multi-level decision feedback equalization for saturation recording

2160 IEEE TRANSACTIONS ON MAGNETICS. VOL. 29. NO. 3, JULY 1993

Multi-level Decision Feedback Equalization for Saturation Recording

John G. Kenney, Member IEEE, L. Richard Carley, Senior Member, IEEE, and Roger W. Wood

Abstract-Fixed-delay tree search with decision feedback (FDTSIDF) has been proposed for retrieving data from hard disk drives. One problem with the algorithm as originally posed is that 4 additions and 1 multiplication sets the critical path delay through the detector. This paper explores the decision space in FDTSIDF using linear discriminants. On recording channels using 2/3(1, 7) run-length limited coding, a detector achieving the performance of FDTSIDG can be implemented with a 2-tap transversal filter. The feedback loop can be re- arranged so that this transversal filter no longer resides in the forward path of the feedback loop. Instead its transfer function is incorporated into the specification of the forward and backward equalizers. This modification leads to the simpler architecture of decision feedback equalizer (DFE), where the slicer performs binary decisions on a multi-level signal. Implemen- tation issues pertaining to phase detection, gain detection, dc detection and adaptive equalization using a least-means squared technique will also be addressed. Simulation results demon- strate adaptive equalization, where the desired model is generated by a 3-tap transversal filter consisting of only adders, delays, multiplexers and 1 programmable coefficient.

I. INTRODUCTION HE dramatic increases in data storage density on hard T disk drives and on tape recorders have come largely

from improvements in heads and storage media, and in the interface between them. Nevertheless, a significant part of the increase can also be attributed to advances in signal processing. At high densities, the readback signals suffer severely from bit crowding or intersymbol interference (ISI) and also have very poor signal to noise ratio (SNR). Conventional readback processing, which in- volves run-length limited (RLL) codes with peak detection and some readback equalization, becomes inadequate at very high densities. As technology allows even more complex functions to be implemented, it is clear that signal processing is going to play an increasingly important role in storage systems.

Manuscript received July 9, 1992; revised March 9, 1993. This work was supported in part by the National Science Foundation

under grant no. ECD-8907068. The government has certain rights to this material.

J. G. Kenney is with the Department of ECE, Oregon State University, Cowallis, OR 97330.

L. R. Carley is with the Department of ECE, Carnegie Mellon Univer- sity, Pittsburgh, PA 15218.

R. W. Wood is with the IBM Corporation, Magnetic Recording Institute, Advanced Recording & Data Detection, 5600 Cottle Rd F84/025, San Jose, CA 95193.

IEEE Log Number 9209188.

A convenient measure of recording density is PW50/ T -the pulse width measured at 50% amplitude divided by the bit period of customer data. Peak detection systems using an RLL code and some amplification of high fre- quency energy are viable up to a density of l.SPWSO/T. At very high densities (PW50/T = 2.5 would be consid- ered an extreme for a hard disk drive) such an approach becomes inadequate. Detection strategies presently under examination provide SNR gains that could lead to a 30- 40% increase in linear storage density just by changing the signal processing [9].

A recent alternative to peak detection in disk systems is Class IV partial-response -signalling followed by maximum-likelihood sequence detection (PR4 + ML) [ 11, [2], [3]. Class IV partial-response signalling equalizes the channel to a fixed-response (1 - D2), where D refers to a unit delay. Maximum-likelihood sequence detection (MLSD) is implemented as a Viterbi decoder [4]. There are two shortcomings of this approach. The choice of the PR4 (1 - D2) is not a good match to high density channels or to channels using minimum run-length constraints. In such cases, higher order PR polynomials must be used leading to an exponential growth in the complexity of the Viterbi detector. Second, the Viterbi algorithm does not guarantee decisions on every clock cycle which forces the clock recovery circuitry to use an auxiliary phase detector and the automatic gain control to use an auxiliary gain detector.

Pate1 [7] proposed a detection scheme for channels with a minimum run length constraint of (d = 1)-that is transitions are separated by at least two bit intervals. Instead of equalizing to a PR4 transfer function, the channel is equalized to an enhanced class IV partial response (EPR4) transfer function-(1 + D - @ - D3). This scheme ob- tains near maximum likelihood performance without Viterbi computations by addressing only the most likely error events. This leads to a detector which requires 15 adders and a fairly simple finite state machine. However, the optimality of this equalization target extends only over a limited range of densities.

A second detection strategy having wide usage in digital communications is Decision Feedback Equalization (DFE) [ 101. DFE uses a whitened matched filter to shape the IS1 so that it is causal. Maximum SNR at the decision rule is achieved when the noise at the output of the whitened matched filter is perfectly decorrelated. A feedback filter using past decisions cancels the causal ISI. The

0018-9464/93$03.00 0 1993 IEEE

KENNEY et al . : DFE FOR SATURATION RECORDING 2161

complexity of DFE increases linearly with the length of the ISI, in contrast to MLSD whose complexity increases exponentially. However, the decision rule in DFE uses less of the signal energy, thus reducing SNR and increas- ing the probability of a detection error.

Enhanced DFE was reported by Wood for a channel with no minimum constraint, d = 0 [16]. The forward filter equalizes the playback signal to a simple target response which is close to optimal at PW5O/T = 2.25 (assuming a rate 8/9 code). The feedback filter is used to cancel all but the first two samples of the input bit response. A 2-dimensional quantizer makes a binary decision based on the outputs of a 2-tap matched filter and a 2-tap auxiliary filter. This detector achieves comparable performance to EPR4-ML over a narrow range of densities.

Moon proposed an alternative data recovery scheme called Fixed-Delay Tree Search with Decision Feedback (FDTWDF) [5], [9]. FDTS/DF uses a fixed-depth maximum likelihood decision rule implemented as a tree search detector. Equalization in FDTS/DF is similar to equalization in Enhanced DFE, with the exception that the forward filter in FDTS/DF can be chosen to optimize SNR over all densities [ 101. In choosing this equalization strategy, FDTS/DF attempts to balance the performance advantages of MLSD with the simplified architecture of linear DFE.

FDTS/DF performs comparably to PR4 + ML on channels with no minimum run-length constraint [6], [9]. Channels incorporating RLL codes with no minimum run- length constraint are the most efficient at storing data, when the limiting factor is linear ISI. However, when the limiting factor is write nonlinearity, minimum run-length constraints may provide higher storage density [8]. On channels with a minimum run-length constraint of 1 or higher, FDTS/DF has been shown both through theoret- ical development and simulation to perform significantly better than PR4+ML [5], [6]. Upon closer examination of the implementation details for FDTSIDF, serious ques- tions as to its practicality arise. It will be shown that the number of sequential operations during a data detection period is 1 multiplication and 4 additions for a tree of depth 1; this is due to the minimum likelihood sequence detector in the feedback loop.

The decision space of the tree-search detector using linear discriminants is explored [ 111. More importantly through, on high density channels with a d = 1 minimum run-length constraint, this paper demonstrates that the FDTS/DF detector can be implemented as a multi-level decision feedback equalizer (DFE). Channels where a d = 1 constraint is used are interesting because the added space between transitions reduces nonlinear IS1 thereby making the assumption of a linear channel up to higher densities valid. The gain in density is achieved at the ex- pense of higher sampling rates for the same information rate.

Section I1 presents background information on the problem of IS1 in high density data storage. In addition,

FDTS/DF is described from an algorithmic and implementation standpoint. Section I11 describes the tree-search detector for a tree of depth 2 in terms of linear discriminants. Linear discriminants, which are 2-tap finite impulse response filters, lead to practical implementations of FDTSIDF. In Section IV, the most significant contri- bution of this work, which is a detector for a channel using a 2/3(1,7) ' RLL code, will be described. Using simple linear system theory, we will show that the detector can be transformed into multi-level DFE. Multi-level DFE, as presented in this paper, should not be confused with DFE on a pulse amplitude modulated signal. Instead DFE is used to achieve a form of partial response equalization at the slicer. The slicer is a simple threshold detector. We will derive the transformation from FDTS/DF to multi-level DFE. Analytical expressions for the performance gain of multi-level DFE over conventional DFE will be provided. Equalization issues for this detector will be addressed; specifically, applying conventional techniques for designing DFE to multi-level DFE and adaptive signal processing for multi-level DFE. Section VI1 will present conclusions and areas for future work.

11. BACKGROUND A linear system model for the magnetic recording chan-

nel is shown in Fig. 1. The input data stream ak is written at a sample rate of 1 / T and has values of & 1. The (1 - D ) block converts ak into a ternary signal xk, which has values of { -2, 0, +2}. The isolated transition response of the head is represented by h(t) . Commonly it is mod- eled as a Lorentzian pulse

A 1 + (2t/PW50>* (1) h(t) =

where A is a gain factor and PW50 is the half height width of the pulse resulting from a transition; density is often expressed in terms of PW50. Finally, the channel has an additive noise source n ( t ) , which is assumed to be both Gaussian and white. The output of the channel is z ( t ) .

A good starting point for describing FDTS/DF is DFE. The block diagram for DFE is shown in Fig. 2. The input signal, z ( t ) , is placed through a matched filter h(-t) , sampled, and then passed through a transversal filter wk. This sequence of filters shapes the IS1 of the channel so that it is causal. In DFE, there is a second transversal filter in a feedback loop called the feedback filter, bk. The impulse response of the feedback filter is the same as that of the equalized channel from ak to rk. The correctly detected data dk is convolved with impulse response of the feedback filter to cancel the causal ISI. Fig. 3 shows a simplified discrete-time description of the magnetic recording channel, where B(D) is the D-transform of bk. Maximum signal-to-noise ratio (SNR) is achieved when

'The form of an RLL code is R (d, k ) . R is the rate of the code-a rate 2/3's indicates that there are 2 user data bits stored in 3 symbols on the disk. d is the minimum number of clocks between adjacent transition. k is the maximum number of clocks between adjacent transitions.

1

2162 IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 3, JULY 1993

*+Tfl+& Fig. 1 . Magnetic recording channel model.

Fig. 2. Decision feedback system with noise whitening matched filter.

I + d

Fig. 3. Simplified discrete-time description of the magnetic recording channel for DFE.

the noise, nk at the output of the transversal filter is uncorrelated. When this objective is achieved the matched filter followed by the transversal filter is called a whitened matched filter [4]. Assuming a noiseless signal and per- fect equalization, the signal at the input of the decision rule (vk in Fig. 3) is + 1 , corresponding to the possible values of the input alphabet, ak.

In DFE, all of the IS1 in the signal r, is removed by a feedback filter, leaving an isolated symbol upon which to perform detection. The feedback filter in DFE is a finite length transversal filter with coefficients B = [b , , b2, . . . , bN]. Instead of equalizing the channel to an impulse 6(n) , as in DFE, FDTS/DF equalizes to a time-domain response: 6 (n + 1) + b, 6 (n). This can also be represented with a D-transform: (D-' + b,). This impulse response can be achieved by removing the coefficient bl from the backward equalizer. The coefficients in the modified backward equalizer for FDTS/DF are B' = [b2, . . . , bN], A maximum likelihood sequence detector, implemented as a fixed-depth tree-search detector, is used as the decision rule in FDTS/DF. A fixed-depth tree is used instead of the Virerbi algorithm, because the feedback filter needs decisions on every clock cycle.

Let us consider the tree of depth 2 shown in Fig. 4. A transition metric is computed for every branch of the tree. The transition metric at time k is calculated by taking the mean-squared error (MSE) between the received symbol and an ideal noiseless value yl

(2) In a tree of depth 2 , the ideal noiseless values are each a linear combination of the first two coefficients of backward equalizer (bo and b, ) where bo is constrained to be

x; = (Vk - yi)2

m'k / = l o r 3 .

Fig. 4. Tree of depth 2

1.0 [9], [12]. Thus the ideal values corresponding to the last branches in the tree (listed from top to bottom) are:

The transition metric computed in (2) is summed with the appropriate accumulated path metric to form the update accumulated path metric. In this example, the updated version of the accumulated path metric is

((1 + b,), (-1 + bl), (1 - b') , ( -1 - bd}.

mi = + j = (1, 2) or (3 , 4) i = (1, 2, 3 , 4 )

If mi or mi has the smallest value than a +1 is detected and {m:, mi} are used as the accumulated path metrics at the next detection period. Otherwise, a - 1 is detected and {mi, ml} are used as the accumulated path metrics at the next detection period. Further descriptions of FDTS/DF can be found in [5], [9].

System Complexity One of the advantages of PR4 + ML is the simplicity

of the circuitry required to compute path metrics. In PR4+ML, the ideal values for y' found in (2) are { -2 , 0, + 2 } [ 2 ] . Equation ( 2 ) can be expanded to

x i = y; - 2yky' + yi2.

Since 0; is common to all path metrics, it may be dropped, leaving a new expression for the path metric:

-2vky' + y'2.

As y' can only be { -2 , 0, 2 } , the value of yi2 can only be (0, 4) . The multiplication found in 2vky' is an addition when y' is -2 or +2. Thus PR4 + ML does not require any multipliers in the detector. In contrast, a generalized partial response polynomial such as (1 + b , D ) , where b, is allowed to vary, would necessitate a multiplier in the detector for computing path metrics.

Now we will consider the implementation complexity of FDTS/DF. Fig. 5 is the block diagram for a direct digital implementation of FDTS/DF incorporating a tree of depth 2. Let us begin by looking at the circuitry involved in computing path metrics. On each clock cycle, a new transition metric, Xi, is computed. The ideal value of the signal must be subtracted from the actual value; this is equivalent to 1 adder. The difference between the received value, vk, and the ideal value, y', is squared. Squaring implies a multiplier or a large lookup table [6]

KENNEY et al . : DFE FOR SATURATION RECORDING 2163

m(k-1)

Fig. 5 . Direct digital implementation of FDTS/DF.

in the critical time path. The final step in computing the path metric is to sum the new transition metric, hi, and the previously accumulated path metric, mi - .

In summary, each path metric is computed with 2 additions and 1 multiplication. These computations must be done for every branch in the tree. Since the detector is intended for playback channels operating in the 50- 100 M Sample/s range, the path metrics have to be computed in parallel. Consequently a tree of depth 2 requires 4 multipliers and 2 adders. In addition to the arithmetic operations required by the path metrics, an adder is used at the summing node of the feedback loop and an adder is also required for metric arbitration. Thus, the total number of operations performed in series during a clock period is 4 additions and 1 multiplication. The total number of pri- mitive arithmetic modules used, exclusive of the ones required by the equalization filters, are 13 adders and 4 multipliers.

111. TREE SEARCH DETECTOR In the previous section, it was shown that a direct dig-

ital implementation of FDTS/DF requires 5 sequential arithmetic operations during a sample period. The following discussion will lead to a greatly simplified architecture. Since the maximum-likelihood detector in FDTS/DF terminates at the root of the tree, accumulated path metrics are not required. Instead, path metrics can be expressed as the sum of a finite and fixed number of transition metrics. For a tree of depth 2, the finite length path metric for node i at time k is

MI = hi + A i - ,

i = (1, 2, 3 , 4 ) , j = (1, 2) or (3, 4)

(3) where the values for j are a function of the previously detected symbol (ik - I . Thus, the four path metrics can be specified as functions of Vk, Vk -

M i = ( ~ k - (1 + b1))2 + ( ~ k - 1 - (1 + bliik-l))2

b l , and cik - I :

(4) M i = (Vk - (1 + b1))2 + (V&l - (1 + bl(ik-,))2

M ; = (Vk - (1 - b1))2 + (Vk- 1 - (-1 + b1Cik-,))2 ( 5 )

(6)

M i = ( ~ k - (-1 - b1))2 + ( ~ k - 1 - (-1 + bl~2k-I))~.

(7)

When Bk - is + 1, j is { 1, 2) . For the other case where ( i k - , is - 1 , j is ( 3 , 4) .

A technique for performing metric arbitration is to show that all of the path metrics from the bottom half of the tree are larger than at least 1 path metric from the top half of the tree. For example, if the inequalities ( M i > Mi) and ( M i > M:) are satisfied, the detected value Bk is +1 re- gardless of value of the path metric M i . The other con- dition under which Bk is detected as a +1 is when (M: > M i ) and (Mi > M i ) ; all other scenarios make Cik = - 1. Inequalities are evaluated using subtraction. Now, a new variable L( i , i’) can be defined for the tree of depth 2

L(i, i’) = - M I

and i = (1, 2)

i’ = ( 3 , 4 ) . (8)

A positive value for L ( i , i’) indicates that the received sequence Vk = { vk - ,, vk} is closer to point i than to point i’ in Euclidean space. The received sequence is equi-dis- tant from both points if L(i, i’) = 0. We will define the line L ( i , i’) = 0 to be L(i , i’). Finally when L(i , i’) is negative, the received sequence is closer to point i’ than to point i.

Using the description for metric arbitration from above, 4 boundaries must be computed: L (1, 3), L (1, 4), L (2, 3) and L (2,4). An example derivation for L (1, 3) follows

L(1, 3) = (Vk - (1 - b1))2 + ( V k - 1

- (-1 + blCik-,))2 - (Vk - (1 + b1))2

- ( q - 1 - (1 + b,(ik-J)2.

L(1, 3) = b l V k + V k - 1 - b , ( l + c i k - , ) .

(9)

(10)

Equation (9) reduces to2

The other 3 metrics are L(1, 4) = (1 + b , ) V k + V k - 1 - bl(ik-1

L(2, 3) = ( -1 + bl)V/( + V k - 1 - b,( l + & I )

L(2,4) = b l V k + V k - 1 + b , ( l - ( i k - 1 ) . (11)

The equations needed to implement this detector will now be summarized. The inequalities are evaluated using (10) and (1 1). These expressions show that the boundaries depend upon the value of Lik - ,. Arbitration in this detector is performed as:

if ((L(1, 3) > 0) and

( U 1 7 4) > 0))

*A factor of 4 is dropped between (9) and (10) without changing the meaning of the decision boundary as only the sign matters.


or

((L(2, 3) > 0)

(L(2, 4) > 0))

and

then bk =

else bk = -1.

The derivation culminating in (10) and (1 1) suggests a geometric representation of the decision space. For example, when ( 6 k - = +l), the ideal noiseless values Y’ = ( y’, y’) are

Y’ = (1 + bl, 1 + b, ) , Y2 = (1 + b , , -1 + bl), Y 3

= (-1 + b, , 1 - b, ) , Y4 = (-1 + b , , - 1 - b, )

where y’ is the ideal value at time k - 1. These 4 points can be depicted in Cartesian space with vk - I on the hor- izontal axis and vk on the vertical axis (See Fig. 6). The decision boundaries defined by evaluating (10) and (11) with L ( i , i’) are shown as well. The shaded region in this figure is the region in vk = { U k - vk) space where the most likely symbol is - 1.

IV. RLL CONSTRAINED DETECTOR On channels with minimum run-length (RL) con-

straints, such as 2 / 3 (1, 7) coded channels where d = 1, it is possible to remove paths that violate the minimum RL constraint. A d = 1 minimum RL constraint requires 2 consecutive + l ’ s or - 1’s. Consider the tree of depth 2 shown in Fig. 7. When L ? - I = +1, path 3 violates the d constraint, because there is a - 1 surrounded by two + 1’s (i.e { +1, -1, +1}). Thus this path can be removed from the arbitration process. This result is easily extended to the state, bk - = - 1. When bk - I = - 1, path 2 is invalid because the sequence { - 1, + 1, - l} violates the d constraint. These constraints permit the path metric arbitration to be written as

state: hk-1 = + 1

L(1, 4) < 0

L(2, 4) < 0

L i k = -1

and

then

else bk = +1

L(1, 4) = (1 + b1)vk + vk-1 - bl

L(2, 4) = bluk + V k - 1

state: dk- = -1

L(1, 3) 3 0

Fig. 6. Decision space, &,-, = +l.

ak., 4 = +1 1

4

Fig. 7. Tree of depth 2 with (d = 1) minimum run-length violation ‘A.

and

L(1,4) > 0

then

bk = +1 else

6 k = -1

L(1, 3) = b l V k -k V k - I

L(1, 4) = (1 + bl )Vk + v k - 1 + bl.

A geometric representation of state i i k - = + 1 is shown in Fig. 8. From the arbitration scheme described above, we see that the discriminant L(2, 4) when the detector is in state bk- = +1 is equivalent to the discriminant L( 1, 3) when the detector is in state i i k - = - 1 [ 131.

A final point should be made regarding the value of b, . A spectral factorization technique described in Lee [lo] was used to determine the causal impulse response of the magnetic recording channel. Fig. 9 shows the value of b1 for symbol densities beginning at 1.33 user bitslPW50 and ending with 3.0 user bitslPW50.

V. Low PROBABILITY BOUNDARIES Although a boundary might be active, if the probability

of it being exercised is minimal, then it may be removed from the detector without sacrificing performance. The scale drawing for the linear discriminants in Fig. 10 where bl = 0.5 shows that most of the detection errors result from a noisy sequence that is ideally Y2 or Y4 crossing

1

KENNEY et al.: DFE FOR SATURATION RECORDING 2165

+1

Fig. 8. 6k-l = + I .

1 .o 2.0 3:O

D.cl.lty

Fig. 9. Coefficient b, as a function of user bit density.

L (2,4). By omitting L ( 1 , 4 ) from the detector, the hashed region in Fig. 1 0 is decided as a - 1 instead of a + 1. The error source arises from a noisy sequence that should be detected as Y' being detected as Y4 instead. However, it should be noted that Y4 has additional noise margin, because its decision space now includes the hashed region.

The probability of an event falling in the hashed region is the difference between the P(L(2, 4) c 01 Y ' , ( ik- = +1) and P ( L ( 1 , 4) c OIY', = +l), for values of v k - I -b;. Bounds on vk are specified as linear functions of vk - I . The upper bound g, (y - and lower bound g 2 ( v k - J are determined by solving for vk in lines

Fig. 10. Modified detector with L(1 , 4) linear discriminant.

The probability that a sequence V, falls into the hashed region in Fig. 1 0 can be determined by evaluating the following double integral:

gZ(Vk - I)

P ( e ) = s" j f(vk-1, 2 4 ) h k dUk-1 (12) gl(Vk - I)

wheref(vk - I , vk) is the 2-dimensional probability density function of the noise. It is assumed that the noise is Gaussian, stationary and uncorrelated. For an ideal value of Y' with d k - = + 1, the mean value for both vk and vk-' is (1 + bl). Thus, the 2-dimensional P ( e ) in (12) with a Gaussian distribution is

To make the analysis pertinent, the P (e) for ( 1 3) should be compared with the overall probability of error, which can be approximated by the most-likely error event at the SNR typical in magnetic recording channels. As Fig. 1 0 shows, the most likely symbol to be detected incorrectly is Y4. The probability that a noisy version of Y4 is de-

L(1, 4) and L(2, 4), respectively:

(-%-I + bi) (1 + b')

-vk - I and &(vk- 1) = ~

b' *

gl (vk - I ) =


tected incorrectly is approximately

The information needed to determine the relative im- portance of L (1,4) is contained in (13) and (14). The two unknowns in both of these equations are b, , which is a function of storage density, and a2, which is a function of SNR3. At an user bit density of 1.8 bitslPW50 and an SNR of 13 dB, the probability of error expressed in (13) is at least 10 orders of magnitude lower than the overall probability-of-error for the channel, which is approximated by (14). The probability-of-error from the major error source increases slightly with symbol density. More importantly though, the error due to the L ( 1 , 4) discriminant being removed from the detector falls off consider- ably as the density is increased. The value of bl for the Lorentzian shown in Fig. 9, increases linearly with density, causing the intersection of the lines L(l , 4) and L(2, 4) to be pushed farther away from Y'. At-an user bit density of 2.5 bitslPW50, the probability of being in the hashed region is This indicates that the L (1, 4) discriminant can be safely dropped from the detector.

while the overall error-rate is

VI. MULTI-LEVEL DFE In the previous section, it was shown that FDTS/DF

can be implemented with one linear discriminant on channels with a d = 1 minimum run-length constraint. The resulting detector is shown in Fig. 1 1 . In Fig. 12, an equivalent implementation of Fig. 11 is shown except copies of the linear discriminant Bo(D) = (b, + D) are now in the paths of the forward and backward equalizer paths.4

With Bo(D) placed in the forward path of the feedback loop, the signal at the input of the decision rule is Vk = f i k+ I +, bl hk + nk. The D-tranlsform of vk is V(D) = (D-' + b,)A(D) + N ( D ) , where A ( D ) is the D-transform of the received signal and N(D) is the D-transform of the additive white noise, nk. With Bo(D) incorporated in the forward and backward equalizers, the signal to the comparator has a D-transform

Q(D) = V(D)Bo(D). (15) Equation (1 5) can be expanded to

Q(D) = (D-' + bl)(bl + D)A(D) + (b, + D)N(D).

(16)

In a tree of depth 2, there are 8 permutations of the binary variable bk = (6, - ,, hk, hk - The resulting detector is one in which binary decisions are made on a multi-level signal.

By examining invalid sequences, a greater appreciation for the performance gains of FDTS/DF with branch pruning can be developed. To facilitate the ensuing explana- tion, bl is chosen to have a value of 1.0. This roughly corresponds to a data density of 2.0 bitslPW50. The two invalidsequencesofhkare {-1, +1, -1) and {+l , -1, +1}. In both cases the value of d k is 0. This causes the decision spaces for { hk = + l} and { hk = - l} (i.e., the second term in (17)) to merge. Without branch pruning, hk- I must be incorporated into the computation of linear discriminants which increases the circuit complexity of the detector. Note, this should be expected as without the d-constraint we could not have dropped either decision boundary.

Next, the SNR of multi-level DFE can be expressed relative to the SNR of conventional DFE which has signal levels of + 1. The minimum Euclidean distance, dmin, between distinct sequences in DFE is 2, leading to the following expression:

or

1 SNR = 7

U

where u2 is the variance of the additive white Gaussian noise.

The closest ideal value of Qk to the threshold at 0 is k { 1 + b:} (e.g., cik = { + I , +I , -I}), yielding a minimum Euclidean distance between symbols o f

(19) dmin = 2(1 + b:)

In multi-level DFE, the noise at the decision rule, which is defined to be n;, has been filtered through Bo(D). The time-domain expression for the noise in multi-level DFE as a function of the noise in conventional DFE is

(17) ~ [ n ; ~ ] = ~ [ ( b , nk + nk- (21)

'SNR = power in an isolated pulsehoise power. Equation (21) can be expanded to: 41n practice the forward and backward equalizers would be implemented

with one transversal filter for each, instead o f the 2 depicted in Figure 12. = b:E[ni] 4- 2bl E[nknk - 11 11. (22)

1

2167 KENNEY et 01.: DFE FOR SATURATION RECORDING

Fig. 1 1 . FDTS/DF with 1 linear discriminant.

User Bll Dcnsily (PWSO)

Fig. 13. SNR gain of multi-level DFE over bi-level DFE for channel using Y - G - H X ~ A C ~ T ~ 2/3(1, 7) RLL code.

B '(0) B '(0) Equations (26) and (27) show a simple relationship between the filters in DFE and the filters in multi-level DFE.

The first design technique described by Salz [ 151 entails Fig. 12. Multi-level DFE.

Given that nk is independent and identically distributed, the (E[nknk- , ] = 0) and the (E[n:] = E[n: - , ] = a2) , the noise variance in (22) is expressed as

E[ni2] = (b: + 1)02. (23) The SNR at the input to the comparator can be written as

where dki, is shown in (19) and the E[ni2] is stated in (23). SNR in multi-level DFE is

The gain in SNR of FDTS/DF over DFE, which is computed by dividing SNRf from (25) by SNR from (18) , is (1 + b;). Fig. 13 shows the performance advantage of multi-level DFE over conventional DFE.

Equalization Techniques for determining tap weights in DFE have

been extensively researched and documented [ l o ] , [ 141, [ 151. This knowledge can be exploited by establishing a relationship between the tap weights in DFE and the tap weights in multi-level DFE. We will begin by assuming that the tap weights have been determined using a mean- squared error technique [ 151 ; the forward equalizer coefficients are W = [wo, - * , w M ] and the backward equalizer coefficients are B = [ b , , - - , b , ] . Previously, it was stated that the backward equalizer in FDTS/DF is the backward equalizer from DFE, B, without the coefficient b , ; that is the backward equalizer in FDTS/DF is B' = [b2, - * - , b N ] . FDTS/DF uses the same forward equalizer as DFE.

Let us refer to Fig. 12, where the forward equalizer is shown to be the cascade of W ( D ) and Bo(D):

inverting an autocorrelation matrix. This autocorrelation matrix is generated from the isolated transition response of the magnetic recording channel. Unfortunately, this method assumes both a linear channel and white spectrum on both the input signal, ak, and noise, nk. This is not an accurate model of a magnetic recording channel, where correlation is added to the signal by the RLL coder, and some of the noise sources are known to be correlated with the transition spacing [ 5 ] . Most importantly, matrix in- version is highly susceptible to numerical problems. For these reasons, recursive algorithms utilizing the statistics of the channel in determining tap weights are preferred. The least mean-square (LMS) algorithm is a commonly used scheme for adapting equalization filters [ 141.

Adaptive algorithms update coefficients at each clock cycle. To acknowledge the time-varying nature of the coefficients, the forward equalizer is stated as W ; = [w;),k, wi,k, - - * , W L , k ] and the backward equalizer is

known algorithm [14] so we will just write the recursive expressions used to adjust the tap weights in the forward and backward equalizers. The forward equalizer is adapted using

stated as B ; = [b;,k, bi,k, * ' * , bh,k]. LMS is a Well

w;+I = w; + 2pwSkEk

where S, is a vector of input signals (see Fig. 12), p, sets the speed of adaptation, and E k is an error signal driving the adaptation. In a similar manner, the tap weights in the backward equalizer are updated using

BL+l = B ; + 2pbdk'k

where Ak is a vector of detected symbols and pb sets the adaptation rate for the backward equalizer.

Up to this point, the adaptation of multi-level DFE is identical to DFE. In fact, the major difference between the two detectors is the way in which the error signal Ek

is computed. Adaptive algorithms such as LMS compute the error signal as the difference between the actual input,

W ' ( D ) = W(D)BO(D). (26) which in this case is qk and some desired signal Qk - (Ek

= qk - &). Data recovery schemes using adaptation ex-

the desired signal in DFE is g k = Lik.

the backward is the cascade Of B o ( D ) press the desired signal as a function of the detected data; and B 1 (D):

B'(D) = B'(D)BO(D). (27) Under the assumption that there have been no detection

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2168 IEEE TRANSACTIONS ON MAGNETICS, VOL. 29. NO. 3. JULY 1993

errors and the equalizers have removed all ISI, the sequence at the input to the comparator is found in (17). The error portion of qk is (b, nk + nk - I) . This noise component can be removed from qk to define the desired impulse response, Qk:

Qk = bl&k+l + (1 + b:)dk + bldk-1. (28)

A data-driven adaptive equalizer in multi-level DFE uses the following error function:

Ek = qk - (bl ( i k + I + ( 1 + bi)dk + bl dk - 1). (29) An optimum value for b1 exists, if the noise is independent and identically distributed (Le., E[nk, nk- I ] = 0). Consequently, bl can also be adapted. The gradient for bl is slightly different than the gradients for W; and B;.

LMS uses a mean-square error criterion to update the coefficients of its equalizer. The cost function for multi- level DFE is derived from (29):

E (b1.k) = E[€; ] = E[(qk - bl,k(dk- I + dk- 1)

- ( l + b:,k)dk)21. (30)

The gradient of this cost function with respect to bl is

v [ ( b ~ , k ) = -2E[(dk-1 + dk-1 + 2bl,kCik)Ek]. (31) In LMS, the expected value E [ . ] is approximated as the value within the brackets. With this in mind, the adaptation on b1 is written as

bl,k+l = b1.k - 26bl(dk+I + d k - 1 + 2bl.kbk)Ek (32)

where Ek is specified in (29) , and is the rate of adaptation for b l , k .

One small implementation detail regarding LMS in multi-level DFE is that the error signal, ek , used in adapting filter coefficients needs a detected value from the next symbol cycle, dk + I . To accommodate this noncausal term, delay must be added to the coefficient update equations as shown below for the backward equalizer.

(33)

LMS Results The expressions for LMS in multi-level DFE were ver-

ified by simulating the playback channel shown in Fig. 14. Since aliasing in the channel is less than 1% at user bit densities greater than 2.0PW50 [17] , a baud rate sampled channel with band-limit white noise can be used. The forward and backward equalizers each had 10 taps. Sim- ulations were performed using a linear model of the channel incorporating a Lorentzian step response and additive white gaussian noise.

The first step in the simulation is to adapt the equalizers and bl using known correct data. The coefficient bl was tested for steady-state, which would indicate convergence in the equalizers. Steady-state was specified as the coefficients b, maintaining a value within f.005 of its average value for 10000 clock cycles. This tolerance was chosen empirically.

I "' 1

Fig. 14. Discrete-time model of the playback channel.

The SNR which is specified as

SNR = hi la : all k

was scanned from 7.25 dB to 14.15 dB. The variance ai is the noise power of nk. Pseudo-random sequences were used at the input source, ak. To acquire a statistically significant sample, 500 errors were counted before terminat- ing the simulation. The error-rate is computed by dividing the number of errors (e.g., 500) by the number of samples that were used to achieve that number of errors. Fig. 15 shows the log (error-rate) versus the SNR at a user bit density of 2.5 bitslPW50. These results agreed fairly closely with Moon's simulation of FDTSIDF at this density. The average value for bl was 1.25. The smallest value forb, was 1.2445 at an input SNR of 7.25 dB. The highest value for bl was 1.2544 at an input SNR of 14.15 dB. bl was observed to increase monotonically with SNR.

Modijied Error Expression The error expression described in (29) cannot be simply

constructed because of the b: term. This term causes bl to appear in the gradient described in (32). At the proposed symbol densities, bl ranges between 1.0 and 2.0 hinting that ( 1 + b:) may be approximated as 2bl . The revised expression for the ideal value at the input of the quantizer is

(34)

This expression can be further simplified by noting that 6 , is just a scaling factor, which can be set to unity. Thus the emor expression for updating the coefficients of the equalizers is

Qk = bldk+I + 2bl&k + b1dk-l.

Ek = qk - (dk+l + 2 4 + dk-1). (35)

The above simplification can now be examined according to its impact on SNR.

We use the model shown in Fig. 16 for determining the SNR of this system, The desired signal at the input of the quantizer has a D-transform ( 1 + 2 0 + @). In this case dmin is 4, as the closest levels in Euclidean space leading to different decisions are -2 and +2. The noise n; at the input of the slicer has a power spectral density

0 + 2 + 0-1) (D-' + 2 + 0) @2, ( l + b l D 1 + b1D-' s, 1,' (0) =

(36) The noise component of the SNR is +,,,,,#(O), which can be computed by taking the inverse Fourier transform of

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KENNEY et a / . : DFE FOR SATURATION RECORDING 2169

7 9 1 1 13 15 SNR (dB)

Fig. 15. Error-rate performance of multi-level. DFE for a symbol density 2.5PW.50 after adapting the forward and backward equalizer with LMS.

Fig. 16. Channel model for the modified error expression.

Sn,,t(ei") at time n = 0. The resulting SNR is (4/4ntn,(0)). This SNR can now be compared to the SNR of the original system which is computed in (25). Over the range of bit densities from 2.0PW50 to 2.6PW50 the value of bl can be seen in Fig. 12 to fall between 0.9 and 1.6. The loss in SNR in this range of densities using the modified error expression found in Eqn. (35) is less than 0.1 dB.

The expression in (35) was verified by simulating the detector on the playback channel shown in Fig. 14. Sim- ulations were performed at a higher user bit density of 2.5 bitslPW50. The coefficients in the forward equalizer, backward equalizer were adapted until steady-state was achieved. Figure 17 shows the error-rate as a function of SNR at a density 2.5 bitslPW50; the gradient functions stated in (29) and (35) were used. As expected there is little noticeable degradation between the two systems, thereby demonstrating the accuracy of the modified error expression described in (35).

Phase, Gain and dc Detection One particularly attractive feature of FDTSIDF is that

decisions are made on every clock cycle. Thus the phase and gain feedback loops can be designed knowing that the time lag between sampling and detection is constant. In contrast, PR4 + ML requires an auxiliary detector, because the Viterbi detector provides decisions in bursts. This section will show that the error expression found in (35) can also be used for phase detection, gain detection and dc offset.

Phase detection in FDTS/DF is performed when there is a trajectory on qk. A positive-going trajectory is defined in terms of ideal signals as

B k - l B k B k f l

while a negative-going trajectory is specified as

B k - 1 > B k > B k + l .

6 8 10 12 14

SNR (dB)

Fig. 17. Comparison of error rate for the detector with the error function determined from the whitened matched filter and the linear approximation to this error function at a user bit density of 2.5PW50.

Fig. 18. Sampling of signal into the decision rule.

It is implied in the above inequalities that Qk must have a value of +2 or -2. Phase detection is not performed on every sample. For descriptive purposes, let us assume that the forward equalizer is a continuous-time filter and that sampling is done in front of the decision rule as shown in Fig. 18. If qt is sampled late on a positive-going trajectory, q k will be greater than Bk. Likewise, if q, is sampled early, then q k will be less than ijk.

Timing recovery can be accomplished with a stochastic gradient algorithm, namely LMS. The mean-squared phase error can be expressed as:

[ ( T k ) = E[ek(Tk)2] = E [ ( q k ( T k ) - 4k)'I- (37)

where q k (Tk) emphasizes the dependence on timing phase. The objective is to minimize the phase error which is described by

The expectation in (37) has been omitted so that the phase- error is in a form that is utilizable by LMS. The error Ek

found in (38) can be determined according to (35). As- suming a symmetrical channel, the magnitude of the slope for a negative transition is equal to the magnitude of the slope for a positive transition. Thus the magnitude of the slope can be ignored when computing phase error. How- ever, it must be known when designing the loop filter in the phase-locked loop. Further discussions of phase- locked loops for digital communications can be found in Lee [lo].

An LMS technique can also be used to adjust the variable gain amplifier (VGA) in the automatic gain control. To reflect the dependence of the mean-squared error upon the gain of the channel we define the function:

t (Ak) = E[€k(Ad2] = E[(qk(Ak) - Bd21. (39)

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 3, JULY 1993 2170

+bz I

41-1 I

t -m

I - 1

Fig. 19. Phase and gain error in multi-level DFE.

The estimate of the gain error, which is determined by taking the partial derivative of (Ak) with respect to Ak is:

To simpfify the above expression, we estimate q k (Ak) with its ideal values & which are { -4, -2, 2, 4 ) .

On a sample by sample basis, a gain error cannot be distinguished from a phase error. However, over time, phase errors and gain errors are independent. A positive- going transition through 2 begins at -2 and ends at 4. When the gain of q k is too low, ck(Ak) is negative. This would also yield a negative phase error indicating that qt was sampled too early. A negative-going transition through 2 begins at 4 and ends at -2. The gain error will be the same-in this example negative-since the gain is independent of time. This time the phase error is multi- plied by - 1, because the slope of the signal is negative. A positive phase error indicates that the signal was sampled too late. This cancels the phase error from the positive going trajectory which indicated premature sampling.

The final piece of information that can be derived for the error signal c k is whether there is a dc offset in the signal at the input to the decision rule. In this case dc offset is computed by putting Ek through a low-pass filter. When there is a positive dc offset in the playback signal, Ek will be consistently positive.

Digital Implementation of Multi-level DFE Figure 19 shows a block diagram for multi-level DFE,

including the error circuitry. The error is computed using the expression for the desired signal found in (34) with b, = 1. The sequential delay after moving b2 into the path of the forward equalizer is a select and an addition. State Dependent Sequence Detection also has a select and an addition establishing its critical path [7]. The added hardware of multi-level DFE over conventional DFE is a 3-tap transversal filter versus a 1-tap transversal filter.

Each tap consists of an adder, a delay, and a multiplexer. Additional hardware includes a 6-bit A/D converter to achieve the desired error-rates [6] and D/A converters for the gain control, clock recovery and dc offset circuitry.

VII. CONCLUSION System complexity issues with the original description

of FDTS/DF were addressed. An impediment to a practical realization of a detector using FDTS/DF is that 4 additions and 1 multiplication must be evaluated in one clock cycle. In contrast, PR4 + ML requires no multipliers in its Viterbi detector, and has a critical path delay consisting of an add, a compare and a select.

We showed that the tree-search detector with a depth of 1 could be analyzed using linear discriminants. In de- veloping a geometric description of the tree-search detector, we showed that the tree-search algorithm could be replaced by a 2-tap FIR filter on channels using a run- length code with a minimum run-length constraint d = 1. Nevertheless, this architecture requires a multiplication within the feedback loop, which increases the critical time path.

By applying simple linear system principles, we showed that the transversal filter could be removed from the forward path of the feedback loop and placed in series with the original forward and backward equalizers of FDTS/ DF. This modification reduced FDTS/DF to multi-level DFE, making binary decisions on a signal which is ideally 4-level.

As multi-level DFE is a new technique for recovering data from disk drives, an error signal was presented for performing dc detection, gain detection, phase detection and adaptive equalization. It was shown that LMS can be used to adapt the forward equalizer, backward equalizer, and the transversal filter used to construct the desired impulse response. Moreover, the transversal filter generat- ing the desired impulse response was controlled by 1 programmable coefficient and did not require multipliers as its input were previous binary data decisions.

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REFERENCES F. Dolivo “Signal processing for high-density digital magnetic recording,” Proc. VLSI and Microelectronic Applications in Intelligent Peripherals and their Interconnection Networks, Hamburg, West Germany, pp. 1/91-96, May 1989. R. W. Wood and D. A. Petersen, “Viterbi detection of class IV partial response on a magnetic recording channel,” IEEE Trans. Comm., vol. COM-34, no. 5 , pp. 454-461, May 1986. T. D. Howell, D. P. McCown, T. A. Diola, Y. Tang, K. R. Hense, and R. L. Gee, “Error rate performance of experimental gigabit per square inch recording components,” IEEE Trans. Magn., vol. 26, no. 5, pp. 2298-2302, Sept. 1990. G. D. Forney, “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 363-378, May 1972. J. J. Moon, Pragmatic sequence detection for storage channels, Ph.D. dissertation, Dept . of Electrical & Computer Engineering, Carnegie Mellon University, Apr. 1990. L. R. Carley and J. G. Kenney, “Comparison of computationally efficient forms of FDTSlDF against PR4-ML,” IEEE Trans. Magn., vol. 27, no. 6, Nov. 1991. “Method and apparatus for processing sample values in a coded signal processing channel,” U.S. Patent 4,945,538 (1990). P. H. Seigel, “Recording codes for digital magnetic storage,” IEEE Trans. Mugn., vol. MAG-21, no. 5, pp. 1344-1349, Sept. 1985. I. J. Moon and L. R. Carley, “Performance comparison of detection methods in magnetic recording,” IEEE Trans. Magn., vol. 26, no.

E. A. Lee and D. G. Messerschmitt, Digital Communications, Bos- ton: Kluwer Academic Publisher, 1988. Duda and Hart, “Pattern classification and scene analysis,” New York: Wiley, 1973. J. G. Kenney, P. A. McEwen, and L. R. Carley, “Evaluation of magnetic recording detection schemes for thin film media,” IEEE Trans. Magn., vol. 27, no. 6, Nov. 1991. J. G. Kenney, “Geometric interpretation of the tree search detector and its hardware implications,” Ph.D. dissertation, Dept. of Electri- cal & Computer Engineering, Carnegie Mellon University, Dec. 1991. S. Qureshi, “Adaptive equalization,” Proc. IEEE, vol. 73, no. 9, pp. 1349-1386, Sept. 1985. J. Salz, “Optimum mean-square decision feedback equalization,”

R. Wood, “Enhanced decision feedback equalization,” IEEE Trans. Magn., vol. 26, no. 5, pp. 2178-2180, Sept. 1990. I . G. Kenney, “Multi-level decision feedback equalization with clock recovery ,” Asilomar Conference on Signal Systems and Computers, Pacific Grove, CA., pp. 945-949, Oct. 1992.

6, pp. 3155-3172, NOV. 1990.

BSTJ, VOI. 52, pp. 1341-1373, Oct. 1973.

John G. Kenney (S’87-M’87-S’87-M’91) received the B.S. degree in 1984 from both Providence College and Columbia University in a com- bined plan engineering program. The M.S. and Ph.D. degrees were received at Camegie Mellon University in 1988 and 1991, respectively.

He is an Assistant Professor of Electrical and Computer Engineering at Oregon State University. Dr. Kenney worked at Codex Corporation in Mansfield, MA as a circuit designer in an advanced development group for TI multiplixers 1984 until 1986. In the fall of 1986, he became a research assistant at Carnegie Mellon University where he studied the design of higher-order multibit noise-shaping coders. His Ph.D. research was in data recovery from magnetic disk drives. His main research interest is the efficient implementation of signal processing algorithms for data recovery.

L. Richard Carley (S’77-M’8l-S’81-M’83-S’84-M’84-SM’90) received the B.S. degree from the Massachusetts Institute of Technology in 1976 and was awarded the Guillemin Prize for the best EE Undergraduate Thesis. He remained at MIT where he received the M.S. degree in 1978 and the Ph.D. in 1984.

He is a Professor of Electrical and Computer Engineering at Carnegie Mellon University. He has worked for MIT’s Lincoln Laboratories and has acted as a consultant in the area of analog circuit design and design automation for Analog Devices and Hughes Aircraft among others. In 1984, he joined Camegie Mellon, and in 1992 he was promoted to Full Professor. His current research interests include the development of CAD tools to support analog circuit design, the design of high performance signal processing ICs employing analog circuit techniques, and the design of low- power high-speed magnetic recording channels.

Dr. Carley received a National Science Foundation Presidential Young Investigator Award in 1985, a Best Technical Paper Award at the 1987 Design Automation Conference, and a Distinguished Paper Mention at the 1991 International Conference on Computer-Aided Design.

Roger W. Wood (A’81-M’90) graduated from University College, Lon- don, in 1972 and, in 1979, received the Ph.D. in Electrical Engineering from the University of British Columbia, Canada. He is manager for Disk Drive Prototyping at the IBM Advanced Magnetic Recording Laboratory in San Jose, California. From 1972-75 he was with the British Telecom Research Department, England, working on microwave radio relay systems. In 1979, he joined Ampex Corporation in Redwood City, California, where he became manager of the Recording Technology Department.

Dr. Wood has worked at IBM on magnetic disk drives since 1986 where he has managed groups in component integration and in advanced channel development before assuming his present position. His research interests include magnetic recording technology and signal processing and detection.

multi-level decision feedback equalization for saturation recording

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