Method and apparatus for calibrating read-head off-track detector

Abstract

In a data storage device having a storage medium, wherein data is written to tracks on the storage medium, data for each track including a preamble, and wherein the preamble in any current track is orthogonal to the preamble in any track adjacent to the current track, and wherein data accumulated in a multi-dimensional space is representative of a relationship between signals from the current track and signals from at least one adjacent track, the relationship between the signals from the current track and the signals from the at least one adjacent track is calibrated by, for each respective position out of a plurality of positions in the multi-dimensional space, accumulating a plurality of data values for the respective position, and determining, from the plurality of data values for the respective position, an inverted covariance matrix without performing a matrix inversion operation, either during run-time or prior to run-time.

Claims

1. A method of operating a data storage device having a storage medium, wherein data is written to tracks on the storage medium, data for each track including a preamble, and wherein the preamble in any current track is orthogonal to the preamble in any track adjacent to the current track, and wherein data accumulated in a multi-dimensional space is representative of a relationship between signals from the current track and signals from at least one adjacent track, the method comprising: calibrating, in control circuitry of the data storage device, the relationship between the signals from the current track and the signals from the at least one adjacent track by, for each respective position out of a plurality of positions in the multi-dimensional space: accumulating, in the control circuitry, a plurality of data values for the respective position; and determining in the control circuitry, from the plurality of data values for the respective position, an inverted covariance matrix without performing a matrix inversion operation.

2. The method of claim 1, wherein the method is performed continually during run-time of the data storage device.

3. The method of claim 2 wherein the determining, in the control circuitry, an inverted covariance matrix, comprises: determining, in the control circuitry, a respective mean of each respective one of the plurality of data values; determining, in the control circuitry, a respective noise component relative to each respective mean; correlating, in the control circuitry, each respective noise component for each respective mean to each other respective noise component for each other respective mean; determining, in the control circuitry, a respective error between each respective noise component and each other respective noise component; and determining, in the control circuitry, elements of the inverted covariance matrix from respective errors and respective noise components.

4. The method of claim 3 wherein the determining a respective mean of each respective one of the plurality of data values comprises adaptively accumulating the mean.

5. The method of claim 3 wherein the correlating comprises adaptively filtering each respective noise component against each other respective noise component.

6. The method of claim 5 wherein the adaptively filtering comprises applying an adaptive least-mean-squares filter.

7. The method of claim 5 wherein the determining elements of the inverted covariance matrix from respective errors and respective noise components further comprises factoring in coefficients of the filter.

8. The method of claim 7 wherein the determining elements of the inverted covariance matrix from respective errors and respective noise components comprises taking a product of each respective error and its respective noise component.

9. The method of claim 3 wherein the determining elements of the inverted covariance matrix from respective errors and respective noise components comprises taking a product of each respective error and its respective noise component.

10. The method of claim 1 wherein the method is performed prior to run-time of the data storage device.

11. The method of claim 10 wherein the determining an inverted covariance matrix comprises: accumulating, in the control circuitry, each respective one of the plurality of data values, and determining a respective mean from each respective accumulated data value; determining, in the control circuitry, respective variance values by accumulating multiplied respective ones of the respective data values; and determining, in the control circuitry, elements of the inverted covariance matrix from predetermined relationships among the respective variance values.

12. The method of claim 11 wherein the determining respective variance values further comprises subtracting products of the respective means from multiplied respective ones of the respective data values.

13. The method of claim 10 further comprising computing a determinant of a covariance matrix from the predetermined relationships among the respective variance values.

14. A data storage device comprising: a storage medium, wherein data is written to tracks on the storage medium, data for each track including a preamble, and wherein the preamble in any current track is orthogonal to the preamble in any track adjacent to the current track, and wherein data accumulated in a multi-dimensional space is representative of a relationship between signals from the current track and signals from at least one adjacent track; and circuitry operable to calibrate the relationship between the signals from the current track and the signals from the at least one adjacent track by, for each respective position out of a plurality of positions in the multi-dimensional space: accumulating a plurality of data values for the respective position; and determining, from the plurality of data values for the respective position, an inverted covariance matrix without performing a matrix inversion operation.

15. The data storage device of claim 14 wherein the circuitry operable to calibrate the relationship between the signals from the current track and the signals from the at least one adjacent track is operable to run continually during run-time of the data storage device.

16. The data storage device of claim 15 wherein the circuitry operable to calibrate the relationship between the signals from the current track and the signals from the at least one adjacent track comprises adaptive filter circuitry that correlates each respective noise component in a plurality of respective noise components against each other respective noise component.

17. The data storage device of claim 16 wherein the adaptive filter circuitry comprises a respective adaptive filter circuit for each respective noise component, operable to correlate that respective noise component against each other respective noise component.

18. The data storage device of claim 17 wherein the circuitry operable to calibrate the relationship between the signals from the current track and the signals from the at least one adjacent track by determining an inverted covariance matrix comprises: a subtractor that determines a respective error between each respective noise component and each other respective noise component; a respective multiplier that determines a respective product of a respective noise component and error output by each respective adaptive filter circuit; and an averaging circuit to determine an expected value of the respective product.

19. The data storage device of claim 14 wherein the circuitry operable to calibrate the relationship between the signals from the current track and the signals from the at least one adjacent track is operable to prior to run-time of the data storage device.

20. The data storage device of claim 19 wherein the circuitry operable to calibrate the relationship between the signals from the current track and the signals from the at least one adjacent track comprises: circuitry operable to accumulate each respective one of the plurality of data values, and to determine a respective mean from each respective accumulated data value; circuitry operable to determine respective variance values by accumulating multiplied respective ones of the respective data values; and circuitry operable to determine elements of the inverted covariance matrix from predetermined relationships among the respective variance values.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Further features of the disclosure, its nature and various advantages, will be apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings, in which like reference characters refer to like parts throughout, and in which:

(2) FIG. 1 is a side elevational view of a portion of a disk drive with which the present disclosure may be used;

(3) FIG. 2 is a plan view of the disk drive portion of FIG. 1, taken from line 2-2 of FIG. 1;

(4) FIG. 3 is a schematic representation of an example of adjacent data tracks with an off-track read head;

(5) FIG. 4 is a schematic representation of a controller for the drive of FIGS. 1 and 2;

(6) FIG. 5 shows an arrangement of three tracks having three orthogonal preamble patterns, with two read heads arranged relative to those tracks;

(7) FIG. 6 is a graphical representation of the outputs of one of the two read heads of FIG. 5;

(8) FIG. 7 is a graphical representation of the outputs of another of the two read heads of FIG. 5;

(9) FIG. 8 is a two-dimensional plot of calibration samples according to an implementation of the subject matter of this disclosure;

(10) FIG. 9 is a schematic representation of circuitry in the read channel of a disk drive according to an implementation of the subject matter of this disclosure;

(11) FIG. 10 is a timing diagram of the operation of the circuitry of FIG. 9 according to an implementation of the subject matter of this disclosure;

(12) FIG. 11 is a graphical representation of an example of probability distributions in two clusters of calibration samples in a plot such as that of FIG. 8;

(13) FIG. 12 is a schematic representation of a maximum likelihood detector circuit according to a first variant of a ratio-based implementation of the subject matter of this disclosure;

(14) FIG. 13 is a schematic representation of a maximum likelihood detector circuit according to a second variant of a ratio-based implementation of the subject matter of this disclosure;

(15) FIG. 14 is a schematic representation of a maximum likelihood detector circuit according to a first variant of a magnitude-based implementation of the subject matter of this disclosure;

(16) FIG. 15 is a schematic representation of a filter circuit according to implementations of this disclosure;

(17) FIG. 16 is a flow diagram of an implementation of an online method of calibrating an off-track detector in a storage device according to this disclosure; and

(18) FIGS. 17A and 17B (hereinafter collectively referred to as FIG. 17) is a flow diagram of an implementation of an off-line method of calibrating an off-track detector in a storage device according to this disclosure.

DETAILED DESCRIPTION

(19) FIGS. 1 and 2 show an example of a disk drive 100 with which the present disclosure may be used. In this particular example, disk drive 100 has three platters 101, 102, 103, although any number of platters may be included in a disk drive with which the present disclosure may be used. As shown, each platter 101, 102, 103 has, on each of its upper and lower surfaces 111, 112, a coating 110 made from a material in which data can be stored, e.g., magnetically. The present disclosure also is relevant to a disk drive in which one or more platters includes coating 110 on only one of its surfaces, but such a disk drive would store less data in the same volume than a disk drive with two-sided platters. The platters 101-103 are mounted on a rotatable spindle 104. Motor 105 rotates spindle 104 to rotate platters 101-103, for example, in the direction of arrow A (FIG. 2). Although motor 105 is shown connected directly to spindle 104, in some cases motor 105 may be located off-axis of spindle 104 and would be connected to spindle 104 through belts or gears (not shown).

(20) Read/write head assembly 120 includes an actuator 121 that bears arms 122-125, one of which is disposed adjacent to each surface 111, 112 of a platter 101, 102, 103 that has a memory storage coating 110. In this example, with heads on both surfaces of each of arms 123, 124, that amounts to four arms 122-125, but in the single-sided platter example discussed above, there would be only three arms. In other examples, the number of arms would increase or decrease along with the number of platters.

(21) Each arm 122-125 bears, at or near its end furthest from actuator 121, and on both its upper and lower surfaces in the case of arms 123, 124, a plurality of read heads/sensors and write heads. In this case, two sensors 131, 132 are shown, and will be used to represent read sensors, although it would normally at least be expected that each set of one or more read sensors has a companion write head (not shown). It should be noted that FIGS. 1 and 2 are schematic only and not to scale. For example, the spindle diameter may be larger by comparison to the disk diameter.

(22) A motor 126, commonly referred to as a voice-coil motor, rotates actuator 121 back and forth, for example, along the directions of arrow B (FIG. 2) to move the heads 131, 132 along the path indicated by dashed arrow 201, although arms 122-125 normally cannot point directly at the center of the disk. The motion of actuator 121 thus changes both the radial and circumferential positions of heads 131, 132, but the circumferential positional change is relatively unimportant insofar as the platters are rotating. The motion of actuator 121 thus is used to control the radial position of heads 131, 132.

(23) The location on surface 111 of platter 101 (the other surfaces are similar) of the aforementioned wedges is shown in FIG. 2. Each servo wedge 200 includes data identifying it by wedge, or sector, number (to give an angular, tangential or circumferential position) and by data representing, at each point along a radius of the platter, the distance from spindle 104, although sometimes some of this information is omitted from some of the wedges.

(24) As noted above, as areal densities for magnetic data storage continue to increase, data tracks are being written in an overlapping or shingled fashione.g., in two-dimensional magnetic recording (TDMR)and reading is being performed using read heads or sensors having dimensions comparable to the track width. Therefore, it is important to know whether, and by how much, a read head is deviating from its nominal position relative to a track. If an off-track condition is detected, system parameters can be optimized for better performance based on the amount by which the heads are off-track. For example, filter parameters, as well as the amount of gain for each head, can be adjusted based on the amount by which the heads are off-track.

(25) As shown in commonly-assigned U.S. Pat. No. 10,008,228, which is hereby incorporated by reference herein in its entirety, adjacent tracks may be written with orthogonal preamblesi.e., preambles that are mutually exclusive, such as those described in commonly-assigned U.S. Pat. No. 9,153,264, which is hereby incorporated by reference herein in its entirety. Specifically, the preambles may be chosen to be a pair of periodic patterns such that the inner product of the two patterns in question, when represented as sequences of write current polarities {1,1}, over a window chosen to contain an integer number of periods of both patterns, is zero. For example, a 2T pattern (110011001100 . . . ) may be written to the preambles on each even-numbered track, while a 3T pattern (111000111000111000 . . . ) may be written to the preambles on each odd-numbered track, although any pair of periods where one period is not a multiple of the other may be selected. There may be other patterns as well, such as a 4T pattern (111100001111000011110000 . . . ). Any pair of such orthogonal patterns may be used for adjacent tracks in accordance with this disclosure. These patterns may be described in terms of tones (i.e., sinusoidal signals at the fundamental frequencies of the respective patterns).

(26) A pair of tracks 301, 302 with orthogonal preambles 311, 312 as described above is shown in FIG. 3. Although tracks 301, 302 are shown as being straight, in the case of a rotating medium such as a disk drive platter, tracks 301, 302 actually would be curved. FIG. 3 also shows a read head 300 which is in an off-track condition; although read head 300 nominally should be aligned with track 301, read head 300 as shown is mainly over track 301, but is partially over track 302 as well. The portions of tracks 301, 302 that are shown would be outside the servo wedges 200i.e., preambles 311, 312 are interspersed among the user data outside the servo wedges 200 (e.g., at the beginnings of at least some data packets). Normally, each preamble is followed by one of sync marks 321, 322. However, in some cases, extra preambles may be inserted without sync marks.

(27) Read head 300 is coupled to a read channel 401 of a hard drive controller 400 (FIG. 4). In addition, an unseen write head is coupled to a write channel 402 of hard drive controller 400. Hard drive controller 400 also includes a processor 410 and a memory 411, as well as a connection 412 to a host processor (not shown). Memory 411 may be used to store the position error signal (PES) data that indicates the track position offsets. A servo control loop in hard drive controller 400 uses the PES data to keep read head 300 (as well as the unseen write head) on track.

(28) Commonly-assigned U.S. Pat. No. 9,424,880, which is hereby incorporated by reference herein in its entirety, describes using Discrete Fourier Transforms to measure the tones in the preambles of two adjacent tracks to determine a ratio representing off-track condition.

(29) FIG. 5 shows an arrangement 500 of tracks having preambles with 2T, 3T and 4T patterns. Two read heads H1 (501) and H2 (502) are positioned above track 511, which, in this example, has preambles with a 2T pattern. Read head H1 (501) has a nominal position toward track 512 which has preambles with a 3T pattern, while read head H2 (502) has a nominal position toward track 513 which has preambles with a 4T pattern.

(30) FIGS. 6 and 7 show the outputs of the two read heads. As seen in FIG. 6, when read head H1 (501) is in its nominal position over track 511 (0% off-track), it has a high 2T output 601 and almost no 3T or 4T outputs 602, 603. As read head H1 (501) moves in the negative off-track direction toward track 512, the 4T signal contribution 603 remains negligible, while the 2T signal contribution 601 decreases and the 3T signal contribution 602 increases, to the point that the 3T signal contribution 602 may exceed the 2T signal contribution 601 at 40% displacement toward track 512. Similarly, as read head H1 (501) moves in the positive off-track direction toward track 513, the 3T signal contribution 602 remains negligible, while the 2T signal contribution 601 decreases and the 4T signal contribution 603 increases, but in this example, in this direction, because the nominal position of read head H1 (501) is closer to track 512, at +40% displacement toward track 513, the 2T signal contribution 601 remains higher than the 4T signal contribution 603.

(31) As seen in FIG. 7, when read head H2 (502) is in its nominal position over track 511 (0% off-track), it has a high 2T output 701 and almost no 3T or 4T outputs 702, 703. As read head H2 (502) moves in the positive off-track direction toward track 513, the 3T signal contribution 702 remains negligible, while the 2T signal contribution 701 decreases and the 4T signal contribution 703 increases, to the point that the 4T signal contribution 703 may exceed the 2T signal contribution 701 at +40% displacement toward track 513. Similarly, as read head H2 (502) moves in the negative off-track direction toward track 512, the 4T signal contribution 703 remains negligible, while the 2T signal contribution 701 decreases and the 3T signal contribution 702 increases, but in this example, in this direction, because the nominal position of read head H2 (502) is closer to track 513, at 40% displacement toward track 512, the 2T signal contribution 701 remains higher than the 3T signal contribution 702.

(32) Although FIGS. 5-7 show an example in which three preamble patterns are used, the same principles apply if only two alternating preamble patterns (e.g., 2T, 3T, 2T, 3T, etc., or 3T, 4T, 3T, 4T, etc.) are used. For example, in the alternating 2T/3T case, the graphs in FIGS. 6 and 7 would show contributions from a 2T signal and two 3T signals (which could be labelled, e.g., 3T.sub.right and 3T.sub.left). More generally, one can consider three signals T.sub.center, T.sub.right and T.sub.left, which can be abbreviated T.sub.c, T.sub.r and T.sub.l.

(33) A disk can be calibrated by purposely placing the read heads H1 (501) and H2 (502) at an on-track position and known off-track positions and taking multiple samples at each position. The samples can be plotted in a two-dimensional space representing ratios of magnitudes of preamble signal contributions from the right and left tracks (T.sub.r and T.sub.l) to the magnitude of the preamble signal contribution from the center track (T.sub.c). The ratio m.sub.Tr/m.sub.Tc of the magnitude of the preamble signal contribution from the right track (T.sub.r) to the magnitude of the preamble signal contribution from the center track (T.sub.c) would be one axis of the two-dimensional space, while the ratio m.sub.Tl/m.sub.Tc of the magnitude of the preamble signal contribution from the left track (T.sub.l) to the magnitude of the preamble signal contribution from the center track (T.sub.c) would be another axis of the two-dimensional space. Different clusters of samples would represent different off-track and on-track (i.e., 0% off-track) positions. This is performed for each track of the disk. During actual operation, a similar sample would be taken and, depending on which cluster that sample fell in in the two-dimensional plot for the track in question, would indicate the degree to which the read heads are off-track.

(34) One way of determining which cluster such an actual operational sample belonged to may be determined by finding a minimum distance from the sample, plotted in the same two-dimensional space as the calibration samples, to a representative point within each cluster, which may, for example, be the geometric center of each cluster. The determination of the minimum distance could be a simple Euclidean computation, or something more complicated, as described below.

(35) The discussion below assumes 2T/3T/4T patterns as shown in FIG. 5, but that assumption is only for illustrative purposes. Any two-pattern or three-pattern arrangement can be used, as long as the various patterns are orthogonal to each other, including, e.g., 2T/3T, 3T/4T, and also 2T/4T (any harmonics resulting from a 2T/4T pattern have been found to be negligible).

(36) A two-dimensional plot 800 as described above, for a 2T/3T/4T case, is shown in FIG. 8. In this case, samples were taken for nine head positionson-track, and off-track by 10%, 20%, 30% and 40% in each direction. Each head position resulted in a cluster 801, 802, 803, 804, 805, 806, 807, 808 or 809 of points, where each point represents a pair of signal ratios. The abscissa 810 represents the ratio of signals from read head H1 (501), r.sub.1=m.sub.3T/m.sub.2T, while the ordinate 820 represents the ratio of signals from read head H1 (501), r.sub.2=m.sub.4T/m.sub.2T. A respective point 811, 812, 813, 814, 815, 816, 817, 818 and 819 may be selected as a representative point for each cluster, to which actual data points may be compared as discussed below.

(37) The dashed lines 821, 822, 823, 824, 825, 826, 827 and 828 represent boundaries between clusters and therefore between degrees (in this example, from 40% to +40% in 10% increments) to which the read heads 501, 502 are on-track or off-track. An actual data point is said to indicate that the read heads are on-track or off-track to that degree based on which of representative points 811, 812, 813, 814, 815, 816, 817, 818 and 819 it is closest to. Several variants are discussed below for determining the minimum distance from a data point to one of representative points 811, 812, 813, 814, 815, 816, 817, 818 and 819.

(38) FIG. 9 is a schematic representation of circuitry 900 in the read channel of a disk drive for processing signals from read heads 501, 502, including off-track detection circuitry 950. Signals 901, 902 from read heads 501, 502, respectively, are input to respective analog front ends (AFEs) 951, 952, and then to respective analog-to-digital converters 911, 912.

(39) In the orientation shown in FIG. 5, read heads 501, 502 move down relative to the disk storage medium (although it is actually the disk storage medium that moves in the other direction). Because read head H1 (501) is ahead of read head H2 (502), the output of analog-to-digital converter 911 is passes through delay line 921 to give the signal from read head H2 (502), as output by analog-to-digital converter 912, time to catch up to the signal from read head H1 (501), so that both signals represent the same location on the disk storage medium. The amount of delay introduced by delay line 921 may be programmable to account, for example, for the physical distance between the two read heads 501, 502. Where such a variable delay line 921 is provided, the delay may be programmed by inputting the delay value at 931. Zero-phase-start (ZPS) timing loop 903 controls the sampling times of analog-to-digital converters 911, 912, so that both analog-to-digital converters 911, 912 sample their respective signals at the same phase (notwithstanding the delay separately introduced after sampling by delay line 921).

(40) Sampled (and delayed in the case of signal 941) signals 941, 942 are filtered by, e.g., a two-dimensional finite impulse response (FIR) filter 960, including respective FIR filters 961, 962, which are added at 904 to provide a set FIR samples 905 to be decoded. The filter taps 916 for FIR filters 961, 962 are loaded from a look-up table (FIR LUT) 906 based on the off-track information 937 derived from signals 941, 942 by off-track detector circuitry 907, which includes Discrete Fourier Transform (DFT) compute engine 917 and Off-Track Estimator 927. The output 937 of off-track detector circuitry 907 is normally selected by multiplexer 908 to control selection of filter taps 916. Alternatively, by asserting control signal 918, a user can inputeither manually, or as the output of other circuitry (not shown)a particular LUT selection signal 947 to select the filter taps 916.

(41) The timing 1000 of the operation of the circuitry in FIG. 9 is shown in FIG. 10. A Read operation begins with the assertion of R-Gate signal 1001, at which time a previous selection of FIR filter taps 1002 are in effect. After a defined interval 1003, which may be programmed by the user, ZPS timing loop 903 sends a signal 1004 setting the sampling phase. At the same time, DFT computations begin in DFT compute engine 917 on samples from each of read heads 501, 502 from analog-to-digital converters 911, 912. DFT computation 1005 for trailing read head H2 (502) continues for a predetermined interval 1006 following timing signal 1004, while DFT computation 1007 for leading read head H1 (501) continues for a longer interval 1008 to compensate for the delay introduced by delay line 921. The previous FIR filter taps 1002 remain in effect during those time intervals, and during a further interval 1009 (related to computation latency) following completion of DFT computation 1007 for leading read head H1 (501), to allow for loading of new FIR filter taps 1012. Following the end of interval 1009, further samples are processed using FIR filter taps 1012.

(42) Off-Track Estimator 927 operates as a maximum likelihood detector, computing the minimum distance in the two-dimensional space of FIG. 8 from a data sample, determined by DFT Compute Engine 917, to one of the calibrated reference parameters (i.e., one of representative points 811, 812, 813, 814, 815, 816, 817, 818, 819), as discussed above. The subject matter of this disclosure encompasses at least two implementations of a maximum likelihood detector.

(43) In a first implementation of a maximum likelihood detector according to the subject matter of this disclosure, ratios are computed in a manner similar to that described above in connection with FIG. 8, and those ratios are processed to determine the minimum distance to one of the calibrated reference parameters, which determines the degree to which the read heads in question are off-track.

(44) In order to understand the operation of such an implementation, it should be recognized that the clusters 801, 802, 803, 804, 805, 806, 807, 808, 809 of calibration data points are not identical in shape, which affects the accuracy of the boundaries 821, 822, 823, 824, 825, 826, 827 and 828. Therefore, the minimum distance determination is probabilistici.e., a determination that a measured point is closer to a first reference point than to a second reference point reflects a greater probability that the off-track position is the one associated with the first reference point than that the off-track position is the one associated with the second reference point.

(45) Referring to FIG. 11, if a first cluster 1101 has a distribution .sub.1 with a center point .sub.1, and a second cluster 1102 has a distribution .sub.2 with a center point .sub.2, then assuming that the off-track probability p.sub.1 represented by first cluster 1101 is:
p.sub.1=(2.sub.1.sup.2).sup.0.5e.sup.(r.sup.1.sup.).sup.2.sup./(2.sup.1.sup.2.sup.)
and that the off-track probability p.sub.2 represented by second cluster 1102 is:
p.sub.2=(2.sub.1.sup.2).sup.0.5e.sup.(r.sup.2.sup.).sup.2.sup./(2.sup.2.sup.2.sup.)
then placing the boundary at the intersection A of the two distributions .sub.1 and .sub.2 means that off-track probability p.sub.1 is more likely than off-track probability p.sub.2 if:
.sub.1.sup.2(r.sub.1).sup.2.sub.2.sup.2(r.sub.2).sup.2>2.sub.1.sup.2.sub.2.sup.2 ln(.sub.1/.sub.2)
However, an assumption that the boundary is located at B, halfway between the center points .sub.1 and .sub.2, is accurate only when .sub.1=.sub.2i.e., when the distributions are identical.

(46) Thus, in a first variant of a first implementation using ratios, the probability p(r/i) that for track r, the degree to which the heads are off-track is i (where i=1 to 9, representing off-track conditions from 40% to +40%) is:
p(r/i)=((2).sup.k|C.sub.i|)e.sup.(r.sup.i.sup.).sup.T.sup.C.sup.i.sup.1.sup.(r.sup.i.sup.),
where:

(47) $r=\left[\begin{array}{c}{r}_1\\ r_2\end{array}\right]\mathrm{and}{}_i=\left[\begin{array}{c}{}_{i,1}\\ {}_{i,2}\end{array}\right]$
and C.sub.i is the covariance matrix that accounts for the accurate shape of each distribution .sub.i:

(48) $C_i=E\left[\left(r-{}_i\right){\left(r-{}_i\right)}^T\right]=\left[\begin{array}{cc}{C}_{i,11}& C_{i,12}\\ C_{i,21}& C_{i,22}\end{array}\right].$
Therefore, , the degree to which the heads are off-track, may be given by:

(49) $\hat{i}=\underset{i}{\min }\left(\ln \left(.Math.C_i.Math.\right)+{\left(r-{}_i\right)}^T{C}_i^{-1}\left(r-{}_i\right)\right)$

(50) A maximum likelihood detector circuit 1200 for determining according to this variant of a ratio-based implementation of the subject matter of this disclosure is shown in FIG. 12. Although FIG. 12 shows two paths 1201, 1202 for computing the i=1 case and the i=9 case, additional paths for i=2 through i=7 cases also are present. r is input to all paths 1201, 1202 at 1203. .sub.i is input to respective adder/subtractor 1204 which computes r.sub.i. The respective inverse matrix C.sup.1.sub.i is input to respective multiplier 1205, which computes (r.sub.i).sup.TC.sub.i.sup.1 including the transposition of r.sub.i. Each of those results is then multiplied by r.sub.i at respective multiplier 1206. The multiplications are matrix multiplications and in this implementation, respective multipliers 1205, 1206 multiply one element at a time and the results are accumulated in respective accumulators 1207. Respective adder 1208 adds in the natural logarithm of the determinant of the respective matrix C.sub.i. The same occurs in every path 1201, 1202 and the minimum of all paths is determined at 1209, yielding at 1210.

(51) A second variant of a ratio-based implementation assumes that the clusters are identical (i.e., .sub.1=.sub.2=). It can then be shown that the covariance matrix C.sub.i reduces to:

(52) $C_i=E\left[\left(r-{}_i\right){\left(r-{}_i\right)}^T\right]=\left[\begin{array}{cc}{}^2& 0\\ 0& {}^2\end{array}\right]$
so that reduces to:

(53) $\hat{i}=\underset{i}{\min }\left({\left(r_1-{}_{i,1}\right)}^2+{\left(r_2-{}_{i,2}\right)}^2\right)$
which in matrix form is merely

(54) $\hat{i}=\underset{i}{\min }\left({\left(r-{}_i\right)}^2\right).$

(55) A maximum likelihood detector circuit 1300 for determining according to this variant of a ratio-based implementation of the subject matter of this disclosure is shown in FIG. 13. Although FIG. 13 shows two paths 1301, 1302 for computing the i=1 case and the i=9 case, additional paths for i=2 through i=7 cases also are present. r is input to all paths 1301, 1302 at 1303. .sub.i is input to respective adder/subtractor 1304 which computes r.sub.i. r.sub.i is multiplied by itself at respective multiplier 1305 to yield (r.sub.i).sup.2. The multiplications are matrix multiplications and in this implementation, each of respective multipliers 1305 multiplies one element at a time and the results are accumulated in respective accumulators 1307. The same occurs in every path 1301, 1302 and the minimum of all paths is determined at 1309, yielding at 1310.

(56) Circuit 1300 thus includes fewer components than circuit 1200, but is less accurate for the reasons discussed in connection with FIG. 11, because the distributions .sub.1 and .sub.2 are usually not identical.

(57) In a second implementation of a maximum likelihood detector according to the subject matter of this disclosure, the minimum distance to one of the calibrated reference parameters, which determines the degree to which the read heads in question are off-track, is determined directly from the individual DFT magnitudes, rather than from ratios of those magnitudes. This requires larger matrix calculations.

(58) In such an implementation, the probability p(x/i) that for track x, the degree to which the heads are off-track is i is:

(59) $p\left(x/i\right)=\frac{1}{\sqrt{{\left(2\right)}^k.Math.C_i.Math.}}{e}^{-\frac{{\left(x-{}_i\right)}^T{C}_i^{-1}\left(x-{}_i\right)}{2}}$
where x is a matrix of the individual magnitudes rather than the ratios of those magnitudes:

(60) $x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]\mathrm{and}{}_i=\left[\begin{array}{c}{}_{i,1}\\ {}_{i,2}\\ {}_{i,3}\\ {}_{i,4}\end{array}\right]\mathrm{and}\text{:}$$C_i=E\left[\left(x-{}_i\right){\left(x-{}_i\right)}^T\right]=\left[\begin{array}{cccc}{C}_{i,11}& C_{i,12}& C_{i,13}& C_{i,14}\\ C_{i,21}& C_{i,22}& C_{i,23}& C_{i,24}\\ C_{i,31}& C_{i,32}& C_{i,33}& C_{i,34}\\ C_{i,41}& C_{i,42}& C_{i,43}& C_{i,44}\end{array}\right]$
Therefore, , the degree to which the heads are off-track, may be given by:

(61) $\hat{i}=\underset{i}{\min }\left(\ln \left(.Math.C_i.Math.\right)+{\left(x-{}_i\right)}^T{C}_i^{-1}\left(x-{}_i\right)\right).$

(62) A maximum likelihood detector circuit 1400 for determining according to one variant of a magnitude-based implementation of the subject matter of this disclosure is shown in FIG. 14. Although FIG. 14 shows two paths 1401, 1402 for computing the i=1 case and the i=9 case, additional paths for i=2 through i=7 cases also are present. x is input to all paths 1401, 1402 at 1403. .sub.i is input to respective adder/subtractor 1404 which computes x.sub.i. The respective inverse matrix C.sup.1.sub.i is input to respective multiplier 1405, which computes (x.sub.i).sup.TC.sub.i.sup.1 including the transposition of x.sub.i. That result is then multiplied by x.sub.i at respective multiplier 1406. The multiplications are matrix multiplications and in this implementation, respective multipliers 1405, 1406 multiply one element at a time and the results are accumulated in respective accumulators 1407. Respective adder 1408 adds in the natural logarithm of the determinant of the respective matrix C.sub.i. The same occurs in every path 1401, 1402 and the minimum of all paths is determined at 1409, yielding at 1410.

(63) A second variant of a magnitude-based implementation of the subject matter of this disclosure has multiple modes. In a first mode, equal distributions are assumed as in the second variant of the first implementation, so that:

(64) 0$C_i=\left[\begin{array}{cccc}{}^2& 0& 0& 0\\ 0& {}^2& 0& 0\\ 0& 0& {}^2& 0\\ 0& 0& 0& {}^2\end{array}\right]$
which reduces circuit 1400 in manner similar to the way circuit 1200 is reduced to circuit 1300.

(65) In other modes of this variant, different diagonals are assumed to be all zeroes. Thus, in a second mode, only the main diagonal has non-zero values but the values are not all the same:

(66) $C_i=\left[\begin{array}{cccc}{C}_{i,11}& 0& 0& 0\\ 0& C_{i,22}& 0& 0\\ 0& 0& C_{i,33}& 0\\ 0& 0& 0& C_{i,44}\end{array}\right]$
In a third mode, the main diagonal and two side diagonals have non-zero values:

(67) $C_i=\left[\begin{array}{cccc}{C}_{i,11}& C_{i,12}& 0& 0\\ C_{i,21}& C_{i,22}& C_{i,23}& 0\\ 0& C_{i,32}& C_{i,33}& C_{i,34}\\ 0& 0& C_{i,43}& C_{i,44}\end{array}\right]$
In a fourth mode, the main diagonal and four side diagonals have non-zero values:

(68) $C_i=\left[\begin{array}{cccc}{C}_{i,11}& C_{i,12}& C_{i,13}& 0\\ C_{i,21}& C_{i,22}& C_{i,23}& C_{i,24}\\ C_{i,31}& C_{i,32}& C_{i,33}& C_{i,34}\\ 0& C_{i,42}& C_{i,43}& C_{i,44}\end{array}\right]$
In a fifth mode, alternating diagonals have non-zero values:

(69) $C_i=\left[\begin{array}{cccc}{C}_{i,11}& 0& C_{i,13}& 0\\ 0& C_{i,22}& 0& C_{i,24}\\ C_{i,31}& 0& C_{i,33}& 0\\ 0& C_{i,42}& 0& C_{i,44}\end{array}\right]$
While the circuitry needed for the second through fifth modes is more like circuitry 1400 than circuitry 1300, the number of matrix elements to be multiplied and accumulated is smaller.

(70) It can be shown that using all of the magnitudes to compute the minimum distance, rather than taking their ratios first, results in more accurate determinations, because some information is lost in the taking of the ratios.

(71) Either using ratios, as described in connection with FIG. 12, or the individual magnitudes as described in connection with FIG. 14, involves inverting a matrix, which is computationally intensive, and would require large and complex circuitry. However, in accordance with implementations of the subject matter of this disclosure, the inverted matrix may be found without complex calculations, and therefore with simplified circuitry. Implementations of the subject matter of this disclosure may be performed onlinei.e., continually during operation of a disk drive, or offlinee.g., in a single calibration operation at the time of manufacture.

(72) In accordance with implementations of the subject matter of this disclosure, and taking the implementations described above as a starting point, circuitry is be provided that processes the input signals x in a manner that directly outputs a matrix that is the same as C.sub.i.sup.1, without having to perform the inversion operation.

(73) Although implementations of the subject matter of this disclosure can operate on the two-input example of FIG. 12 or the four-input example of FIG. 14, or any other number of inputs, the discussion that follows will be presented in the context of the four-input example of FIG. 14.

(74) Thus, considering the example of FIG. 14, the x values are the input samples and the values are the centerpoints of the distributions 801-809. The values are obtained by continually obtaining new inputs and accumulating and averaging those inputs. The differences between the x values and the values are noise:

(75) $\left[\begin{array}{c}{x}_1-{}_{i,1}\\ x_2-{}_{i,2}\\ x_3-{}_{i,3}\\ x_4-{}_{i,4}\end{array}\right]=\left[\begin{array}{c}{n}_1\\ n_2\\ n_3\\ n_4\end{array}\right]$
The noise can be whitened by predicting each noise term as a function of the other noise terms:

(76) $\left[\begin{array}{c}{\hat{n}}_1\\ {\hat{n}}_2\\ {\hat{n}}_3\\ {\hat{n}}_4\end{array}\right]=\left[\begin{array}{ccc}{f}_{12}& f_{13}& f_{14}\\ f_{21}& f_{23}& f_{24}\\ f_{31}& f_{32}& f_{34}\\ f_{41}& f_{42}& f_{43}\end{array}\right]\left[\begin{array}{cccc}{n}_2& n_1& n_1& n_1\\ n_3& n_3& n_2& n_2\\ n_4& n_4& n_4& n_3\end{array}\right]$
where each f.sub.ij is a filter coefficient representing the correlation of the i.sup.th and j.sup.th noise terms to each other.

(77) Various whitening filters can be used, such as a minimum-mean-square-error (MMSE) filter or a least-mean-square (LMS) filter. For purposes of this discussion a least-mean-square filter can be provided as shown in FIG. 15. In filter 1500, one noise component n.sub.1 is being filtered against components n.sub.2, n.sub.3, n.sub.4. Indeed, for four components, four filters 1500 are provided. In a second filter 1500 (not shown), noise component n.sub.2 is filtered against components n.sub.1, n.sub.3, n.sub.4. In a third filter 1500 (not shown), noise component n.sub.3 is filtered against components n.sub.1, n.sub.2, n.sub.4. And in a fourth filter 1500 (not shown), noise component n.sub.4 is filtered against components n.sub.1, n.sub.2, n.sub.3.

(78) In each filter 1500, e.sub.i (i=1, . . . , j) is the error resulting from subtracting at 1501, from n.sub.1, the product of each n.sub.j (ji) and the respective filter coefficient f.sub.ij. On each iteration t (arrival of a new sample), each coefficient f.sub.ij adapts:
f.sub.ij.sup.(t+1)=f.sub.ij.sup.(t).Math.n.sub.j.Math.e.sub.i
where is the adaptation bandwidth. The filter is adaptive to minimize e.sub.i.

(79) The full set of whitened noise samples can be written as:

(80) $\left[\begin{array}{c}{n}_1\\ n_2\\ n_3\\ n_4\end{array}\right]-\left[\begin{array}{c}{\hat{n}}_1\\ {\hat{n}}_2\\ {\hat{n}}_3\\ {\hat{n}}_4\end{array}\right]=\left[\begin{array}{cccc}1& -f_{12}& -f_{13}& -f_{14}\\ -f_{21}& 1& -f_{23}& -f_{24}\\ -f_{31}& -f_{32}& 1& -f_{34}\\ -f_{41}& -f_{42}& -f_{43}& 1\end{array}\right]\left[\begin{array}{cccc}{n}_1& n_1& n_1& n_1\\ n_2& n_2& n_2& n_2\\ n_3& n_3& n_3& n_3\\ n_4& n_4& n_4& n_4\end{array}\right]$
and the variance of each whitened noise sample can be written as:
.sub.i.sup.2=E[(n.sub.i{circumflex over (n)}.sub.i).sup.2].
Thus the covariance matrix of DFT noise can be written in terms of expected values as:

(81) $C_i=E\left[\left(x-{}_i\right){\left(x-{}_i\right)}^T\right]=\left[\begin{array}{cccc}{C}_{i,11}& C_{i,12}& C_{i,13}& C_{i,14}\\ C_{i,21}& C_{i,22}& C_{i,23}& C_{i,24}\\ C_{i,31}& C_{i,32}& C_{i,33}& C_{i,34}\\ C_{i,41}& C_{i,42}& C_{i,43}& C_{i,44}\end{array}\right]=[\begin{array}{cccc}E\left[n_1{n}_1\right]& E\left[n_1{n}_2\right]& E\left[n_1{n}_3\right]& E\left[n_1{n}_4\right]\\ E\left[n_2{n}_1\right]& E\left[n_2{n}_2\right]& E\left[n_2{n}_3\right]& E\left[n_2{n}_4\right]\\ E\left[n_3{n}_1\right]& E\left[n_3{n}_2\right]& E\left[n_3{n}_3\right]& E\left[n_3{n}_4\right]\\ E\left[n_4{n}_1\right]& E\left[n_4{n}_2\right]& E\left[n_4{n}_3\right]& E\left[n_4{n}_4\right]\end{array}]$

(82) The optimum prediction filter F, is the matrix of all of the f.sub.ij,

(83) $\left[\begin{array}{cccc}1& f_{21}& f_{31}& f_{41}\\ f_{12}& 1& f_{32}& f_{42}\\ f_{13}& f_{23}& 1& f_{43}\\ f_{14}& f_{24}& f_{34}& 1\end{array}\right]$
The f.sub.ij values in the i.sup.th column may be written as 13 matrix F.sub.i. Using F.sub.1 as an example:
F.sub.1=R.sub.1.sup.1P.sub.1
where

(84) 0$R_1=\left[\begin{array}{c}E\left[n_2{n}_2\right]\\ E\left[n_3{n}_2\right]\\ E\left[n_4{n}_2\right]\end{array}\begin{array}{c}E\left[n_2{n}_3\right]\\ E\left[n_3{n}_3\right]\\ E\left[n_3{n}_3\right]\end{array}\begin{array}{c}E\left[n_2{n}_4\right]\\ E\left[n_3{n}_4\right]\\ E\left[n_4{n}_4\right]\end{array}\right]\mathrm{and}{P}_1=\left[\begin{array}{c}E\left[n_1{n}_2\right]\\ E\left[n_1{n}_3\right]\\ E\left[n_1{n}_4\right]\end{array}\right].$

(85) The covariance matrix C can be rewritten in the form

(86) $C=\left[\begin{array}{cc}E\left[n_1{n}_1\right]& P_1^T\\ P_1& R_1\end{array}\right]$
in which C, a 44 matrix, is decomposed into a 11 matrix, a 31 matrix, a 13 matrix and a 33 matrix. The desired inverted covariance matrix C.sup.1 can be similarly decomposed:

(87) $C^{-1}=\left[\begin{array}{cc}{A}_{11}& B_{13}^T\\ B_{31}& K_{33}\end{array}\right]$

(88) By definition, CC.sup.1=I, where I is the identity matrix. This yields two equations in two unknowns:

(89) $\begin{array}{c}.Math.\left[\begin{array}{cc}E\left[n_1{n}_1\right]& P_1^T\\ P_1& R_1\end{array}\right]\left[\begin{array}{cc}{A}_{11}& B_{13}^T\\ B_{31}& K_{33}\end{array}\right]=\left[\begin{array}{cc}1& {\underset{\_}{0}}_{13}^T\\ {\underset{\_}{0}}_{31}& I_{33}\end{array}\right]\\ .Math.E\left[n_1{n}_1\right].Math.A+P_1^T.Math.B=1\mathrm{and}{P}_1.Math.A+R_1.Math.B=\underset{\_}{0}\end{array}$
From the Second Equation:

(90) $P_1.Math.A+R_1.Math.B=\underset{\_}{0}.Math.\frac{B}{A}=-R_1^{-1}.Math.P_1=-F_1$
From the First Equation:

(91) $\begin{array}{c}E\left[n_1{n}_1\right].Math.A+P_1^T.Math.B=1.Math.E\left[n_1{n}_1\right].Math.A-P_1^T.Math.F_1.Math.A=1\\ .Math.\left(E\left[n_1{n}_1\right]-P_1^T.Math.F_1\right).Math.A=1\\ A=\frac{1}{\left(E\left[n_1{n}_1\right]-P_1^T.Math.F_1\right)}\end{array}$

(92) Continuing:

(93) $\begin{array}{c}E\left[n_1.Math.e_1\right]=E\left[n_1.Math.\left(n_1-\hat{n}\right)\right]=E\left[n_1{n}_1\right]-E\left[n_1{\hat{n}}_1\right]\\ =E\left[n_1{n}_1\right]-E\left[n_1.Math.\left(\left[\begin{array}{ccc}{n}_2& n_3& n_4\end{array}\right].Math.\left[\begin{array}{c}{f}_{12}\\ f_{13}\\ f_{14}\end{array}\right]\right)\right]\\ =E\left[n_1{n}_1\right]-E\left[\begin{array}{ccc}{n}_1{n}_2& n_1{n}_3& n_1{n}_4\end{array}\right].Math.F_1\\ =E\left[n_1{n}_1\right]-P_1^T.Math.F_1\end{array}$
Substituting Yields:

(94) $A=\frac{1}{E\left[n_1.Math.e_1\right]}\mathrm{and}B=-\frac{F_1}{E\left[n_1.Math.e_1\right]}$

(95) The same can be done for the other columns, yielding:

(96) $C_i^{-1}={E\left[\left(x-{}_i\right){\left(x-{}_i\right)}^T\right]}^{-1}=\left[\begin{array}{cccc}\frac{1}{E\left[n_1.Math.e_1\right]}& \frac{-f_{12}}{E\left[n_2.Math.e_2\right]}& \frac{-f_{13}}{E\left[n_3.Math.e_3\right]}& \frac{-f_{14}}{E\left[n_4.Math.e_4\right]}\\ \frac{-f_{21}}{E\left[n_1.Math.e_1\right]}& \frac{1}{E\left[n_2.Math.e_2\right]}& \frac{-f_{23}}{E\left[n_3.Math.e_3\right]}& \frac{-f_{24}}{E\left[n_4.Math.e_4\right]}\\ \frac{-f_{31}}{E\left[n_1.Math.e_1\right]}& \frac{-f_{32}}{E\left[n_2.Math.e_2\right]}& \frac{1}{E\left[n_3.Math.e_3\right]}& \frac{-f_{34}}{E\left[n_4.Math.e_4\right]}\\ \frac{-f_{41}}{E\left[n_1.Math.e_1\right]}& \frac{-f_{42}}{E\left[n_2.Math.e_2\right]}& \frac{-f_{43}}{E\left[n_3.Math.e_3\right]}& \frac{1}{E\left[n_4.Math.e_4\right]}\end{array}\right]$
without having to perform a matrix inversion operation. Instead, for each column, one instance of circuit 1500 of FIG. 15 can be used. Each time a sector is read, a new set of values is derived. The adaptive LMS filter will continually provide the f.sub.ij values, while the product n.sub.i.Math.e.sub.i determined at 1502 can be averaged at 1503 to provide E[n.sub.i.Math.e.sub.i].

(97) Once C.sub.i.sup.1 has been found, |C| can be found as 1/|C.sub.i.sup.1|

(98) Although the values have been assumed to be constant in the discussion thus far, in fact the values also can be updated continually. Recalling that each value represents the mean of one of the clusters 801-809 of on-track/off-track calibration values, by brute force the mean could be updated by multiplying the former mean value by the number of previous samples, adding the new sample value, and dividing by the increased number of samples. However, such an operation is computationally intensive in hardware, particularly to perform the division operation.

(99) Therefore, in accordance with implementations of the subject matter of this disclosure, each mean is adaptively updated, recognizing that as the number of samples grows, the effect of the newest sample will be small:
.sub.ij.sup.(i+1)=(1).Math..sub.ij.sup.(t)+.Math.x.sub.j
where is the adaptation bandwidth, and may be on the order of 2.sup.9. Thus the new value .sub.ij.sup.(t+1) is nearly identical to old value .sub.ij.sup.(t), including a contribution of (12.sup.9) times the old value, and a contribution of only 2.sup.9 times the new value. These values converge after about 2,000 samples. With each sample representing one sector, and each track having 256 sectors, 2,000 samples are obtained after fewer than four revolutions. With a platter rotational speed of 7,200 RPM, four revolutions occur in about 33.3 ms.

(100) Once the parameters are calibrated, circuits 1500 also can be used to measure the actual off-track error:

(101) $\begin{array}{c}\hat{i}=\underset{i}{\min }\left(\ln \left(.Math.C_i.Math.\right)+{\left(x-{}_i\right)}^T{C}_i^{-1}\left(x-{}_i\right)\right)\\ \begin{array}{c}{\left(x-{}_i\right)}^T{C}_i^{-1}\left(x-{}_i\right)=\left[\begin{array}{cccc}{n}_1& n_2& n_3& n_4\end{array}\right].Math.\\ [\begin{array}{cccc}\frac{1}{E\left[n_1.Math.e_1\right]}& \frac{-f_{12}}{E\left[n_2.Math.e_2\right]}& \frac{-f_{13}}{E\left[n_3.Math.e_3\right]}& \frac{-f_{14}}{E\left[n_4.Math.e_4\right]}\\ \frac{-f_{21}}{E\left[n_1.Math.e_1\right]}& \frac{1}{E\left[n_2.Math.e_2\right]}& \frac{-f_{23}}{E\left[n_3.Math.e_3\right]}& \frac{-f_{24}}{E\left[n_4.Math.e_4\right]}\\ \frac{-f_{31}}{E\left[n_1.Math.e_1\right]}& \frac{-f_{32}}{E\left[n_2.Math.e_2\right]}& \frac{1}{E\left[n_3.Math.e_3\right]}& \frac{-f_{34}}{E\left[n_4.Math.e_4\right]}\\ \frac{-f_{41}}{E\left[n_1.Math.e_1\right]}& \frac{-f_{42}}{E\left[n_2.Math.e_2\right]}& \frac{-f_{43}}{E\left[n_3.Math.e_3\right]}& \frac{1}{E\left[n_4.Math.e_4\right]}\end{array}].Math.\\ \left[\begin{array}{c}{n}_1\\ n_2\\ n_3\\ n_4\end{array}\right]\\ =\left[\begin{array}{cccc}\frac{e_1}{E\left[n_1.Math.e_1\right]}& \frac{e_2}{E\left[n_2.Math.e_2\right]}& \frac{e_3}{E\left[n_3.Math.e_3\right]}& \frac{e_4}{E\left[n_4.Math.e_4\right]}\end{array}\right].Math.\\ \left[\begin{array}{c}{n}_1\\ n_2\\ n_3\\ n_4\end{array}\right]\\ =\underset{i=1}{\overset{4}{.Math.}}\frac{n_i.Math.e_i}{E\left[n_i.Math.e_i\right]}\end{array}\end{array}$
This is the same calculation that is performed in each circuit 1500.

(102) A method 1600 according to this implementation is diagrammed in FIG. 16. At 1601, for each of four (or two) samples per cluster 801-809, the noise is computed by subtracting the accumulated mean from the current value. At 1602, each noise term is predicted, preferably using an adaptive least-mean-squares filter, from the other noise terms, and the error between the actual and predicted noise is determined and used to adapt the filter. At 1603, inverted covariance matrix entries are derived from the filter coefficients and the expectations of the products of each respective noise term and its respective error, and method 1600 ends.

(103) In accordance with another implementation of the subject matter of this disclosure, calibration parameters can be determined off-linee.g., once at the time of manufacture, in the manner described below. Such calibration can be performed with about 10,000 or fewer samples, which can be obtained in less than 167 ms.

(104) As noted above, the error to be determined is

(105) 0$\hat{i}=\underset{i}{\min }\left(\ln \left(.Math.C_i.Math.\right)+{\left(x-{}_i\right)}^T{C}_i^{-1}\left(x-{}_i\right)\right)$
which involve one constant, ln(|C.sub.i|), four means, .sub.i (i=1, . . . , 4), and the entries in the covariance matrix, C.sub.i. For simplicity, one can consider the aforementioned fifth mode in which alternating diagonals have non-zero values:

(106) $C_i=\left[\begin{array}{cccc}{C}_{i,11}& 0& C_{i,13}& 0\\ 0& C_{i,22}& 0& C_{i,24}\\ C_{i,31}& 0& C_{i,33}& 0\\ 0& C_{i,42}& 0& C_{i,44}\end{array}\right]$
Because C.sub.i,13=C.sub.i,31 and C.sub.i,24=C.sub.i,42, there are only six entries to be determined.

(107) The four means, .sub.i (i=1, . . . , 4), can be determined by accumulating the samples x using the same circuit of FIG. 14 that is used at run-time to make the actual off-track determination, and dividing by the number of samples, N, which may be obtained by counting servo read-gate (R-Gate) signals. The correlation matrix entries are then x.sub.1.sup.2, x.sub.2.sup.2, x.sub.3.sup.2, x.sub.4.sup.2, x.sub.1x.sub.3, x.sub.2x.sub.4, which may be obtained by setting the non-zero terms of the correlation matrix in FIG. 14 to 1, and separately accumulating (not shown) the six outputs of the second multiplier 1406, from which the means are then subtracted (not shown):
E[(x.sub.i.sub.i)(x.sub.j.sub.j)]=E[x.sub.ix.sub.j)].sub.i.sub.j.

(108) If the variance entries are denoted v.sub.i (i=1, . . . , 6), then the covariance matrix C is

(109) $\left[\begin{array}{cccc}{v}_1& 0& v_5& 0\\ 0& v_2& 0& v_6\\ v_5& 0& v_3& 0\\ 0& v_6& 0& v_4\end{array}\right]$

(110) C.sup.1 can be found as follows:

(111) By definition, C.Math.C.sup.1=I:

(112) $\left[\begin{array}{cccc}{v}_1& 0& v_5& 0\\ 0& v_2& 0& v_6\\ v_5& 0& v_3& 0\\ 0& v_6& 0& v_4\end{array}\right].Math.C^{-1}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
By a series of operations, C can be transformed into I, and by performing the same operations on I, the equation continues to hold:

(113) $\left[\begin{array}{cccc}{v}_1& 0& v_5& 0\\ 0& v_2& 0& v_6\\ v_5& 0& v_3& 0\\ 0& v_6& 0& v_4\end{array}\right].Math.C^{-1}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\frac{v_1}{v_5}& 0& 1& 0\\ 0& \frac{v_2}{v_6}& 0& 1\\ 1& 0& \frac{v_3}{v_5}& 0\\ 0& 1& 0& \frac{v_4}{v_6}\end{array}\right].Math.C^{-1}=\left[\begin{array}{cccc}\frac{1}{v_5}& 0& 0& 0\\ 0& \frac{1}{v_6}& 0& 0\\ 0& 0& \frac{1}{v_5}& 0\\ 0& 0& 0& \frac{1}{v_6}\end{array}\right]\left[\begin{array}{cccc}\frac{v_1}{v_5}-\frac{v_5}{v_3}& 0& 0& 0\\ 0& \frac{v_2}{v_6}-\frac{v_6}{v_4}& 0& 0\\ 0& 0& \frac{v_3}{v_5}-\frac{v_5}{v_1}& 0\\ 0& 0& 0& \frac{v_4}{v_6}-\frac{v_6}{v_2}\end{array}\right].Math.C^{-1}=\left[\begin{array}{cccc}\frac{1}{v_5}& 0& -\frac{1}{v_3}& 0\\ 0& \frac{1}{v_6}& 0& -\frac{1}{v_4}\\ -\frac{1}{v_1}& 0& \frac{1}{v_5}& 0\\ 0& -\frac{1}{v_2}& 0& \frac{1}{v_6}\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].Math.C^{-1}=[\begin{array}{cccc}\frac{v_3}{\left(v_1{v}_3-v_5^2\right)}& 0& -\frac{v_5}{\left(v_1{v}_3-v_5^2\right)}& 0\\ 0& \frac{v_4}{\left(v_2{v}_4-v_6^2\right)}& 0& -\frac{v_6}{\left(v_2{v}_4-v_6^2\right)}\\ -\frac{v_5}{\left(v_1{v}_3-v_5^2\right)}& 0& \frac{v_1}{\left(v_1{v}_3-v_5^2\right)}& 0\\ 0& -\frac{v_6}{\left(v_2{v}_4-v_6^2\right)}& 0& \frac{v_2}{\left(v_2{v}_4-v_6^2\right)}\end{array}]$

(114) The left side of the final equation is I.Math.C.sup.1 which, by definition, is C.sup.1. Accordingly, the matrix on the right side of the final equation also is C.sup.1. To simplify the notation, one can substitute:
P=(v.sub.1v.sub.3v.sub.5.sup.2)
Q=(v.sub.2v.sub.4v.sub.6.sup.2)
yielding:

(115) $C^{-1}=\left[\begin{array}{cccc}\frac{v_3}{P}& 0& -\frac{v_5}{P}& 0\\ 0& \frac{v_4}{Q}& 0& -\frac{v_6}{Q}\\ -\frac{v_5}{P}& 0& \frac{v_1}{P}& 0\\ 0& -\frac{v_6}{Q}& 0& \frac{v_2}{Q}\end{array}\right]$

(116) This offline calibration implementation may be performed separately for each potential on-track/off-track position. In the context of the implementations described above, calibration would be performed for each of the nine clusters 801-809 representing nine potential on-track/off-track positions.

(117) In this implementation, for each position to be calibrated, 11 values are accumulatedfour x values representing the two-dimensional position of each cluster in the two orthogonal domains, the squares of each of those x values and two cross-multiplications of the x values for deriving the diagonals of the covariance matrix as described above, and N (the number of R-Gate signals, which is the number of samples).

(118) The four x values from which the means .sub.i are derived may be referred to as M.sub.i (i=1, . . . , 4). The squared and cross-multiplied x values may be referred to as A.sub.i (i=1, . . . , 6). The means and variances may then be derived as follows:

(119) $\begin{array}{c}{}_i=\frac{M_i}{N}\\ v_i=\frac{A_i}{N}-{}_i^{2}i\left[0,3\right],v_5=\frac{A_5}{N}-{}_1{}_3,v_6=\frac{A_6}{N}-{}_2{}_4\end{array}$
From these, as noted above, one can determine C and C.sup.1:

(120) $C=\left[\begin{array}{cccc}{v}_1& 0& v_5& 0\\ 0& v_2& 0& v_6\\ v_5& 0& v_3& 0\\ 0& v_6& 0& v_4\end{array}\right]$$C^{-1}=\left[\begin{array}{cccc}\frac{v_3}{P}& 0& -\frac{v_5}{P}& 0\\ 0& \frac{v_4}{Q}& 0& -\frac{v_6}{Q}\\ -\frac{v_5}{P}& 0& \frac{v_1}{P}& 0\\ 0& -\frac{v_6}{Q}& 0& \frac{v_2}{Q}\end{array}\right]$$P=\left(v_1{v}_3-v_5^2\right),Q=\left(v_2{v}_4-v_6^2\right)$
The determinant of the covariance matrix C is:
|C|=(v.sub.1v.sub.3v.sub.5.sup.2)(v.sub.2v.sub.4v.sub.6.sup.2)=PQ
and therefore:
b=ln(|C|)=ln(PQ)
The final calibrated parameters for each cluster 809-809 are: 1. four means, .sub.i 2. six entries in inverse co-variance matrix C.sup.1:

(121) $\left[\frac{v_3}{P},\frac{v_4}{Q},\frac{v_1}{P},\frac{v_2}{Q},-\frac{v_5}{P},-\frac{v_6}{Q}\right],$
and 3. one constant term b,
for a total of 11 calibrated parameters per on-track/off-track position.

(122) A method according to this implementation is diagrammed in FIG. 17. At 1701, N samples are sent; N may be about 10,000. At 1702, four (or two; see above) values (e.g., Discrete Fourier Transform values) M.sub.1, M.sub.2, M.sub.3, M.sub.4, as well as squares A.sub.1, A.sub.2, A.sub.3, A.sub.4, of those values, and two cross-multiplication terms, A.sub.5, A.sub.6, are accumulated from each sample. At 1703, for the current on-track/off-track condition being calibrated,

(123) calibrated means are determined as .sub.i=M.sub.i/N, (i=1, . . . , 4), and calibrated variances are determined as:

(124) $v_i=\frac{A_i}{N}-{}_i^{2}i\left[0,3\right],v_5=\frac{A_5}{N}-{}_1{}_3,v_6=\frac{A_6}{N}-{}_2{}_4$
At 1704, the covariance matrix C and its inverse C.sup.1 are determined as:

(125) 0$C=\left[\begin{array}{cccc}{v}_1& 0& v_5& 0\\ 0& v_2& 0& v_6\\ v_5& 0& v_3& 0\\ 0& v_6& 0& v_4\end{array}\right]C^{-1}=\left[\begin{array}{cccc}\frac{v_3}{P}& 0& -\frac{v_5}{P}& 0\\ 0& \frac{v_4}{Q}& 0& -\frac{v_6}{Q}\\ -\frac{v_5}{P}& 0& \frac{v_1}{P}& 0\\ 0& -\frac{v_6}{Q}& 0& \frac{v_2}{Q}\end{array}\right]$
where
P=(v.sub.1v.sub.2v.sub.5.sup.2),Q=(v.sub.2v.sub.4v.sub.6.sup.2).
At 1705, the constant b is determined as b=ln(PQ).

(126) At 1706 it is determined whether there are additional on-track/off-track conditions to be calibrated. If not, method 1700 ends. If at 1706, there are additional on-track/off-track conditions to be calibrated, then flow returns to 1701.

(127) As used herein and in the claims which follow, the construction one of A and B shall mean A or B.

(128) It will be understood that the foregoing is only illustrative of the principles of the invention, and that the invention can be practiced by other than the described embodiments, which are presented for purposes of illustration and not of limitation, and the present invention is limited only by the claims which follow.