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Maximum Likelihood Decoding for Multi-Level Cell Memories with Scaling and Offset Mismatch

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Reliability is a critical issue for modern multi-level cell memories. We consider a multi-level cell channel model such that the retrieved data is not only corrupted by Gaussian noise, but hampered by scaling and offset mismatch as well. We assume that the intervals from which the scaling and offset values are taken are known, but no further assumptions on the distributions on these intervals are made. We derive maximum likelihood (ML) decoding methods for such channels, based on finding a codeword that has closest Euclidean distance to a specified set defined by the received vector and the scaling and offset parameters. We provide geometric interpretations of scaling and offset and also show that certain known criteria appear as special cases of our general setting.
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Maximum Likelihood Decoding for Multi-Level
Cell Memories with Scaling and Offset Mismatch
Renfei Bu
Applied Mathematics Dept., Optimization Group
Delft University of Technology
Delft, Netherlands
R.Bu@tudelft.nl
Jos H. Weber
Applied Mathematics Dept., Optimization Group
Delft University of Technology
Delft, Netherlands
J.H.Weber@tudelft.nl
Abstract—Reliability is a critical issue for modern multi-level
cell memories. We consider a multi-level cell channel model such
that the retrieved data is not only corrupted by Gaussian noise,
but hampered by scaling and offset mismatch as well. We assume
that the intervals from which the scaling and offset values are
taken are known, but no further assumptions on the distributions
on these intervals are made. We derive maximum likelihood (ML)
decoding methods for such channels, based on finding a codeword
that has closest Euclidean distance to a specified set defined by
the received vector and the scaling and offset parameters. We
provide geometric interpretations of scaling and offset and also
show that certain known criteria appear as special cases of our
general setting.
Index Terms—multi-level cell memories, maximum likelihood
decoding, Euclidean distance, Pearson distance, scaling and offset
mismatch
I. INTRODUCTION
As the on-going data revolution demands storage systems
that can store large quantities of data, multi-level cell mem-
ories are gaining attention. A multi-level cell is a memory
element capable of storing more than a single bit of informa-
tion, compared to a single-level cell which can store only one
bit per memory element [1]. For example, in multi-level cell
NAND flash technology, information is stored by introducing
more voltage levels that are used to represent more than one
bit [2].
It is obvious that, as the number of levels increases, the
storage capacity of multi-level cell memories is enhanced.
However, due to the increase in the per-cell storage density, the
reliability of multi-level cell memories experiences a diverse
set of short-term and long-term variations.
Unpredictable stochastic errors are exacerbated with the
short-term variation. For example, random errors occur in the
programming/reading process, and sometimes it is hard to
initialize a cell with the exact voltage. As a result, error cor-
recting techniques are usually considered and applied in multi-
level cell memories, such as BCH codes [3], Reed-Solomon
codes [4], LDPC codes [5], trellis coded modulation [6], and
so on.
In the long term, the performance of multi-level cell mem-
ories degrades with age. As documented in [7], the number
of electrons of a cell decreases and some cells even become
defective over time. The amount of electron leakage depends
on various physical parameters, e.g., the device’s temperature,
the magnitude of the charge, the quality of the gate oxide or
dielectric, and the time elapsed between writing and reading
the data. It is hard to precisely model these long-term effects
on multi-level cell memories. In this paper, we focus on the
mean change over time, while variance issues were discussed
in [8].
Scaling and offset can weaken the cell’s state strength by
moving its level closer to the next reference voltage. Various
techniques have been proposed to improve the detector’s
resilience to scaling and offset mismatch. Estimation of the
unknown shifts may be achieved by using reference cells,
but this is very expensive with respect to redundancy. Also,
coding techniques can be applied to strengthen the detector’s
reliability in case of scaling and offset mismatch; these include
rank modulation [9], balanced codes [10], and composition
check codes [11]. However, these methods often suffer from
large redundancy and high complexity.
Immink and Weber [12] advocate the use of Pearson dis-
tance decoding instead of traditional Euclidean distance de-
coding, in situations which require resistance towards scaling
and/or offset mismatch. We use the same channel model as
used in [12]: besides the noise, which varies from symbol
to symbol, a multiplicative factor aand an additive term b
specify the scaling and offset mismatch, respectively, which
are assumed to be constant within one block of code symbols,
but may be different for the next block. Even though this
model neglects certain aspects of multi-level cell memories,
such as inter-cell coupling or dependent noise, it still captures
key properties of the data corruption process in multi-level cell
memories.
The contribution of this work is two-fold. Firstly, in Sec-
tion III, we derive a maximum likelihood (ML) decoding
criterion for multi-level cell channels with Gaussian noise and
also suffering from the scaling aand the offset b, which are
known to be within certain ranges, specifically 0< a1
aa2and b1bb2. The ML decoding criterion will
also be illustrated with geometric interpretations. Secondly,
the proposed ML criterion provides a general framework,
including the scaling-only case and the offset-only case. Some
known criteria [13] [14] are shown to be special cases of this
framework for particular a1,a2,b1, and b2settings.
978-1-5386-8088-9/19/$31.00 ©2019 IEEE
This paper aims to generalize ML decoding for multi-
level cell channel with Gaussian noise and scaling and offset
mismatch. We start by providing the multi-level cell channel
model in Section II, starting with several definitions and end-
ing with the Euclidean distance-based and Pearson distance-
based decoding criteria. In Section III, we show how to achieve
ML decoding for this channel. We continue in Section IV
considering several special cases, which relate to known results
in this area. We wrap up the paper with some comments and
ideas for future work in Section V.
II. PRELIMINARIES AND CHA NN EL MO DE L
We start by introducing some notations. For any vector u=
(u1, u2, . . . , un)Rn, let
¯u =1
n
n
X
i=1
ui
denote the average symbol value, let
σu= n
X
i=1
(ui¯u)2!1/2
denote the unnormalized symbol standard deviation, and let
kuk= n
X
i=1 |ui|2!1/2
denote the (Euclidean) norm. We write hu,vifor the standard
inner product (the dot product) of two vectors uand v, i.e.,
hu,vi=
n
X
i=1
uivi=kukkvkcos θ,
where θis the angle between uand v. Note that hu,ui=
kuk2.
Consider transmitting a codeword x= (x1, x2, . . . , xn)
from a codebook Sover the q-ary alphabet Q={0,1, . . . , q
1},q2, where nis a positive integer. This is based on the
fact that each cell is initialized with one of a finite discrete
set of voltages. The transmitted symbols xiare distorted by
additive noise vi, by a factor a > 0, called scaling/gain, and
by an additive term b, called offset, i.e., the received symbols
riread
ri=a(xi+vi) + b,
for i= 1, . . . , n. The parameters viRare zero-mean i.i.d.
Gaussian noise samples with variance of σ2R, that is, the
noise vector vhas distribution
φ(v) =
n
Y
i=1
1
σ2πev2
i/(2σ2).(1)
The scaling and offset (unknown to both the sender and the
receiver) may slowly vary in time due to various factors in
multi-level cells. So we assume they may differ from codeword
to codeword, but do not vary within a codeword. The received
vector when a codeword xis transmitted is
r=a(x+v) + b1,(2)
where 1= (1,1,...,1) is the real all-one vector of length n.
A. Euclidean Distance-Based Decoding
A well-known decoding criterion upon receipt of the vector
ris to choose a codeword ˆx ∈ S which minimizes the
(squared) Euclidean distance between the received vector r
and codeword ˆx, i.e.,
Le(r,ˆx) = krˆxk2=
n
X
i=1
(riˆxi)2.(3)
It is known to be ML with regard to handling Gaussian
noise, but not optimal in situations which require resistance
towards scaling and/or offset mismatch.
B. Pearson Distance-Based Decoding
The Pearson distance measure [12] naturally lends itself
to immunity to scaling and/or offset mismatch. The Pearson
distance between the received vector rand a codeword ˆx ∈ S
is defined as
Lp(r,ˆx) = 1 ρr,ˆx,(4)
where ρr,ˆx is the Pearson correlation coefficient
ρr,ˆx =r¯r1,ˆx ¯
ˆx1
σrσˆx
.(5)
A Pearson decoder chooses a codeword which minimizes
this distance. As shown in [12], a modified Pearson distance-
based criterion leading to the same result in the minimization
process reads
L0p(r,ˆx) =
n
X
i=1
(riˆxi+¯
ˆx)2,(6)
if there is no scaling mismatch, i.e., a= 1. Use of the Pearson
distance requires that the set of codewords satisfies certain
special properties [12].
A geometric meaning for Pearson distance is provided
in [14]. Since the offset bchanges the mean of a vector, it
seems reasonable to consider normalized vectors ˆx ¯
ˆx1 and
r¯r1 rather than ˆx and r. On the other hand, scaling a vector
of mean 0 by aonly changes its standard deviation by a factor
of a. So it seems reasonable to scale the normalized vectors so
that they have standard deviation 1. It is not difficult to show
that this is ρr,ˆx.
III. MAX IM UM LIKELIHOOD DECODING
If a vector ris received, optimum decoding must determine
a codeword ˆx ∈ S maximizing P(ˆx |r). If all codewords are
equally likely to be sent, then, by Bayes Theorem, this scheme
is equivalent to maximizing P(r|ˆx ), that is, the probability
that ris received, given ˆx is sent.
From (2), we know v= (rb1)/a ˆx when aand bare
fixed, and since ais nonzero, the likelihood P(r|ˆx )in this
case is
φ((rb1)/a ˆx).
Here, we consider the situation that the scaling and the offset
take their values within certain ranges, specifically 0< a1
aa2and b1bb2, but do not make any further
o
r
A
B
D
C
1
R
2
R
3
R
4
R
5
R
6
R
7
R
8
9
U R
Fig. 1. Subdivision of U0={cr+d1|c, d R}.
assumptions on the distributions on these intervals. Thus, in
order to achieve ML decoding, the criterion to maximize
among all candidate codewords ˆx is
max
0<a1aa2,b1bb2
φ((rb1)/a ˆx).(7)
Since the logarithm function is strictly increasing on the
positive real numbers and φis a positive function, an equiv-
alent formulation of the problem is to find ˆx S that
maximizes
max
0<a1aa2,b1bb2
log φ((rb1)/a ˆx).
Since
log φ((rb1)/a ˆx) = nlog(σ2π)
1
2σ2
n
X
i=1
((rib)/a ˆxi)2(8)
has a component nlog(σ2π)that is independent of ˆx and
r, and since 1
2σ2is a positive constant, a maximum likelihood
decoder finds a codeword ˆx that minimizes
min
0<a1aa2,b1bb2
n
X
i=1
((rib)/a ˆxi)2,
i.e., it minimizes the squared Euclidean distance between the
candidate codeword ˆx and the points in
U={(rb1)/a|0< a1aa2, b1bb2},
which is a subset of the subspace
U0={cr+d1|c, d R}
in Rn.
The squared Euclidean distance between a vector ˆx and the
set Uis defined as
Le(U, ˆx) =
n
X
i=1
(piˆxi)2,
where p= (p1, p2, . . . , pn)is the point in Uthat is closest
to ˆx. The most likely candidate codeword xofor a received
vector has the smallest Le(U, ˆx), that is
xo= arg min
ˆx∈S
Le(U, ˆx).(9)
Hence, ˆx ∈ S closest to Uis chosen as the ML decoder output.
In order to calculate Le(U, ˆx)for a codeword ˆx, we first
find the point in U0that is closest to ˆx and then check if this
point is in U. Applying the first derivative test gives that the
closest point in U0to ˆx is p0=c0r+d01with
c0=hr,ˆxi − n¯r¯
ˆx
hr,ri − n¯r2
and
d0=hr,ri¯
ˆx − hr,ˆxi¯r
hr,ri − n¯r2.
In Fig. 1, we depict the subset Uin gray when a1<1< a2
and b1<0< b2. Four vertices A,B,C,Dare also shown
in the picture:
A= (rb11)/a1,
B= (rb21)/a1,
C= (rb21)/a2,
D= (rb11)/a2.
Perpendicular lines (blue dash) in U0to sides of Uthrough ver-
tices are pictured in Fig. 1. These perpendicular lines and the
sides of Useparate U0into 9 subsets, namely, R1, R2, . . . , R9.
For instance, the perpendicular lines to side BC and BC itself
form the boundaries of R5. We use the notation R9in Fig. 1
for the subset Ufor clerical convenience.
Theorem 1. If p0is in the subset Ri,i= 1,...,9, then the
closest point in Uto ˆx is
p=
hrb11,ˆxi
krb11k2(rb11)if i= 1,
hrb21,ˆxi
krb21k2(rb21)if i= 5,
(r(¯r a1¯
ˆx)1)/a1if i= 3,
(r(¯r a2¯
ˆx)1)/a2if i= 7,
Aif i= 2,
Bif i= 4,
Cif i= 6,
Dif i= 8,
p0if i= 9.
(10)
The ML decoding criterion is minimizing Le(p,ˆx)among all
candidate codewords.
o
r
ˆ
x
M
(a)
o
r
ˆ
x
M
(b)
o
r
ˆ
x
M
(c)
Fig. 2. The distance of a candidate codeword ˆx to the subset
{r/a |0< a1aa2}: three cases in (11), (a) hr,ˆxi>hr,ri/a1, (b)
hr,ˆxi<hr,ri/a2and (c) otherwise, assuming a1<1< a2.
Proof. If p0is in the subset R1, maximizing (7) is equivalent
to minimizing the smallest squared Euclidean distance from
the codeword ˆx to the line segment
AD ={(rb11)/a|0< a1aa2},
which is shown in Fig. 1. Let θbe the angle between ˆx and
rb11. The point on AD closest to ˆx is p=α(rb11)
with
α= (kˆxkcos θ)/krb11k=hrb11,ˆxi/krb11k2.
Similarly, when p0is in the subset R5, the point on BC =
{(rb21)/a|0< a1aa2}closest to ˆx is p=α(rb21)
with
α=hrb21,ˆxi/krb21k2.
If p0is in the subset R3, the point pUthat is closest to
ˆx must be on the line segment
AB ={(rb1)/a1|b1bb2},
which is shown in Fig. 1. The point on AB that is closest to
ˆx is p= (rβ1)/a1, with β=¯r a1¯
ˆx,which follows from
the first derivative test. The proof is similar when p0is in the
subset R7, with the line segment CD taking the role of AB.
If p0is in the subset R2, then the closest point in Uto ˆx is
the vertex A= (rb11)/a1, as can be observed from Fig. 1.
Similar results are found for the situations that p0is in the
subset R4,R6, and R8, where the closest point in Uto ˆx is
B,C, and D, respectively.
Obviously, the closest point in Uto ˆx is p0itself when p0
is in the subset R9=U.
IV. SPE CI AL CA SE S
Several special values of a1,a2,b1and b2are considered,
leading to typical cases for maximizing (7); these include the
scaling-only and offset-only cases. Not only ML decoding
criteria are discussed, but also conventional decoding criteria
as introduced in Section II.
A. Scaling-Only Case
In the scaling-only case, i.e., b= 0, we simply have
r=a(x+v),
where the scaling, a, is unknown to both sender and receiver.
In Theorem 2 of [13], the following ML criterion was
presented for the case that there is bounded scaling (0< a1
aa2) and no offset mismatch (b= 0):
La1,a2(r,ˆx) =
Le(r/a1,ˆx)if hr,ˆxi>hr,ri/a1,
Le(r/a2,ˆx)if hr,ˆxi<hr,ri/a2,
kˆxk2hr,ˆxi
krk2
otherwise.
(11)
This result can also be simply found from the general frame-
work presented in the previous section, by setting b1=b2= 0
in Theorem 1. Note that this gives indeed that p=r/a1if
p0R2R3R4, which corresponds to the situation that
kˆxkcos ϕ
krk=hr,ˆxi
hr,ri>1/a1,
where ϕis the angle between ˆx and r. Similarly, note that
p=r/a2if p0R6R7R8, which corresponds to the
situation that
kˆxkcos ϕ
krk=hr,ˆxi
hr,ri<1/a2.
Finally, note that p=hr,ˆxi
krk2rif p0R1R5R9, which
corresponds to the ‘otherwise’ case in (11), and that
Le(p,ˆx) = Le hr,ˆxi
krk2r,ˆx!=kˆxk2hr,ˆxi
krk2
.
In Fig. 2, we draw the three cases in (11), where the subset
{r/a |0< a1aa2}is a line segment in the direction of
r. The circle points are the closest points on this line segment
to ˆx.
Next, we consider the situation that a10and a2
, i.e., the only knowledge on the gain ais that it is
a positive number, without further limitations. The subset
{r/a |aR, a > 0}is a ray from the origin in the direction
of r. In this case, it follows from the above that ML decoding
can be achieved by minimizing
La(r,ˆx) = (kˆxk2hr,ˆxi
krk2
if hr,ˆxi
hr,ri>0,
kˆxk2otherwise.
(12)
One reason for this choice is that it behaves well with respect
to an affine scaling function (a > 0), since
La(r,ˆx) = La(r/a, ˆx).
That is, scaling a vector rby adoes not change the angle ϕ
between ˆx and r.
o
r
ˆ
x
(a)
o
r
ˆ
x
(b)
Fig. 3. The distance of a candidate codeword ˆx to the line segment {r
b1|b1bb2}: two cases in (13), (a) ¯r ¯
ˆx < b1and (b) ¯r ¯
ˆx > b2,
assuming b1<0< b2.
B. Offset-Only Case
In the offset-only case, i.e., a= 1, we simply have
r=x+v+b1,
where the offset bis unknown to both sender and receiver.
In Theorem 1 of [13], the following ML criterion was
presented for the case that a= 1 and b1bb2:
Lb1,b2(r,ˆx) =
Le(rb11,ˆx)if ¯r ¯
ˆx < b1,
Le(rb21,ˆx)if ¯r ¯
ˆx > b2,
Le(r(¯r ¯
ˆx)1,ˆx)otherwise.
(13)
This result also follows from the general setting presented in
the previous section, by substituting a1=a2= 1. Note that
the first case in (13) corresponds to the situation that p0
R1R2R8, the second case to p0R4R5R6, and
the last case to p0R3R7R9.
We illustrate the first two situations of Lb1,b2(r,ˆx)in Fig. 3
and the last one in Fig. 4, where {rb1|b1bb2}is
shown by a line segment passing through rwith direction 1.
The point in {rb1|b1bb2}that is closest to ˆx is rb11
or rb21for the situations in Fig. 3. For the ‘otherwise’ case
in (13), we consider in Fig. 4 the normalized vectors ˆx ¯
ˆx1
and r¯r1 rather than ˆx and r.
By letting b1→ −∞ and b2→ ∞, we obtain from (13)
that the criterion
Lb(r,ˆx) = Le(r(¯r ¯
ˆx)1,ˆx)
=
n
X
i=1
(riˆxi+¯
ˆx)2n¯r2
=L0p(r,ˆx)n¯r2,
when there is no knowledge at all of the magnitude of the
offset [13]. Noting that the last term n¯r2is irrelevant in the
minimization process, we conclude that the modified Pearson
criterion L0p(r,ˆx)achieves ML decoding in this case.
C. Unbounded Scaling and Offset Case
In this subsection, an ML decoding criterion derived by
Blackburn [14] for the situation when both the scaling aand
o
r
ˆ
x
ˆ ˆ
x x1
r r1
Fig. 4. The distance of a candidate codeword ˆx to the line segment {r
b1|b1bb2}for the ‘otherwise’ case in (13), assuming b1<0< b2.
the offset bare unbounded (a10,a2→ ∞,b1→ −∞,
b2→ ∞) is reconsidered as a special case of the results
presented in Section III. In [14], Blackburn shows that an ML
decoder chooses a codeword ˆx minimizing
lr(ˆx) = σ2
ˆx(1 ρ2
r,ˆx)when ρr,ˆx >0,
σ2
ˆx otherwise.(14)
His argument was that when the scaling factor aand the
offset term bare fully unknown, except for the sign of a,
then maximizing (7) is equivalent to minimizing the smallest
squared Euclidean distance from the codeword ˆx to the subset
U+={(rb1)/a|a, b R, a > 0},
which is a half-subspace of U0Rn. Note that when a10,
a2→ ∞,b1→ −∞,b2→ ∞, our Uis indeed equal to
Blackburn’s set U+. Note that p0=c0r+d01is either in
R9=U=U+or in R7. By (5), c0and d0can be rewritten
as
c0=ρr,ˆxσˆx
σr
(15)
and
d0=¯
ˆx c0¯r.(16)
In case p0R9, which happens if and only if ρr,ˆx >0,
then Theorem 1 says p=p0=c0r+d01. Note that
Le(c0r+d01,ˆx)
=
n
X
i=1
[c0ri+d0ˆxi]2
=
n
X
i=1 c0(ri¯r)xi¯
ˆx)2
=
n
X
i=1 hc2
0(ri¯r)22c0(ri¯r)(ˆxi¯
ˆx) + (ˆxi¯
ˆx)2i
=c2
0σ2
r2c0ρr,ˆxσrσˆx +σ2
ˆx
=ρr,ˆxσˆx
σr2
σ2
r2ρr,ˆxσˆx
σrρr,ˆxσˆx σr+σ2
ˆx
=σ2
ˆx(1 ρ2
r,ˆx),
which is indeed the same as in (14) when ρr,ˆx >0.
10 11 12 13 14 15 16 17 18 19
SNR (dB) = -20log10
10-3
10-2
10-1
100
WER
Pearson Distance Decoding
Euclidean Distance Decoidng
ML Decoding
Fig. 5. Word error rate (WER) against signal-to-noise ratio (SNR) when
q= 4,n= 8,a= 1.07, and b= 0.07.
In case p0R7, then Theorem 1 says p=¯
ˆx1 since
a2→ ∞. Hence,
Le(p,ˆx) = Le(¯
ˆx1,ˆx) = σ2
ˆx.
This shows that Blackburn’s criterion (14) indeed appears as
a special case of our general setting.
D. Simulation Results
Thus far, we have discussed ML decoding for Gaussian
noise channels with scaling and offset mismatch, and have
mentioned that Euclidean distance decoding is ML decoding
for Gaussian noise channels in Section II, while the Pearson
distance criterion (4) is optimal for channels with scaling and
offset mismatch, due to its intrinsic immunity to both scaling
and offset mismatch.
Figure 5 shows simulation results of Pearson distance de-
coding, Euclidean distance decoding, and ML decoding (14)
when q= 4 and n= 8. The word error rate (WER) of
10,000 trials is shown as a function of the signal-to-noise
ratio (SNR =20 log10 σ). Results are given for 2-constrained
codes [12], [15], while a= 1.07 and b= 0.07. The simula-
tions indicate that for this case Pearson distance decoding has
a comparable performance as ML decoding, while Euclidean
distance decoding performs considerably worse.
V. CONCLUSION
We have derived a maximum likelihood decoding criterion
for multi-level cell memories with Gaussian noise and scaling
and/or offset mismatch. In our channel model, scaling and
offset are restricted to certain ranges, 0< a1aa2and
b1bb2, which is a generalization of several prior art
settings. For instance, by letting a10,a2→ ∞,b1
−∞,b2→ ∞, we obtain the same ML decoding criterion as
proposed by Blackburn for the case of unbounded gain and
offset. We also provided geometric interpretations illustrating
the main ideas.
Scaling and offset mismatch are important issues in multi-
level cell memories, but not the only ones. As future work,
one could try to derive ML decoding criteria for multi-level
cell memories for which the channel model includes dependent
noise and/or inter-cell interference as well.
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