IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 6, JUNE 2010
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Pattern-Dependent Noise Predictive Soft Detection in the Post-Processor With Error Filters Ivana Djurdjevic1 , Bruce A. Wilson2 , and Travis R. Oenning2 Applied Micro Circuits Corporation, Sunnyvale, CA 94098 USA Hitachi Global Storage Technologies, San Jose, CA 95135 USA This paper investigates a pattern-dependent noise predictive soft detection method for channel architectures that are based on a long target response and post-processing rather than a short target response and base-line wander compensation. We utilize properties of the autoregressive pattern-dependent noise model to compare tentative Viterbi sequence with alternative sequences in the post-processor and efficiently compute soft information. Even though the post-processor cannot consider all possible sequences like trellis-based detectors can, we demonstrate for a short target response that our post-processing solution does not experience any loss in performance compared to the optimal maximum a posteriori trellis-based soft detector. The complexity of the computations in the post-processor grows only linearly with the target length, as opposed to the exponential growth in complexity in trellis-based detectors. In this way we can efficiently perform nearly optimal pattern-dependent soft detection in the post-processor for a very long target response without base-line wander compensation. Index Terms—Base-line wander compensation, pattern-dependent autoregressive noise model, perpendicular magnetic recording, post-processing, soft detection.
I. INTRODUCTION
A
PERPENDICULAR recording read-back signal has valuable content at low frequencies, but typically has a notch at DC. A short DC-free target attenuates valuable signal components close to DC and causes performance loss. There are two approaches that enable effective use of low-frequency signal components. In both approaches a high-pass filter (HPF) with a notch very close to DC is used, resulting in a long tail in a response. One solution is to apply baseline wander compensation (BLC) and eliminate the long tail in a response [1]. In this case, a trellis-based detector for a short DC target can be used. Another solution is to use a post-processor for a very long DC-free target response and correct the tentative decision of the Viterbi algorithm for a short DC-free target response [2], [3]. Unlike in trellis-based detectors, the complexity of a post-processor grows only linearly with target length. The noise in magnetic recording read channels exhibits strong pattern dependency, mainly due to transition jitter noise. Long bit sequences of the same polarity are mostly “quiet”, while patterns with many transitions tend to be very noisy. The common model for magnetic recording noise is pattern-dependent auat the detector input toregressive model [4]–[6]. The noise is assumed to be a pattern-dependent autoregressive (Markov) process: (1)
represents a sequence of recorded bits that impact parameters of the noise statistics at time . The noise predictors and standard deviation of the white Gaussian noise part depend on the recorded pattern . The number of noise predictors and the length of the pattern are model parameters. A pattern
Manuscript received October 31, 2009; revised January 09, 2010; accepted February 01, 2010. Current version published May 19, 2010. Corresponding author: I. Djurdjevic (e-mail:
[email protected]). Digital Object Identifier 10.1109/TMAG.2010.2043227
The optimal values of and required to accurately represent pattern-dependent noise can vary. The complexity of any pattern-dependent detector increases as the pattern length and the number of noise predictors increase. However, sufficient accuracy is typically achieved for relatively small values of and . The values of the noise predictors and the white Gaussian noise standard deviation are obtained as a least squares solution during training. Additional improvement in a detector performance can be achieved by extending it to include pattern dependency of the noise. The optimal detectors for the autoregressive pattern-dependent noise model are trellis-based maximum likelihood (ML) or maximum a posteriori (MAP) detectors, with branch computation modified to account for the pattern dependency of the noise [4]–[7]. Due to exponential growth in complexity with target length, trellis-based detectors can be used only for short DC targets and well-designed BLC. Efficient hard and soft detection in a post-processor have been extensively investigated in [2], [3], [8], [9]. The goal of this work is improving the performance of the soft post-processor by accurate and simple computation of soft bit information that includes both long target response and pattern-dependent noise.
II. SOFT POST-PROCESSING DETECTION A sequence of bits has a metric associated with it that represents a measure of a posteriori probability of that se, quence. In this work we assume that where is the received sequence at the detector input. Trellisbased detectors consider every possible bit sequence in a computationally efficient way and select the one with the best metric. For very long targets however, trellis-based algorithms become too complex to implement. In channel architectures with a postprocessor, a post-processor receives a sequence at the output of the Viterbi algorithm that uses a short target response approximation. A post-processor considers improved metric of a longer and more accurate target response, and computes the difference
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between the improved metric of the tentative Viterbi sequence and candidate alternative sequences: (2) Any alternative sequence in (2) differs from the tentative sequence by an error event . We assume that bits take values , and that an error event components take values . The number of positions where an error event has non-zero values is very small compared to the sequence length. The position where an error event has its first non-zero value is the position where an error event starts. The position where an error event has its last non-zero value is the position where an error event ends. Two error events that have the same non-zero sequence starting at different locations are said to be error events and of the same type. For example, are two error events of the same type starting at different bit positions. An error event type is and denoting only labeled using symbols in the set values from the starting to the ending position. For example, and has the label . In a post-proerror event type of cessor we consider a set of most common error event types for , perpendicular recording, e.g., starting at every bit position. The length of the error event is equal to the length of its error event type label. The difference in between tentative and alternative sequence can be metric efficiently computed using error filters and matched filter metric is also called an error value as [3]. The difference metric it is associated with a particular error event. Note that not every error event type can occur at every location in a bit sequence. An . error value of an error event that is not plausible is which leaves the Also, we have to consider the error event tentative sequence unchanged at every bit position and has error . The soft post-processor (PP) detection in [9] value uses the computed error values to determine soft information, i.e., log-likelihood ratio (LLR) of a bit at position as follows:
(3) is a set of all error events that result in a bit at position where to be 0 and is a set of all error events that result in bit at position to be 1. III. PATTERN-DEPENDENT NOISE-PREDICTIVE SOFT DETECTION IN THE POST-PROCESSOR (SOFT PDNP-PP) In the proposed PDNP soft detector in a post-processor, the LLR computation remains the same as in (3). The main problem solved in this paper is accurate and efficient computation of the to simultaneously take into account patdifference metric tern-dependent noise and long target response. Fig. 1(a) illustrates a general block diagram of the equalization and soft post-processing detection. The received digital signal is equalized using a digital finite impulse response (DFIR) filter and adaptive equalization. Since the post-processor and Viterbi use different target responses, two different equalizers are applied. The sequence at the output of the second DFIR is denoted as and represents the first input to the soft
Fig. 1. Examples of equalization and soft post-processing detection block diagrams.
post-processor, in our case soft PDNP-PP. The second input to the post-processor is tentative Viterbi output . Frequently, the Viterbi target response and the post-processor target response have a common factor. If this is the case, the received digital signal can be equalized to a single response, i.e., a common factor, and additional filtering can be applied to further shape the input of the Viterbi and the post-processor. This type of architecture can simplify equalization and is depicted in Fig. 1(b). More details on equalization as well as examples of long and short equalization targets used in this work can be found in [3]. Note, however, that in [3] implementation oriented to a matched filter metric [10], [11], is described, whereas we are focused on Euclidean metric implementation. Therefore, all targets in [3] represent matched filtered targets of the ones used in this work. A specific short equalization target example used in this work is [1, 1]. The shaping filter at the Viterbi input is a convolution of and , where and are pattern independent factor noise predictors determined in optimization. The is necessary due to a notch at DC. The shaping filter at the post-processor input is very long, for example more than 20 as a factor. As shown in taps long, and also contains [3], this filter can be fully determined with only one parameter which is optimized during training. In the soft PDNP-PP, autoregressive pattern-dependent noise model is applied to the received signal and the knowledge of the set of most likely error events is utilized to accurately and . In this efficiently compute the set of difference metrics way, LLR can be provided and a tentative Viterbi output can be corrected. Details on the difference metrics computation are given in the remainder of this section. Incorporating input LLR is not discussed in this paper, but the basic principle is the same as in [9]. Fig. 2 depicts the problem we are trying to solve. Assume that the sequence length is . The point in an -dimensional space represents an ideal long target response corresponding to a tentative sequence, i.e., convolved with the long target . Similarly, the point represents an ideal long target response convolved corresponding to an alternative sequence, i.e., with the long target . The point represents a received equalized sequence at the detector input.
DJURDJEVIC et al.: PATTERN-DEPENDENT NOISE PREDICTIVE SOFT DETECTION IN THE POST-PROCESSOR WITH ERROR FILTERS
Fig. 2. Three points in an n-dimensional space corresponding to tentative, alternative and received sequences.
Assuming the autoregressive pattern-dependent noise model, the metrics of tentative and alternative sequences are given as follows:
(4) The matrices and are autocorrelation matrices of the -dimensional noise vectors corresponding to the two bit sequences. In the case of pattern-dependent autoregressive noise model with a small number of predictors and short memory, the inverse matrices of these two matrices posses several important properties that allow us to perform an error value computation in a relatively easy way on a small number of received samples. In Fig. 3 we depict these properties. It can be shown and can be directly computed from the that matrices noise predictor values. Also, the matrices and have non-zero main diagonal values, a small number of non-zero values around the main diagonal, and all zeros elsewhere. Finally, the two matrices are the same almost everywhere and the size of the sub-matrix where they are different, highlighted in , does not depend on the target length, but gray and denoted on the number of noise predictors, the length of the pattern and on the error event length. These numbers are small compared to the target length. The number of noise predictors and the pattern length that achieve a good trade-off between performance and complexity are commonly determined in simulation. are always suffiWe have observed that values ciently large to obtain good performance gain. As a result of the above described properties, a difference metric or an error value can be expressed as follows:
(5)
where
, and . In (5), the error value computation is broken into two parts. A matrix is called discrepancy matrix and a constant is called a discrepancy constant. A discrepancy matrix and a discrepancy constant depend on the underlying error event and the tentative bit sequence surrounding the underlying error event.
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Fig. 3. The inverses of autocorrelation matrices corresponding to tentative and alternative sequences.
What is important to note is that the number of different discrepancy matrices and constants as well as the size of a discrepancy matrix is independent of the target length. For example, if and the number of we use model with pattern length , for the shortest error event type there noise predictors are 16 discrepancy matrices and discrepancy constants and the size of each discrepancy matrix is 4 4, regardless of the target appears complex length. Hence, even though the term to compute as it includes matrix multiplication, it involves a small number of samples independent of target length. The term does not depend on the alternative sequence and we need to use only one set of noise predictors corresponding to the tentative sequence. This simplifies computation significantly as a can be pre-computed. The term large part of represents the energy in the long target response of an error event after whitening using the set of noise predictors corresponding to the tentative sequence, whereas the term represents a scalar product of and after both vectors have been whitened with the set of noise predictors corresponding to the tentative sequence. Even though a vector is very long, we only need to consider its components at pohas non-zero components. Since error sitions where events are typically short, the number of non-zero components of vector is proportional to the target length. Further simplification is possible using ideas given in [10] and [11]. Hence, the does not include matrix multipliactual computation of cation, but it does involve a larger number of samples proportional to the target length. In this way we can implement an exact PDNP difference metric computation in a post-processor based on error filters with relatively small complexity that grows only linearly with target length. IV. SOFT PDNP-PP PERFORMANCE The performance gain of the proposed soft pattern-dependent noise-predictive post-processing detector (soft PDNP-PP) is shown in Fig. 4. The magnetic recording read-back signal is generated assuming a hyperbolic tangent transition response. The percentage of jitter noise power is defined with respect to an all transition pattern. The user bit density is defined as where is the time required for transition response to rise from 1/2 to 1/2 of a maximum amis the uncoded bit duration. Simple inner itplitude and
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length 2. With HPF turned on, a part of the valuable signal content is inevitably lost. Also, we have to use a long target in the post-processing. Nevertheless, the soft PDNP-PP achieves close to 0.9 dB over soft PP. Note that PDNP-MAP for target of length 23 is prohibitively complex to implement. REFERENCES
Fig. 4. The performance of the PDNP soft detector in the post-processor based on error filters.
erative code and outer Reed–Solomon code are used [3]. The soft PP detector does not account for any pattern dependency of the noise. In the simulated environment, we can safely turn off a HPF and apply PDNP-MAP together with a short target response. Accounting for pattern dependency of the noise and including valuable signal components at low frequencies provides approximately 1.1 dB gain. The PDNP-MAP detector uses noise predictor and pattern length . Similar gain can be achieved with the proposed soft PDNP-PP under noise predictor and with patthe same conditions, with tern length . The match between PDNP-MAP and soft PDNP-PP results indicates that soft PDNP-PP is correctly implemented and well-designed. Both for PDNP-MAP detection and for soft PDNP-PP detection, the equalization target is of
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