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We consider the transmission and storage of encoded strings of symbols over a noisy channel, where dynamic threshold detection is proposed for achieving resilience against unknown scaling and offset of the received signal. We derive simple rules for dynamically estimating the unknown scale (gain) and offset. The estimates of the actual gain and offset so obtained are used to adjust the threshold levels or to re-scale the received signal within its regular range. Then, the re-scaled signal, brought into its standard range, can be forwarded to the final detection/decoding system, where optimum use can be made of the distance properties of the code by applying, for example, the Chase algorithm. A worked example of a spin-torque transfer magnetic random access memory (STT-MRAM) with an application to an extended (72, 64) Hamming code is described, where the retrieved signal is perturbed by additive Gaussian noise and unknown gain or offset.
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1
Dynamic Threshold Detection Based on Pearson
Distance Detection
Kees A. Schouhamer Immink, Kui Cai, and Jos H. Weber
Abstract—We consider the transmission and storage of encoded
strings of symbols over a noisy channel, where dynamic threshold
detection is proposed for achieving resilience against unknown
scaling and offset of the received signal. We derive simple rules
for dynamically estimating the unknown scale (gain) and offset.
The estimates of the actual gain and offset so obtained are used to
adjust the threshold levels or to re-scale the received signal within
its regular range. Then, the re-scaled signal, brought into its
standard range, can be forwarded to the final detection/decoding
system, where optimum use can be made of the distance proper-
ties of the code by applying, for example, the Chase algorithm.
A worked example of a spin-torque transfer magnetic random
access memory (STT-MRAM) with an application to an extended
(72, 64) Hamming code is described, where the retrieved signal
is perturbed by additive Gaussian noise and unknown gain or
offset.
Index Terms—Constrained coding, storage systems, non-
volatile memories, Pearson distance, Euclidean distance, channel
mismatch, Pearson code.
I. INTRODUCTION
In mass data storage devices, the user data are translated
into physical features that can be either electronic, magnetic,
optical, or of other nature. Due to process variations, the
magnitude of the physical effect may deviate from the nominal
values, which may affect the reliable read-out of the data.
We may distinguish between two stochastic effects that de-
termine the process variations. On the one hand, we have
the unpredictable stochastic process variations, and on the
other hand, we may observe long-term effects, also stochastic,
due to various physical effects. For example, in non-volatile
memories (NVMs), such as floating gate memories, the data
is represented by stored charge. The stored charge can leak
away from the floating gate through the gate oxide or through
the dielectric. The amount of leakage depends on various
physical parameters, for example, the device temperature, the
magnitude of the charge, the quality of the gate oxide or
dielectric, and the time elapsed between writing and reading
the data.
Spin-torque transfer magnetic random access memory (STT-
MRAM) [1] is another type of emerging NVMs with nanosec-
ond reading/writing speed, virtually unlimited endurance, and
Kees A. Schouhamer Immink is with Turing Machines Inc, Willem-
skade 15d, 3016 DK Rotterdam, The Netherlands. E-mail: immink@turing-
machines.com.
Kui Cai is with Singapore University of Technology and Design (SUTD),
8 Somapah Rd, 487372, Singapore. E-mail: cai kui@sutd.edu.sg.
Jos Weber is with Delft University of Technology, Delft, The Netherlands.
E-mail: j.h.weber@tudelft.nl.
This work is supported by Singapore Agency of Science and Technology
(A*Star) PSF research grant, and Singapore Ministry of Education Academic
Research Fund Tier 2 MOE2016-T2-2-054.
zero standby power. In STT-MRAM, the binary input user
data is stored as the two resistance states of a memory
cell. Process variation causes a wide distribution of both the
low and high resistance states, and the overlapping between
the two distributions results in read errors. Furthermore, it
has been observed that with the increase of temperature,
the low resistance hardly changes, while the high resistance
decreases, leading to a drift of the high resistance to the low
resistance [2], which may lead to a serious degradation of the
data reliability for conventional detection.
The probability distribution of the recorded features changes
over time, and specifically the mean and the variance of
the distribution may change. The long-term effects are hard
to predict as they depend on, for example, the (average)
temperature of the storage device. An increase of the variance
over time may be seen as an increase of the noise level of the
storage channel, and it has a bearing on the detection quality.
The mean offsets can be estimated using an aging model,
but, clearly, the offset depends on unpredictable parameters
such as temperature, humidity, etc, so that the prediction is
inaccurate. Various techniques have been advocated to improve
the detector resilience in case of channel mismatch when the
mean and the variance of the recorded features distribution
have changed.
For example, estimation of the unknown offsets may be
achieved by using reference cells, i.e., redundant cells with
known stored data. The method is often considered too ex-
pensive in terms of redundancy, and alternative methods with
lower redundancy have been sought for.
Also, coding techniques can be applied to alleviate the
detection in case of channel mismatch. Specifically balanced
codes [3], [4], [5] and composition check codes [6], [7] prefer-
ably in conjunction with Slepian’s optimal detection [8] have
been shown to offer solace in the face of channel mismatch.
These coding methods are often considered too expensive in
terms of coding hardware and redundancy when high-speed
applications are considered.
Immink and Weber [9] advocated detectors that use the
Pearson distance instead of the traditional Euclidean distance
as a measure of similarity. The authors assume that the
offset is constant (uniform) for all symbols in the codeword.
In [10], it is assumed that the offset varies linearly over the
codeword symbols, where the slope of the offset is unknown.
The error performance of Pearson-distance-based detectors is
intrinsically resistant to both offset and gain mismatch.
Although minimum Pearson distance detection restores the
error performance loss due to channel mismatch without too
much redundant overhead, it is, however, an important open
problem to optimally combine it with error correcting codes.
2
Source data are usually encoded to improve the error reliabil-
ity, which means that the codewords have good (Hamming)
distance properties using structures such as, for example,
Hamming or BCH codes. Exhaustive optimal detection of
such codes is usually an impracticality as it requires the
distance comparison of all valid codewords. The celebrated
Chase algorithm [11] has been recommended as it enables
the trading of decoder complexity versus error performance
of conventional error correcting codes. The Chase algorithm
makes preliminary hard decisions of reliable symbols based
on a given threshold level. The Chase algorithm reduces the
exhaustive search of all symbols in the codeword to only a
small number of unreliable symbols. In case of channel mis-
match, however, due to incorrectly tuned threshold levels, the
hard decisions made are unreliable, and the Chase algorithm
fails to deliver reliable detection.
In this paper, we present new dynamic threshold detection
techniques used to estimate the channel’s unknown gain and
offset. The estimates of the actual gain and offset so obtained
are used to scale the received signal or to dynamically adjust
the threshold levels on a word-by-word basis. Then, the cor-
rected signal, brought into its standard range, can be forwarded
to the final detection/decoding system, where optimum use can
be made of the distance properties of the code.
We set the scene in Section II with preliminaries and a
description of the mismatched channel model. In Section III,
we analyze the case where it is assumed that only the offset
is unknown and the gain is known. In Section IV, we discuss
the general case, where both gain and offset are unknown. In
Section V, we study the principal case of our paper, where it
is assumed that an error correcting code is applied to improve
the error performance of the channel. We start by showing
that channel mismatch has a detrimental effect on the error
performance of the extended Hamming code decoded by a
Chase decoder. We show that the presented dynamic threshold
detector (DTD) restores the error performance close to the
situation with an ideal, well-informed, receiver. Section VI
concludes the paper.
II. PRELIMINARIES AND CHANNEL MODEL
We consider a communication codebook, S⊆ Qn, of
selected codewords x= (x1, x2, . . . , xn)over the binary
alphabet Q={0,1}, where n, the length of x, is a positive
integer. We pursued the binary case here as it is the most
important in storage practice. The number of computations
grows rapidly for larger alphabets, see [9], (37), which may
complicate the detector design. The codeword, xS, is
translated into physical features, where logical ‘0’s are written
at an average (physical) level b0and the logical ‘1’s are
written at an average (physical) level 1 + b1, where b0and
b1R. Both b0and b1are average deviations, or ‘offsets’,
from the nominal levels, and are relatively small with respect
to the assumed unity difference (or amplitude) between the two
physical signal levels. The offsets b0and b1may be different
for each codeword, but do not vary within a codeword. For
unambiguous detection, the average of the physical levels
associated with the logical ‘0’s, b0, is assumed to be less than
that associated with the ‘1’s, 1 + b1. In other words, we have
the premise
b0<1 + b1.(1)
Assume a codeword, x, is sent. The symbols of the received
vector r= (r1, . . . , rn)are distorted by additive noise and
given by
ri=xi+f(xi;b0, b1) + νi,(2)
where we define the switch function
f(x;b0, b1) = (1 x)b0+xb1,
and x∈ {0,1}is a dummy integer. We assume that the
received vector, r, is corrupted by additive Gaussian noise
ν=(ν1, . . . , νn), where νiRare zero-mean independent
and identically distributed (i.i.d) noise samples with normal
distribution N(0, σ2). The quantity σ2Rdenotes the noise
variance. We may rewrite (2) and obtain
ri=axi+b+νi,(3)
where
b=b0and a= 1 + b1b0.(4)
The mean levels, b0and b1, may slowly vary (drift) in time
due to charge leakage or temperature change. As a result, the
coefficient, a= 1 + b1b0, usually called the gain of the
channel, and the offset, b=b0, are both unknown to sender
and receiver. From the premise (1) we simply have a > 0.
Note that in [9] the authors study a slightly different channel
model, ri=a(xi+νi) + b, where also the noise component,
νi, is scaled with the gain a.
We start, in the next section, with the simplest case, namely
the offset only case, a= 1.
III. OFFS ET-ON LY CASE
In the offset-only case, b0=b1=band a= 1, we simply
have
ri=xi+b+νi,(5)
where the quantity, b, is an unknown (to both sender and re-
ceiver) offset. For detection in the above offset-only situation,
Immink and Weber [9] proposed the modified Pearson distance
instead of the Euclidean distance between the received vector
rand a candidate codeword ˆ
xS. The modified Pearson
distance is defined by
δ(r,ˆ
x) =
n
i=1
(riˆxi+ ˆx)2,(6)
where we define the mean of an n-vector of reals zby
z=1
n
n
i=1
zi.(7)
For clerical convenience we drop the variable rin (6). A
minimum Pearson distance detector operates in the same way
as the traditional minimum Euclidean detector, that is, it
outputs the codeword xo‘closest’, as measured in terms of
Pearson distance, to the received vector, r, or in other words
xo= arg min
ˆ
xS
δ(ˆ
x).(8)
3
Immink and Weber showed that the error performance of the
above detection rule is independent of the unknown offset
b. The evaluation of (8) is in principle an exhaustive search
for finding xo, but for a structured codebook, S, the search
is much less complex. We proceed our discussion with the
definition of a useful concept.
Let Swdenote the set of codewords of weight w, that is,
Sw={x∈ Qn:
n
i=1
xi=w}, w = 0, . . . , n.
A set Swis often called a constant weight code of weight w.
We study examples, where the codebook, S, is the union of
|V|constant weight codes defined by
S=
wV
Sw,(9)
where the index set V⊆ {0,1, . . . , n}.
After working out (6), we obtain
δ(ˆ
x) =
n
i=1
(riˆxi)2+nˆx(2rˆx),(10)
where the first term is the square of the Euclidean distance
between rand ˆ
x, and the second term, nˆx(2rˆx), makes the
distance measure, δ(ˆ
x), independent of the unknown offset b.
The exhaustive search (8) can be simplified by the following
observations. The decoder hypothesizes that xSw. Then we
have
δ(ˆ
xSw) =
n
i=1
(riˆxi)2+w2rw
n.(11)
Since (8) is a minimization process, we may delete irrelevant
(scaling) constants, and obtain
δ(ˆ
xSw) =
n
i=1
r2
i2
n
i=1
ˆxiri+
n
i=1
ˆx2
i
+w2rw
n
w1+2rw
n2
n
i=1
ˆxiri.(12)
The symbol is used to denote equivalence of the expressions
(11) and (12) deleting (scaling) constants irrelevant to the
minimization operation defined in (8). Note that the term
w1+2rw
n
depends on the number of ‘1’s, w, of ˆ
xand, thus, not on the
specific positions of the ‘1’s of ˆ
x. The only degree of freedom
the detector has for minimizing δ(ˆ
xSw)is permuting the
symbols in ˆ
xfor maximizing the inner product n
i=1 ˆxiri.
Slepian [8] showed that the inner product n
i=1 ˆxiri,ˆ
x
Sw, is maximized by pairing the largest symbol of rwith the
largest symbol of ˆ
x, the second largest symbol of rwith the
second largest symbol of ˆ
x, etc.
To that end, the nreceived symbols, ri, are sorted, largest
to smallest, in the same way as taught in Slepians prior art.
Let (r
1, r
2, . . . , r
n)be a permutation of the received vector
(r1, r2, . . . , rn)such that r
1r
2. . . r
n. Then, since the
wlargest received symbols, r
i,1iw, are paired with
‘1’s (and the smallest symbols r
i,w+ 1 inwith ‘0’s),
we obtain
δw=w1+2rw
n2
w
i=1
r
i
=
w
i=1 2(r
ir) + n+ 1 2i
n,(13)
where for convenience we use the short-hand notation
δw= min
ˆ
x
δ(ˆ
xSw).
Since, as is immediate from (13), δ0=δn= 0, the detector
cannot distinguish between the all-‘0’ or the all-‘1’ codewords.
For enabling unique detection one of the two (or both)
codewords must be barred from the code book S. In other
words, either V⊆ {1, . . . , n}or V⊆ {0,1, . . . , n 1}.
Such constrained codes, S, called Pearson codes, have been
described in [9]. In order to reduce computational load, we
may rewrite (13) in recursive form, and obtain for 1wn,
δw=δw12(r
wr) + n+ 1 2w
n,(14)
where we initialize with δ0= 0. The value wVthat
minimizes δwis denoted by ˆw, or
ˆw= arg min
wV
δw.(15)
Once we have obtained ˆw, we may obtain an estimate of
the sent codeword, x, by applying Slepian’s algorithm, and,
subsequently we find an estimate of the offset, b. The estimate
of the offset, denoted by ˆ
b, is obtained by averaging (5), or
ˆ
b=1
n
n
i=1
(riˆxi) = ¯rˆw
n.(16)
The retrieved vector, r, is re-scaled by subtracting the esti-
mated offset, ˆ
b, so that
ˆri=riˆ
b=ri¯rˆw
n,1in, (17)
where ˆ
rdenotes the corrected vector. Note that we can,
instead of re-scaling the received signal as done above, adjust
the threshold levels used in a Chase decoder to discriminate
between reliable and unreliable symbols. For asymptotically
small noise variance, σ2, we may assume with high probability
that ˆw=xi, so that the variance of the offset estimate, ˆ
b,
can be approximated by
E[(bˆ
b)2]σ2
n, σ 1,(18)
where E[] denotes the expectancy operator. The next example
illustrates the detection algorithm.
Example 1: Let n= 6,x= (110010),σ= 0.125, and
offset b= 0.2. The received word is r=(1.194, 1.233, -
0.024, 0.331, 1.402, 0.263), and after sorting we have r=
4
(1.402, 1.233, 1.194, 0.331, 0.263, -0.024). We simply find
r= 0.733. The next table shows δwversus wusing (14).
w r
wδw
1 1.402 0.505
2 1.233 1.005
3 1.194 1.761
4 0.331 1.123
5 0.263 0.682
60.024 0.000
We find ˆw= 3. The estimated offset equals
ˆ
b=rˆw/n = 0.733 3/6 = 0.233.
Example 2: Let, S,neven, be the union of two constant
weight codes, that is,
S=Sw0Sw1,(19)
where w0=n
21and w1=n
2+ 1. We find from (13) that
δw0=2
w0
i=1
r
i+w01+2rw0
n
and
δw1=2
w1
i=1
r
i+w11 + 2rw1
n,
so that
δw1δw0=2(r
n
2+r
n
2+1)+4r. (20)
We define the median of the received vector, ˜r, as the average
of the two middle values (neven) [12], that is,
˜r=1
2(r
n
2+r
n
2+1).(21)
The receiver decides that ˆw=w1if
δw1δw0<0,
or, equivalently, if
˜r > r. (22)
In the next section, we take a look at the general case where
we face both gain and offset mismatch, a̸= 1 and b̸= 0.
IV. PEARSON DISTANCE DETECTION
We consider the general situation as in (3) where the
symbols of the received vector r= (r1, . . . , rn)are given
by
ri=axi+νi+b, (23)
where both quantities a,a > 0, and bare unknown. Immink
and Weber proposed the Pearson distance as an alternative
to the Euclidean distance in case the receiver is ignorant of
the actual channel’s gain and offset [9]. The Pearson distance
between the n-vectors rand ˆ
xis defined by
δp(ˆ
x) = 1 ρr,ˆ
x,(24)
where
ρr,ˆ
x=n
i=1(rir)(ˆxiˆx)
σrσˆx
(25)
is the Pearson correlation coefficient. The (unnormalized)
variance of the vector zis defined by
σ2
z=
n
i=1
(ziz)2.(26)
A minimum Pearson distance detector operates in the same
way as the minimum Euclidean detector, that is, it outputs
the codeword xo‘closest’, as measured in terms of Pearson
distance, to the received vector, or in other words
xo= arg min
ˆ
xS
δp(ˆ
x).(27)
The minimum Pearson distance detector estimates the sent
codeword x, and implicitly it offers an estimate of the gain, a,
and offset, b, using (23). We start by evaluating (24) and (27).
Since (27) is a minimization process, we may delete irrelevant
(scaling) constants, and obtain
δp(ˆ
x)≡ − 1
σˆx
n
i=1
rixiˆx).(28)
As in the previous section, we consider a code S=wVSw,
where the index set V⊆ {0,1, . . . , n}. Let ˆ
xSw, then
δp(ˆ
xSw)1
ww2
nwr
n
i=1
riˆxi.(29)
Note that δp(ˆ
xSw)is undefined for w= 0 and w=n,
and we must bar both the all-‘0’ and all-‘1’ words from Sfor
unique detection. Clearly, V⊆ {1, . . . , n 1}.
Except for the inner product riˆxi, the above expression
depends on the number of ‘1’s, w, of ˆ
xand, thus, not on the
specific positions of the ‘1’s of ˆ
x. For maximizing the inner
product riˆxiwe must pair the wlargest symbols riwith
the w1’s of ˆ
x. Let (r
1, r
2, . . . , r
n)be a permutation of the
received vector (r1, r2, . . . , rn)such that r
1r
2. . . r
n.
Since the w1’s are paired with the largest symbols, r
i,1
iw, we have [8]
δp,w =1
ww2
n
w
i=1
(r
ir),(30)
where δp,w is a short-hand notation of min ˆ
xδp(ˆ
xSw). The
detector evaluates δp,w for all wV. Define
ˆw= arg min
wV
δp,w.(31)
The decoder decides that the ˆwlargest received signal ampli-
tudes, r
i,1iˆware associated with a ‘one’, and nˆw
smallest received signal amplitudes, r
i,ˆw+ 1 inare
associated with a ‘zero’.
The estimates of the gain, ˆa, and offset, ˆ
b, of the received
vector rare found by using (4). Let ˆ
b0and ˆ
b1denote the
estimates of b0and b1, respectively. Then we find
ˆ
b0=1
nˆw
n
i= ˆw+1
r
i
and
ˆ
b1=1 + 1
ˆw
ˆw
i=1
r
i,
5
TABLE I
SIM ULATI ON S RES ULTS O F 105SAM PL ES F OR σ= 0.1AN D n= 6. TH E
VALUE S IN PAR EN THE SE S ARE C OM PUT ED U SIN G (35) AND (36),
RE SPE CT IVE LY.
w σ2
ˆ
b,w2σ2
ˆa,w 2
1 0.201 (0.200) 1.201 (1.200)
2 0.250 (0.250) 0.745 (0.750)
3 0.333 (0.333) 0.668 (0.667)
4 0.497 (0.500) 0.751 (0.750)
5 1.011 (1.000) 1.198 (1.200)
so that, after using (4),
ˆa= 1 + ˆ
b1ˆ
b0=1
ˆw
ˆw
i=1
r
i1
nˆw
n
i= ˆw+1
r
i(32)
and
ˆ
b=ˆ
b0=1
nˆw
n
i= ˆw+1
r
i.(33)
The normalized vector ˆ
ris found after scaling and offsetting
with the estimated gain, ˆa, and offset, ˆ
b, that is,
ˆri=riˆ
b
ˆa,1in. (34)
After the above normalization, the normalized vector, ˆ
r, is
corrected to its standard range, and may be forwarded to the
second part of the decoder, where the vector is processed,
decoded, and quantized.
The variance of the estimates ˆaand ˆ
bdepend on the numbers
of 1’s and 0’s in the sent codeword x. For asymptotically small
noise variance, σ2, so that we may assume that with high
probability ˆw=xi, the variance of the offset, b, denoted
by σ2
ˆ
b,w, can be approximated by
σ2
ˆ
b,w = E[(bˆ
b)2] = E
b1
nˆw
n
i= ˆw+1
r
i2
=1
nwσ2, σ 1.(35)
Similarly, the variance of the estimate of the gain a, denoted
by σ2
ˆa,w , is given by
σ2
ˆa,w = E[(aˆa)2] = n
w(nw)σ2, σ 1.(36)
The above findings are intuitively appealing as they show that
the quality of the estimate of the quantities, aand b, depends
on the numbers, nwand w, of ‘0’s and ‘1’s in the sent
codeword, respectively. We have verified the above estimator
quality using computer simulations. Results of our simulations
are collected in Table I, where we assumed the case σ=
0.1and n= 6. We are now considering the general case
of uncoded i.i.d input data, so that the sent codeword does
not have a specified weight. The codeword’s weight is in the
range {1, . . . , n 1}. For the i.i.d. case, the variance of the
TABLE II
SIM ULATI ON S RES ULTS O F 105SAM PL ES F OR σ= 0.1. TH E VALUE S IN
PARE NT HES ES A RE CO MPU TE D USI NG ( 37) A ND (38), R ESP EC TIV ELY.
n σ2
ˆ
b2σ2
ˆa2
8 0.297 (0.296) 0.5919 (0.5919)
16 0.135 (0.135) 0.2700 (0.2699)
32 0.064 (0.065) 0.1293 (0.1293)
64 0.031 (0.032) 0.0634 (0.0635)
128 0.017 (0.016) 0.0314 (0.0315)
estimations ˆaand ˆ
b, denoted by σ2
ˆaand σ2
ˆ
b, can be found as
the weighted average of σ2
ˆa,w and σ2
ˆ
b,w, or
σ2
ˆ
b=σ2
2n2
n1
w=1 n
w1
nw(37)
and
σ2
ˆa=σ2
2n2
n1
w=1 n
wn
w(nw).(38)
Results of computations and simulations are shown in Table II,
where we assumed the case σ= 0.1. We have computed
the relative variance of the estimators σ2
ˆ
b2and σ2
ˆa2for
different values of the noise level, σ, and observed that (37)
and (38) are accurate up to a level where the detector is close
to failure (word error rate >0.1).
In the next section, we show results of computer simulations
with the newly developed DTD algorithms applied to the
decoding of an extended Hamming code.
V. APP LI CATI ON T O AN E XT EN DED HAMMING CODE
Error correction is needed to guarantee excellent error
performance over the memory’s life span. To be compatible
with the fast read access time of STT-MRAM, the error
correction code adopted needs to have a low redundancy of
around ten percent and it must have a short codeword length.
A (71, 64) regular Hamming code is used for Everspins 16
MB MRAM, where straightforward hard decision detection is
used [13]. Cai and Immink [14] propose a (72, 64) extended
Hamming code with a two-stage hybrid decoding algorithm
that incorporates hard decision detection for the first-stage plus
a Chase II decoder [11] for the second stage of the decoding
routine.
In the next subsection, we show, using computer simula-
tions, that the application of DTD in the above scenario offers
resilience against unknown charge leakage or temperature
change. We show results of computer simulations with the
(72, 64) Hamming code, which is applied to a simple channel
with additive noise.
A. Evaluation of the Hamming code
An (n, n r)Hamming code is characterized by two
positive integer parameters, rand n, where the redundancy
r > 1is a design parameter and n,n2r1is the length of
the code [13]. The payload is of length nr. The minimum
Hamming distance of a regular Hamming code equals dH= 3.
6
An extended Hamming code is a regular (n, n r)Hamming
code plus an overall parity check. The minimum Hamming
distance of an extended Hamming code equals dH= 4.
The word error rate of binary words transmitted over an
ideal, matched, channel, using a Hamming code under max-
imum likelihood soft decision decoding, denoted by WERH,
equals (union bound estimate)
WERHAH(n, r)QdH
2σ, σ 1,(39)
where AHdenotes the average number of codewords at
minimum distance dH, and the Q-function is defined by
Q(x) = 1
2π
x
eu2
2du. (40)
For a regular Hamming code, we have
AH(n, r) = n(n1)
6, n = 2r1.(41)
For a shortened Hamming code, n < 2r1, since the weight
distribution of many types of linear codes, including Hamming
codes, is asymptotically binomial [15] for n1, we can use
the approximation
AH(n, r)n
31
2r,(42)
and for an extended Hamming code (only even weights)
AH(n, r)n
41
2r1.(43)
Exhaustive optimal detection of long Hamming codes, such
as the extended (72,64) is an impracticality as it requires
the distance comparison of 264 valid codewords. Sub-optimal
detection can be accomplished with, for example, the well-
known Chase algorithm [11], [14].
The Chase algorithm selects Tof the least reliable bits
by selecting the symbols, ri, having least absolute channel
value with respect to the decision level. The remaining nT
symbols, that is the most reliable ones, are quantized. Then,
the Tunreliable symbols are selected, using exhaustive search,
in such a way that the word so obtained is a valid codeword
of the Hamming code at hand and that the word minimizes
the Euclidean distance to the received vector r. The error
performance of the Chase algorithm is worse than the counter-
part error performance of the full-fledged maximum likelihood
detector given by (39). The loss in performance depends on
the parameter T.
As the parameter Tdetermines the complexity of the search,
it is usually quite small in practice. The majority of symbols
are thus quantized using hard decision detection, where a pre-
fixed threshold is used. The error performance of the Chase
decoder depends therefore heavily on the accuracy of the
threshold with respect to mismatch of the gain and offset
of the signal received. This means that the Chase decoder
loses a major part of its error performance in case of channel
mismatch.
Using computer simulations, we computed the error perfor-
mance of the Chase decoder in the presence of offset or gain
mismatch versus the noise level 20 log10 σ. We simulated
the error performance of an extended (72,64) Hamming
code decoded by the Chase decoder, where we selected, in
Figure 1, the offset mismatch case, a= 1 and b= 0.15.
Figure 2 shows the gain mismatch case, a= 0.85 and b= 0
(b0= 0,b1=0.15). Both diagrams show the significant
loss in performance due to channel mismatch. Combinations of
offset and gain mismatch give similar devastating results [9].
The word error rate found by our simulations of the ideal
channel (without mismatch), is quite close to the theoretical
performance given by the union bound estimate (39) and (43).
In order to improve the detection quality, we applied DTD,
as presented in the previous sections, followed by (standard)
Chase decoding. Before discussing the simulation results,
we note two observations. The all-‘1’ word is not a valid
codeword, and the all-‘0’ word is a valid codeword of the
(72,64) Hamming code. The probability of occurrence of the
all-‘0’ word, assuming equiprobable codewords is 264
1019, which is small enough to be ignored for most practical
situations. The weight of a codeword of an extended Hamming
code is even, so that the number of evaluations of δwor δp,w ,
see (13) and (30), can be reduced.
Figure 1 shows the word error rate in case DTD is applied in
the offset-only case, a= 1 and b= 0.15. We notice that DTD
restores the error performance close to the error performance
of the ideal offset-free situation.
Figure 2, the gain mismatch case, shows that the error
performance with DTD (Curve 2) is worse than that of the
ideal case, a= 1, without applying DTD (Curve 4). This can
easily be understood: in case a= 0.85 (b0= 0, b1=0.15),
the average levels, b0and 1 + b1, of the recorded data, xi, are
closer to each other than in the ideal case, a= 1. Curve 3
shows that the error performance with DTD is close to the
situation, where the receiver is informed about the actual gain,
a= 0.85. This gives a fairer comparison, and we observe that
the WER of DTD almost overlaps with the simulated, matched
channel, performance. This demonstrates the efficacy of DTD
for the case of a= 0.85 (b0= 0, b1=0.15).
Figure 3 shows the WER as a function of the offset
mismatch, b, where a= 1 and 20 log σ= 15, using
a Chase decoder, T= 4. The error performance of the
DTD is unaffected by the offset mismatch, b, and the error
performance is close to the performance without mismatch.
Figure 4 shows the WER as a function of b1, where the
gain a= 1 + b1,20 log σ= 15.5, and b0= 0, using a
Chase decoder, T= 4. Curve 3 shows the situation where
the receiver is informed about the actual gain (no mismatch),
and we infer that the error performances of a receiver of the
matched channel and a receiver of the mismatched channel
combined with DTD are very similar.
Above we have shown simulation results of dynamic thresh-
old detection used in conjunction with an extended Hamming
code and a Chase decoder. We remark that although in this
paper we exemplify DTD detection on an extended Hamming
code, the hybrid DTD/decoding algorithm is a general tool
that can be applied to other (extended) BCH codes, LDPC,
polar codes, etc., for applications in both data storage and
transmission systems.
7
13 13.5 14 14.5 15 15.5 16
10−6
10−5
10−4
10−3
10−2
10−1
100
−20 logσ
WER
a=1
1 b=0.15, w/o DTD
2 b=0.15, with DTD
3 b=0, w/o DTD
4 b=0, with DTD
5 union bound
Fig. 1. Word error rate (WER) of the extended (72,64) Hamming
code with and without dynamic threshold detection (DTD), and with
and without an offset, b= 0.15, using a Chase decoder, T= 4. The
union bound estimate to the word error rate for the ideal channel,
a= 1 and b= 0, given by (39), is plotted as a reference (Curve 5).
13 13.5 14 14.5 15 15.5 16 16.5 17
10−6
10−5
10−4
10−3
10−2
10−1
100
−20 logσ
WER
1 a=0.85 (b0=0, b1=−0.15), w/o DTD
2 a=0.85 (b0=0, b1=−0.15), with DTD
3 a=0.85 (b0=0, b1=−0.15), with known
a, b0, b1 for detection/decoding
4 a=1 (b0=0, b1=0), w/o DTD
5 a=1 (b0=0, b1=0), with DTD
6 union bound
Fig. 2. Word error rate (WER) of the extended (72,64) Hamming
code with and without dynamic threshold detection (DTD), and with
and without a gain mismatch, a= 0.85 (b0= 0,b1=0.15), using
a Chase decoder, T= 4. The union bound estimate, Curve 6, to the
word error rate for the ideal channel, a= 1 and b= 0, given by
(39), is plotted as a reference. Curves 2 and 3 show that the error
performance with DTD is close to the situation, where the receiver
is informed about the actual gain, a= 0.85.
VI. CONCLUSIONS
We have considered the transmission and storage of encoded
strings of binary symbols over a storage or transmission
channel, where a new dynamic threshold detection system
has been presented, which is based on the Pearson distance.
Dynamic threshold detection is used for achieving resilience
against unknown signal-dependent offset and corruption with
additive noise. We have presented two algorithms, namely a
first one for estimating an unknown offset only and a second
one for estimating both unknown offset and gain. As an
0 0.05 0.1 0.15 0.2
10−5
10−4
10−3
10−2
10−1
b
WER
−20 logσ = 15
1 w/o DTD
2 with DTD
Fig. 3. Word error rate (WER) of the extended (72,64) Hamming
code with and without dynamic threshold detection (DTD), versus
the offset mismatch b, where a= 1 and 20 log σ= 15, using a
Chase decoder, T= 4.
−0.2 −0.15 −0.1 −0.05 0
10−5
10−4
10−3
10−2
10−1
b1
WER
−20 logσ = 15.5
a=1+b1, b0=0
1 w/o DTD
2 with DTD
3 with known a, b0, b1 for detection/decoding
Fig. 4. Word error rate (WER) of the extended (72,64) Hamming
code with and without dynamic threshold detection (DTD), versus
the gain mismatch a= 1 + b1,b0= 0, where 20 log σ= 15.5,
using a Chase decoder, T= 4. Curve 3 shows the situation where
the receiver is informed about the actual gain.
example to assess the benefit of the new dynamic threshold
detection, we have investigated the error performance of an
extended (72,64) Hamming code using a Chase decoder. The
Chase algorithm makes hard decisions of reliable symbols that
are above or below a given threshold level. In case of channel
mismatch, however, due to incorrectly tuned threshold levels,
the hard decisions made are unreliable, and as a result the
Chase algorithm fails. We have shown that the error perfor-
mance of the extended Hamming code degrades significantly
in the face of an unknown offset or gain mismatch. The
presented threshold detector dynamically adjusts the threshold
levels (or re-scales the received signal), and improves the error
performance by estimating the unknown offset or gain, and
8
restores the performance close to the performance without mis-
match. A worked example of a Spin-torque transfer magnetic
random access memory (STT-MRAM) with an application to
an extended (72, 64) Hamming code has been described, where
the retrieved signal is perturbed by additive Gaussian noise and
unknown gain or offset.
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Kees A. Schouhamer Immink (M’81-SM’86-F’90)
received his PhD degree from the Eindhoven Uni-
versity of Technology. He was from 1994 till 2014
an adjunct professor at the Institute for Experimental
Mathematics, Essen-Duisburg University, Germany.
In 1998, he founded Turing Machines Inc., an
innovative start-up focused on novel signal pro-
cessing for solid-state (Flash) memories, where he
currently holds the position of president. Immink
designed coding techniques of digital audio and
video recording products such as Compact Disc, CD-
ROM, DCC, DVD, and Blu-ray Disc. He received a Knighthood in 2000, a
personal Emmy award in 2004, the 2017 IEEE Medal of Honor, the 1999
AES Gold Medal, the 2004 SMPTE Progress Medal, the 2014 Eduard Rhein
Prize for Technology, and the 2015 IET Faraday Medal. He received the
Golden Jubilee Award for Technological Innovation by the IEEE Information
Theory Society in 1998. He was inducted into the Consumer Electronics Hall
of Fame, elected into the Royal Netherlands Academy of Sciences and the
(US) National Academy of Engineering. He received an honorary doctorate
from the University of Johannesburg in 2014. He served the profession as
President of the Audio Engineering Society inc., New York, in 2003.
Kui Cai received her B.E. degree in information
and control engineering from Shanghai Jiao Tong
University, Shanghai, China, and joint Ph.D. degree
in electrical engineering from Technical University
of Eindhoven, The Netherlands, and National Uni-
versity of Singapore. Currently, she is an Associate
Professor with Singapore University of Technology
and Design (SUTD). She received 2008 IEEE Com-
munications Society Best Paper Award in Coding
and Signal Processing for Data Storage. She is
an IEEE senior member, and served as the Vice-
Chair (Academia) of IEEE Communications Society, Data Storage Technical
Committee (DSTC) during 2015 and 2016. Her main research interests are
in the areas of coding theory, information theory, and signal processing for
various data storage systems and digital communications.
Jos H. Weber (S’87-M’90-SM’00) was born in
Schiedam, The Netherlands, in 1961. He received
the M.Sc. (in mathematics, with honors), Ph.D.,
and MBT (Master of Business Telecommunications)
degrees from Delft University of Technology, Delft,
The Netherlands, in 1985, 1989, and 1996, respec-
tively. Since 1985 he has been with the Faculty of
Electrical Engineering, Mathematics, and Computer
Science of Delft University of Technology. Cur-
rently, he is an associate professor in the Department
of Applied Mathematics. He is the chairman of the
WIC (Werkgemeenschap voor Informatie- en Communicatietheorie in the
Benelux) and the secretary of the IEEE Benelux Chapter on Information
Theory. He was a Visiting Researcher at the University of California at Davis,
USA, the University of Johannesburg, South Africa, the Tokyo Institute of
Technology, Japan, and EPFL, Switzerland. His main research interests are in
the area of channel coding.
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