# Noise-predictive maximum-likelihood detection

Noise-Predictive Maximum-Likelihood (NPML) is an advanced digital signal-processing method suitable for magnetic data storage systems that operate at high linear recording densities. It is used for reliable retrieval of data recorded in the magnetic medium.

Data are read back as a weak and noisy analog signal by the read head, and NPML aims at minimizing the influence of noise in the detection process. Therefore, it allows recording data at higher areal densities than other detection schemes, such as peak detection, partial response maximum likelihood (PRML), and extended partial-response maximum likelihood (EPRML) detection.[1]

Although advances in head and media technologies have historically been the driving forces behind the increases in the areal recording density, digital signal processing and coding established themselves as cost-efficient techniques for enabling additional substantial increases in areal density while preserving the high reliability of hard disk drive (HDD) systems.[1] Accordingly, the deployment of sophisticated detection schemes based on the concept of noise prediction are of paramount importance in the HDD industry.

## Principles

In general, NPML refers to a family of sequence-estimation data detectors, which arise by imbedding a noise prediction/whitening process[2][3][4] into the branch metric computation of the Viterbi algorithm, which is a well known data detection technique for communication channels that exhibit intersymbol interference (ISI) with finite memory.

Reliable operation of the prediction/whitening process is in general achieved by using hypothesized decisions associated with the branches of the Trellis on which the Viterbi algorithm operates as well as tentative decisions corresponding to the path memory associated with each trellis state. The NPML detectors can thus be viewed as a family of reduced-state sequence-estimation detectors offering a range of implementation complexities. The complexity is essentially governed by the number of detector states, which is equal to 2K, 0 ≤ KM, with M denoting the maximum number of controlled ISI terms introduced by the combination of a partial-response shaping equalizer and the noise predictor. By judiciously choosing the parameter K, practical NPML detectors can be devised for the magnetic recording channel that provide a substantial performance improvement over PRML and EPRML detectors in terms of error rate and/or linear recording density[2][3][4]

Assuming that there is neither noise enhancement nor noise correlation, the PRML sequence detector performs maximum-likelihood sequence estimation. But, as the operating point moves to higher linear recording densities, this assumption no longer holds and there is a loss of optimality associated with linear partial-response (PR) equalization, which enhances noise and renders it correlated. Clearly, a very close match between the desired target polynomial and the physical channel will guarantee that this loss is minimal. An effective way to achieve near optimal performance independently of the operating point—in terms of linear recording density—and the noise conditions is via noise prediction. In particular, the power of a stationary noise sequence ${\displaystyle n(D)}$, where the ${\displaystyle D}$ operator corresponds to a delay of one bit interval, at the output of a PR equalizer can be minimized by using an infinitely long predictor. A linear predictor with coefficients {pl}, l = 1, 2,…,[needs copy edit] operating on the noise sequence ${\displaystyle n(D)}$ will produce the estimated noise sequence ń(D). Then, the prediction-error sequence given by e(D) = n(D) - ń(D) = n(D) (1 - P(D))[needs copy edit] is white with minimum power. The optimum predictor P(D) = p1D + p2D2 + …[needs copy edit], or the optimum noise-whitening filter W(D) = 1 - P(D)[needs copy edit], is the one which minimizes the prediction error sequence e(D)[needs copy edit] in a mean-square sense [2][3][4][5][6]

An infinitely long predictor filter would lead to a sequence detector structure that requires an unbounded number of states. Therefore, finite-length predictors that render the noise at the input of the sequence detector approximately white are of interest. Generalized PR shaping polynomials of the form G(D) = F(D) × W(D)[needs copy edit], where F(D)[needs copy edit] is a polynomial of order S and the noise-whitening filter W(D)[needs copy edit] has a finite order of L, give rise to NPML systems when combined with sequence detection[2][3][4][5][6] In this case, the effective memory of the system is limited to M = L + S, requiring a 2L+S[needs copy edit]-state NPML detector if no reduced-state detection is employed.

As an example, if F(D) = 1 - D2[needs copy edit] then this corresponds to the classical PR4 signal shaping. Using a whitening filter W(D)[needs copy edit], the generalized PR target becomes G(D) = (1 - D2) × W(D)[needs copy edit], and the effective ISI memory of the system is limited to M = L + 2 symbols. In this case, the full-state NMPL detector performs maximum likelihood sequence estimation (MLSE) using the 2L+2-state trellis corresponding to G(D)[needs copy edit].

The NPML detector is efficiently implemented by using the Viterbi Algorithm, which recursively computes the estimated data sequence. [2][3][4][5][6]

â(D) = arg mina(D) ǁz(D) - a(D)G(D2[needs copy edit], where a(D)[needs copy edit] denotes the binary sequence of recorded data bits and z(D) the signal sequence at the output of the noise whitening filter W(D)[needs copy edit].

Reduced-state sequence-detection schemes[7][8][9] have also been studied extensively for application in the magnetic-recording channel [2][4] and the references therein. For example, it can readily be seen that the NPML detectors with generalized PR target polynomials G(D) = F(D) × W(D)[needs copy edit] can be viewed as a family of reduced-state detectors with embedded feedback. These detectors also exist in a form in which the decision-feedback path can be realized by simple table look-up operations, whereby the contents of these tables can be updated as a function of the operating conditions.[2] Analytical and experimental studies have shown that a judicious tradeoff between performance and state complexity leads to practical schemes with considerable performance gains. Thus, reduced-state approaches are promising for increasing the linear density even further.

Depending on the surface roughness and particle size, particulate media might exhibit nonstationary data-dependent transition or medium noise rather than colored stationary medium noise. Improvements on the quality of the readback head as well as the incorporation of low-noise preamplifiers may render the data-dependent medium noise a significant component of the total noise affecting the performance of the magnetic-recording system. Because medium noise is correlated and data-dependent, information about the noise and data patterns in past samples can provide information about the noise in the current sample. Thus, the concept of noise prediction for stationary Gaussian noise sources developed in [2][6] can be naturally extended to the case where the noise characteristics depend highly on the local data patterns.[1][10][11][12] By modeling the data-dependent noise as a finite-order Markov process, the optimum MLSE for channels with ISI has been derived.[11] In particular, it has been shown that when the data-dependent noise is conditionally Gauss–Markov, the branch metrics can be computed from the conditional second-order statistics of the noise process. In other words, the optimum MLSE can be implemented efficiently by using the Viterbi algorithm, in which the branch-metric computation involves data-dependent noise prediction.[11] Because the predictor coefficients and prediction error both depend on the local data pattern, the resulting structure has been called data-dependent NPML detector. [1][12][13] The reduced-state sequence detection schemes discussed above in connection with NPML detection can also be applied to data-dependent NPML, providing a significant reduction of implementation complexity.

Finally, NPML and its various forms also represent the core read-channel and detection technology used in recording systems employing advanced error-correcting codes that lend themselves to soft decoding, such as low-density parity check (LDPC) codes. For example, if noise-predictive detection is performed in conjunction with a maximum a posteriori (MAP) detection algorithm such as the BCJR algorithm[14] then NPML and NPML-like detection allow the computation of soft reliability information on individual code symbols, while retaining all the performance advantages associated with noise-predictive techniques. The soft information generated in this manner is used for soft decoding of the error-correcting code. Moreover, the soft information computed by the decoder can be fed back again to the soft detector to improve detection performance. In this way it is possible to iteratively improve the error-rate performance at the decoder output in successive soft detection/decoding rounds.

## History

In past three decades, several digital signal-processing and coding[disambiguation needed] techniques were introduced into hard disk drives to improve the drive error-rate performance for operation at ever higher areal densities as well as for reducing the manufacturing and servicing costs. In the early 1990s, partial-response class-4[15][16][17] (PR4) signal shaping in conjunction with maximum-likelihood sequence detection, eventually known as PRML technique [15][16][17] replaced the peak detection systems that used run-length-limited (RLL) (d,k)-constrained coding. This development also paved the way for future applications of advanced coding and signal-processing techniques [1] in magnetic data storage.

NPML detection was first described in 1996 [4][18] and eventually found wide application in the read channel design of HDDs. The “noise predictive” concept was later extended to handle not only autoregressive (AR) noise processes but also autoregressive moving-average (ARMA) stationary noise processes [2] The concept was also extended to include a variety of non-stationary noise sources, such as head, transition jitter and media noise;[10][11][12] it was applied with great success to the design of various post-processing schemes[19][20][21] for further improvement of the error rate performance. Today noise prediction is used as an integral part of the metric computation in a wide variety of iterative detection/decoding schemes.

The pioneering research work on partial-response maximum-likelihood (PRML) and noise-predictive maximum-likelihood (NPML) detection and its impact on the industry were recognized in 2005 by the prestigious European Eduard Rhein Foundation Technology Award.

## Applications

The NPML technology and its reduced complexity variants were first introduced into IBM’s line of hard disk drive products in the late 1990s, a business that was acquired by Hitachi Global Storage Technology[22] (HGST) in 2002. Eventually, noise-predictive detection became a de facto standard and in its various instantiations became the core technology of the read channel module in HDD systems.[23][24]

In 2010, NPML was introduced into IBM’s Linear Tape Open (LTO) tape drive products and in 2011 also into IBM’s enterprise-class tape drives.

## References

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