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Paper text:
longitudinal recording has been the standard method in magnetic recording for decades .
recent research has shown that this method seems to reach its physical limits in the near future due to the superparamagnetic effects .
on the other hand , the technique which has been known prior to the longitudinal recording 
perpendicular recording , has recently been the centre of research attention .
perpendicular magnetic recording offers promising increased in recording densities , up to 1 terabit per square inch seems feasible  @xcite .
as the areal density is increased , however , the signal processing aspects of magnetic recording becomes more difficult .
sources of distortion including media noise , electronics and head noise , jitter noise , inter - track interference , thermal asperity , partial erasure and dropouts become more apparent and unless appropriate mitigation techniques are present , signals can not be retrieved reliably from the recording media .    since the discovery of turbo codes
, soft - decision iterative decoding has been shown to be able to provide significant coding gain over the conventional detection method on magnetic recording .
the utilisation of iterative decoding on the concatenation of partial - response ( pr ) channel and powerful error - correcting codes such as low - density parity - check ( ldpc ) and turbo codes has been proposed in many literatures .
iterative decoding is a reduced - complexity method to achieve the optimum solution  the maximum - likelihood solution and as such , iterative decoding is sub - optimal .    in this paper , we present a method to improve the sub - optimality of the iterative decoding and demonstrate its applications to perpendicular magnetic recording in the presence of media noise .
the improved method , which is known as the received - vector - coordinate - modification ( rvcm ) algorithm , follows on the previous works of the authors  @xcite,@xcite,@xcite , @xcite,@xcite .
this method is similar to the works of  @xcite and  @xcite .
this paper also investigates the use of signal - to - noise ratio ( snr ) mismatch  @xcite to mitigate the effect of media noise .
the rest of the paper is organised as follows .
section  [ sec : channel ] describes the perpendicular recording channel used .
the description of the rvcm algorithm is outlined in section  [ sec : rvcm ] and the performance of this algorithm is demonstrated in section  [ sec : performance ] .
section  [ sec : conclusions ] concludes this paper .
[ fig : block - diagram ] shows the block diagram of the perpendicular recording system model used in this paper .
the user data , denoted as @xmath0 , is a sequence of of input symbols taking values of @xmath1 .
some error - protection redundancy is added to the sequence @xmath0 by the error - correcting - codes ( ecc ) encoder forming codeword sequence @xmath2 . to simulate the write current ,
the sequence @xmath2 is mapped to @xmath3 according to @xmath4 operation .
the scaling factor of @xmath5 is to ensure the transition takes values of @xmath6 .
we assume that the read head produces zero voltage in the region of magnetic transitions and some voltage in the region of constant magnetic polarity .
we approximate the single - transition step response , denoted as @xmath7 , using the hyperbolic - tangent function  @xcite : @xmath8 where @xmath9 is the saturation level or the amplitude from zero to peak ( normalised to unity ) and @xmath10 is the time taken for @xmath7 to go from @xmath11 to @xmath12 .
it is assumed that @xmath13 and @xmath10 are normalised to the symbol period , @xmath14 . throughout the paper , it is assumed that @xmath15 .
we define the response of two adjacent transitions ( dibit - response ) @xmath16 as : @xmath17 and the readback signal @xmath18 is simply the convolution of @xmath2 and @xmath16 plus some noise : @xmath19 where @xmath20 is the overall noise in the recording system which consists of media , jitter and electronic noise , i.e. @xmath21 .    the media noise , @xmath22 originates from the imperfections of the media and its effect is significant in the magnetic transition regions .
typically , media noise is approximately four times the electronic noise at transition regions . in our system model
, we consider the media noise as additive - white - gaussian - noise ( awgn ) with mean of @xmath23 and variance of @xmath24 , which exists in the transition region only . as
shown in fig .
[ fig : block - diagram ] , the media saturation noise depends on the saturation level and it is evaluated as @xmath25 . unlike media noise which is media dependent , the jitter noise @xmath26 is due to timing imperfection only . to model the sampling jitter noise , @xmath26 , the @xmath27th order taylor approximation is used . the jitter estimation block shown in fig .
[ fig : block - diagram ] is done with @xmath28th order taylor series expansion of @xmath7 .
the jitter probability density function is assumed to be uniform , limited by a maximum value .
the electronic noise , @xmath29 is awgn with mean of @xmath23 and variance of @xmath30 .
the recording system in fig .
[ fig : block - diagram ] caters for many different simulation cases with varying degree of electronics , media and sampling jitter noise .
we define the channel snr as : @xmath31    the noisy readback signal is equalised to @xmath32 pr target which is only optimal for electronic noise at the considered @xmath10  @xcite .
it serves for comparison purposes only .
the maximum - a - posteriori ( map ) decoder of the pr channel exchanges extrinsic with the ecc decoder to deliver solution which is used for performance evaluation .
it has been shown that the rvcm algorithm provides considerable coding gain for parallel and serial concatenated turbo codes  @xcite,@xcite and ldpc codes  @xcite .
the algorithm can be applied directly to perpendicular recording and is described briefly below .
let @xmath33 denote an @xmath34tuple vector at the output of the map decoder , that is the a - posteriori probability ( app ) of the map decoder .
let @xmath35 denote the reliability sequence of @xmath36 , where @xmath37 .
assume that @xmath38 is an integer where @xmath39 and @xmath40 is a vector of length @xmath38 .
* step 1 . * : :    store the vector @xmath41 , let the integer    @xmath42 be initialised to @xmath23 .
* step 2a . * : :    set @xmath43 and    @xmath44 .
restart the iterative    decoder , store the decoded vector    ( @xmath45 ) . *
* : :    set @xmath46 .
restart the iterative    decoder , store the decoded vector    ( @xmath47 ) and restore    @xmath48 , i.e.    @xmath49
. * step 3 . * : :    if @xmath50 then set @xmath51 and    continue to * step 2*. otherwise , stop the algorithm and from the list    of all decoded vectors    @xmath52 ,    @xmath53 , choose a    decoded vector that has the minimum euclidean distance .
it is assumed    that the iterative decoder always outputs a codeword .    from the steps above ,
it is clear that the complexity of the algorithm depends on @xmath38 .
one of the major obstacles concerning the rvcm algorithm is the difficulty in finding the symbol(s ) that , if modified , can converge the iterative decoder to the maximum - likelihood solution  @xcite,@xcite .
these symbols are referred as the critical symbols and their distribution is uniform with no sign of vulnerable or favourite symbol positions .
on the other hand , due to their uniform distribution , it is likely that we can find one of the critical symbols if we confine our search to a small group , i.e. keeping the value of @xmath38 low . in this way
, we can reduce the computational complexity for the price of sub - optimum performance . as we will show later that
, the gain obtained by confining @xmath38 to a small value is still significant compared to the performance of the standard iterative decoder .
there are various methods for selecting the critical symbols , see  @xcite for details . in this paper
, we restrict the selection to one method only , that is the reliability of the app at the output of the map decoder .
we evaluate the performance of the rvcm algorithm on some short - block length turbo and ldpc codes . fig .
[ fig : turbo - sm0 ] and  [ fig : turbo - sm4 ] show the error rate performance of the turbo code under the standard iterative and rvcm decoders .
the turbo code considered is the @xmath54 tail - biting turbo code , where @xmath55 , @xmath56 and @xmath57 with @xmath58interleaver .
significant improvement is noticed and the increase in performance gets better as snr increases .
it is worth noting that the results in fig .
[ fig : turbo - sm4 ] were obtained using snr mismatch technique . with this technique ,
the improvement in performance over standard iterative decoder is even greater as snr increases . in the presence of media noise ,
snr mismatch methods do not provide the same performance as observed with electronics noise only  @xcite , however better targets for media noise are being investigated by the authors .
error performance of rvcm decoder on the turbo code in the presence of electronic noise only , width=220 ]    error performance of rvcm decoder on the turbo code in the presence of electronic and media noise , width=220 ]    error performance of rvcm decoder on the @xmath59 cyclic ldpc code in the presence of electronic and media noise , width=220 ]    performance of the @xmath60 cyclic and @xmath61 codes in the presence of electronic and media noise , width=220 ]    similar performance improvement is observed for the ldpc codes , see fig .
[ fig : gf2 - 127 - 84 ] .
the @xmath59 cyclic ldpc code , which has minimum - distance of @xmath62 , was constructed using a method described in  @xcite . as mentioned earlier , the rvcm algorithm allows one to trade off the performance against the computational complexity . from fig .
[ fig : gf2 - 127 - 84 ] , despite the performance obtained by setting @xmath63 is approximately @xmath64db inferior to that by setting @xmath65 , the coding gain from the standard iterative decoding is significant . for the case of @xmath63 , we select the critical symbols based on the reliability measure at the output of the map decoder . from the vector @xmath41 , we construct a vector @xmath66 of length @xmath38 such that @xmath67 . in fig .
[ fig : gf2 - 255 - 175 ] , we compare the performance of the @xmath60 cyclic code and that of the @xmath68 quasi - cyclic code .
we can see that the rvcm algorithm provides significant gain , within one order of magnitude improvement , over the bp algorithm . at approximately @xmath69 ber ,
the performance of the cyclic code with rvcm is within @xmath70db away from the longer code under bp decoding .
we have shown that the application of the rvcm algorithm to perpendicular magnetic recording shows promising results .
simulation results show that improvement of within one order of magnitude is possible .
short block length offers an attractive error - correction scheme in which rvcm algorithm can be fully exploited by setting @xmath65 .
a bank consisting of @xmath71 parallel rvcm decoders can be built on chips and the decoding of short - block length data has low latency .
the performance of longer block - length codes , up to a certain error - rate , can be outperformed by the application of rvcm algorithm to shorter codes .
the exact point , at which the longer codes start to perform better , depends on the code structure .
we also extended our investigations on using some non binary cyclic ldpc codes  @xcite and we observe similar improvement as in the binary cases .
further investigations in identifying the critical symbols will allow the application of rvcm algorithm to long powerful codes .
the authors would like to thank prof .
barry k. middleton of university of manchester for the channel noise discussion .
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an algorithm of improving the performance of iterative decoding on perpendicular magnetic recording is presented . 
 this algorithm follows on the authors previous works on the parallel and serial concatenated turbo codes and low - density parity - check codes . 
 the application of this algorithm with signal - to - noise ratio mismatch technique shows promising results in the presence of media noise . 
 we also show that , compare to the standard iterative decoding algorithm , an improvement of within one order of magnitude can be achieved .