Learning While Transmitting: Pilotless Polar Coded Modulation for Short Packet Transmission

1 Learning While T ransmitting: Pilotless P olar Co ded Mo dulation for Short P ac k et T ransmission Geon Choi, Memb er, IEEE and Namy o on Lee, Senior Memb er, IEEE A bstr act —Short pac k ets mak e c hannel learning ex- p ensiv e. In pilot-aided transmission ( P A T ), a non- negligible fraction of the pac k et is consumed b y pi- lots, creating a direct pre-log loss and tightening the reliabilit y margin needed for ultra-reliable low-latency comm unication. W e prop ose a pilot-free p olar-co ded framew ork that replaces explicit pilots with c o de d pi- lots . The message is carried by tw o p olar-co ded seg- men ts: a quadrature phase shift keying ( QPSK ) segmen t that is deco dable without c hannel state information ( CSI ), and a higher-order quadrature amplitude mo d- ulation ( QAM ) segmen t that pro vides high sp ectral eciency . The receiv er emplo ys hybrid de c o ding : it rst join tly infers CSI during successiv e-cancellation-based deco ding of the QPSK segment by exploiting QPSK phase-rotation inv ariance together with polar frozen- bit constrain ts; the deco ded QPSK symbols then act as implicit pilots for coherent detection and deco ding of the QAM segment. The split also mak es rate adap- tation practical b y conning the symmetry/ frozen- bit requiremen ts for phase resolution to the QPSK segmen t, enabling puncturing and shortening without breaking the pilot-free mec hanism. F or multi-block fad- ing, we optimize the split and co de parameters via den- sit y ev olution with Gaussian approximation ( DEGA ); for higher-order mo dulation, we use bit-interlea ved co ded modulation capacity approximation to obtain equiv alen t channel parameters. Incorp orating channel- estimation error v ariance into the DEGA-based analy- sis, sim ulations o v er practical multi-block block-fading c hannels sho w gains up to 1 . 5 dB o v er P A T in the short- blo c klength regime. I. Introduction Ultra-reliable lo w-latency comm unication ( URLLC ) c hanges the op erating p oint of wireless design. When the pac k et is short and the error probability target is stringent, there is no longer room to aver age out ineciencies: every c hannel use sp ent on ov erhead directly reduces the infor- mation that can b e delivered within the latency budget. This regime is increasingly central in emerging closed-loop applications —rob ot control for ph ysical articial in telli- gence, autonomous driving, and industrial automation — where the con trol cycle itself allocates only a small num ber of channel uses p er up date. P olar co des, in tro duced b y Arıkan [ 1 ], ha ve b ecome a leading co ding solution for this short-blo ck, high-reliability regime. They are the rst family of c hannel co des pro v en to The authors are with the Department of Electrical Engineer- ing, Pohang Univ ersit y of Science and T echnology ( POSTECH ), Pohang 37673, South Korea ( e-mail: geon.choi@postech.ac.kr; nylee@postech.ac.kr ). ac hiev e the capacity of symmetric binary-input memory- less channels under low-complexit y successive cancellation ( SC ) deco ding, and their structure supports ecien t list deco ding and practical implementations. These adv an- tages led to their adoption in the 5G New Radio ( NR ) standard [ 2 ]. Beyond standardization, a large bo dy of w ork has improv ed nite-length p erformance through pre- transform and concatenation ideas, including cyclic redun- dancy chec k ( CRC ) -aided deco ding, dynamic frozen bits, p olarization-adjusted conv olutional ( P AC ) co de-type con- structions, and learning-assisted metho ds [ 3 ] – [ 9 ]. In short, from a c o ding p ersp ectiv e, the ingredients for URLLC are strong [ 10 ] – [ 14 ]. The more subtle b ottleneck is often not coding, but c oher enc e . Consider the basic blo ck-fading picture: within a coherence interv al of T c hannel uses, the c hannel is ( ap- pro ximately ) constan t but unknown. Pilot-assisted trans- mission ( P A T ) spends τ of these T uses on pilots, leaving only T − τ uses for data [ 15 ] – [ 18 ]. The o v erhead creates a fundamen tal pre-log penalty: even b efore accoun ting for estimation error, the eective pa yload is reduced by a factor (1 − τ / T ) . When T is small —as in fast fading, high mobility , or short pack ets—this p enalty is no longer negligible. Moreov er, in the very regime where reliability m ust b e highest, imp erfect channel estimates can turn the remaining data sym bols into a mismatch problem, comp ounding the loss. This motiv ates pilot-free ( or pilot- minimized ) designs that embed channel inference into the co ded transmission itself, aiming to approach coherent p erformance without explicit pilot o v erhead [ 19 ] – [ 24 ]. A. R elate d W orks Pilot-free p olar-co ded comm unication has developed along several directions, each trading o generality , com- plexit y , and robustness. A rst direction uses c o de c onstr aints as implicit r ef- er enc es . Y uan et al. [ 25 ] prop osed a t wo-stage sc heme that eliminates explicit pilots by exploiting frozen-bit constrain ts for join t c hannel estimation and deco ding: candidate phase rotations are ev aluated through successiv e cancellation list ( SCL ) deco ding and lik eliho o d compar- ison. The approach ac hieves near-coheren t p erformance for quadrature phase shift k eying ( QPSK ), but scales p o orly to multi-block fading b ecause the n um b er of phase com binations grows rapidly , and extending to higher- order mo dulation is challenging due to amplitude-related am biguities. 2 A second direction resolv es phase ambiguit y b y en- for cing e quivarianc e . Phase-equiv ariant p olar co des freeze rotation-discriminating bits to lev erage co de automor- phisms and disambiguate unknown phase without pi- lots [ 26 ]. While eective for QPSK and 16-quadrature am- plitude mo dulation ( QAM ), these constructions are fragile under the rate-matching op erations required in practice: puncturing/shortening can disrupt frozen-bit patterns and symmetry , undermining phase resolution. A third direction introduces emb e dde d pilot p ositions . Systematic p olar codes with selected p ositions used for c hannel estimation were dev elop ed in [ 27 ], enabling opera- tion in m ulti-carrier and Doppler c hannels. Ho w ever, these reserv ed p ositions ultimately do not carry data, so ov er- head is not fully conv erted into co ding gain. Hybrid p olar enco ding [ 28 ] creates dynamic pilots by mixing systematic and non-systematic segmen ts, enabling blind estimation follo w ed b y coheren t decoding; current formulations are mainly tailored to real-v alued fading with binary phase shift k eying ( BPSK ) and do not directly address unknown phase osets in complex baseband channels. These w orks rev eal a recurring design tension: a practi- cal pilot-free scheme must ( i ) supp ort higher-order mo du- lation while resolving phase ambiguities, ( ii ) remain com- patible with exible rate-matching and arbitrary co de lengths, and ( iii ) keep complexity manageable in m ulti- blo c k fading. Existing metho ds typically satisfy only a subset of these requirements. B. Overview of A ppr o ach W e prop ose a pilot-free p olar-co ded transmission frame- w ork that resolv es the ab ov e tension b y separating t wo roles that are usually coupled: channel anchoring and sp e ctr al eciency . The core idea is a c o de-splitting arc hi- tecture: a QPSK-mo dulated segment provides a constant- amplitude anchor that enables blind c hannel inference using the algebraic structure of p olar co des, while a higher-order QAM segment carries the bulk of the in- formation at high sp ectral eciency . The receiv er rst deco des the QPSK segmen t to reco ver the relev ant c hannel state information ( CSI ), then reuses the decoded bits as implicit pilots to enable coherent detection and deco ding of the QAM segmen t. Importantly , b y conning the phase- resolution constrain ts to the QPSK portion, the ov erall design remains compatible with practical rate-matching. C. Contributions Our main contributions are summarized as follows: • Code-splitting architecture: W e develop a pilot- free polar-co ded framework that partitions each code- w ord into a QPSK segmen t for blind c hannel estima- tion and a higher-order QAM segmen t for sp ectral eciency , supporting arbitrary blo ck lengths. • Hybrid deco ding with implicit pilots: W e design an ecient tw o-phase receiv er that rst resolves CSI through deco ding of the QPSK segmen t, then uses the reco vered bits as implicit pilots for coherent QAM detection and deco ding. • Rate-matc hing integration: The split lo calizes the delicate frozen-bit/ symmetry constrain ts to the QPSK component, enabling puncturing/ shortening ( e.g., quasi-uniform rate-matching ) without breaking phase-am biguit y resolution. • Multi-block fading optimization: W e pro vide a systematic optimization framew ork for multi- ple fading blo cks by extending densit y ev olution with Gaussian appro ximation ( DEGA ) to the code- splitting setting. F or higher-order modulation, we construct a binary input–additive white Gaussian noise ( BI-A W GN ) approximation c hannel and apply a bit-interlea ved coded mo dulation ( BICM ) capacity- matc hing metho d to determine equiv alen t channel parameters. • Analysis and v alidation: W e incorp orate channel- estimation error v ariance into the DEGA-based anal- ysis and v alidate the mo del through extensive simu- lations across diverse fading proles and mo dulation orders, demonstrating gains up to 1 . 5 dB o ver P A T in practical multi-block blo ck-fading channels. The remainder of this pap er is organized as follows. Section I I presents the channel mo del and p olar prelim- inaries. Section I I I describ es the prop osed co de-splitting arc hitecture and enco ding pro cedure. Section IV develops the hybrid deco ding algorithm with implicit pilot gener- ation. Section V presen ts the DEGA- and BICM-based optimization framew ork. Section VI reports sim ulation results. Section VI I concludes the pap er. I I. System Model and Back ground A. System Mo del W e consider a pack et of length N t transmitted ov er a blo c k-fading channel with coherence time N c . The pack et exp eriences B independent fading blo c ks, where B = N t / N c . The channel co ecient for the b th fading blo ck is denoted as h b = | h b | e j ϕ b . Let the transmitted pac ket b e represented as x = [ x 1 , x 2 , . . . , x B ] , where x b ∈ C N c for b ∈ { 1 , 2 , . . . , B } . The corresp onding received signal vector y = [ y 1 , y 2 , . . . , y B ] is giv en by y b = h b x b + v b , ( 1 ) where v b ∈ C N c denotes the additive white Gaussian noise ( A W GN ) vector with indep endent and iden tically distributed elements following v b ∼ C N ( 0 , σ 2 I ) . Throughout this pap er, w e denote the i th elemen t of v ector x as x i and the i th element of vector x k as x k,i . B. Pilot-Aide d Communic ations W e b egin b y briey reviewing the pilot-aided communi- cation framework. In such systems, the transmitted sym- b ol v ector x ∈ C N t is partitioned into B + 1 comp onents: pilot symbols x ( b ) p ∈ C N ( b ) p for the b th blo ck fading, and 3 data symbols x d ∈ C N d , where P B b =1 N ( b ) p + N d = N t . The pilot symbols, typically pseudo-random QPSK sequences of length N ( b ) p , are kno wn a priori to the receiver and facilitate c hannel estimation. The data symbols, of length N d , conv ey K information bits. The pilot symbols are inserted b et ween the data sym b ols, so that symbol vector x ( b ) p exp eriences the b th blo ck fading channel. Assuming N s -ary QAM with constellation X N s , eac h sym b ol x i ∈ X N s represen ts n s = log 2 ( N s ) consecutiv e co dew ord bits. 1 The constellation is normalized such that 1 N s P x ∈X N s | x | 2 = 1 , and Gray lab eling is employ ed throughout [ 30 ]. Giv en a total pack et size N c , pilot length N p , and modu- lation size N s , the eectiv e codeword length is M = n s N d . A ccordingly , the transmitter enco des K information bits m ∈ F K 2 in to a codeword c ∈ F M 2 with nominal code rate R = K / M . How ever, due to the pilot ov erhead, the eectiv e co de rate b ecomes R p eff = K N c = K N p + N d = (1 − α ) n s R, ( 2 ) where α = N p / N c denotes the pilot ov erhead fraction. C. Polar Co des and R ate-Matching P olar co des are dened by a transform matrix F N = F ⊗ n 2 where N = 2 n and F 2 =  1 0 1 1  . The co de is charac- terized by an information index set I ⊆ { 0 , 1 , . . . , N − 1 } , while the frozen index set F is the complement of I . The generator matrix consists of the ro ws of F N indexed by I . Polar co des achiev e arbitrary co de rates R = K / N by setting |I | = K . The inheren t constrain t of Arıkan ’ s construction limits p olar co de lengths to p ow ers of t w o, i.e., M = 2 n . T o supp ort arbitrary codeword lengths M 6 = 2 n , rate- matc hing techniques are employ ed: • Shortening and Puncturing: Used to reduce co de length from a larger mother co de of size N > M . • Extension and Rep etition: Used to increase co de length from a smaller mother co de of size N < M . In 5G NR systems, quasi-uniform puncturing and short- ening with sub-blo ck interlea ving are employ ed for rate- matc hing, ensuring compatibility while maintaining go o d p erformance [ 2 ]. Shortening remov es the last N − M in terlea v ed co deword bits, while puncturing remov es the rst N − M bits. Rep etition extends the rst M − N in terlea v ed c odeword bits. I I I. Pilot-Free Polar-Coded Modula tion In this section, we presen t a rate-matching, pilot-free p olar-co ded transmission framework designed for ecient short-pac k et comm unication o ver fading channels. The core of our approach is a co de-splitting architecture that 1 W e assume the codeword bits are interlea ved appropriately for the bit-in terlea ved co ded modulation ( BICM ) scheme [ 29 ]. 𝐅 𝑵 ! 𝐅 𝑵 " Rate - matching Rate - matching QPSK 𝑁 " - QAM Po la r - transform Modulation 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 m → F K 2 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 m 1 → F K 1 2 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 c 0 → F M 0 2 AAAFXHicbdTLb9MwHAdwb7QwNgYbQ1yQkMVA4hBVSdnG4LT3+9GN9bGtpXJct7OaxJHtAJ2VK1f+Ea7wF3DmzoW/BaetZHcjUqWfPv7m50cq+3FAhXTdP2Pjd3L5u/cm7k9OPZh++Ghm9nFFsIRjUsYsYLzmI0ECGpGypDIgtZgTFPoBqfrd9Wy8+olwQVl0JnsxaYSoE9E2xUhqas5AVffbEKdND9ZpBOshkle+r7bSZvGjOmx6aXNm3i24/QfeLrxhMb8y9+39k9+/npeas7mX9RbDSUgiiQMkxKXnxrKhEJcUBySdrCeCxAh3UYdc6jJCIREN1d9KCl9pacE24/oXSdhX+w2FQiF6oa+T2VLFzbEM/zd2mcj2ckPRKE4kifBgonYSQMlgdi6wRTnBMujpAmFO9VohvkIcYalPb2SWrLdkLBAjW1GSdq8HklUB9TniPRUzQbOTplHHgRgF2IGIc/ZZFEIiUTo50qLLJPmSWYu09ffsb14xjqIOFVepOt1eS1VxcdHx3hUdNx2NdTghkYl5C0tO0V12lm/m4oTH+isMU+6SAxfeOFDHh/PW8apS/b+BXq1aTdMUDnjN4jXD6xavG96weEPzsPemxZsmvWXxluFti7cN71i8Y3rvWrxr0nsW7xnet3jf8IHFB6b3ocWHJn1k8ZHhY4uPDZcsLpneJxafmPSpxaeGP1j8wfCZxWemd9nisklXLK4YrlpcNVyzuGZ6n1t8btIXFl8Yvrb4WrO+V7ybt8jtolIseEuFxRN9wRTB4JkAz8AL8Bp44C1YATugBMoAg6/gO/gBfub+5nP5qfz0IDo+NnxnDow8+af/AMnf4/I= c 1 → F M 1 2 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 x 0 → X N 0 c N s 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 x 1 → X N 1 c 4 Π Fig. 1. Enco ding structure of the proposed code-splitting metho d. Information bits are partitioned into t wo sub-messages that are independently encoded and modulated using dierent schemes. Π is an in terlea v er for BICM [ 29 ]. separates the transmission into m ultiple SC-deco dable 2 segmen ts. Consider the transmission of K information bits o ver N t c hannel uses, comprised of B indep endent fading blocks with coherence time N c sym b ols. The prop osed scheme is parameterized by { N b c , K b } B b =0 , satisfying the constraints P B b =0 N b c = N t and P B b =0 K b = K . The o v erall enco ding structure, illustrated in Fig. 1, follows a multi-path design in which the information message m ∈ F K 2 is partitioned in to B + 1 sub-messages, m b ∈ F K b 2 , whic h are pro cessed indep enden tly as follow: 3 • Component 0: The sub-message m 0 is enco ded with a polar code C 0 , resulting in a co deword c 0 with length M 0 = N 0 c log 2 N s . W e use conv en tional rate-matc hing tec hniques. F ollowing interlea ving, the co deword bits are mo dulated with N s -QAM symbol vector x 0 of length N 0 c . The symbol vector x 0 is further divided in to x 0 = [ x 0 , 1 , x 0 , 2 , . . . , x 0 ,B ] . • Components i ( i = 1 , 2 , . . . , B ) – co ded-pilot: Eac h sub-message m i is enco ded using p olar co de C i with the constrain t { N i − 2 , N i − 1 } ⊆ F i , pro- ducing co deword c i of length M i = 2 N i c . W e use con v en tional rate-matc hing tec hniques, but exclude shortening. The co dew ord is then QPSK-mo dulated to generate symbol vector x i of length N i c , whic h serv es as the channel estimation component for the i th fading blo ck with c hannel coecient h i . The complete transmitted signal consists of co ded-pilot comp onen ts x i for c hannel estimation and higher-order mo dulated sub-v ectors x 0 ,i for data transmission, arranged as x = [ x 1 , x 0 , 1 , x 2 , x 0 , 2 , . . . , x B , x 0 ,B ] . Consequently , each sub-v ector pair [ x b , x 0 ,b ] exp eriences the b th fading blo ck with channel co ecient h b . The eectiv e co de rate of the prop osed pilot-free p olar- co ded transmission scheme is giv en by R e = K M = n s α 0 R 0 + B X b =1 2 α b R b , ( 3 ) where M = P B b =0 M b , α b = N b c / N t and R b = K b / M b . This eective co de rate formulation naturally includes 2 W e use the term SC-decodable to indicate co des able to be decoded with SC t ypes of decoder with low complexity 3 The framew ork naturally accommodates outer codes suc h as CRC by treating the concatenated message [ m , p crc ] as the information sequence. 4 Mag. Estim . 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 y 1 → C N 1 c 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 y 0 → C N 0 c Π !" Demapper Decoding Re - encoding Channel Estim . Demapper Decoding Phase Estim . 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 ˆ m 1 → F K 1 2 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 ˆ x 1 AAAFYHicbdTJbhMxGAdwtzS0lKUt5QQSsmiROIyimdCWwqn7vqRttrYTIo/jJFZmk+0BUmtegAuvwhXOnLlz5UnwJJHstIwU6dPP//m8TGQv9ikXtv1nbPzeRO7+5NSD6YePHj+ZmZ17WuFRwjAp48iPWM1DnPg0JGVBhU9qMSMo8HxS9bqb2Xj1E2GcRmFJ9GJSD1A7pC2KkVDUmF10O0hI12vBIG3Y0KUhdAMkOp4nd9JG4aM8bNhpY3bBztv9B94tnGGxsDb/7cOz379eFhtzE4tuM8JJQEKBfcT5tWPHoi4RExT7JJ12E05ihLuoTa5VGaKA8LrsbyeFr5U0YSti6hcK2FfzDYkCznuBp5LZUvntsQz/N3adiNZqXdIwTgQJ8WCiVuJDEcHsbGCTMoKF31MFwoyqtULcQQxhoU5wZJast4gin49sRQravRlIVvnUY4j1ZBxxmp02DdsWxMjHFkSMRZ95PiACpdMjLbqRIF8ya5KW+qb9zcuIobBNeSeV57sbqSwsL1vO+4Jlp6OxNiMk1DFnacUq2KvW6u1cnLBYfYVhyl6x4NJbC6r4cF4Xr0vZ/xuo1cr1NE3hgDcM3tC8afCm5i2DtxQPe28bvK3TOwbvaN41eFfznsF7uve+wfs6fWDwgeZDgw81Hxl8pHsfG3ys0ycGn2g+NfhUc9Hgou59ZvCZTp8bfK75wuALzSWDS7p32eCyTlcMrmiuGlzVXDO4pntfGnyp01cGX2m+MfhGsbpXnNu3yN2iUsg7K/nlM3XBFMDgmQIvwCvwBjjgHVgDe6AIygCDr+A7+AF+TvzNTeVmcnOD6PjY8J15MPLknv8De5bluQ== ˆ m 0 → F K 0 2 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 | ˆ h | 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 ˆ ω o ! set 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 | ˆ h | e j ˆ ω offset 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 1 N 1 c N 1 c → 1 ! i =0 y 1 ,i ˆ x ↑ 1 ,i , Fig. 2. Deco ding o w chart sho wing sequential processing of QPSK and QAM p ortions with blind channel estimation. the pilot-aided case as a special instance. By selecting K b = 0 and N b c = N ( b ) p for b ≥ 1 , w e recov er the eective co de rate for conv entional pilot-aided transmission, i.e., R eff = R p eff . Thus, the proposed framew ork generalizes the standard pilot-aided comm unication system. F urthermore, b y appropriately c hoosing the parameters ( K b , N b c ) for b ≥ 0 , the eectiv e co de rate can b e exibly optimized to achiev e a desired reliability target or spectral eciency constrain t. In addition, the prop osed method can ac hieve higher sp ectral eciency compared to pilot-aided communica- tion. When the deco ding of y b ≥ 1 is successful, the receiver can leverage the deco ded information ˆ x b ≥ 1 for enhanced c hannel estimation. Since these successfully deco ded sym- b ols serv e as kno wn reference signals, they eectiv ely func- tion as pilot sym bols while maintaining channel estima- tion qualit y equiv alent to that of conv entional pilot-based metho ds. Ho wev er, unlike traditional pilot sc hemes where dedicated symbols carry no information, our approac h enables x b ≥ 1 to simultaneously transmit data bits and pro vide channel estimation capability . IV. Hybrid Decoding In this section, we present a hybrid deco ding metho d. F or simplicit y , w e restrict our atten tion to the single blo ck- fading channel case with B = 1 . The deco ding pro cedure op erates in t wo stages. First, the co ded-pilot comp onent y 1 is deco ded in a blind fashion, yielding the decoded message ˆ m 1 . This message is then re-enco ded to generate a pilot sequence for estimating the channel coecient h 1 . Second, using the estimated channel, the higher-order mo dulated comp onen t y 0 is deco ded in a coheren t manner. A. Blind De c o ding for x 1 W e now describ e the deco ding pro cess for m 1 , whic h lev erages both the constant-amplitude prop ert y of QPSK mo dulation and the algebraic structure of p olar codes. As illustrated in Fig. 2, the prop osed blind deco ding pro ce- dure consists of three main stages: ( i ) channel magnitude estimation, and ( ii ) c hannel phase estimation. 1 ) Channel Magnitude Estimation: F or constan t- amplitude mo dulation such as QPSK where | x 1 ,i | 2 = 1 , c hannel magnitude estimation leverages the relationship | y 1 ,i | 2 = | h | 2 | x 1 ,i | 2 + | v 1 ,i | 2 + 2 Re ( h ∗ x ∗ 1 ,i v 1 ,i ) . ( 4 ) Since the noise components v 1 ,i are i.i.d. with zero mean, a v eraging ov er all received symbols yields 1 N 1 c N 1 c − 1 X i =0 | y 1 ,i | 2 ≈ | h | 2 + σ 2 , ( 5 ) pro viding the magnitude estimate | ˆ h | = v u u u t max   0 , 1 N 1 c N 1 c − 1 X i =0 | y 1 ,i | 2 − σ 2   , ( 6 ) where the max ( · , 0) op eration ensures non-negative esti- mates in the presence of estimation errors. 2 ) Channel Phase Estimation: Channel phase estima- tion is more challenging due to the rotational symmetry inheren t in QPSK mo dulation. The c hannel phase e j ϕ ro- tates all receiv ed sym bols by ϕ radians. How ever, the four- fold rotational symmetry of QPSK creates an inherent am biguit y in phase estimation, as rotations by multiples of π /2 cannot b e distinguished without additional infor- mation. Due to this phase ambiguit y , the estimated phase ˆ ϕ can b e decomp osed as ˆ ϕ = m π 2 + ˆ ϕ oset , ( 7 ) where m ∈ { 0 , 1 , 2 , 3 } represents integer am biguit y and ˆ ϕ oset ∈ [0 , π /2) is the fractional oset. W e adopt the estimation strategy from [ 26 ], inv olving t wo steps: esti- mating the fractional oset without c hannel deco ding, then resolving the integer ambiguit y using the p olar code structure. F ractional Phase Oset Estimation: F or fractional oset estimation, the metho d employs the Viterbi and Viterbi phase estimation ( VVPE ) algorithm [ 31 ]. The VVPE algorithm is designed for M-PSK mo dulation and op erates b y rotating all received symbols to a com- mon reference p oint, thereb y eliminating the mo dulation- dep enden t phase v ariations. F or QPSK ( M = 4) , the k ey insight is that raising eac h sym b ol to the fourth p ow er eliminates the data-dep endent phase v ariations: 4 ϕ = 4 m π 2 + 4 ϕ oset = 4 ϕ oset , ( 8 ) since 4 mπ /2 = 2 mπ is a m ultiple of 2 π . The fractional phase oset is estimated as [ 32 ] ˆ ϕ oset = 1 4 tan − 1    P N 1 c − 1 i =0 Im  y 4 1 ,i | y 1 ,i | 3  P N 1 c − 1 i =0 Re  y 4 1 ,i | y 1 ,i | 3     − π 4 . ( 9 ) In teger Phase Ambiguit y Resolution: T o resolve the in teger ambiguit y m , the approach exploits the al- gebraic properties of p olar co des under phase rotation. Consider a p olar codeword c = [ c 0 , c 1 , c 2 , c 3 , . . . , c N − 1 ] . 5 Phase rotations by multiples of π /2 transform the co de- w ord according to sp ecic patterns: c 0 = [ c 0 , c 1 , c 2 , c 3 , . . . , c N − 2 , c N − 1 ] , ( 10 ) c π /2 = [¯ c 1 , c 0 , ¯ c 3 , c 2 , . . . , ¯ c N − 1 , c N − 2 ] , ( 11 ) c π = [¯ c 0 , ¯ c 1 , ¯ c 2 , ¯ c 3 , . . . , ¯ c N − 2 , ¯ c N − 1 ] , ( 12 ) c 3 π /2 = [ c 1 , ¯ c 0 , c 3 , ¯ c 2 , . . . , c N − 1 , ¯ c N − 2 ] , ( 13 ) where ¯ c = c ⊕ 1 ( binary addition ) and c ϕ denotes the co dew ord obtained after phase rotation by ϕ . These transformations can b e decomp osed in to a p er- m utation follow ed by a translation. The p ermutation σ is dened as σ : [ c 0 , c 1 , c 2 , c 3 , . . . ] 7→ [ c 1 , c 0 , c 3 , c 2 , . . . ] , ( 14 ) whic h sw aps adjacent pairs of bits. The rotated codewords can then b e expressed as c π /2 = σ ( c 0 ) ⊕ [1 , 0 , 1 , 0 , . . . , 1 , 0] , ( 15 ) c π = σ ( c 0 ) ⊕ [1 , 1 , 1 , 1 , . . . , 1 , 1] , ( 16 ) c 3 π /2 = σ ( c 0 ) ⊕ [0 , 1 , 0 , 1 , . . . , 0 , 1] , ( 17 ) where ⊕ denotes bit-wise X OR op eration. A ccording to [ 33 , Denition 6 ], the p erm utation σ cor- resp onds to an ane transformation of the monomial represen tation that preserv es the p olar co de structure, making it an automorphism for p olar co des following the standard partial order. 4 By our assumption that { N − 2 , N − 1 } ⊆ F , these p ositions are normally set to zero in the original co deword c 0 . How ever, under phase rotation, these p ositions take on sp ecic v alues that uniquely iden tify the rotation angle: c π /2 : [ u N − 2 , u N − 1 ] = [1 , 0] , ( 18 ) c π : [ u N − 2 , u N − 1 ] = [0 , 1] , ( 19 ) c 3 π /2 : [ u N − 2 , u N − 1 ] = [1 , 1] . ( 20 ) These relationships enable in teger phase am biguit y reso- lution b y treating the last tw o frozen p ositions as informa- tion bits during deco ding, then using the deco ded v alues to determine the rotation angle. B. Coher ent De c o ding for x 0 The blind deco ding architecture enables the receiver to estimate the c hannel without explicit pilots b y treating the deco ded codeword ˆ c 1 as an eectiv e sequence of pilot sym- b ols. Let ˆ x 1 denote the corresp onding QPSK-mo dulated sym b ol vector. The receiver computes the maximum lik e- liho o d channel estimate as ˆ h = 1 N 1 c N 1 c − 1 X i =0 y 1 ,i ˆ x ∗ 1 ,i , ( 21 ) with estimation v ariance σ 2 ˆ h = σ 2 / N 0 c . 4 The p ermutation σ corresp onds to ane transform p 7→ Ap + b of vector of monomials p = [ p 0 , p 1 , . . . , p log 2 N − 1 ] with A = I and b = [0 , 0 , . . . , 0 , 1] ⊤ . The translation corresp onds to ipping some message bits as p er ( 18 ) - ( 20 ) . This c hannel estimate is then used for soft demo dulation of the remaining data symbols. Specically , for each bit k of sym bol i , the receiv er computes the log-likelihoo d ratio ( LLR ) as LLR i,k = log P x ∈K k, 1 exp  − | y 0 ,i − ˆ hx | 2 σ 2 e  P x ∈K k, 0 exp  − | y 0 ,i − ˆ hx | 2 σ 2 e  , ( 22 ) where K k, 0 and K k, 1 denote the sets of constellation p oints with the k -th bit equal to 0 and 1, resp ectiv ely , and σ 2 e = σ 2 ˆ h + σ 2 captures the aggregate uncertaint y from b oth channel estimation and thermal noise. Armed with these bit-wise LLRs, the receiv er proceeds with p olar deco ding to recov er the information bits in m 0 . C. Complexity A nalysis W e analyze the computational complexity of the pro- p osed pilot-free scheme and compare it with con v en tional pilot-aided transmission. The complexity is ev aluated in terms of the n um b er of arithmetic op erations required for enco ding and deco ding. 1 ) Enc o ding Complexity: The enco ding complexit y of the prop osed scheme is iden tical to that of con v en- tional p olar codes. Each component co de C i requires O ( N i log 2 N i ) op erations for p olar enco ding, where N i is the mother co de length. Since encoding op erations are p erformed indep endently for eac h component, the total enco ding complexity is O ( N 0 log 2 N 0 + N 1 log 2 N 1 ) . ( 23 ) This complexity is comparable to con v en tional pilot-aided sc hemes using p olar codes with similar total co dew ord lengths. 2 ) De c o ding Complexity: The deco ding complexity con- sists of three main comp onents: blind channel estimation for the QPSK segment, blind decoding of C 1 , and coherent deco ding of C 0 . Channel Magnitude Estimation: Computing the c hannel magnitude estimate in ( 6 ) requires N 1 c additions for the summation, one square ro ot operation, and several scalar m ultiplications. The complexit y is therefore O ( N 1 c ) . F ractional Phase Oset Estimation: The VVPE algorithm in ( 9 ) inv olv es computing the fourth p ow er of eac h receiv ed sym bol, follow ed b y summation and arctan- gen t op erations. This requires O ( N 1 c ) multiplications and additions, plus one arctangent computation. The ov erall complexit y remains O ( N 1 c ) . Blind Deco ding of C 1 : Resolving the integer phase am biguit y requires one full polar deco ding pass of comp o- nen t co de C 1 . This introduces an additional O ( N 1 log 2 N 1 ) op erations. F ollowing successful decoding, w e m ust reco v er the original message by removing the translation v ector and inv erting the p ermutation σ . The translation remo v al requires N 1 binary XOR operations at the frozen p osi- tions { N 1 − 2 , N 1 − 1 } , while the p ermutation inv ersion is accomplished through the polar transform, requiring O ( N 1 log 2 N 1 ) op erations. Since the deco ding complexity 6 dominates these p ost-pro cessing steps, the total complex- it y for blind deco ding of C 1 is O ( N 1 log 2 N 1 ) . Channel Re-enco ding and Estimation: After de- co ding C 1 , the deco ded message ˆ m 1 is re-enco ded to gen- erate the pilot sequence ˆ x 1 . This re-enco ding step requires O ( N 1 log 2 N 1 ) operations. The maxim um likelihoo d chan- nel estimate in is then computed via correlation, requiring N 1 c complex multiplications and additions. The ov erall complexit y for this stage is O ( N 1 log 2 N 1 + N 1 c ) . Coheren t Deco ding of C 0 : Once the channel estimate ˆ h is obtained, comp onent co de C 0 is deco ded using stan- dard SC or SCL deco ding. The LLR computation in ( 22 ) for higher-order QAM requires ev aluating exponentials o v er all constellation p oints for each received sym b ol. F or N s -QAM, each symbol requires O ( N s ) op erations, resulting in O ( N 0 c N s ) complexity for demapping. The sub- sequen t p olar decoding requires O ( N 0 log 2 N 0 ) operations. F or practical mo dulation orders ( N s ≤ 64 ), the deco ding complexit y dominates, yielding an ov erall complexity of O ( N 0 log 2 N 0 ) for this stage. 3 ) Over al l Complexity: Com bining all stages, the total deco ding complexity of the prop osed scheme is O ( N 0 log 2 N 0 + N 1 log 2 N 1 ) . ( 24 ) Compared to pilot-aided transmission using a single p olar co de of length N P A T with complexity O ( N P A T log 2 N P A T ) , the prop osed sc heme in troduces an additional decoding pass for the QPSK segment. How ever, since N 1  N 0 in typical congurations optimized for sp ectral eciency , the additional complexity o verhead is mo dest. F urther- more, the elimination of explicit pilots allo ws for smaller total co deword lengths while achieving equiv alen t or su- p erior p erformance, p otentially osetting the complexity increase. F or SCL deco ding with list size L , the complexity scales linearly with L , yielding O ( L ( N 0 log 2 N 0 + N 1 log 2 N 1 )) . ( 25 ) In practical URLLC scenarios where small list sizes ( L ≤ 8 ) suce for the target reliability , this complexit y re- mains manageable and compatible with lo w-latency re- quiremen ts. V. Code Optimiza tion A. BICM channel mo del Consider the complex-A W GN channel W : X → C with conditional probability density function W ( y | x ) = 1 π σ 2 e − | y − x | 2 σ 2 ( 26 ) where X is the constellation symbol set with p o wer nor- malization P x ∈X | x | 2 = |X | , and the signal-to-noise ratio is given by SNR = 1/ σ 2 . Consider a constellation of size |X | = 2 n s . F or each sym b ol x i ∈ X , the corresp onding binary sequence b i = ( b i, 0 , b i, 1 , . . . , b i,n s − 1 ) ∈ F n s 2 is determined b y the bit- lab eling function ψ : F n s 2 → X such that ψ : b i 7→ x bi2de ( b i ) , ( 27 ) 𝐅 𝑵 ! 𝐅 𝑵 " Rate - matching Rate - matching QPSK 𝑁 " - QAM 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 m 0 → F K 0 2 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 m 1 → F K 1 2 Π … … 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 W 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 W Par al le l BI - AW GN s w ith differen t SNR 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 ˆ W (1) 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 ˆ W (0) 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 ˆ W (2) Density Evolution 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 P error 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 K 0 ,K 1 AAAF0HicbdTbbtMwGAfwdKwwBowNLkDixmJCGhBVSenKEELa+dQdunY9bEupHNftrCZxZLuwLooQXPKGvAGPgZNWstsRKdKnn//57NhR3NAjXFjWn8zMvdns/QdzD+cfPX6y8HRx6Vmd0wFDuIaoR1nThRx7JMA1QYSHmyHD0Hc93HD7W8l44xtmnNDgXAxD3PJhLyBdgqCQ1F78VWpbXx0uIDNBqW2PSvAFODTEDArKAujjdxFkPcf0SRC3I/kAeJ9kZaoUg8jhXVA9qcTAQR0qgONDce26fmTHK+V2OooZo0yOexg4OOTEo8Fn2cBKZ3wL2ovLVs5KL3C3sMfFsjG+yu2l2aLToWjg40AgD3J+ZVuhaMlFCoI8HM87A45DiPqwh69kmbwCb0XpZsXgjZQO6FIm70CAVPUnIuhzPvRdmUxehU+PJfi/sauB6K61IhKEA4EDNJqoO/CAoCDZedAhDCPhDWUBESNyrQBdQwaRkOczMUvSW1Dq8YlXiQTp344kqTziMsiGUUg5Sc6SBD0TIOghE0C54d95zscCxvMTLeTBTEmfCnyTWAd35TeUbkdEGQx6hF/HUWVvM47yq6um/SlvWvFkrMcwDlTMLhTNvLVmrk3nwgEL5bmMU1bRBIUPJpDx8bwO2oii9MOR64824lh+LClvarypeEvjLcXbGm9LHvfe0XhHpXc13lW8p/Ge4n2N91XvA40PVPpQ40PFJY1Lio80PlK9jzU+VukTjU8Un2p8qriscVn1PtP4TKUrGlcUVzWuKj7X+Fz1rmlcU+m6xnXFDY0bipsaN1XvC40vVPpS40vFtxrfSpZ/Gnv6v3K3qOdzdjG3elZYXi+M/zlzxivjtbFi2MZHY93YN8pGzUDG38xC5kXmZbaSvcn+yP4cRWcy42eeGxNX9vc/dRYIoQ== K ω 0 ,K ω 1 = arg min K 0 + K 1 = K SNR · ( P error → ω ; K 0 ,K 1 ) Fig. 3. Co de parameter optimization procedure. where bi2de ( b i ) = P n s − 1 j =0 b i,j 2 j con v erts the binary vector to its decimal equiv alent. Instead of op erating on symbols x i directly , we dene the equiv alent channel ¯ W as ¯ W ( y | b i ) = W ( y | ψ ( b i )) . This c hannel can b e decomp osed in to n s binary input channels ¯ W ( k ) : F 2 → C , with probabilit y densit y function ¯ W ( k ) ( y | b i,k ; b i,k − 1 , . . . , b i, 0 ) ( 28 ) = 1 Z X b ∈ F n s − i − 1 2 W ( y | ψ ([ b , b i,k , b i,k − 1 , . . . , b i, 0 ])) ( 29 ) where b i,k − 1 , . . . , b i, 0 represen t genie-aided bits corre- sp onding ϕ ( b i ) and Z is the normalization factor. In the BICM channel mo del, the channel ¯ W ( k ) is ap- pro ximated by ˜ W ( k ) : F 2 → C with densit y function ˜ W ( k ) ( y | b i,k ) ( 30 ) = 1 Z X b b ∈ F n s − i − 1 2 X b a ∈ F i 2 W ( y | ψ ([ b b , b i,k , b a ])) . ( 31 ) In terms of mutual information, the BICM c hannel induces some loss in ac hiev able rate: I ( W ) = I ( X ; Y ) = I ( B 0 , . . . , B n s − 1 ; Y ) ( 32 ) = n s − 1 X k =0 I ( B k ; Y | B 0 , . . . , B k − 1 ) ( 33 ) = n s − 1 X k =0 I ( ¯ W ( k ) ) ( 34 ) ≥ n s − 1 X k =0 I ( B k ; Y ) = n s − 1 X k =0 I ( ˜ W ( k ) ) ( 35 ) where X , Y , and B k are random v ariables corresp onding to x , y , and b i , I ( W ) is symmetric mutual information of c hannel W and I ( X ; Y ) is mutual information of random v ariables X and Y . According to [ 29 ], this appro ximation results in minimal loss of channel capacity when carefully designed bit-to-symbol mappings are emplo yed, suc h as Gra y mapping. Since w e use a demapp er that treats bits em b edded in the same symbol as indep endent, the BICM c hannel mo del is well-suited for our error probability analysis. 7 B. BI-A WGN A ppr oximation The parallel channel ˜ W ( k ) is non-Gaussian channel in general. In our co de design, we approximate the channel ˆ W ( k ) as binary input A W GN c hannel with conditional probabilit y density function ˆ W ( k ) ( y | b ) = 1 p 2 π σ 2 k e − ( y − (1 − 2 b )) 2 2 σ 2 k , ( 36 ) where σ 2 k is corresp onding Gaussian noise v ariance and equiv alent SNR is 1/ σ 2 k . T o compute the equiv alen t SNR, we match the m utual information of the tw o channels as I ( ˆ W ( k ) ) = I ( ˜ W ( k ) ) . ( 37 ) In tuitiv ely , this matc hing pro cess preserv es the av erage qualit y of LLR v alues for each c hannel output. In our equiv alent BI-A W GN framework, the original sym b ol- input channel is transformed into parallel bit-input chan- nels with Gaussian-approximated outputs, where the mu- tual information matching criterion ensures that the av- erage information conten t av ailable for deco ding is main- tained despite the non-Gaussian to Gaussian approxima- tion. This approach enables p olar co de construction and p erformance analysis. Fig. 3 summarizes the equiv alent c hannel mo del. C. Err or Pr ob ability A nalysis Supp ose the blo ck error probability of the i th com- p onen t code C i is P ( i ) error . An error occurs if at least one comp onen t co de fails. Thus, the ov erall error probability P error is computed as P error = 1 − (1 − P (0) error )(1 − P (1) error ) . ( 38 ) T o compute the error probability of each comp onent co de, w e adopt the density ev olution framework with Gaussian approximation, abbreviated as DEGA [ 34 ], [ 35 ]. Consider a p olar co de with mother co de length N . F or simplicit y , we ignore the in terleav er in the follo wing ex- planation. The i th co deword bit c i = c n s k + j is mo dulated into the k th transmitted sym b ol and transmitted ov er the j th equiv alen t BI-A W GN channel ˆ W ( j ) . The channel log- lik eliho o d ratio ( LLR ) for the i th co dew ord bit is dened as L ( c ) i = log P ( y k | c i = 0) P ( y k | c i = 1) . ( 39 ) The successiv e cancellation ( SC ) deco der uses the chan- nel LLRs ( L ( c ) 0 , L ( c ) 1 , . . . , L ( c ) N − 1 ) to compute the bit LLRs ( L ( u ) 0 , L ( u ) 1 , . . . , L ( u ) N − 1 ) , dened as L ( u ) i = log P ( y , u i − 1 0 | u i = 0) P ( y , u i − 1 0 | u i = 1) . ( 40 ) In our error analysis, we fo cus on the propagation from c hannel LLRs to bit LLRs, which is directly used to compute the co de block error probabilit y . W e b egin b y reviewing the LLR propagation pro cess during SC deco ding. 1 ) LLR Pr op agation Pr o c ess in SC De c o ding: SC decod- ing can b e understoo d as a message passing algorithm that follo ws a sp ecic sc heduling p olicy for processing elemen ts. Pro cessing Elemen ts: T o explain the deco ding pro- cess, w e in tro duce the processing element, which is a fundamen tal comp onent of the p olar transform. Each pro- cessing elemen t takes a pair of scalar inputs ( U 1 , U 2 ) and pro duces a pair of scalar outputs ( U 1 ⊕ U 2 , U 2 ) , where ⊕ denotes the X OR op eration. The p olar transform consists of log 2 ( N ) levels, and each lev el con tains N /2 pro cessing elemen ts arranged in parallel. Lab eling and Message Flow: W e assign the lab el ( d, j ) to the j th pro cessing element at the d th level, where d ∈ { 0 , 1 , . . . , log 2 ( N ) − 1 } and j ∈ { 0 , 1 , . . . , N /2 − 1 } . Eac h pro cessing elemen t handles t w o t ypes of messages with k ∈ { 1 , 2 } indicating the upp er or low er p osition: • Hard-decision bits: U d, → j,k ( inputs from left ) and U d j,k ( outputs to right ) • LLR messages: L d, ← j,k ( inputs from righ t ) and L d j,k ( outputs to left ) Hard-decision bits and LLR messages ow in opp osite directions through the p olar deco ding tree. Hard-decision bits propagate from the decoded information bits ( leftmost lev el ) bac k tow ard the channel ( rightmost level ), hence the → notation for inputs. Conv ersely , LLR v alues propagate from the c hannel outputs ( righ tmost level ) tow ard the information bits ( leftmost level ), hence the ← notation for LLR inputs. T o establish the corresp ondence b etw een pro cessing ele- men ts and codeword bits, we introduce a mapping function α d : ( j, k ) 7→ i that maps pro cessing elemen t position ( j, k ) at level d to co dew ord bit index i : α d ( j, k ) =  j 2 n − d − 1  2 n − d +  j −  j 2 n − d − 1  2 n − d − 1  + ( k − 1)2 n − d − 1 ( 41 ) The systematic interconnection b etw een pro cessing el- emen ts at dierent levels is dened by the permutation relationships: U d, → α − 1 d ( α d − 1 ( j,k )) = U d − 1 j,k ( 42 ) L d +1 , ← α − 1 d +1 ( α d ( j,k )) = L d j,k , ( 43 ) where α − 1 d denotes the in v erse mapping that conv erts co dew ord bit index back to pro cessing elemen t position at level d . Message Updates: Each pro cessing unit uses LLR messages from the righ t ( higher-indexed levels ) to compute LLR outputs tow ard the left ( lo w er-indexed levels ): L d +1 , ← α − 1 d +1 ( α d ( j, 1)) = L d j, 1 = f ( L d, ← j, 1 , L d, ← j, 2 ) , ( 44 ) L d +1 , ← α − 1 d +1 ( α d ( j, 2)) = L d j, 2 = g ( L d, ← j, 1 , L d, ← j, 2 , U d, → j, 1 ) , ( 45 ) where the functions f and g are dened as f ( L a , L b ) = log  1 + e L a + L b e L a + e L b  , ( 46 ) g ( L a , L b , u ) = L b + (1 − 2 u ) L a . ( 47 ) 8 Similarly , each pro cessing unit propagates hard-decision bits from the left ( lo w er-indexed levels ) tow ard the right ( higher-indexed levels ): U d − 1 , → α − 1 d − 1 ( α d ( j, 1)) = U d j, 1 = U d, → j, 1 ⊕ U d, → j, 2 , ( 48 ) U d − 1 , → α − 1 d − 1 ( α d ( j, 2)) = U d j, 2 = U d, → j, 2 . ( 49 ) The message up dates are p erformed only when all re- quired messages are av ailable. The hard-decision bits U n, → j,k are initialized as U n, → j,k =      0 , α n ( j, k ) ∈ F 0 , α n ( j, k ) ∈ I and L n j,k > 0 1 , α n ( j, k ) ∈ I and L n j,k < 0 . ( 50 ) The LLR v alues are initialized as L 0 , ← j,k = L ( c ) α 0 ( j,k ) . 2 ) Statistic al Char acterization: Instead of computing exact LLR v alues, DEGA mo dels the LLR messages as Gaussian random v ariables and trac ks their means µ d j,k and v ariances ( σ d j,k ) 2 : L d, ← j,k ∼ N ( µ d j,k , ( σ d j,k ) 2 ) . ( 51 ) In the BICM channel model with BI-A W GN appro xima- tion, we assume that each co deword bit c i is transmitted through an A WGN c hannel with dieren t c hannel quali- ties. Due to channel symmetry , the v ariance of the LLR is determined by its mean through the relation ( σ d j,k ) 2 = µ d j,k . This appro ximation greatly simplies the analysis while pro viding accurate estimates of the deco ding error probabilit y under SC deco ding. DEGA is equiv alent to p erforming SC deco ding under the assumption that all bits are frozen ( i.e., U d, → j,k = 0 ), but with mo died message passing rules and initializ ation pro cedures. W e rst introduce the equiv alen t c hannel mo del and present the initialization pro cess based on this mo del, follo w ed b y the mo died message passing rules for statistical tracking. Co dew ord Bits to Symbols: Consider the co deword c = ( c 0 , c 1 , . . . , c M − 1 ) with length M and a constellation of size 2 n s . Before sym b ol mapping, the co deword bits are t ypically interlea v ed as ( c 0 , c 1 , . . . , c M − 1 ) 7→ ( c π (0) , c π (1) , . . . , c π ( M − 1) ) , ( 52 ) where π ( · ) is the in terleaving permutation. The in terlea v ed bits are then group ed in to n s distinctiv e sets according to their bit p ositions within eac h symbol: B i = { c π ( i ) , c π ( n s + i ) , c π (2 n s + i ) , . . . , c π (( B − 1) n s + i ) } , ( 53 ) where B = M / n s is the n umber of transmitted symbols and i ∈ { 0 , 1 , . . . , n s − 1 } . The symbol mapp er ϕ : B 0 × B 1 × · · · × B n s − 1 → X tak es n s bits from the co deword to form the j th transmitted sym bol: ϕ : c j 7→ x bi2de ( c j ) , ( 54 ) where c j = ( c π ( j n s ) , c π ( j n s +1) , . . . , c π (( j +1) n s − 1) ) and the sym b ol is transmitted through the A W GN channel. BI-A W GN Channel Mo del: According to the BI- A WGN approximation mo del introduced in Section V-B, the bits in set B i are treated as b eing transmitted through an indep enden t equiv alent A W GN c hannel ˆ W ( i ) with noise v ariance σ 2 i . The input-output relationship of the i th equiv alent c hannel is ˆ W ( i ) : B i → R , x 7→ y , ( 55 ) with probability density function ˆ W ( i ) ( y | x ) = 1 p 2 π σ 2 i e − ( y − x ) 2 2 σ 2 i . ( 56 ) LLR Initialization: Using the equiv alent channels ˆ W ( i ) , we can compute the av erage LLR v alues from the c hannel observ ations. F or the zero-co dew ord transmission assumption, the a v erage LLR for bits transmitted through c hannel ˆ W ( i ) is 2 σ 2 i . T o establish the corresp ondence b e- t w een polar deco ding lab els and channel outputs, we initialize the LLR means at the rightmost level ( lev el n = log 2 ( N ) ). If the co dew ord bit corresponding to p osition ( j, k ) at lev el n is transmitted through channel ˆ W ( p ) , we set: µ n j,k = 2 σ 2 p , π ( α n ( j, k )) ∈ Z + p. ( 57 ) Statistical Message Up dates: The up date rules di- rectly follow from the SC deco ding message up date rules. A t eac h processing elemen t, equation ( 44 ) can be rewritten as tanh  1 2 L d +1 , ← α − 1 d +1 ( α d ( j, 1))  = tanh  1 2 L d, ← j, 1  tanh  1 2 L d, ← j, 2  . ( 58 ) Under the Gaussian approximation where L d, ← j,k ∼ N ( µ, 2 µ ) , w e deriv e the mean up date equations by taking exp ectations. The exp ectation can be expressed using the function ψ ( · ) , dened as ψ ( µ ) = E  tanh  X 2  , ( 59 ) where X ∼ N ( µ, 2 µ ) . This function can b e appro ximated as [ 36 ] ψ ( x ) ≈ ( 1 − exp ( − 0 . 4527 x 0 . 86 + 0 . 0218) , 0 < x ≤ 10 , 1 − p π x  1 − 10 7 x  e − x 4 , x > 10 . ( 60 ) F or the second output corresp onding to equation ( 45 ) , under the assumption that all bits are frozen ( i.e., U d, → j, 1 = 0 ), the update simplies to a direct addition of the input means. Therefore, at eac h processing element, the Gaussian distribution parameters are up dated according to: µ d +1 α − 1 d +1 ( α d ( j, 1)) = ψ − 1 ( ψ ( µ d j, 1 ) · ψ ( µ d j, 2 )) , ( 61 ) µ d +1 α − 1 d +1 ( α d ( j, 2)) = µ d j, 1 + µ d j, 2 , ( 62 ) σ d j,k = q 2 µ d j,k . ( 63 ) These up date rules enable ecien t trac king of LLR statistics throughout the p olar decoding tree without requiring actual message computations, signicantly re- ducing the computational complexity of error probability analysis. 9 3 ) Blo ck Err or Pr ob ability of SC De c o ding: The SC deco ding error probability for C i can b e approximated follo wing [ 34 ], [ 35 ] with the mo died channel initialization as: P ( i ) error = 1 − Y j,k : α n ( j,k ) ∈I i 1 − Q r µ n j,k 2 !! , ( 64 ) where I i denotes the set of information bit indices for C i , and µ n j,k represen ts the nal LLR mean at the leftmost lev el ( lev el n = log 2 ( N ) ) corresponding to the information bit decisions. The pro duct is tak en ov er all pro cessing elemen t positions ( j, k ) at level n whose corresp onding co dew ord bit indices α n ( j, k ) b elong to the information set I i . F or the proposed co de-splitting sc heme, w e compute the error probabilities with appropriate c hannel modications: • F or C 1 : Standard DEGA initialization at level 0 using equiv alent c hannel v ariances σ 2 p • F or C 0 : Mo died initialization at level 0 using σ 2 p + σ 2 ˆ h to account for channel estimation error v ariance σ 2 ˆ h The ov erall error probability is then computed as: P error = 1 − (1 − P (0) error )(1 − P (1) error ) . ( 65 ) This approac h captures the trade-o b etw een the im- pro v ed channel estimation quality ( due to the additional reference symbols from successfully deco ded C 1 ) and the reduced co de rate for C 0 , enabling accurate prediction of the ov erall system p erformance under the prop osed co de- splitting scheme for higher-order QAM. D. Par ameter Optimization for Co de-Splitting Design Giv en the total num b er of c hannel uses N c and the allo cation ( N 0 c , N 1 c ) where N 0 c + N 1 c = N c , w e assume that N 1 c sym b ols use QPSK mo dulation while N 0 c sym b ols emplo y higher-order QAM mo dulation. W e further assume that the rate-matching metho d and mother co de lengths N i for eac h comp onent co de are predetermined. Under these constrain ts, we present a systematic approach to determine the optimal information bit allo cation ( K 0 , K 1 ) for the comp onen t co des C 0 and C 1 , summarized in Fig. 3. The optimization pro cedure searc hes o ver all feasible com binations of ( K 0 , K 1 ) that satisfy the constraints K i ≤ N i and K 0 + K 1 = K , where K is the total num b er of information bits. F or each combination, we compute the BLER using the error probabilit y analysis framework presen ted in the previous section. The searc h algorithm proceeds as follo ws: First, w e dene a design SNR range [ SNR min , SNR max ] and start the search from SNR min with sequential SNR incremen ts. F or each SNR v alue, w e ev aluate the ov erall BLER for all feasible ( K 0 , K 1 ) combinations using the error probabilit y form ula P error = 1 − (1 − P (0) error )(1 − P (1) error ) . Among all com binations, we select the ( K ∗ 0 , K ∗ 1 ) that ac hiev es the minim um BLER. If this minimum BLER falls b elow the target threshold, we terminate the search and adopt the corresp onding parameters for co de de sign. Otherwise, we pro ceed to the next SNR v alue. The algorithm nds the optimal parameter pair ( K ∗ 0 , K ∗ 1 ) that achiev es the target BLER at the minimum required SNR under SC decoding. When cyclic redundancy chec k ( CR C ) co des with K crc bits are emplo yed for error detection, the total information bits are adjusted to K + K crc , and the same optimization pro cedure applies with the mo died constraint K 0 + K 1 = K + K crc . Although the error probabilit y analysis is deriv ed for SC deco ding, the optimized parameters can be directly applied to SCL deco ding, which typically achiev es b etter p erformance due to its enhanced error correction capabil- it y . VI. Simula tion Resul ts W e ev aluate the prop osed co ded-pilot scheme through Mon te Carlo sim ulations under block-fading c hannels. The c hannel coecient h = e j ϕ has unit magnitude with uniformly distributed phase ϕ ∼ U (0 , 2 π ) . T o enable fair comparison across modulation orders, w e x the total n um b er of channel uses: for a given sym bol coun t, 4-QAM transmits ov er x c hannel uses, while 16-QAM uses x /2 and 64-QAM uses x /4 , matc hing t ypical sp ectral eciency congurations in 5G NR c hannels [ 30 ]. All schemes transmit K information bits per pack et with an 11-bit CRC for error detection following the 5G NR standard [ 2 ]. F or the pilot-aided baseline, w e optimize pilot allocation by ev aluating candidate lengths of 4, 8, 16, and 32 symbols p er pack et. The pilot length minimizing the required SNR to achiev e BLER of 10 − 3 is selected for eac h conguration. The prop osed co ded-pilot scheme uses the same candidate set { 4 , 8 , 16 , 32 } for N 1 selection, with co de parameters optimized according to the metho d in Section V. Rate-matc hing via puncturing ( for R < 0 . 55 ) or shortening ( for R ≥ 0 . 55 ). Sub-block interlea ving follow 5G NR recommendations. A. V alidation of The or etic al A nalysis W e rst v alidate the theoretical BLER approximation deriv ed in Section V-C by comparing it against Monte Carlo simulations. SCL deco ding with list size L = 1 is employ ed. Fig. 4 and Fig. 5 show results for single blo c k fading ( B = 1 ) and three blo ck fading ( B = 3 ), resp ectiv ely , with M = 600 coded bits and pilot co de length M 1 = 32 ( equiv alen tly N 1 c = 32 for 4-QAM, N 1 c = 16 for 16-QAM, N 1 c = 8 for 64-QAM ). F or eac h mo dulation order, w e examine three code rates: R ∈ { 0 . 25 , 0 . 5 , 0 . 75 } bits p er channel use, corresp onding to K ∈ { 150 , 300 , 450 } information bits. The theoret- ical curves ( dashed lines ) closely match the n umerical Mon te Carlo results ( solid lines ) across all congurations at medium to high co de rates ( R ≥ 0 . 5 ). The approximation main tains accuracy o v er the BLER range from 10 − 4 to 10 − 1 , v alidating the analytical framework for b oth single and m ultiple blo c k-fading scenarios. A t the lo w est code rate ( R = 0 . 25 ), a noticeable gap emerges b etw een theo- retical and n umerical curv es, particularly at lo w SNR. This 10 0 2 4 6 8 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Numerical Theoretical 6 8 10 12 14 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Numerical Theoretical 7.5 10.0 12.5 15.0 17.5 20.0 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Numerical Theoretical ( a ) 4-QAM ( b ) 16-QAM ( c ) 64-QAM Fig. 4. Comparison b etw een theoretical BLER appro ximation ( dashed lines ) and Monte Carlo sim ulation ( solid lines ) for single blo ck fading ( B = 1 ) with M = 600 total co ded bits and M 1 = 32 co ded-pilot bits. Three co de rates are shown: R ∈ { 0 . 25 , 0 . 5 , 0 . 75 } bits/ channel use. The theoretical analysis accurately predicts performance across all mo dulation orders and co de rates. 0 2 4 6 8 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Numerical Theoretical 6 8 10 12 14 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Numerical Theoretical 7.5 10.0 12.5 15.0 17.5 20.0 22.5 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Numerical Theoretical ( a ) 4-QAM ( b ) 16-QAM ( c ) 64-QAM Fig. 5. Comparison b etw een theoretical BLER approximation and Monte Carlo simulation for three block fading ( B = 3 ) with M = 600 coded bits and M b = 16 co ded bits p er block. The theoretical mo del maintains accuracy in multi-block fading scenarios across dierent modulation orders and code rates. discrepancy suggests that the approximation may ov eres- timate p erformance at low SNR, p otentially resulting in sub optimal co de parameter selection by the optimization algorithm. Nev ertheless, the theoretical mo del correctly captures the qualitative b ehavior and pro vides accuracy acceptable for design purp oses, enabling reliable parame- ter optimization at practical op erating points for moderate to high co de rates. B. Performanc e Comp arison with Pilot-Aide d Scheme W e compare the proposed co ded-pilot scheme against the pilot-aided baseline using SCL deco ding with list size L = 8 . T wo congurations are ev aluated in Fig. 6 and Fig. 7: M = 240 coded bits with B = 1 , and M = 720 co ded bits with B = 3 , resp ectively . F or eac h total blo c klength M , three message lengths are tested: K ∈ { 60 , 120 , 180 } for M = 240 , and K ∈ { 180 , 360 , 540 } for M = 720 , yielding eective co de rates of appro ximately 0.25, 0.5, and 0.75. The p erformance comparison reveals distinct trends dep ending on the num b er of fading blo cks. F or the three blo c k-fading case ( B = 3 , Fig. 7 ), the prop osed scheme consisten tly outperforms the pilot-aided baseline across all mo dulation orders and co de rates. The gain is particularly pronounced at higher co de rates. F or instance, in the 16- QAM conguration at K = 540 , the proposed sc heme ac hiev es approximately 1.5 dB gain at BLER of 10 − 3 . Similar substantial gains are observ ed in the 64-QAM conguration at K = 360 . In contrast, for the single block-fading case ( B = 1 , Fig. 6 ), the proposed scheme demonstrates p erformance adv antages primarily at medium to high co de rates ( R ≥ 0 . 5 ). A t the lo w est rate ( R = 0 . 25 , K = 60 ), the pilot- aided baseline achiev es comparable or slightly b etter p er- formance. This suggests that with a single fading blo ck, the o verhead of splitting the code into tw o comp onen ts b ecomes less justied when ample coding resources are a v ailable at low rates. Ho wev er, at higher rates suc h as K = 120 for 16-QAM, the prop osed sc heme achiev es appro ximately 1 dB gain at BLER of 10 − 3 , demonstrating clear b enets when co ding resources b ecome constrained. The performance improv ement in fav orable regimes stems from tw o factors: ( i ) the co ded-pilot sc heme elimi- nates dedicated pilot o v erhead by embedding channel esti- mation information within the co deword, and ( ii ) the joint optimization of co de parameters ( M 0 , K 0 ) and ( M 1 , K 1 ) balances detection reliabilit y and coding eciency . The b enets are more pronounced in multi-block fading ( B = 3 ) where channel estimation information can b e exploited across m ultiple coherence interv als, and at higher co de rates where sp ectral eciency becomes critical. 11 −2 0 2 4 6 8 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Proposed Pilot -aided 4 6 8 10 12 14 16 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Proposed Pilot -aided 7.5 10.0 12.5 15.0 17.5 20.0 22.5 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Proposed Pilot -aided ( a ) 4-QAM ( b ) 16-QAM ( c ) 64-QAM Fig. 6. BLER p erformance comparison between proposed co ded-pilot sc heme ( solid lines ) and pilot-aided baseline ( dashed lines ) for single block fading ( B = 1 ) with M = 240 coded bits. Three message lengths K ∈ { 60 , 120 , 180 } corresp onding to co de rates { 0 . 25 , 0 . 5 , 0 . 75 } are ev aluated. The proposed scheme achiev es consistent SNR gains across all modulation orders, with larger gains at higher co de rates. −2 0 2 4 6 8 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Proposed Pilot -aided 4 6 8 10 12 14 16 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Proposed Pilot -aided 5.0 7.5 10.0 12.5 15.0 17.5 20.0 SNR (dB) 10 −4 10 −3 10 −2 10 −1 BLER Proposed Pilot -aided ( a ) 4-QAM ( b ) 16-QAM ( c ) 64-QAM Fig. 7. BLER performance comparison for three blo ck fading ( B = 3 ) with M = 720 co ded bits and message lengths K ∈ { 180 , 360 , 540 } . The proposed co ded-pilot scheme maintains p erformance adv an tages o ver the pilot-aided baseline at longer blo cklengths, demonstrating scalability across dieren t pac k et sizes and mo dulation orders. C. R ate Al lo c ation Str ate gy A nalysis Fig. 8 illustrates ho w the rate allocation betw een the tw o comp onen t co des v aries with system parameters. The rate ratio R 1 / R 0 ( where R 1 is the coded-pilot rate and R 0 is the main code rate ) reveals distinct trends dep ending on total blo c klength M , co ded-pilot length M 1 , and mo dulation order. Fig. 8 ( a ) sho ws the rate ratio as a function of total blo c klength M . Increasing M from 100 to 1000 causes R 1 to decrease while R 0 remains relativ ely stable. This indicates that channel estimation b ecomes the b ottleneck for short pack ets, requiring higher pilot co de rates. As M gro ws, the system can aord lo w er R 1 while main taining adequate detection p erformance. The rate ratio decreases from appro ximately 0.6 to 0.2, demonstrating that the eect of pilot sym b ols b ecomes m uch smaller at longer blo c klength. Fig. 8 ( b ) plots the rate ratio v ersus co ded-pilot length M 1 . As M 1 increases from 8 to 160, the pilot co de rate R 1 increases for b oth puncturing and shortening. This reects the need to impro ve pilot detection reliability when more resources are allo cated to c hannel estima- tion. Simultaneously , R 0 decreases to comp ensate for the reduced main co deword length, prev en ting performance degradation in data detection. The rate ratio R 1 / R 0 gro ws from approximately 0.2 to 0.8, sho wing that longer pilot co des require higher relativ e rates to main tain balanced error protection. Fig. 8 ( c ) sho ws ho w the rate ratio v aries with mo du- lation order. Higher-order mo dulation increases the rate ratio R 1 / R 0 . F or 4-QAM, the ratio is approximately 0.1, while for 64-QAM it rises to 3. This trend arises b ecause eac h pilot symbol o ccupies a channel use that could oth- erwise carry more data co ded bits at higher mo dulation orders. F or instance, in 64-QAM, each pilot symbol dis- places 6 co ded bits from the main co de, compared to only 2 bits in 4-QAM. T o oset this increased opp ortunit y cost, R 1 m ust b e raised accordingly . VI I. Conclusion This work shows that pilot ov erhead is not a law of nature—it is a design c hoice. By folding channel learning in to the co deword itself, we av oid spending dedicated sym b ols on pilots while still acquiring the c hannel reli- ably . The key is the dual-pac ket structure architecture: a QPSK p ortion that simultaneously carries information and enables blind channel estimation, follo wed by a higher- order QAM p ortion that conv erts the recov ered channel kno wledge into sp ectral eciency . This separation of roles —learn while you talk, then talk faster once y ou kno w —is what mak es pilot-free op eration practical without constraining mo dulation order or codeword length. 12 200 400 600 800 1000 M 0.2 0.4 0.6 0.8 1.0 R 1 / R 0 Shortening Puncturing 20 40 60 80 100 120 140 N 1 0.2 0.4 0.6 0.8 R 1 / R 0 Shortening Puncturing 2 (4-QAM) 4 (16-QAM) 6 (64-QAM) Modulation Order (bits/symbol) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 R 1 / R 0 Shortening Puncturing ( a ) V arying M ( b ) V arying N 1 ( c ) V arying QAM order Fig. 8. Rate allo cation ratio R 1 / R 0 between co ded-pilot and main data co de as a function of system parameters. ( a ) Ratio decreases with total blo cklength M ( xed M 1 = 32 , K = 0 . 5 M , 4-QAM ). ( b ) Ratio increases with pilot co de length M 1 ( xed M = 600 , K = 300 , 4-QAM ). ( c ) Ratio increases with modulation order ( xed M = 600 , K = 150 , M 1 = 32 ). Green circles represen t shortening and red squares represent puncturing for rate-matching of C 0 . The net eect is simple: more of the pack et is used for pa yload, not b o okkeeping. In short pack ets, that accoun t- ing matters. Simulations conrm that reclaiming pilot sym b ols translates into a tangible p erformance gain —up to ab out a 1.5 dB co ding adv an tage ov er conv entional pilot-aided baselines—while preserving robust channel ac- quisition. The broader message is that, for ultra-reliable lo w-latency links, the right question is not ho w to optimize pilots, but how to design co des and modulation so that pilots b ecome unnecessary . References [ 1 ] E. Arıkan, “ Channel p olarization: A metho d for constructing capacity-ac hieving codes for symmetric binary-input memory- less c hannels, ” IEEE T rans. Inf. The ory , vol. 55, no. 7, pp. 3051– 3073, 2009. [ 2 ] 3GPP , “ NR ; multiplexing and channel co ding, ” TS 38.212, Rel. 16 , 2020. [ 3 ] K. Niu and K. 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