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Design of Multiple Trellis-Coded Multi-H CPM Based on Super Trellis

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Design of Multiple Trellis-Coded Multi-H CPM Based on Super Trellis 1130 LIU XIAN, LIU AIJUN, ZHANG BANGNING, PAN KEGANG, DESIGN OF MULTIPLE TRELLIS-CODED MULTI-H CPM … Design of Multiple Trellis-Coded Multi-h CPM Based on Super Trellis LIU Xian, LIU Aijun, ZHANG Bangning, PAN Kegang Institute of Communications Engineering, PLA University of Science and Technology, Nanjing 210007, China [email protected] Abstract. It has been shown that the multiple trellis code h schemes change the modulation index in a cyclical fash- can perform better than the conventional trellis code over ion for successive symbol intervals. Bhumi A. Dave and AWGN channels, at the cost of additional computations per Raveendra K. Rao described a class of CPM signals re- trellis branch. Multiple trellis coded multi-h CPM schemes ferred to as generalized asymmetric multi-h phase-coded have been shown in the literature to have attractive power- modulation which adaptively changed modulation indices bandwidth performance at the expense of increased by a function of the current symbol, previous symbol and receiver complexity. In this method, the multi-h format is phase state and hence could achieve significant amount of made to be associated with the specific pattern and increase in minimum distance as compared to ordinary repeated rather than cyclically changed in time for succes- multi-h CPM signaling [5]. sive symbol intervals, resulting in a longer effective length Trellis coded multi-h CPM schemes using a Viterbi of the error event with better performance. It is well known decoder has been investigated to achieve further coding that the rate (n-1)/n multiple trellis codes combined with gains [6], [7], in which trellis paths remained unmerged 2n-level CPM have good power-bandwidth performance. In longer and offered generally improved minimum Euclidean this paper, a scheme combining rate 1/2 and 2/3 multiple free distance. This scheme results in a kind of concatenated trellis codes with 4- and 8-level multi-h CPM is shown to code and provides better performance with properly chosen have better power-bandwidth performance over the upper modulation indices [8–11]. A nonlinear CPM technique is bound than the scheme with single-h. presented in [12] which achieves the maximal constraint length allowed by the number of states and achieves attrac- tive minimum distances compared with other existing CPM Keywords signaling formats. However, maximization of the constraint length does not ensure the maximization of the minimum Multiple trellis code, multi-h continuous phase distance and hence this technique does not guarantee the modulation, merged encoder, super trellis. highest achievable minimum distance. This technique also has the disadvantage of using a large number of modula- tion indices. 1. Introduction The multiple trellis code, wherein more than one channel symbol per trellis branch is transmitted, could With the adoption of MIL-STD-188-181B on 20 usually provide superior performance [13]. Such mecha- March 1999 [1], Multi-h Continuous Phase Modulation nism accommodates more freedom for the assignment of (CPM) became a mandatory requirement of US. Military signal points to transmitted symbols, and thus for many UHF SATCOM terminals. A dual-h CPM scheme (with cases, produce better codes. Although the multiple trellis two distinct modulation indices) was selected as a Tier-II code is a powerful code, the branch extension operations waveform in IRIG-106 Aeronautical Telemetry (ARTM) required for the transition between states are more complex standard in 2004 because of its superior spectral efficiency than that in the conventional trellis code. and error performance as compared to the legacy PCM/FM (Pulse Code Modulation/Frequency Modulation) wave- In this paper, we propose a design procedure based on forms [2]. super trellis of the merged encoder combined by multiple trellis encoder and CPE (Continuous Phase Encoder). Sys- In these multi-h CPM schemes, a properly chosen cy- tem performance in AWGN channel is analyzed based on clic set of modulation indices results in delayed merging of the minimum squared Euclidean free distance dominated neighboring phase trellis paths and therefore, provides by the minimal error event of the merged encoder. a larger minimum Euclidean distance than conventional single-h CPM schemes. The error performance for multi-h The remainder of this paper is organized as follows. CPM schemes on AWGN and flat fading channels were System model and the merged encoder constructed by presented in [3] and [4] respectively. However, most multi- multiple trellis encoder and CPE are given in Section 2. RADIOENGINEERING, VOL. 21, NO. 4, DECEMBER 2012 1131 Super trellis structure of the merged encoder is presented in and Stüber showed that the tilted phase representation of Section 3. Section 4 presents the design procedure of the multi-h CPM signals results in a periodic phase trellis in multiple trellis coded multi-h CPM system to achieve supe- [15], which enables the decomposition of multi-h CPM rior performance. Performance analysis is described in modulator as a cascade of a periodic recursive CPE and Section 5 and numerical results and discussion are given in a memoryless modulator. Section 6. Finally, conclusions are drawn in Section 7. First, the complex envelope of a multi-h CPM waveform with H different modulation indices is given by E 2. System Model (2) stjtt,exp,,0UU 0 T 2.1 Multiple Trellis Coded Multi-h CPM where E is the energy per symbol, 0 is a constant phase System Description term which can be ignored if phase synchronization is assumed. The time varying phase t,U , also called the The multiple trellis coded multi-h CPM system con- excess phase, is defined as sidered in this paper is shown in Fig. 1. At epoch n, the source information bit bn is input to a multiple trellis en- . (3) thUMqtnTt,2U π n 21n ,0 H coder whose output is code symbol Un from the M -ary n0 alphabet 0,1, ,M 1 . The code symbol Un is fed into Here () denotes the modulo H operation and the modula- H CPE whose output n is input to memoryless modulator tion index for each symbol interval is chosen cyclically (MM) which completely specifies the physical phase and from the set {h0,h1,…,hH – 1}, q()is the phase shaping func- therefore, completely specifies the output signal. The two tion. In order to ensure a finite number of states in the sub-encoders construct the merged encoder, and the analy- decoding trellis, modulation indices are restricted to the set sis of overall error event is based on such merged encoder. of rational numbers with a common denominator P, such that hi = Ki/P. The denominator P is always chosen such that gcd(K0, K1,…, KH – 1, P) = 1, where gcd() denotes the bn U greatest common divisor of these arguments. n n st The excess phase in (3) can be written as nL L1 thUMhUMqtniT,U π ini212iniπ 21 HH ii00 bˆ Uˆ ˆ n n n rt (4) Substituting = t - nT such that the interval t [nT, (n + 1)T] is equivalent to [0, T], (4) becomes Fig. 1. Transmitter/Receiver structure of multiple trellis coded multi-h CPM system. nL L1 nT,2U π hini Uini 4π h U q iT HH For AWGN channels, the received signal is ii00 LnL1 2πM 1 hUqiTMni ni π 1 h i rt st nt (1) H H ii00 (5) where n(t) is a Gaussian random process with power spec- Second, we define the tilted phase for tral density N0 W/Hz. The received signal is fed into t [nT, (n + 1)T] as matched filter whose output ˆn is input to Viterbi de- n1 modulator. The output of demodulator Uˆ is input to πM 1htnTn n H . ttMh,,UUπ 1 i ˆ H Viterbi decoder whose output bn is the recovered informa- i0 T tion bits. (6) Note that the sum of second and third term is actually 2.2 Decomposition of Multi-h CPM the negative of the lowest trajectory of the excess phase trellis. When written in terms of , where [0, T], and Rimoldi presented a tilted phase description of the substituting nT,U from (5), we get single-h CPM signals in [14] by shifting the carrier fre- nL L1 quency from f to f = f - h(M – 1)/2T, where is the 0 1 0 h nT,2U π hini Uini 4π h U q iT HH modulation index, T is the symbol interval. The resulting ii00 LL11 πMh1 n modified phase (tilted phase) trellis is time invariant, and H 2πMhqiTMh1 ni π 1 ni HH the single-h CPM modulator can be interpreted as a cas- ii01T cade of a linear CPE and a memoryless modulator. Saleem (7) 1132 LIU XIAN, LIU AIJUN, ZHANG BANGNING, PAN KEGANG, DESIGN OF MULTIPLE TRELLIS-CODED MULTI-H CPM … All the terms in nT,U are periodic in n and 3. Super Trellis Structure of Merged depend only on the time translated variable . So the tilted phase transformation turns out to be a periodic function of Encoder time with period H . Because of the existence of periodicity in the state On the time interval, t [nT, (n + 1)T], (7) can be machine, the trellis structure of CPE decomposed by multi- expressed as h CPM becomes complex, and much more difficult to draw with several indices. For simplicity, the super trellis struc- nL L1 ture with single-h CPM is presented and the minimum thURtnThUqtniT,2U πiniini4π HH ii00 error event based on super trellis is analyzed firstly, then (8) the analysis will be extended to the design of multiple trellis coded multi-h CPM system. where Rt is a data independent term, and is defined as LL11 πMht1 n 3.1 Construction of Super Trellis H RtMhπ 12 ni π MhqiT 1 ni HH ii10T 00 00 0 0 0 (9) 21 21 11 The tilted phase is generated from the input symbols 30 CPE+MM 03 03 22 22 by the CPE of (8).
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