Apparatus and method for noise-predictive maximum likelihood detection

This application is a 371 of PCT/IB97/00554 filed May 15, 1998.

1. Field of the Invention

Present invention relates to data detection apparatus and methods. It is in particular applicable to direct access storage (DASD) devices using partial-response (PR) signaling and maximum likelihood sequence detection (MLSD).

2. Background Information

In DASD systems as well as in data transmission systems, the signal read from the storage device or received at the output of a transmission channel has to be converted in the receiver into a sequence of symbols which most likely represents the data (symbol) sequence initially stored or transmitted by the sender, despite interference between adjacent symbols, and despite added noise.

The optimum receiver for detecting an uncoded data sequence in the presence of inter-symbol interference (ISI) and noise consists of a whitened-matched filter followed by a Viterbi detector which performs maximum likelihood sequence detection on the ISI trellis. In PRML (Partial-Response Maximum Likelihood) systems the composite noise at the input of the detector is not white, resulting in sub-optimal performance. This sub-optimality is more pronounced at high recording densities since the linear PR4 (PR class 4) equalizer tends to enhance the electronic noise component of the composite noise at the input of the PRML detector. In addition to colored noise, nonlinear distortion due to non-linear bit-shift and head asymmetry could further degrade performance of PRML systems.

International Patent Application PCT/WO97/11544 (IBM) discloses a scheme for data detection in a direct access storage device, called NPML (Noise-Predictive Maximum Likelihood detection), that arises by imbedding a noise prediction/whitening filter into the branch metric computation of the Viterbi detector. It has been shown via simulations that the NPML detectors offer substantial performance gains over the PRML detectors.

However, implementation of the imbedded predictor/whitener as a bank of finite-impulse-response (FIR) filters or equivalent table look-up operations, e.g. by means of random access memory (RAM), demands a certain level of complexity of the NPML hardware realization. A further difficulty with presently available implementation technologies is the time constraint imposed by high speed clock requirements.

Application of the Viterbi algorithm to channels represented by an IIR (Infinite Impulse Response) discrete-time equivalent model has been studied in a publication by A. Duel Hallen et al. “Delayed decision-feedback sequence estimation,” IEEE Trans. Commun., COM-37, pp.428-436, May 1989. The approach in that reference assumes a decision feedback equalization prefilter whose feedback filter is an IIR filter combined with Viterbi detection. In contrast, the present invention considers a forward partial-response or generalized partial-response linear equalizer and a noise whitening IIR predictor imbedded into the Viterbi detector.

It is a main object of present invention to provide a more efficient and simpler realization of the noise prediction/whitening mechanism for a NPML detector.

It is a further object of the invention to devise a partial-response maximum likelihood detection scheme that utilizes past decisions from the path memory of the Viterbi detector for the noise prediction process.

Another object of the invention is to enable additional mechanisms for a maximum likelihood detector which allow compensation of reading head asymmetries and of DC offset in the receiver.

The invention provides a partial-response signaling, maximum likelihood detector with an infinite impulse response (IIR) noise predictor and whitening filter imbedded in the Viterbi detector (INPML). As an additional feature, a DC (zero frequency) notch filter may be provided for DC offset compensation. Furthermore, an additional compensation method can be provided that is imbedded in the branch metric computation of the INPML detector for dynamic signal-asymmetry and DC offset compensation.

Advantages of the invention are that, besides its simplicity in implementation, it does not compromise performance of the detector. It has the further important advantage that it can be “piggy-backed” on existing PRML systems so that there is no need for development and implementation of an entirely new channel architecture which would be a complex and costly task. The addition of a DC-notch filter or DC compensation method renders the INPML detector completely immune to DC offset, tolerant and robust against various types of non-linearity and in particular head asymmetry, and tolerant to thermal asperities, which is a further impairment in digital magnetic recording systems.

In the following, embodiments of the invention are described with reference to the drawings.

FIG. 1 is a block diagram of a PRML detector in which present invention can be implemented.

FIG. 2A is a block diagram of the maximum likelihood detector according to the invention with a single-pole imbedded IIR noise predictor.

FIG. 2B shows an alternative, equivalent implementation of the IIR noise predictor for the inventive maximum likelihood detector of FIG. 2 A.

FIG. 3 shows the 4-state trellis for the INPML detector of FIG. 2 A.

FIG. 4 is a block diagram of an alternative maximum likelihood detector in accordance with the invention in which an imbedded IIR noise predictor with 2 poles and 2 zeros is provided.

FIG. 5 shows the bit error probability for various ML detectors including the INPML detector of the invention for a channel with the normalized linear recording density PW50/T≅2.86.

FIG. 6 shows the relative SNR for a given bit error probability, also for various ML detectors including INPML (4 states), as a function of PW50/T.

FIG. 7A is a block diagram of an inventive INPML detector with an additional, imbedded DC-notch filter.

FIG. 7B schematically shows a first arrangement of an INPML detector with a cascaded DC-notch filter (between equalizer and detector).

FIG. 7C schematically shows a second arrangement of an INPML detector with a cascaded DC-notch filter (preceding the equalizer).

FIG. 8 illustrates a RAM based dynamic non-linearity estimation to be used for DC offset and head asymmetry compensation.

FIG. 9 shows the bit error probability for the INPML detector with and without asymmetry estimation (channel PW50/T≅2.86).

FIG. 1 is the block diagram of a known PRML channel architecture for a data recording system. Such a system was also described in the above mentioned International Patent Application PCT/W097/11544.

Customer data I n are written in the form of binary digits α n Ε{−1, +1 } by write head 15 on the disk 11 after being encoded in an encoder 12 by a rate-8/9 RLL (Run Length Limited) code and serialized in a serializer 13 and precoded in a precoder 14 by a operation where D is the unit delay operator and ⊕ means addition modulo 2 . When receiving the customer data from said disk 11 , an analog signal r(t) is generated by the read head 15 and provided at the read head's output. The signal r(t) is then applied via arm electronic circuitry 16 to a variable gain amplifier (VGA) circuit 17 . The output signal of the VGA circuit 17 is first low-pass filtered using an analog low-pass filter 18 (LPF) and then converted to a digital form X n by an analog-to-digital (A/D) converter 19 . The functions of the A/D converter 19 and VGA amplifier 17 are controlled by the timing recovery and gain control loops 20 / 26 and 21 / 26 , respectively. The analog low-pass filter 18 is preferably a filter which boosts the higher frequencies to avoid saturation of the A/D converter 19 and to support the digital equalizer 22 . The digital samples x n at the output of the A/D converter 19 (line labeled A in FIG. 1) are first shaped to PR4 signal samples by the digital equalizer 22 (line labeled B in FIG. 1) and are then passed on to a maximum likelihood detector circuitry 10 (ML detector) in the form of digital noisy samples y n . The output data of the ML detector 10 (i.e. the final decisions on the line labeled C in FIG. 1 ), after inverse precoding by means of an inverse precoder 23 performing a I ⊕D 2 operation, are fed via a deserializer 24 to a rate 9/8 decoder 25 for the rate-8/9 RLL code which delivers the retrieved customer data Î n-d . The inverse precoder function 23 following the ML detector in FIG. 1 can be a separate functional block (as shown) or it can be imbedded in the trellis (survivor path memory) of the ML detector.

In the above mentioned International Patent Application PCT/WO97/11544, the maximum likelihood detector 10 was a noise-predictive maximum likelihood detector (NPML detector) which uses a finite impulse response (FIR) filter/predictor to estimate the noise contents of the input signals.

Present invention provides an improvement for the maximum likelihood detector portion of a PRML data recording channel. Its principles are shown in the block diagram of FIG. 2A which represents the ML detector portion 10 of FIG. 1 . The PR4-equalized samples y n appearing on line 30 (marked B) are modified by the noise prediction process according to the invention, fed as samples Z n to the Viterbi detector 31 which on its output 32 delivers the recovered data symbols {circumflex over (α)} n-d , which are the delayed final decisions (marked C) of the Viterbi detector. Preliminary symbols (path history decisions) {circumflex over (α)} n are delivered on an output 33 of the Viterbi detector. These preliminary symbols are fed to a reconstruction means 34 having transfer function I-D 2 , to obtain on line 29 a reconstructed sample sequence ({circumflex over (α)} n -{circumflex over (α)} n-2 ). This reconstructed sample sequence is subtracted in subtracting means 39 from the noisy equalized samples y n appearing on branch line 38 , to obtain noise samples Ŵ n on line 28 . These noise samples Ŵ n are fed to noise predictor circuit 35 which is an infinite impulse response (IIR) filter. The predicted noise samples {overscore (W)} n appearing on output line 36 are combined in circuit 37 with the noisy samples y n to provide samples Z n for input to the Viterbi detector 31 . Note that the noise prediction and metric update operations are performed separately for each state using decisions from the corresponding path memory. These per state operations in the resulting INPML detector are symbolized in FIG. 2 by double signal flow lines.

The IIR predictor 35 of present preferred embodiment is a single-pole device. It receives the noise samples Ŵ n on line 28 , which then are delayed in unit delay means 40 . Its output, after combination with a delayed noise signal (output of unit delay means 43 ) in adder 41 and multiplication by a coefficient a in multiplication means 42 , provides the desired noise signal {overscore (W)} n on line 36 . The IIR noise predictor 35 executes the transfer function αD/(I—αD).

Thus, the noise predictor is an infinite impulse response device which is imbedded (for each state) into the maximum likelihood detection arrangement, as shown in FIG. 2 A. This arrangement will be termed INPML detector in the following (for “Infinite impulse response Noise Predictive Maximum Likelihood” detector). Its operation is described in the sequel.

Let y n be the output of the PR4 digital equalizer, line labeled B in FIG. 1 and FIG. 2 A. This output then consists of a PR4 data signal, with noiseless nominal values of −2, 0, +2, and colored noise, i.e.,

Yn=α n -α n-2 +W n   (1)

where α n ε{1, +1} denotes the encoded/precoded data sequence written on the magnetic medium with a rate 1/T and w n represents the colored noise sequence at the output of the digital partial-response equalizer (T is the channel encoded bit time interval and subscript n indicates time instant nT). The power of the colored noise component w n in (1) can be reduced by noise prediction.

FIG. 2A shows the basic concept of a single coefficient IIR predictor imbedded into the Viterbi detector. The basic principle can be explained as follows. Let E     ( D ) = 1 1 - α     D , α < 1 ( 2 )

denote the D-transform of the single-pole IIR prediction error (whitening) filter. Then, assuming ideal path history decisions {circumflex over (α)} n , i.e., ŵ n =w n on line 28, the D-transform of the whitened noise sequence at the output of E(D) is given by e     ( D ) = 1 1 - α     D     w     ( D ) = [ 1 + P     ( D ) ]     w     ( D ) = w     ( D ) + w _     ( D ) , ( 3 )

with P     ( D ) = α     D 1 - α     D . ( 4 )

w(D) in (3) represents the D-transform of the colored noise sequence in (1) and {overscore (e)} denotes predicted value through the single-pole prediction filter. In the time domain {overscore (w)}(D) satisfies the following recursive equation

{overscore (w)} n =α({overscore (w)} n-1 +w n-1 )=α{overscore (w)} n-1 +α(y n-1 −α n-1 +α n-3 )=αe n-1 ·  (5)

The single-pole IIR predictor filter 35 described by (4) and (5) and shown in FIG. 2A can also be implemented as shown in FIG. 2 B. This IIR predictor 35 b also receives noise samples {overscore (w)} n on line 28. They are combined in adder 41 with predicted noise samples {overscore (w)} n , then delayed by unit delay element (D) 40 b and then multiplied in multiplication means 42 by a coefficient α, to provide predicted noise samples {overscore (w)} n on output line 36 .

Equation (5) and PR4 signalling define a 4-state ML detector which encompasses a single-pole IIR predictor. If s j , j=0, 1, 2, 3, denotes the j-th state (α n-1 , α n-2 ) of the 4-state PR4 trellis then, in view of (1) and (3), the branch metric at time nT of the Viterbi detector corresponding to the state transition s j →S k takes the form

λ n (S j , s k )=[y n −α n +α n-2 (S j )+{overscore (w)} n (S j )] 2 =[e n (S j , S k )] 2 ,  (6)

where α n and α n-2 (s j ) are determined by the hypothesized state transition s j →S k and {overscore (w)} n (S j ) is the predicted noise sample associated with state s j . In particular, the reconstructed PR4 samples (α n-1 −α n-3 ) in (5) are replaced in practice by ({circumflex over (α)} n-1 −{circumflex over (α)} n-3 ) which are obtained from the path memory of the Viterbi detector associated with state s j .

The 4-state INPML detector with a single-pole IIR predictor/whitening filter operates on the trellis shown in FIG. 3 . Every state at time instant nT is associated with

State Metrics: J n (0), J n (1), J n (2), J n (3),

Noise Estimates: {overscore (W)} n (0), {overscore (W)} n (1), {overscore (W)} n (2), {overscore (W)} n (3).

The branch metrics shown on the trellis branches of FIG. 5 are computed as follows: λ n     ( 0 , 0 ) =  e n     ( 0 , 0 )  2 , where     e n     ( 0 , 0 ) = y n + w ~ n     ( 0 ) 

 λ n     ( 1 , 0 ) =  e n     ( 1 , 0 )  2 , where     e n     ( 1 , 0 ) = y n + 2 + w ~ n     ( 1 ) 

 λ n     ( 2 , 1 ) =  e n     ( 2 , 1 )  2 , where     e n     ( 2 , 1 ) = y n + w ~ n     ( 2 ) 

 λ n     ( 3 , 1 ) =  e n     ( 3 , 1 )  2 , where     e n     ( 3 , 1 ) = y n + 2 + w ~ n     ( 3 ) 

 λ n     ( 0 , 2 ) =  e n     ( 0 , 2 )  2 , where     e n     ( 0 , 2 ) = y n - 2 + w ~ n     ( 0 ) 

 λ n     ( 1 , 2 ) =  e n     ( 1 , 2 )  2 , where     e n     ( 1 , 2 ) = y n + w ~ n     ( 1 ) 

 λ n     ( 2 , 3 ) =  e n     ( 2 , 3 )  2 , where     e n     ( 2 , 3 ) = y n - 2 + w ~ n     ( 2 ) 

 λ n     ( 3 , 3 ) =  e n     ( 3 , 3 )  2 , where     e n     ( 3 , 3 ) = y n + w ~ n     ( 3 ) ( 7 )

The state metrics and noise estimates at the next time instant (n+I)T are computed as follows: J n + 1     ( 0 ) = min     { J n     ( 0 ) + λ n  ( 0 , 0 ) , J n     ( 1 ) + λ n     ( 1 , 0 ) } w ~ n + 1     ( 0 ) = { α     e n     ( 0 , 0 ) or α     e n     ( 1 , 0 )     Depending     on     whether     state     0        or     state     1    was     the     previous     “ winning ”     state 

 J n + 1     ( 1 ) = min     { J n     ( 2 ) + λ n     ( 2 , 1 ) , J n     ( 3 ) + λ n     ( 3 , 1 ) } 

    w ~ n + 1     ( 1 ) = { α     e n     ( 2 , 1 ) or α     e n     ( 3 , 1 )     Depending     on     whether     state     2     or     state     3    was     the     previous     “ winning ”     state 

 J n + 1     ( 2 ) = min     { J n     ( 0 ) + λ n  ( 0 , 2 ) , J n     ( 1 ) + λ n     ( 1 , 2 ) } 

 w ~ n + 1     ( 2 ) = { α     e n     ( 0 , 2 ) or α     e n     ( 1 , 2 )     Depending     on     whether     state     0     or     state     1    was     the     previous     “ winning ”     state 

 J n + 1     ( 3 ) = min     { J n     ( 2 ) + λ n  ( 2 , 3 ) , J n     ( 3 ) + λ n     ( 3 , 3 ) } 

 w ~ n + 1     ( 3 ) = { α     e n     ( 2 , 3 ) or α     e n     ( 3 , 3 )     Depending     on     whether     state     2     or     state     3    was     the     previous     “ winning ”     state

By setting α=0 the INPML detector becomes a conventional 4-state PRML detector.

An alternative embodiment of the noise predictor in the inventive INPML detector is now described with reference to the block diagram of FIG. 4 . In this case the prediction error (whitening) filter includes two poles and two zeros.

The predictor 45 shown in FIG. 4 has the same connections to the other portions of the Viterbi detector as predictor 35 in FIG. 2 A: A line 28 a (28) transferring noise samples ŵ n , and a line 36 a ( 36 ) for the predicted noise samples {overscore (W)} n .

The noise samples ŵ n appearing on input 28 a are twice delayed in unit delay elements 47 and 48 , then multiplied in multiplier 49 by factor [−(α 2 +β 2 )], and then combined in adder unit 50 with “auxiliary” signals on lines 51 , 52 , and 53 to provide predicted noise samples {overscore (w)} n on line 36 a.

The “auxiliary” signals are derived as follows: Signal on line 51 results from multiplying the output of delay unit 47 in multiplier 54 by a factor [−(α 1 +β 1 )] Signal on line 52 results from multiplying predicted noise samples {overscore (W)} n , delayed in unit delay means 55 , by a factor [−α 1 ] in a multiplier 56 . Signal on line 53 results from multiplying predicted noise sample {overscore (W)} n , twice delayed in unit delay means 55 and 57 , by a factor [−α 2 ] in multiplier 58 . IIR noise predictor 45 executes the transfer function

[−(α 1 +β 1 )D−(α 2 +β 2 )D 2 ]/[1+α 1 D+α 2 D 2 ].

Let E     ( D ) = 1 - β 1     D - β 2     D 2 1 + α 1     D + α 2     D 2 , ( 8 )

denote the D-transform of the two-poles, two-zeros IIR prediction error (whitening) filter. Then, the D-transform of the whitened noise sequence at the output of E(D) is given by e     ( D ) = 1 - β 1     D - β 2     D 2 1 + α 1     D + α 2     D 2     w     ( D ) = [ 1 + P     ( D ) ]     w     ( D ) = w     ( D ) + w ~     ( D ) , ( 9 )

with P     ( D ) = - ( α 1 + β 1 )     D - ( α 2 + β 2 )     D 2 1 + α 1     D + α 2     D 2 . ( 10 )

w(D) in (9) represents the D-transform of the colored noise sequence in (1) and {overscore (w)}(D) denotes its predicted value through the 2-poles/2-zeros prediction filter. In the time domain {overscore (w)}(D) satisfies the following recursive equation

{overscore (w)}=−α 1 {overscore (w)} n-1 −α 2 {overscore (w)} n-2 −(α 1 +β 1 ) (y n-1 −α n-1 α n-3 )

−(α 2 +β 2 ) (y n-2 −α n-2 +α n-4 ).

Equation (11) and PR4 signalling can define a 4-state Viterbi detector which encompasses a 2-zeros/2-poles IIR predictor. If s j , j=0, 1, 2, 3, denotes the j-th state of the 4-state PR4 trellis then, in view of ( 1 ) and ( 9 ), the branch metric at time nT of the Viterbi detector corresponding to the state transition s j →S k takes the form

λ n (s j , s k )=[y n −α n α n-2 (s j )+{overscore (w)} n (S j )] 2 =[e n (s j , s k ] 2 ,   (12)

where α n and α n-2 (s j ) are determined by the hypothesized state transition s j →S k and {overscore (w)} n (s j ) is a noise estimate associated with state s j . As in the previous embodiment the branch metrics ( 12 ) for each state transition can now be computed explicitly from expressions ( 7 ) with {overscore (w)} n (s j ) according to ( 11 ).

Following the same approach, the recursive equation of a 1-zero/2-poles IIR predictor filter with the corresponding branch metric equation can be readily derived (β 2 =0 in ( 8 )). Similarly, the recursive equation of a 2-poles IIR predictor filter with the corresponding branch metric equation can also be obtained (β 1 =0 and β 2 =0 in ( 8 )).

For the magnetic recording channel it has been found that in the entire range of 0.5≲PW50/T ≲3.5 where PW50 is the pulse width at the 50% amplitude point of the channel's step response and Tis the duration of the channel encoded bit, IIR predictors with at most 2 poles and 2 zeros offer the best possible performance irrespective of the noise source, i.e., electronic noise or disk noise, or a combination thereof. In particular, for 0.5≲PW50/T ≲2.7 the best possible performance is achieved with an IIR predictor consisting of 2 poles and 1 zero, whereas for 2.7≲PW50/T ≲3.5 best possible performance is achieved with an IIR predictor consisting of 2 poles and 2 zeros. If the dominant noise source is electronic noise then an INPML detector with a single-pole IIR predictor offers already substantial performance gains over PRML detectors in a wide range of linear recording densities.

The error performance of a magnetic recording system employing INPML detection has been studied by computer simulation. FIG. 5 shows bit error probabilities of simulated 4-state INPML and NPML detectors for the channel model with PW50/T≈2.86. Curve 2 shows the performance of INPML4 (4-state INPML) with a single-pole predictor; compared to curves 1 (PRML) and 3 (EPRML), a gain in noise margin of 2 dB and 0.5 dB is obtained, respectively. Curve 4 corresponds to 4-state NPML with N=4 predictor coefficients. Compared to curve 3 an additional gain of 0.4 dB is obtained. Finally, curve 5 corresponds to a 4-state INPML detector with a 2-zeros and 2-poles IIR predictor. Compared to curve 4 an additional 0.2 dB is obtained, and compared to PRML 2.7 dB gain is obtained.

FIG. 6 shows the relative SNR (Signal-to-Noise Ratio) required to achieve a bit error probability of P b =10 −5 for the 4-state NPML with N=4 predictor coefficients, 4-state INPML with a single-pole IIR predictor, and EPRML, all with respect to a PRML system, as a function of the normalized linear density PW50/T for the Lorentzian channel model. Curve 1 corresponds to our reference PRML system. Curve 2 corresponds to the 8-state EPRML system and curves 3 and 4 correspond to 4-state INPML with a single-pole IIR predictor filter and 4-state NPML/N=4, respectively. We observe that the single-pole INPML system is very close in performance to the NPML system and that both provide superior performance compared to all the other systems at any linear recording density.

Further improvement of the invented INPML detector described so far can be achieved by incorporating a DC-notch filter for eliminating sensitivity to DC offset.

The transfer function of a digital notch filter is given by a second order rational function with two parameters that characterize the location and width of the notch (see article by P. Regalia, S. K. Mitra and P. P. Vaidyanathan “The Digital All-Pass Filter: A Versatile Signal Processing Building Block,” PROCEEDINGS of the IEEE , VOL.76, pp. 19-37, January 1988). The sensitivity of INPML detectors to DC offset can be eliminated by utilizing such a filter with a notch at DC. It can be seen that the transfer function of a digital DC-notch filter is given by H     ( D ) = 1 + b 2  [ 1 - D 1 - bD ] , ( 13 )

where b is a constant that determines the width of the notch. For a very narrow notch at DC, b must be chosen very close to unity. In this case, (1+b)/2≈1 and the transfer function of the notch filter simplifies to H     ( D ) = 1 - D 1 - bD . ( 14 )

Digital filters with a notch at DC are very simple to implement and do not require multipliers. Similarly, the transfer function of a DC-notch analog filter is given by H     ( s ) = s s + c , ( 15 )

where c is a constant that determines the width of the notch. For a very narrow notch c must be chosen very close to zero.

The DC-notch filter can be either imbedded into the INPML detector as part of the whitening IIR predictor or cascaded with the INPML detector. FIGS. 7A, 7 B and 7 C show the block diagrams of three possible configurations of an INPML detector with a DC-notch filter. In FIG. 7A, the notch filter 44 a is included in the feedback path between the second output of the Viterbi detector 31 and its input, the path which also contains the IIR predictor 35 . In the configuration of FIG. 7B, the notch filter 44 b is inserted between the digital equalizer 22 and the INPML detector 10 , and thus receives the samples y n at its input. In the case of FIG. 7C, the notch filter 44 c precedes the digital equalizer 22 . Clearly, the same approach can be applied to other DC sensitive detectors such as NPML, non-adaptive DFE (decision feedback equalizer) and FDTS (fixed delay tree search) detectors. Summarizing, a DC-notch filter renders INPML:

completely immune to DC offset,

tolerant to MR (magneto-resistive) head asymmetry,

tolerant to thermal asperities.

An MR head is a nonlinear transducer that introduces asymmetry in the readback signal. Positive peaks can differ in amplitude from negative peaks by as much as 30 %. DC insensitive detectors such as PRML and INPML cascaded with a DC-notch filter are quite tolerant to small amounts of head asymmetry and suffer only a small performance degradation. The high losses in performance due to large amounts of head asymmetry can be recovered by an asymmetry estimation/compensation algorithm which is imbedded into the branch-metric computation of the INPML detector. Consider the following generic signal asymmetry model: f     ( a n , a n - 2 ) = [ F + 2     for     a n = 1     and     a n - 2 = - 1 F 0     for     a n = a n - 2    F - 2     for     a n = - 1     and     a n - 2 = 1

Then, the branch metric computations of the 4-state INPML detector whose trellis is shown in FIG. 3 must be modified as follows: λ n     ( 0 , 0 ) =  e n     ( 0 , 0 )  2 , where     e n     ( 0 , 0 ) = y n - F 0 + w ~ n     ( 0 ) λ n     ( 1 , 0 ) =  e n     ( 1 , 0 )  2 , where     e n     ( 1 , 0 ) = y n - F - 2 + 2 + w ~ n     ( 1 ) λ n     ( 2 , 1 ) =  e n     ( 2 , 1 )  2 , where     e n     ( 2 , 1 ) = y n - F 0 + w ~ n     ( 2 ) λ n     ( 3 , 1 ) =  e n     ( 3 , 1 )  2 , where     e n     ( 3 , 1 ) = y n - F - 2 + w ~ n     ( 3 ) λ n     ( 0 , 2 ) =  e n     ( 0 , 2 )  2 , where     e n     ( 0 , 2 ) = y n - F + 2 + w ~ n     ( 0 ) λ n     ( 1 , 2 ) =  e n     ( 1 , 2 )  2 , where     e n     ( 1 , 2 ) = y n - F 0 + w ~ n     ( 1 ) λ n     ( 2 , 3 ) =  e n     ( 2 , 3 )  2 , where     e n     ( 2 , 3 ) = y n - F + 2 + w ~ n     ( 2 ) λ n     ( 3 , 3 ) =  e n     ( 3 , 3 )  2 , where     e n     ( 3 , 3 ) = y n - F 0 + w ~ n     ( 3 )

The state metrics and feedback estimates at the next time instant (n+1)T are computed as follows: J n + 1     ( 0 ) = min     { J n     ( 0 ) + λ n  ( 0 , 0 ) , J n     ( 1 ) + λ n     ( 1 , 0 ) } w ~ n + 1     ( 0 ) = { α     e n     ( 0 , 0 ) or α     e n     ( 1 , 0 )     Depending     on     whether     state     0        or     state     1    was     the     previous     “ winning ”     state 

 J n + 1     ( 1 ) = min     { J n     ( 2 ) + λ n     ( 2 , 1 ) , J n     ( 3 ) + λ n     ( 3 , 1 ) } 

    w ~ n + 1     ( 1 ) = { α     e n     ( 2 , 1 ) or α     e n     ( 3 , 1 )     Depending     on     whether     state     2     or     state     3    was     the     previous     “ winning ”     state 

 J n + 1     ( 2 ) = min     { J n     ( 0 ) + λ n  ( 0 , 2 ) , J n     ( 1 ) + λ n     ( 1 , 2 ) } 

 w ~ n + 1     ( 2 ) = { α     e n     ( 0 , 2 ) or α     e n     ( 1 , 2 )     Depending     on     whether     state     0     or     state     1    was     the     previous     “ winning ”     state 

 J n + 1     ( 3 ) = min     { J n     ( 2 ) + λ n  ( 2 , 3 ) , J n     ( 3 ) + λ n     ( 3 , 3 ) } 

 w ~ n + 1     ( 3 ) = { α     e n     ( 2 , 3 ) or α     e n     ( 3 , 3 )     Depending     on     whether     state     2     or     state     3    was     the     previous     “ winning ”     state

The values of the nonlinear function ƒ(α n , α n-2 ) can be estimated for a particular head and media combination during manufacturing and then be used as fixed predetermined values in the branch metric computation unit of the INPML detector.

Alternatively, an algorithm that estimates these values dynamically can be used. Dynamic estimation can be achieved by a table look-up stochastic gradient algorithm which updates one memory location per time interval. Consider delayed decisions from the path memory of the INPML detector, say {circumflex over (α)} n-d , {circumflex over (α)} n-2-d . These decisions then form the address to the memory location of a RAM in which the corresponding values are stored. A simple schematic of a RAM with four memory locations containing corresponding values at time nT is shown in FIG. 8 . At the next time instant, (n+1)T, the content of the memory location addressed by ({circumflex over (α)} n-d , {circumflex over (α)} n-2-d ) is updated according to the following algorithm:

{circumflex over (ƒ)} n+1 ({circumflex over (α)} n-d , {circumflex over (α)} n-2-d )={circumflex over (ƒ)} n ({circumflex over (α)} n-d , {circumflex over (α)} n-2-d )+μe n-d ,

e n-d =[y n-d −{circumflex over (ƒ)} n ({circumflex over (α)} n-d , {circumflex over (α)} n-2-d)],

where Y n-d is the delayed, nonlinearly distorted received signal and μ is a small constant. The above algorithm can also be applied to remedy other forms of nonlinearity. In general, the number of symbols which the nonlinear function ƒ(.) depends on determines the size of the RAM.

The following asymmetry model has been used in the simulations: f     ( a n , a n - 2 ) = [ 2 + 4     δ + DC = F + 2     for     a n = 1     and     a n - 2 = - 1 0 + DC = F 0     for     a n = a n - 2 - 2 + 4     δ + DC = F - 2     for     a n = - 1     and     a n - 2 = 1.

This model is basically a second-order nonlinearity coupled with a DC offset. In the simulations δ=0.1, i.e., 33% asymmetry, and DC =0.05 have been chosen.

FIG. 9 shows the error performance of a 4-state INPML detector with a single-pole predictor in the presence of electronic noise and head asymmetry for the channel model with PW50/T≈2.86. The head asymmetry is 33%. Curve 1 shows the performance of INPML without any asymmetry compensation. Curves 2 and 3 correspond to INPML combined with adaptive asymmetry estimation and μ=0.05 and μ=0.005, respectively. Finally, curve 4 shows the performance of INPML with ideal asymmetry estimation/compensation. These performance results show that the dynamic estimation methods can achieve the performance of ideal estimates.

Note that, besides a fully digital implementation, the whole or portions of the elements shown in the INPML detector 10 of FIGS. 2 A/B, FIG. 4 and FIG. 7, can be implemented in analog circuit technology.