in Tables 1 and 2 to the I and Q data sequences produces the identical I and Q baseband transmitted signals to those that would be produced by passing the I and Q IJF encoder outputs of FIG. 5 through the crosscorrelator (half symbol mapping) of the FQPSK (XPSK) scheme. An example of this is shown with reference to FIGS. 6a and 6b. The Q signal must be delayed by T/2 to produce an offset form of modulation. Alternately stated, for arbitrary I and Q data sequences, FQPSK can alternately be generated by the symbol-by-symbol mappings of Tables 1 and 2 as applied to these sequences.

[0079] The mappings of Tables 1 and 2 become a specific embodiment of XTCQM as described herein. First, the I and Q transitions needed for the BCD representations of the indices of s_i(t) and s_j(t) are rewritten in terms of their modulo 2 sum equivalents. That is, using the (0,1) form of the I and Q data symbols, Tables 1 and 2 show that

[0080] with

[0081] resulting in the baseband I and Q waveforms s₁(t)s₁(tnT,) and s_Q(t)s_j(tnT) The signals that are modulated onto the I and Q carriers are y₁(t)s₁(t) and y_Q(t)s_Q(tT/2). Thus, in each symbol interval $(\left(n-\frac{1}{2}\right)T_{s}t\left(n+\frac{1}{2}\right)T_s$

[0082] for y₁(t) and nTt(n1)T, for y_Q(t)), the I and Q channel baseband signals are each chosen from a set of 16 signals, s₁(t),i0.1, . . . , 15 in accordance with the 4-bit BCD representations of their indices defined by (3) together with (4).

[0083] A graphical illustration of the implementation of this mapping is given in FIG. 3, which is a specific embodiment of FIG. 1 with N₁N₂N₃L₁L₂L₃1. The mapping in FIG. 3 can be interpreted as a 16-state trellis code with two binary inputs D_1.n1.D_Qn and two waveform outputs s_i(t).s_j(t) where the state is defined by the 4-bit sequence D_1n,D_1.n1.D_Q.n1.D_Q.n2. The trellis is illustrated in FIG. 7 and the transition mapping is given in Table 3.

TABLE 3
Trellis State Transistions
	Current State	Input	Output	Next State
	0 0 0 0	0 0	0	0	0 0 0 0
	0 0 0 0	0 1	1	12	0 0 1 0
	0 0 0 0	1 0	0	1	1 0 0 0
	0 0 0 0	1 1	1	13	1 0 1 0
	0 0 1 0	0 0	3	4	0 0 0 1
	0 0 1 0	0 1	2	8	0 0 1 1
	0 0 1 0	1 0	3	5	1 0 0 1
	0 0 1 0	1 1	2	9	1 0 1 1
	1 0 0 0	0 0	12	3	0 1 0 0
	1 0 0 0	0 1	13	15	0 1 1 0
	1 0 0 0	1 0	12	2	1 1 0 0
	1 0 0 0	1 1	13	14	1 1 1 0
	1 0 1 0	0 0	15	7	0 1 0 1
	1 0 1 0	0 1	14	11	0 1 1 1
	1 0 1 0	1 0	15	6	1 1 0 1
	1 0 1 0	1 1	14	10	1 1 1 1
	0 0 0 1	0 0	2	0	0 0 0 0
	0 0 0 1	0 1	3	12	0 0 1 0
	0 0 0 1	1 0	2	1	1 0 0 0
	0 0 0 1	1 1	3	13	1 0 1 0
	0 0 1 1	0 0	1	4	0 0 0 1
	0 0 1 1	0 1	0	8	0 0 1 1
	0 0 1 1	1 0	1	5	1 0 0 1
	0 0 1 1	1 1	0	9	1 0 1 1
	1 0 0 1	0 0	14	3	0 1 0 0
	1 0 0 1	0 1	15	15	0 1 1 0
	1 0 0 1	1 0	14	2	1 1 0 0
	1 0 0 1	1 1	15	14	1 1 1 0
	1 0 1 1	0 0	13	7	0 1 0 1
	1 0 1 1	0 1	12	11	0 1 1 1
	1 0 1 1	1 0	13	6	1 1 0 1
	1 0 1 1	1 1	12	10	1 1 1 1
	0 1 0 0	0 0	4	2	0 0 0 0
	0 1 0 0	0 1	5	14	0 0 1 0
	0 1 0 0	1 0	4	3	1 0 0 0
	0 1 0 0	1 1	5	15	1 0 1 0
	0 1 1 0	0 0	7	6	0 0 0 1
	0 1 1 0	0 1	6	10	0 0 1 1
	0 1 1 0	1 0	7	7	1 0 0 1
	0 1 1 0	1 1	6	11	1 0 1 1
	1 1 0 0	0 0	8	1	0 1 0 0
	1 1 0 0	0 1	9	13	0 1 1 0
	1 1 0 0	1 0	8	0	1 1 0 0
	1 1 0 0	1 1	9	12	1 1 1 0
	1 1 1 0	0 0	11	5	0 1 0 1
	1 1 1 0	0 1	10	9	0 1 1 1
	1 1 1 0	1 0	11	4	1 1 0 1
	1 1 1 0	1 1	10	8	1 1 1 1
	0 1 0 1	0 0	6	2	0 0 0 0
	0 1 0 1	0 1	7	14	0 0 1 0
	0 1 0 1	1 0	6	3	1 0 0 0
	0 1 0 1	1 1	7	15	1 0 1 0
	0 1 1 1	0 0	5	6	0 0 0 1
	0 1 1 1	0 1	4	10	0 0 1 1
	0 1 1 1	1 0	5	7	1 0 0 1
	0 1 1 1	1 1	4	11	1 0 1 1
	1 1 0 1	0 0	10	1	0 1 0 0
	1 1 0 1	0 1	11	13	0 1 1 0
	1 1 0 1	1 0	10	0	1 1 0 0
	1 1 0 1	1 1	11	12	1 1 1 0
	1 1 1 1	0 0	9	5	0 1 0 1
	1 1 1 1	0 1	8	9	0 1 1 1
	1 1 1 1	1 0	9	4	1 1 0 1
	1 1 1 1	1 1	8	8	1 1 1 1

[0084] In this table, the entries in the column labeled input correspond to the values of the two input bits D_1.n1,D_Qn that result in the transition. The entries in the column output correspond to the subscripts i and j of the pair of symbol waveforms s_i(t),s_j(t) that are output.

[0085] It is well known that the rate at which the sidelobes of a modulation's power spectral density (PSD) roll off with frequency is related to the smoothness of the underlying waveforms that generate it. That is, a waveform that has more continuous waveform derivatives will hare faster Fourier transform decays with frequency.

[0086] The crosscorrelation mappings of FQPSK is based on a half symbol characterization of the SQORC signal. Hence, there is no guarantee that the slope or any higher derivatives of the crosscorrelator output waveform is continuous at the half symbol transition points. From Equation (2b) and the corresponding illustration in FIG. 4, it can be observed that four out of the sixteen possible transmitted waveforms, namely, s₅(t),s₆(t),s₁₃(t),s₁₄(t) have a slope discontinuity at their midpoint. Thus, for random I and Q data symbol sequences, on the average the transmitted FQPSK waveform will likewise have a slope discontinuity at one quarter of the uniform sampling time instants. Therefore, for a random data input sequence, a discontinuity in slope occurs one quarter of the time.

[0087] Based on the above reasoning, it is predicted that an improvement in PSD rolloff could be obtained if the FQPSK crosscorrelation mapping could be modified so that the firs, derivative of the transmitted baseband waveforms is always continuous. This enhanced version of FQPSK requires a slight modification of the above-mentioned four waveforms in FIG. 4. In particular, these four transmitted signals are redefined in a manner analogous to s₁(t),s₂(t),s₉(t),s₁₀(t), namely $\begin{array}{cc}{s}_5\left(t\right)=\{\begin{array}{c}\sin \frac{t}{T_s}+\left(1-A\right){\sin }^2\frac{t}{T_s},-T_s/2t0\\ \sin \frac{t}{T_s},0t{T}_s/2\end{array},s_{13}\left(t\right)=-s_5\left(t\right)s_6\left(t\right)=\{\begin{array}{c}\sin \frac{t}{T_s},-T_s/2t0\\ \sin \frac{t}{T_s}-\left(1-A\right){\sin }^2\frac{t}{T_s},0t{T}_s/2\end{array},s_{14}\left(t\right)=-s_6\left(t\right)& \left(5\right)\end{array}$

[0088] Note that not only do the signals s₅(t),s₆(t),s₁₃(t),s₁₄(t) as defined in (5) not have a slope discontinuity at their midpoint, or anywhere else in the defining interval. Also, the zero slope at their endpoints has been preserved. Thus, the signals in (5) satisfy both requirements for desired signal set waveforms as discussed in Section 3.1.2. Using (5) in place of the corresponding signals of (2b) results in a modified FQPSK signal that has no slope discontinuity anywhere in time regardless of the value of A.

[0089] FIG. 5 illustrates a comparison of the signal s₀(t) of (5) with that of (2b) for a value of A1/{square root}{square root over (2)}.

[0090] The signal set selected for enhanced FQPSK has a symmetry property for s₀(t)s₃(t) that is not present for s₄(t)s₇(t). In particular, s₁(t) and s₂(t) are each composed of one half of s₀(t) and one half of s₃(t), i.e., the portion of S₁(t) from tT,/2 to t0 is the same as that one half of s₀(t) whereas the portion of s₁(t) from t0 to tT,/2 is the same as that of s₃(t) and vice versa for s₂(t). To achieve the same symmetry property for s₄(t)s₇(t), one would have to reassign s₄(t) as $\begin{array}{cc}{s}_4\left(t\right)=\{\begin{array}{c}\sin \frac{t}{T_s}+\left(1-A\right){\sin }^2\frac{t}{T_s},-T_s/2t0\\ \sin \frac{t}{T_s}-\left(1-A\right){\sin }^2\frac{t}{T_s},0t{T}_s/2\end{array},s_{12}\left(t\right)=-s_4\left(t\right)& \left(6\right)\end{array}$

[0091] This minor change produces a complete symmetry in the waveform set. Thus, it has an advantage from the standpoint of hardware implementation and produces a negligible change in spectral properties of the transmitted waveform. The remainder of the discussion, however, ignores this minor change and assumes the version of enhanced FQPSK first introduced in this section.

[0092] Consider an XTCQM scheme in which the mapping function is performed identically to that in the FQPSK embodiment (i.e., as in FIG. 3) but the waveform assignment is made as follows and as shown in FIG. 9: $\begin{array}{cc}\begin{array}{cc}{s}_0\left(t\right)=s_1\left(t\right)=s_2\left(t\right)=s_3\left(t\right)=1,& -T_s/2t{T}_s/2,\end{array}{s}_4\left(t\right)=s_5\left(t\right)=s_6\left(t\right)=s_7\left(t\right)=\{\begin{array}{cc}-1,& -T_s/2t0\\ 1,& 0t{T}_s/2\end{array}{s}_i\left(t\right)=-s_{i-8}\left(t\right),i=8,9,,15& \left(7\right)\end{array}$

[0093] that is, the first four waveforms are identical (a rectangular pulse) as are the second four (a split rectangular unit pulse) and the remaining eight waveforms are the negatives of the first eight. As such there are only four unique waveforms which are denoted by $c_i\left(t\right){|}_{i=0}^3$

[0094] where c₀(t)s₀(t).c₁(t)s₄(t),c₂(t)s₈(t),c₃(t)s₁₂(t). Since the BCD representations for each group of four identical waveforms the two least significant bits are irrelevant, i.e., the two most significant bits are sufficient to define the common waveform for each group, the mapping scheme can be simplified by eliminating the need for I₀.I₁ and Q₀.Q₁. FIG. 3 shows how eliminating all of I₀.I₁ and Q₀.Q₁ accomplishes multiple purposes. The two encoders can be identical and need only a single shift register stage. Also, the correlation between the two encoders in so far as the mapping of either one's output symbols to both s_t(t) and s_Q(t) has been eliminated which therefore results in what: might be termed a degenerate form of XTCQM.

[0095] The resulting embodiment is illustrated in FIG. 10. Since the mapping decouples the I and Q as indicated by the dashed line in the signal mapping block of FIG. 10, it is sufficient to examine the trellis structure and its distance properties for only one of the two I and Q channels. The trellis diagram for either channel of this modulation scheme would have two states as illustrated in FIG. 11. The dashed line indicates a transition caused by an input 0 and the solid indicates a transition caused by an input 1. Also, the branches are labeled with the output signal waveform that results from the transition. An identical trellis diagram exists for the Q channel.

[0096] This embodiment of XTCQM has a PSD identical to that of the uncoded OQPSK (which is the same as uncoded QPSK) for the transmitted signal. In particular, because of the constraints imposed by the signal mapping, the waveforms C₁(t)s₄(t) and c₃(t)s₁₂(t) can never occur twice in succession. Thus, for any input information sequence, the sequence of signals s_t(t) and s_Q,(t) cannot transition at a rate faster than 1/T,sec. This additional spectrum conservation constraint imposed by the signal mapping function of XTCQM can reduce the coding (power) gain relative to that which could be achieved with another mapping which does not prevent the successive repetition of c₁(t) and c₃(t) However, the latter occurrence would result in a bandwidth expansion by a factor of two.

[0097] If instead of a split rectangular pulse in (7), a sinusoidal pulse were used, namely, $\begin{array}{cc}{s}_4\left(t\right)=s_5\left(t\right)=s_6\left(t\right)=s_7\left(t\right)=\sin \frac{t}{T_s},-T_s/2t{T}_s/2s_i\left(t\right)=-s_{i-8}\left(t\right),i=12,13,14,15& \left(8\right)\end{array}$

[0098] then the same simplification of the mapping function as in FIG. 10 occurs resulting in decoupling of the I and Q channels. The trellis diagram of FIG. 11 can then be used for either the I or Q channel. Once again, this has a PSD identical to that of uncoded SQORC which is the same as uncoded QORC.

[0099] The signal assignment and mapping of FIG. 3 can be simplified such that

[0100] then in the BCD representations for each group of eight identical waveforms the three least significant bits are irrelevant. Only the first significant bit is needed to define the common waveform for each group. Hence, the mapping scheme can be simplified by eliminating the need for I₀,I₁,I₂ and Q₀,Q₁,Q₂. Defining the two unique waveforms c₀(t)s₀(t),c₁(t)s₈(t) obtains the simplified degenerate mapping of FIG. 12 which corresponds to uncoded OQPSK with NRZ data formatting.

[0101] Likewise, if instead of the signal assignment in (9) the relation below is used: $\begin{array}{cc}{s}_0\left(t\right)+s_1\left(t\right)==s_7\left(t\right)\{\begin{array}{cc}-1,& -T_s/2t0\\ 1,& 0t{T}_i/2\end{array}{s}_i\left(t\right)=-s_{i-8}\left(t\right),i=8.9,,15& \left(10\right)\end{array}$

[0102] then the mapping of FIG. 12 produces uncoded OQPSK with Manchester (biphase) data formatting.

[0103] An optimum detector for XTCQM is a standard trellis coded receiver which employs a bank of filters which are matched to the signal waveform set, followed by a Viterbi (trellis) decoder. The bit error probability (BEP) performation of such a receiver can be described in terms of its minimum squared Euclidean distance d_min², taken over all pairs of paths through the trellis. Comparing d_min² for one TCM scheme with that of another scheme or with an uncoded modulation provides a measure of the relative asymptotic coding gain in the limit of infinite E_h/N₀. To compute d_min² for a given TCM (of which XTCQM is one), it is sufficient to determine the minimum Euclidean distance over all pairs of error event paths that emanate from a given state, and first return to that or another state a number of branches later.

[0104] The procedure and actual coding gains that can be achieved relative to uncoded OQPSK are explained with reference to results for the specific embodiments of XTCQM discussed above.

[0105] For conventional or enhanced FQPSK, the smallest length error event for which there are at least two paths that start in one state and remerge in the same or another state is 3 branches. For each of the 16 starting states, there are exactly 4 such error event paths that remerge in each of the 16 end states. FIG. 13 is an example of these error event paths for the case where the originating state is 0000 and the terminating state is 0010.

[0106] The trellis code defined by the mapping in Table 3 is not uniform, e.g., it is not sufficient to consider only the all zeros path as the transmitted path in computing the minimum Euclidean distance. Rather all possible pairs of error event paths starting from each of the 16 states (the first 8 states are sufficient in view of the symmetry of the signal set) and the ending in each of the 16 states and must be considered to determine the pair having the minimum Euclidean distance.

[0107] Upon examination of the squared Euclidean distance between all pairs of paths, regardless of length, it has been shown that the minimum of this distance normalized by the average bit energy which is one half the average energy of the signal (symbol) set, is for FQPSK given by $\begin{array}{cc}\frac{d_{\mathrm{mm}}^2}{2{\stackrel{\_}{E}}_h}=\frac{16\left[\frac{7}{4}-\frac{8}{3}-A\left(\frac{3}{2}+\frac{4}{3}\right)+A^2\left(\frac{11}{4}+\frac{4}{}\right)\right]}{\left(7+2A+15A^2\right)}=1.56& \left(11\right)\end{array}$

[0108] where {overscore (E)}_h denotes the average bit energy of the FQPSK signal set, i.e., one-half the average symbol energy of the same signal set. For enhanced FQ2SK we have $\begin{array}{cc}\frac{d_{\mathrm{mm}}^2}{2{\stackrel{\_}{E}}_h}=\frac{\left(3-6A+15A^2\right)}{\frac{21}{8}-\frac{8}{3}-A\left(\frac{1}{4}-\frac{8}{3}\right)+\frac{29}{8}{A}^2}=1.56& \left(12\right)\end{array}$

[0109] which coincidentally is identical to that for FQPSK. Thus, the enhancement of FQPSK provided by using the waveforms of (5) as replacements for their equivalents in (2b) is significantly beneficial from a spectral standpoint with no penalty in asymptotic receiver performance.

[0110] To compare the performance of the optimum receivers of FQPSK and enhanced FQPSK with that of conventional uncoded offset QPSK (OQPSK) we note for the latter that d_min²/{overscore (E)}_h2 which is the same as that for BPSK. Thus, as a trade against the significantly improved power spectrum afforded by FQPSK and its enhanced version relative to that of OQPSK, an asymptotic loss of only 10 log(1/1.56)1.07 dB is experienced. These results should be compared with the significantly poorer performance of the conventional FQPSK receiver which makes symbol-by-symbol decisions based independently on the I and Q samples, and results in an asymptotic loss in E_h/N₀ performance on the order of 2 to 2.5 dB relative to uncoded OQPSK.

[0111] For the 2-state trellis diagram in FIG. 11, the minimum squared Euclidean distance occurs for an error event path of length 2 branches. Considering the four possible pairs of such paths that eminate from one of the 2 states and remerge at the same or the other state, then for the waveforms of FIG. 9 it is simple to see that d_min²4T. Since the average energy of the signal (symbol) set on the I (or Q) channel is E_uvT_s which is also equal to the average bit energy (since the channel by itself represents only one bit of information), then the normalized minimum squared Euclidean distance is d_min²/2{overscore (E)}_h2 which represents no asymptotic coding gain over OQPSK. At finite values of E_h/N₀ there will exist some coding gain since the commutation of error probability performance takes into account all possible error event paths, i.e., not only those corresponding to the minimum distance. Thus, in conclusion, the trellis coded OQPSK scheme presented here is a method for generating a transmitted modulation with a PSD that is identical to that of uncoded OQPSK and offers the potential of coding gain at finite SNR without the need for transmitting a higher order modulation (e.g., conventional rate trellis coded 8PSK with also achieves no bandwidth expansion relative to uncoded QPSK), the latter being significant in that receiver synchronization circuitry can be designed for a quadriphase modulation scheme.

[0112] Here again the minimum squared Euclidean distance occurs for the same error event paths as described above. With reference to the signal waveform, we now have d_min²3T,. Since the average energy of this signal (symbol) set is E_uv0.75T, which again per channel is equal to the average bit energy, then the normalized minimum squared Euclidean distance is also d_min²/2{overscore (E)}_h2 which again represents no asymptotic coding gain over SQORC. Even though its pulse shaping SQORC has an improved PSD relative to OQPSK, it suffers from a 3 dB envelope fluctuation whereas OQPSK is constant envelope.