3096 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004
Alternating Coordinates Minimization Algorithm
for Estimating Parameters of Partial Erasure
Plus Transition Shift Model
Tsai-Sheng Kao and Mu-Huo Cheng, Member, IEEE
Abstract—The identification of model parameters of a
high-density recording channel is usually difficult and complicated. In this paper, we successfully apply the alternating coordinates minimization (ACM) algorithm for estimating parameters of a partial erasure plus transition shift model (PETSM). The resulting algorithm turns out to iteratively solve two least square problems and is guaranteed to converge. Furthermore, the obtained model for a nonlinear partial response channel is more accurate than conventional models such that the maximum likelihood (ML) detector has better bit error rate (BER) performance without increasing its realization complexity. Computer simulations show that the ACM algorithm can accurately estimate the model parameters and the BER for the detector is significantly improved especially when the transition shift parameter is large.
Index Terms—Alternating coordinates minimization, maximum
likelihood detector, partial erasure ratio, partial response channel, transition shift parameter.
I. INTRODUCTION
N
ONLINEAR distortions are the primary factors to limit the detector performance in high-density magnetic storage [1], [2]; these distortions are mainly the transition shift and the partial erasure. Several models have been presented to char-acterize the nonlinear distortions [3], [4], including the partial erasure plus transition shift model (PETSM) and simple par-tial erasure model (SPEM). However, the model parameters are usually difficult to estimate or measure [5], [6]. Recently, the authors applied the expectation-maximization (EM) algorithm [7] for identifying the parameters of a SPEM, and assumed that the effect of transition shift had been precompensated. This as-sumption makes the EM approach difficult to estimate model parameters directly from the measurement data without proper precompensation.In this paper, the alternating coordinates minimization (ACM) algorithm [8], [9] is successfully applied for estimating parameters of a PETSM, including both nonlinear effects of transition shift and partial erasure. The resulting algorithm turns out to iteratively solve two least square problems and is guaranteed to converge. The obtained model for a nonlinear partial response channel is more accurate than conventional
Manuscript received October 15, 2003; revised December 17, 2003 and De-cember 24, 2003. This work was supported by the National Science Council, Taiwan, under NSC92-2213-E-009-084.
The authors are with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: mhcheng@cc.nctu.edu.tw).
Digital Object Identifier 10.1109/TMAG.2004.830223
models such that the maximum likelihood (ML) detector has better performance without increasing its realization complexity. The algorithm, therefore, enables us to accurately estimate parameters directly from the measurement data and to design a detector with improved performance.
II. CHANNELMODEL
The sampled output, , of the PETSM [4] is given by (1) where express the duration of the channel, is the Lorentzian function, is the transition shift parameter, rep-resents the partial erasure effect, determined as follows:
(2) where and denote the partial erasure ratios. Note that the common setting for has been relaxed and thus the model flexibility is enhanced. The data is obtained by the nonreturn-to-zero-inverted (NRZI) encoding of the plus and minus binary recorded data Thus and may be of values . Since the product
for all can only be either or 0, we denote a switch function by
(3) The PETSM (1) then can be represented in a new formulation
(4) where the channel parameters and
for . Therefore, the parameters of this model (4) consist of , and . The problem here is to find the model parameters for minimizing the following square output error
(5) where is the sampled measurements of a magnetic recording channel and is the number of sample data.
KAO AND CHENG: ALTERNATING COORDINATES MINIMIZATION ALGORITHM 3097
III. ACM ALGORITHM FORESTIMATINGMODELPARAMETERS
A. Formulation of Two “Linear” Equations
Divide the model parameters into two vectors and with representing the channel parameters and representing the partial erasure ratio parameters, as follows:
(6) (7) where the superscript denotes the transpose operation. Denote
and
(8) then the model output is a linear form of the channel param-eters
(9) Similarly, the model output also can be formulated as a linear form of the partial erasure ratio parameters
(10)
where and
(11)
(12)
(13) Note that (9) and (10) look like linear equations but in fact they are nonlinear.
B. ACM Algorithm for Estimating Model Parameters
The ACM algorithm iteratively performs the following two major operations until convergence; the iteration number is de-noted by the subscript . The first operation is given
to solve for minimizing which can be expressed as (14)
where , and
. Since the vector and matrix are evaluated under the condition , the performance in (14) is obviously a quadratic function of and the unique solution of can be obtained
(15)
Fig. 1. (a) Samples of the channel parametersg s and their estimates at the 257th iteration and (b) samples of the channel parametersf s and their estimates at the 257th iteration.
Similarly, the second operation is given , obtained in the previous operation, to solve for minimizing which is (16)
where , and each vector is evaluated
using (8) with . The performance function in (16) is also quadratic and the solution is
(17) Each operation involves a quadratic minimization and solves a unique minimum. Thus, is guaranteed nonincreasing. Furthermore, since is bounded below by zero, the ACM algorithm will always converge. The algorithm here terminates
when the measure, , is less
than a predetermined small value . C. Simulation Example
Let 1000 measurement data be generated as
(18) where is also a Lorentzian function with
, and is the addi-tive white Gaussian noise. The noise variance is set to dB to make the signal-to-noise ratio (SNR), defined
as , equal 20 dB. The channel
lengths are given by The ACM algorithm is initialized with all model parameters set to zeros and the predetermined value . Here, the algorithm terminated at the 257th iteration, and the convergent average square output error is dB. The estimated partial erasure ratios and are respectively 0.7013 and 0.4942, and the obtained model parameters are shown in Fig. 1, which illustrates that the ACM algorithm can accurately estimate the model parameters.
IV. NONLINEAR PR4 CHANNEL: MODELING ANDDETECTORPERFORMANCE
For high-density magnetic storage, the model duration is usually long; this makes the complexity to realize the detector,
3098 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004
TABLE I
TOBTAINEDSDRS INdBOFEACHMODEL FORPR4 CHANNELWITHVARIOUS
TRANSITIONSHIFTVALUES
designed for the obtained model, prohibitively high. Therefore, the magnetic channel is often equalized to a partial response model and then the ML detector is designed to improve the performance in high-density magnetic storage [10], [11]. As-sume that the magnetic channel is equalized to the class-IV partial response (PR4) with minimum bandwidth [2], then its impulse response is given by
(19) If the nonlinear effects of partial erasure and transition shift are considered, the readback sampled signal, , is obtained by
(20) where and represent the effective channel lengths of
, and denotes the noise which is normally colored be-cause of the PR4 equalizer. In this paper, the proposed model for the nonlinear PR4 channel is given by
(21) The parameters in (21) are further esti-mated by the ACM algorithm.
We use the signal-to-distortion (SDR) ratio, defined as , to measure the model capability. The measurement data, for each value of from 0.1 to 0.5, are generated using (20) with , and . For simplicity, white noise is used and the SNR is 20 dB. The resulting SDRs are listed in Table I. The linear superposition model (LSM) results in the lowest SDR because it ignores the nonlinear effects. The SPEM [1] also yields poor performance in SDR when the transition shift parameter is equal to 0.2 or larger. While Ryan’s model [2], because of linearization, produces high SDRs only for small , our model (21) always results in a very high SDR even for as large as 0.5.
The trellis diagram of the proposed model can be derived from (21) and is identical to those in [1] and [2], except that the model output is modified. The Viterbi algorithm is used to realize the ML detector for the detection of . Since the data can be recovered by the relation
when or when . Note
that recovering from may cause error propagation; this effect, however, is minor in our simulation. The bit error rate (BER) performance of for each model under various SNRs
Fig. 2. Bit error rate of a nonlinear PR4 channel for = 0:3.
for is shown in Fig. 2. Hence, the complexity of the ML detector is not increased and the performance is improved because of the increasing model accuracy.
V. CONCLUSION
We have applied the alternating coordinates minimization algorithm for estimating the parameters of a PETSM. This algorithm can also be used to identify the parameters of the nonlinear PR4-equalized channel. The obtained model greatly increases the modeling accuracy and improves the performance of the corresponding ML detector without increasing its re-alization complexity.
ACKNOWLEDGMENT
The authors are grateful to the anonymous reviewers for their comments which greatly improved the quality of the paper.
REFERENCES
[1] I. Lee, T. Yamauchi, and J. M. Cioffi, “Modified maximum likelihood se-quence estimation in a simple partial erasure model,” in Proc. Int. Conf.
Commun., 1994, pp. 245–249.
[2] W. E. Ryan and N. H. Yeh, “Viterbi detector for pr4-equalized magnetic recording channels with transition-shift and partial erasure nonlineari-ties,” IEEE Trans. Magn., vol. 32, pp. 3950–3952, Sept. 1996. [3] R. Hermann, “Volterra modeling of digital magnetic saturation recording
channels,” IEEE Trans. Magn., vol. 26, pp. 2125–2127, Sept. 1990. [4] B. Lin, K. E. Hild, and J. R. Cruz, “Magnetoresistive read/write channel
models,” IEEE Trans. Magn., vol. 35, pp. 4528–4531, Nov. 1999. [5] X. Che and P. A. Ziperovich, “A time-correlation method of calculating
nonlinearities utilizing pseudo-random sequences,” IEEE Trans. Magn., vol. 30, pp. 4239–4241, Nov. 1994.
[6] Y. S. Cho and N. J. Lee, “An estimation technique for nonlinear distor-tion in high-density magnetic recording channels,” IEEE Trans. Magn., vol. 34, pp. 40–44, Jan. 1998.
[7] T. S. Kao and M. H. Cheng, “Expectation and maximization algorithm for estimating parameters of a simple partial erasure model,” IEEE
Trans. Magn., vol. 39, pp. 608–612, Jan. 2003.
[8] I. Csiszárr and G. Tusnády, “Information geometry and alternating minimization procedures,” Statist. Decis., vol. supplement issue, pp. 205–237, 1984.
[9] A. Yeredor, “The extended least squares criterion: minimization algo-rithms and applications,” IEEE Trans. Signal Processing, vol. 49, pp. 74–86, Jan. 2001.
[10] J. G. Proakis, “Equalization techniques for high-density magnetic recording,” IEEE Signal Processing Mag., vol. 15, pp. 73–82, July 1998.
[11] R. W. Wood and D. A. Petersen, “Viterbi detection of class IV partial response on a magnetic recording channel,” IEEE Trans. Magn., vol. 34, pp. 454–461, May 1986.
