Polar Coding for Fading Channels
PPolar Coding for Fading Channels
Hongbo Si, O. Ozan Koyluoglu, and Sriram Vishwanath
Laboratory for Informatics, Networks, and CommunicationsWireless Networking and Communications GroupThe University of Texas at Austin1 University Station, C0806, Austin, TX 78712Email: { sihongbo,ozan } @mail.utexas.edu, [email protected] Abstract —A polar coding scheme for fading channels is pro-posed in this paper. More specifically, the focus is Gaussian fadingchannel with a BPSK modulation technique, where the equivalentchannel could be modeled as a binary symmetric channel withvarying cross-over probabilities. To deal with variable channelstates, a coding scheme of hierarchically utilizing polar codes isproposed. In particular, by observing the polarization of differentbinary symmetric channels over different fading blocks, eachchannel use corresponding to a different polarization is modeledas a binary erasure channel such that polar codes could beadopted to encode over blocks. It is shown that the proposedcoding scheme, without instantaneous channel state informationat the transmitter, achieves the capacity of the correspondingfading binary symmetric channel, which is constructed fromthe underlying fading AWGN channel through the modulationscheme.
I. I
NTRODUCTION
Polar codes are the first family of provably capacity achiev-ing codes for arbitrary symmetric binary-input discrete mem-oryless channels (B-DMC) with low encoding and decodingcomplexity [1] [2]. Channel polarization has then been gener-alized to arbitrary discrete memoryless channels with the sameorder of construction complexity and error probability behav-ior [3]. Moreover, polar codes are also proved to be optimalfor lossy compression with respect to binary symmetric source[4][5], and then further extended to larger source alphabet [6].Polar codes also contribute significantly to non-discreteinput channels. By adopting polar codes as embedded codesat each expanded level, expansion coding scheme [7] achievesthe capacity of additive exponential noise channel in highSNR region with low coding complexity. Besides, a polarcoding scheme achieving capacity for additive Gaussian noisechannel is investigated in [8], which utilizes the polarizationresult for multiple access channel [9]. It has been shownthat the approach of using a multiple access channel with alarge number of binary-input users has much better complexityattributes than the one of using a single-user channel with largeinput cardinality.In this paper, we investigate the polar coding scheme forbinary-input AWGN channel [10][11] in a fading scheme.By adopting BPSK modulation and demodulation technique,additive Gaussian noise fading channel has been boiled downto a binary symmetric channel (BSC) with finite set of tran-sition probabilities according to the channel quality. The keyintuition of the proposed scheme is based on observing the po-larization characteristics of different BSCs. By hierarchically using polar codes, where the transmitter encodes over blocks,it can be proved that the designed coding scheme achieves thecapacity of converted channel (fading BSC).The rest of paper is organized as follows. After introducingthe preliminary results on polar codes and problem backgroundin Section II and III, respectively, the polar coding scheme forfading channels is stated and illustrated in Section IV. Thepaper concludes with a discussion section.II. P
RELIMINARY FOR P OLAR C ODES
The construction of polar code is based on the observa-tion of channel polarization. Consider a binary-input discretememoryless channel W : X → Y , where X = { , } . Define F = (cid:20) (cid:21) . Let B N be the bit-reversal operator defined in [1], where N =2 n . By applying the transform G N = B N F ⊗ n ( F ⊗ n denotesthe n th Kronecker power of F ) to u N , consider transmittingthe encoded output x N through N independent copies of W .Then N new binary-input coordinate channels W ( i ) N : X →Y N × X i − are constructed, where for each i ∈ { , . . . , N } the transition probability is given by W ( i ) N ( y N , u i − | u i ) (cid:44) (cid:88) u i +1: N N − W N ( y N | u N G N ) . Then as N tends to infinity, the channels { W ( i ) N } polarize toeither noiseless or pure-noisy, and the fraction of noiselesschannels is close to I ( W ) , the symmetric mutual informationof channel W [1].To this end, polar codes can be considered as G N -cosetcodes with parameter ( N, K, A , u A c ) , where u A c ∈ X N − K is frozen vector (can be set to all-zero vector for symmetricchannel [1]), and the information set A is chosen as a K -element subset of { , . . . , N } such that the Bhattacharyyaparameters satisfies Z ( W ( i ) N ) ≤ Z ( W ( j ) N ) for all i ∈ A and j ∈ A c .The decoder in polar coding scheme is successive cancela-tion (SC) decoder, which gives an estimate ˆ u N of u N givenknowledge of A , u A c , and y N by computing ˆ u i (cid:44) (cid:26) , if i ∈ A c ,d i ( y N , ˆ u i − ) , if i ∈ A , a r X i v : . [ c s . I T ] A p r n the order i from to N , where d i ( y N , ˆ u i − ) (cid:44) (cid:40) , if W ( i ) N ( y N , ˆ u i − | W ( i ) N ( y N , ˆ u i − | ≥ , , otherwise.It has been proved that by adopting SC decoder, polar codesachieves any rate R < I ( W ) with decoding error scalingas O (2 − N β ) , where β < / . Moreover, the encoding anddecoding complexity of polar codes are both O ( N log N ) .III. P ROBLEM B ACKGROUND
Fading channels characterize the wireless communicationchannels, where the channel states are changing over timesand only available at the decoders. Fading coefficient typicallyvaries much slower than transmission symbol duration inpractice. To this end, a block fading model [12] is proposed,whereby the state is assumed to be a constant over coherencetime intervals and stationary ergodic across fading blocks.Consider the AWGN fading channel, Y b,i = H b,i X b,i + Z b,i , b = 1 , . . . , B, i = 1 , . . . , N, (1)where Z b,i is i.i.d. additive Gaussian noise with variance E Z ; X b,i is channel input with power constraint BN B (cid:88) b =1 N (cid:88) i =1 x b,i ≤ E X ; H b,i is channel gain random variable; N is blocklength; and B is number of blocks. For this moment, H b,i are assumed tobe constant within a block and follow an i.i.d. fading processover blocks. In particular, for the two sates case { h , h } weconsider, omitting the indices, the distribution of H is givenby Pr { H = h } (cid:44) q and Pr { H = h } (cid:44) q = 1 − q .Using BPSK modulation, any codeword produced by en-coder is mapped to signal with element in {−√ E X , + √ E X } .After utilizing a BPSK demodulation at the decoder, theequivalent channel can be formulated as a binary symmetricchannel, with transition probability relating to channel states.More specifically, the converted channel is given by ¯ Y b,i = ¯ X b,i ⊕ ¯ Z b,i , b = 1 , . . . , B, i = 1 , . . . , N, (2)where ¯ X b,i and ¯ Y b,i are both Bernoulli random variables repre-senting channel input and output correspondingly; ¯ Z b,i is i.i.d.channel noise, also distributed as Bernoulli random variable,but related to channel state. More precisely, if H b,i = h , thenPr { ¯ Z b,i = 1 } = 1 − Φ( h √ SNR ) (cid:44) p , (3)and if H b,i = h , thenPr { ¯ Z b,i = 1 } = 1 − Φ( h √ SNR ) (cid:44) p , (4)where Φ( · ) is CDF of normal distribution and SNR = E X /E Z . In other words, the channel can be modeled as W (cid:44) BSC ( p ) with probability q , and as W (cid:44) BSC ( p ) with probability q .The ergodic capacity of the converted channel (fading BSC)is given by [12] C SI-D = q [1 − H ( p )] + q [1 − H ( p )] , (5) I ( W ( π ( i )) N ) M BG p p π ( i )101 N Fig. 1:
Illustration of polarizations for two BSCs.
Values of I ( W ( π ( i )) N ) , the reordered mutual information, are shown forboth polarizations. The blue-solid plot represents the channelwith higher transition probability p , and the red-dashed for p . Three index categories are denoted by G , M , and B inorder.where H ( · ) is the binary entropy function, and SI-D refersto channel state information at the decoder. The capacityachieving input distribution is uniform over { , } . In thispaper, we show a polar coding scheme achieving the capacityof converted fading channel with low encoding and decodingcomplexity, without having instantaneous channel state infor-mation at the transmitter (only the statistical knowledge isassumed).IV. P OLAR C ODING FOR F ADING C HANNEL
A. Intuition
In polar coding for a general B-DMC W , we have seen thechannel can be polarized by transforming a set of independentcopies of given channels into a new set of channels whosesymmetric capacities tend to 0 or 1 for all but a vanishing frac-tion of indices. To this end, an information set A is constructedby picking the indices corresponding to K minimum values of Z ( W ( i ) N ) , which is equivalent to picking those correspondingto K largest values of I ( W ( i ) N ) . In this sense, the constructionof A is deterministic. However, as indicated in [1], the indicesin A are not adjacent. For this, we introduce a permutation π : { , . . . , N } → { , . . . , N } , which reorders all the indicesby the value of I ( W ( i ) N ) ranging from high to low. Note thatthe construction of polar codes already implies the fact thatfor channels of the same type, their permutation mappings arethe same.Another fact about polar codes is that the polarization isuniform [13]. Consider polarizing two B-DMCs, for instanceBSCs with parameters p and p respectively, then the infor-mation sets, denoted by A and A , satisfy A ⊆ A if p ≥ p . (6)In other words, if a particular channel index constructed fromthe worse channel (BSC with larger transition probability)polarizes to be noiseless, so does that of the better channel(BSC with smaller transition probability). Based on thisobservation, when polarizing two W and W with transition ˜ A| v ( |M| ) v (1) ... ˜ u (1) ˜ u ( |M| ) ... × G B = = × G N u (1) ...... u ( B ) ˜ π − Bπ − ...... x (1) x ( B ) |G| |B| N |M| rotate Fig. 2:
Illustration of polar encoder for fading channel with two states.
In this numerical example, we assume N = 16 , B = 8 , |A| = 5 , |G| = 7 , |M| = 5 , and |B| = 4 . Bits in blue are information bits, and ones in white are frozen to zeros. Afterencoding of Phase 1, the codewords are rotated and embed to the messages of Phase 2 to generate the finalized codeword.probabilities defined by (3) and (4) respectively, the indicesafter permutation π can be divided into three categories(illustrated in Fig. 1). Without loss of generality, we assume p ≥ p .1) G : both channels are good, i.e. I ( W ( π ( i ))1 ,N ) → , I ( W ( π ( i ))2 ,N ) → . M : only channel 2 is good, while channel 1 is bad, i.e. I ( W ( π ( i ))1 ,N ) → , I ( W ( π ( i ))2 ,N ) → . B : both channels are bad, i.e. I ( W ( π ( i ))1 ,N ) → , I ( W ( π ( i ))2 ,N ) → . Denote the information sets for two channels as A and A correspondingly, then obviously A = G , and A = G ∪ M .Moreover, we have: |G| = |A | = N [1 − H ( p ) − (cid:15) ] , (7) |M| = |A | − |A | = N [ H ( p ) − H ( p )] , (8) |B| = N − |A | = N [ H ( p ) + (cid:15) ] , (9)where (cid:15) is a arbitrary small positive number.For the fading channel, we consider the transmitter hasno prior knowledge of channel states before transmitting,hence, coding over channels with indices in M is challenging.Observe that for those channels, with probability q theyare nearly noiseless, and with probability q they are purelynoisy. To this end, each channel can be modeled as a binaryerasure channel (BEC) from the viewpoint of blocks, and wedenote this channel as ˜ W . This intuition inspires our designof encoder and decoder for fading channels. B. Encoder
The encoding process of polar coding for fading channelhas two phases, hierarchically using polar codes to generate
N B -length codewords, where N is blocklength and B is thenumber of blocks.
1) Phase 1:
Consider a set of B -length block messages v ( k ) with k ∈ { , . . . , |M|} . For every v ( k ) , construct polar code ˜ u ( k ) , which is G B -coset code with parameter ( B, | ˜ A| , ˜ A , ,where ˜ A is the information set for ˜ W (cid:44) BEC ( q ) , and wechoose | ˜ A| = (1 − q − (cid:15) ) B. (10)In other words, we construct a set of polar codes, where eachcode corresponds to an index in set M , with the same rate − q − (cid:15) , the same information set ˜ A , and the same frozen values0 as well. Mathematically, if denote the reordering permutationfor ˜ W as ˜ π , then ˜ π ( v ( k ) ) = [ v ( k )1 , . . . , v ( k ) | ˜ A| , , . . . , , (11) ˜ u ( k ) = v ( k ) G B . (12)
2) Phase 2:
Consider another set of N -length messages u ( b ) with l ∈ { , . . . , B } . For every u ( b ) , construct po-lar code x ( b ) , which is G N -coset codes with parameter ( N, |G| , G , u ( b ) G c )) , where G is BSC information set with sizegiven by (7). Remarkably, we do not froze all non-informationbits to be 0, but embed the blockwise codewords from Phase1. More precisely, if denote the permutation operator of BSCas π , then π ( u ( b ) ) = [ u ( b )1 , . . . , u ( b ) |G| , ˜ u (1) b , . . . , ˜ u ( |M| ) b , , . . . , , (13) x ( b ) = u ( b ) G N . (14)By collecting all { x ( b ) } B together, the encoder generates andoutputs a codeword with length N B . An example to illustratethis encoding process is shown in Fig. 2.
C. Decoder
After receiving the sequence y NB from channel, thedecoder’s task is trying to make estimates { ˆ v ( k ) } |M| and { ˆ u ( b ) } B , such that the information bits in both sets ofmessages match the ones at the transmitter end with highprobability. Rewrite channel output y NB as a B × N matrix,with row vectors { y ( b ) } B . As that of the encoding process,the decoding process also has two phases: M| ...ˆ u (1) ˆ u ( B ) ...ˆ˜ u (1) ˆ˜ u ( |M| ) B ... Decoder 1Decoder 2Decoder 2 π − ...... y (1) y ( B ) N rotate ...ˆ v (1) ˆ v ( |M| ) ˜ π − | ˜ A| Decoder 3Decoder 3 ... |G| |B| ...
Fig. 3:
Illustration of polar decoder for fading channel with two states.
We use the same parameters as encoder. AfterPhase 1, decoder outputs all estimates { ˆ u ( l ) } B using BSC SC decoder based on channel states, then selected columns ofdecoded results are rotated and delivered as inputs to Phase 2. In the next phase, the decoder basically uses BEC SC decoderto decode ˆ v ( k ) from ˆ˜ u ( k ) for every k ∈ { , . . . , |M|} . Bits in shade represent for erasures.
1) Phase 1:
For every b ∈ { , . . . , B } , decode ˆ u ( b ) from y ( b ) using SC decoder. More precisely, because at the receiverend, channel state is available, then receiver can adopt thecorresponding SC decoder for BSC based on the channelstate observed. Remarkably, for index in M , we decode aserasure, denoted as “e”, for bad channel state. To this end,polar decoder is given by: if the channel state is h , then use Decoder 1 , otherwise use
Decoder 2 , where the two decodersare expressed follows: − Decoder 1: ˆ u ( b ) i (cid:44) , if b ∈ B , e , if b ∈ M ,d ,i ( y ( b ) , ˆ u ( b )1: i − ) , if b ∈ G , in the order i from to N , where d ,i ( y ( b ) , ˆ u ( b )1: i − ) (cid:44) , if W ( i )1 ,N ( y ( b ) , ˆ u ( b )1: i − | W ( i )1 ,N ( y ( b ) , ˆ u ( b )1: i − | ≥ , , otherwise. − Decoder 2: ˆ u ( b ) i (cid:44) (cid:26) , if b ∈ B ,d ,i ( y ( b ) , ˆ u ( b )1: i − ) , if b ∈ G ∪ M , in the order i from to N , where d ,i ( y ( b ) , ˆ u ( b )1: i − ) (cid:44) , if W ( i )2 ,N ( y ( b ) , ˆ u ( b )1: i − | W ( i )2 ,N ( y ( b ) , ˆ u ( b )1: i − | ≥ , , otherwise.After decoding from y ( b ) block by block, the decoder outputa B × N matrix ˆ U with rows { ˆ u ( b ) } B .
2) Phase 2:
Select columns of ˆ U with indices in M afterpermutation π to construct a B ×|M| matrix ˆ˜ U . Consider eachcolumn of ˆ˜ U , denoted by ˆ˜ u ( k ) for k ∈ { , . . . , |M|} , as theinput to decoder in Phase 2. Then receiver aims to decode ˆ v ( k ) from ˆ˜ u ( k ) using SC decoder with respect to ˜ W = BEC ( q ) . More formally, the decoder in Phase 2 is expressed as follow: − Decoder 3: ˆ v ( k ) j (cid:44) (cid:40) , if k ∈ ˜ A c , ˜ d j (ˆ˜ u ( k ) , ˆ v ( k )1: j − ) , if k ∈ ˜ A , in the order j from to B , where ˜ d j (ˆ˜ u ( k ) , ˆ v ( k )1: j − ) (cid:44) , if ˜ W ( j ) N (ˆ˜ u ( k ) , ˆ v ( k )1: j − | W ( j ) N (ˆ˜ u ( k ) , ˆ v ( k )1: j − | ≥ , , otherwise.After Phase 2, the decoder output a |M| × B matrix ˆ V withrows { ˆ v ( k ) } |M| . An example to illustrate the decoding ofboth phases is shown in Fig. 3. D. Achievable Rate
We want to show the rate in proposed polar coding schemeachieves the capacity of converted fading channel given by(5). Intuitively, by using BSC SC decoders corresponding tochannel states, the output from Phase 1 successfully recoversall information bits in { u ( b ) } B . Moreover, for those with in-dices corresponding to M , the decoder could decode correctlyif channel state is h , and set to erasures otherwise. Thus, theinput to decoding Phase 2, vector ˆ˜ u ( k ) can be considered as anoutput of BEC ( q ) , hence BEC SC decoder could decode allinformation bits in v ( k ) correctly for any k ∈ { , . . . , |M|} .More formally, we have the following theorem. Theorem 1.
The proposed polar coding scheme achieves anyrate
R < C
SI-D with arbitrarily small error probability forsufficiently large N and B .Proof: The proof is straightforward by utilizing errorbound from polar coding. In Phase 1 of decoding, the er-ror probability of recovering u ( b ) correctly for each b ∈{ , . . . , B } is given by P ( b )1 ,e = O (2 − N β ) , (15)here β < / . Similarly, in decoding Phase 2, the error prob-ability of recovering v ( k ) correctly for each k ∈ { , . . . , M } is given by P ( k )2 ,e = O (2 − B β ) . (16)Hence, by union bound, the total decoding error probability isupper bounded by P e ≤ B (cid:88) b =1 P ( b )1 ,e + |M| (cid:88) k =1 P ( k )2 ,e = O ( B − N β ) + O ( N − B β ) → , when N and B tend to infinity. In particular, we consider B = o (2 N β ) and N = o (2 B β ) .Moreover, from the analysis, it is evident that all messagesbits in v ( k ) and u ( b ) are decodable, then the achievable ratefor the designed scheme is given by R = 1 N B (cid:110) |M|| ˜ A| + B |G| (cid:111) = 1 N B (cid:110) N [ H ( p ) − H ( p )] B [1 − q − (cid:15) ]+ BN [1 − H ( p ) − (cid:15) ] (cid:111) = q [1 − H ( p )] + q [1 − H ( p )] − δ ( (cid:15) )= C SI-D − δ ( (cid:15) ) , where we have used (7), (8) and (10), and δ ( (cid:15) ) (cid:44) (cid:15) [1 + H ( p ) − H ( p )] → , as (cid:15) → . Thus, any rate
R < C
SI-D is achievable.
E. Complexity Analysis
As we have seen, polar coding schemes for both BSCand BEC have relatively low complexity. Since the proposedpolar coding scheme for fading channel hierarchically utilizespolar codes, the character of low complexity is consequentlyinherited. More precisely, |M| number of B -length polarcodes as well as B number of N -length polar codes areutilized. Hence, the overall complexity of the coding scheme,for both encoding and decoding, is given by |M| · O ( B log B ) + B · O ( N log N ) = O ( N B log(
N B )) . V. D
ISCUSSION
In this section, we generalize the polar coding scheme tofading channels with arbitrary finite number of states. Assumechannel gain H b,i has S states from set { h , . . . , h S } , where S is a positive integer. Assume the distribution of H b,i omittingindices is given by Pr { H = h s } (cid:44) q s , where s ∈ { , . . . , S } .Then the converted channel using BPSK, defined in (2), is stilla BSC, whereas with probability q s , the transition probabilityis given by Pr { ¯ Z = 1 } = 1 − Φ( h s √ SNR ) (cid:44) p s . (17) Denote the converted BSC corresponding to state h s as W s ,then the capacity of converted channel is given by C SI-D = S (cid:88) s =1 q s [1 − H ( p s )] , (18)where − H ( p s ) is the capacity of W s .Observe that when polarizing S BSCs with different tran-sition probabilities, the indices could be divided into S + 1 sets after permutation π . More mixture sets M , . . . , M S − are defined in this case. Without loss of generality, we assume p ≥ p ≥ · · · ≥ p S . Then |M s | = H ( p s ) − H ( p s +1 ) , andfor index in set M s , W , . . . , W s are polarized to be purelynoisy and all others to be noiseless. To this end, we considera BEC with erasure probability e s = (cid:80) st =1 q t to characterizethe polarization result for index in M . (See Fig. 4 for anintuition.) B π ( i )101 NI ( W ( π ( i )) N ) p p p G . . . p S − M M . . . M S − p S Fig. 4:
Illustration of polarizations for S BSCs.
There are S − mixture sets, denoted as M , . . . , M S − .Polar coding scheme designed for this channel is similar.In Phase 1 of encoding, transmitter needs to generate S − sets of polar codes, where each one is G B -coset codes withparameter ( B, | ˜ A s | , ˜ A s , with respect to BEC ( e s ) , and all theencoded codewords are embed into messages for Phase 2. Atthe receiver end, Phase 1 should use one of S SC decoders forBSC to decode ˆ u ( l ) , based on observation of channel states.Then in Phase 2, S − BEC SC decoders are working inparallel to recover the information bits. By adopting this polarcoding scheme, the achievable rate is given by R = 1 N B (cid:40) B |G| + S − (cid:88) s =1 |M s || ˜ A s | (cid:41) = [1 − H ( p ) − (cid:15) ] + S − (cid:88) s =1 [ H ( p s ) − H ( p s +1 )](1 − e s − (cid:15) )= S (cid:88) s =1 q s [1 − H ( p s )] − δ (cid:48) ( (cid:15) ) , where δ (cid:48) ( (cid:15) ) = (cid:15) [1 + H ( p ) − H ( p S )] . Thus, the proposedpolar coding scheme achieves the capacity of channel, and theencoding and decoding complexities are both given by S − (cid:88) s =1 |M s |· O ( B log B )+ B · O ( N log N ) = O ( N B log(
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