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Abstract

We consider noisy communications and storage systems that are hampered by varying offset of unknown magnitude such as low-frequency signals of unknown amplitude added to the sent signal. We study and analyze a new detection method whose error performance is independent of both unknown base offset and offset’s slew rate. The new method requires, for a codeword length n ≥ 12, less than 1.5 dB more noise margin than Euclidean distance detection. The relationship with constrained codes based on mass-centered codewords and the new detection method is discussed.
A new detection method for noisy channels with
time-varying offset
Kees A. Schouhamer Immink, Fellow, IEEE and Jos H. Weber, Senior Member, IEEE
Abstract—We consider noisy communications and storage sys-
tems that are hampered by varying offset of unknown magnitude
such as low-frequency signals of unknown amplitude added to
the sent signal. We study and analyze a new detection method
whose error performance is independent of both unknown base
offset and offset’s slew rate. The new method requires, for a
codeword length n12, less than 1.5 dB more noise margin than
Euclidean distance detection. The relationship with constrained
codes based on mass-centered codewords and the new detection
method is discussed.
Keywordschannel mismatch, constrained code, Pearson
code, Pearson distance, slew rate, varying offset.
I. INTRODUCTION
Euclidean-distance-based detection of transmitted or stored
encoded data signals is optimal in the presence of white
Gaussian noise, but its error performance is vulnerable in the
presence of channel mismatch, such as offset of unknown
magnitude. Unknown offset magnitude variations may be
caused by a variety of interference sources. For example,
in optical disc recording, scratches and finger prints on the
disc [1] cause low-frequency varying offset in the read-out
signal.
In nonvolatile memories (NVMs) data are represented by
stored charge [2, 3, 4]. The stored charge can leak away
from the floating gate through the gate oxide or through
the dielectric. The amount of leakage, called drift, depends
on various physical parameters, such as, for example, the
device temperature and the time elapsed between writing and
reading [5].
In [6, 7], the authors assume that the unknown offset mis-
match can be approximated by a zeroth-order, constant, term
for all symbols in a codeword. In this model, the offset term
may vary from word to word, but is fixed within a codeword.
The authors advocate detection based on the Pearson distance,
which is resilient to unknown, but constant within a codeword,
offset and gain (scaling) of the received signal [6, 7, 8].
The zeroth-order model can be overly simplistic in specific
communications channels, where the offset or low-frequency
interference may vary so rapidly that the basic premise that
the offset is constant within a codeword is false.
A low-frequency varying offset can be segmented into an
approximately piecewise linear function of time, where the
‘pieces’ have a length equal to the codeword length. As
Kees A. Schouhamer Immink is with Turing Machines Inc, Willem-
skade 15d, 3016 DK Rotterdam, The Netherlands. E-mail: immink@turing-
machines.com.
Jos H. Weber is with Delft University of Technology, Delft, The Nether-
lands. E-mail: j.h.weber@tudelft.nl.
discussed in [9], memory cells of nonvolatile data storage
products that are closer to warmer spots lose their data charge
more rapidly than memory cells closer to colder spots, so that
offset loss is not constant within a codeword [4]. Evidently,
the (varying) offset cannot be considered to be equal for all
symbols in a codeword, and alternative detection methods have
been sought for.
It is assumed in this paper that the unknown time-varying
offset can be approximated by a word-wise first-order term
that varies linearly over the codeword symbols, where both the
base offset and offset’s slew rate (the offset’s first-order rate
of change) are unknown. Both unknown terms, the base offset
and offset’s slew rate, may vary from codeword to codeword,
but are fixed within a codeword. The quest for advanced
detection techniques that are immune to unknown, first-order,
offset variation is not new. Skachek and Immink [9] introduced
mass centered codewords whose detection is independent
of both unknown base offset and offset’s slew rate. They
concluded that the redundancy of their scheme is prohibitively
large for many applications. Bu and Weber [10] also addressed
a channel model where the offset varies within a codeword.
They introduced Pearson-distance-based detection in conjunc-
tion with a difference operator and a pair-constrained code.
Their adopted code has significantly less redundancy than
the previously proposed mass-centered codes [9]. However,
it requires a 3 dB higher noise margin, which makes it less
suitable for noise-dominant channels.
Alternative solutions are wanted that are less costly in terms
of noise figure or redundancy. To that end, we propose and
analyze a detection method based on a new distance measure,
whose error performance is independent of both unknown base
offset and offset’s slew rate. The rate of the requisite binary
constrained code is very high as only one codeword must
be barred from the repertoire of 2npossible codewords. It
requires less than 1.5 dB more noise margin than Euclidean
distance detection, or less than 1 dB with respect to Pearson
distance detection, both for n= 12.
The paper is organized as follows. In Section II, we start
with preliminaries, a description of the adopted channel model,
and a description of the properties of prior art minimum
Pearson distance detection. In Section III, we propose a novel
detection method that improves the detector’s resilience in case
the received signal is distorted by changing offset. We analyze
the error performance of the new detection method, and offer
results of simulations. Receiver complexity is discussed in
Section IV. Section V furnishes the conclusions of our paper.
II. PRELIMINARIES,CHANNEL MODEL,PR IO R ART
Let Sbe a codebook of selected binary codewords x=
(x1, . . . , xn), where the integer ndenotes the codeword length,
and xi∈ {0,1}. The received signal is denoted by the vector
rhaving real entries ri, which is defined by
ri=xi+νi+Ii, i = 1,2, . . . , n. (1)
The (real) variables νidenote zero-mean white additive Gaus-
sian noise samples with variance σ2, for example caused
by thermal noise. The (real) variable Iidenote time-varying
interfering offset. It is assumed that the interfering offset, Ii,
can be approximated by a word-wise linear function waveform
(ramp), which is denoted by
Ii=b0+b1i, i = 1,2, . . . , n, (2)
where the (real) coefficients b0and b1denote the unknown
base offset and the unknown offset’s slew rate, respectively.
Both offset and offset’s slew rate are assumed to be constant
within a codeword; they may vary from codeword to code-
word.
A. Motivating example
The error performance of Euclidean detection is seriously
deteriorated in the face of relatively small mismatch. In
order to demonstrate this, Figure 1 shows, for n= 12, the
word error rate (WER) versus signal-to-noise ratio, SNR =
20 log(σ)(dB), of in Curve (c) Euclidean detection, ideal
noisy channel without mismatch, i.e. b0=b1= 0, in Curve (a)
noisy channel with offset mismatch, b0= 0.1and b1= 0.01.
The diagram clearly shows that the error performance of
Euclidean detection is seriously degraded by a small offset.
Curve (b) is obtained using a novel detection scheme, whose
error performance is independent of both channel mismatch
terms b0and b1; the new method is described in Section III. We
may observe that in the matched case, the error performance
of the new method is inferior to that of Euclidean detection,
but in case of mismatch the situation changes, and the new
method has a superior error performance. The development
and analysis of the new method is the main topic of our paper,
and described in Section III and further.
B. Prior art, constant offset
In order to make the error performance independent of
unknown (base) offset mismatch, Immink and Weber [6]
introduced the (modified) Pearson distance between two n-
vectors. Let x,ˆ
x∈ S, be two n-vectors, where Sis the
set of chosen codewords. For the base offset mismatch case,
Ii=b0, they proposed the distance measure between the
received vector rand ˆ
x
δ0(r,ˆ
x) =
n
X
i=1 riˆxi+ ˆx2,(3)
where y=1
nPn
i=1 yidenotes the average of the entries of
an n-vector y. Since δ0(r,0) = δ0(r,1), the receiver cannot
distinguish between the all-0 word and the all-1 word, denoted
by 0and 1, respectively. The ambiguity can be remedied by
17 17.5 18 18.5 19 19.5 20
SNR
10-6
10-5
10-4
10-3
10-2
10-1
WER
(a)
(c)
(b)
Fig. 1. Word error rate, WER versus signal-to-noise ratio, SNR =
20 log(σ)(dB). Curve (a) shows Euclidean detection with channel
mismatch, b0= 0.1,b1= 0.01, Curve (b) shows the new detection
method, which is detailed in Section III, and Curve (c) shows
Euclidean detection for the ideal AWGN channel, no mismatch,
b0=b1= 0. All curves for n= 12.
arbitrarily excluding one of them, say the word 0, so that
S={0,1}n\ {0}.
A minimum Pearson distance detector outputs the codeword
xo= arg min
ˆx∈S
δ0(r,ˆ
x).(4)
By substituting ri=xi+νi+b0into (3), we can easily verify
that the outcome of (4) is independent of b0for codewords in
S. Note that offset may vary from codeword to codeword, but
not within a codeword. In many practical situations of interest,
however, the offset within a codeword is not constant, but
slowly varying, i.e, in our model b16= 0. In the next section,
we develop a new detection method that can cope with varying
offset Ii=b0+b1i,b16= 0.
III. NEW DISTANCE MEASURE
In the vein of (3), we establish a distance measure that is
independent of the varying offset term Ii=b0+b1i.
A. Definition of distance measure
Define
δ(r,ˆ
x) =
n
X
i=1
(riˆxi+ϕi(ˆ
x))2,(5)
where the proposed ˆ
x-dependent term is
ϕi(ˆ
x) = β0(ˆ
x) + β1(ˆ
x)i. (6)
The coefficients β0(ˆ
x)and β1(ˆ
x)are to be determined to
ensure that the outcome of the detection is independent of the
unknown offset parameters b0and b1. The decoded codeword,
xo, is found, as in (4), by the minimization process
xo= arg min
ˆx∈S
δ(r,ˆ
x).(7)
After substituting (1) into (5), we have
δ(r,ˆ
x) =
n
X
i=1
(xi+νiˆxi+ϕi(ˆ
x))2
+ 2
n
X
i=1
(ˆxi+ϕi(ˆ
x)) Ii
+
n
X
i=1
[I2
i+ 2Ii(xi+νi)].(8)
The first term is independent of the offset term Ii. The third
term is independent of ˆ
x, and therefore irrelevant in view of
the minimization process (7). The second term
2
n
X
i=1
(ˆxi+ϕi(ˆ
x)) (b0+b1i),(9)
is independent of the unknown variables b0and b1if we choose
the coefficients β0(ˆ
x)and β1(ˆ
x)in such a way that, see (6),
n
X
i=1
(ˆxi+β0(ˆ
x) + β1(ˆ
x)i)=0 (10)
and n
X
i=1
i(ˆxi+β0(ˆ
x) + β1(ˆ
x)i)=0.(11)
After substituting the well-known expressions for Pik,k=
1,2, we obtain two equations for the unknown coefficients,
β0(ˆ
x)and β1(ˆ
x):
β0(ˆ
x) + n+1
2β1(ˆ
x) = ζ0(ˆ
x)
n+1
2β0(ˆ
x) + (n+1)(2n+1)
6β1(ˆ
x) = ζ1(ˆ
x),(12)
where the zeroth and first moment of the codeword ˆ
xare
defined by
ζ0(ˆ
x) = 1
n
n
X
i=1
ˆxiand ζ1(ˆ
x) = 1
n
n
X
i=1
iˆxi.(13)
Solving the linear system (12) in the unknown coefficients,
β0(ˆ
x)and β1(ˆ
x), yields
β0(ˆ
x)=2(2n+ 1)ζ0(ˆ
x)3ζ1(ˆ
x)
n1(14)
and
β1(ˆ
x) = 6(n+ 1)ζ0(ˆ
x)+2ζ1(ˆ
x)
n21,(15)
which establishes with (5) and (6) the new detector algorithm.
In the next subsection, we analyze the error performance of
the new detection method based on (5).
B. Analysis of the error performance
We adopt here the same set of codewords, S={0,1}n\{0},
which is used in conjunction with the prior art modified
Pearson distance detector [6]. Let x∈ S be the sent codeword,
and let ˆ
x∈ S,ˆ
x6=x. In view of (7), δ(r,ˆ
x)can be
rewritten as an equivalent expression, which is convenient
for the computation of the error performance. The detector’s
performance is independent of a term c1+c2i, (c1and c2
arbitrary constants), thus we may delete Ii=b0+b1ior sub-
tract ϕi(x) = β0(x) + β1(x)iwithout effect on the outcome
of (7). Then, exploiting the linearity of the expressions (13),
(14), (15) in ϕi, we derive from (8) after deleting irrelevant
terms:
δ(r,ˆ
x)
n
X
i=1
(xiˆxi+ϕi(ˆ
x) + νi)2
n
X
i=1
(xiˆxi+ϕi(ˆ
x)ϕi(x) + νi)2
=
n
X
i=1
(xiˆxiϕi(xˆ
x) + νi)2.(16)
where the equivalence symbol denotes that the expressions
on both sides of yield xoafter the minimization (7). Let
e=xˆ
x, then we obtain
δ(r,ˆ
x)
n
X
i=1
(eiϕi(e) + νi)2.(17)
The detector errs, if it restores ˆ
xinstead of the sent x, that
is, if
δ(r,ˆ
x)< δ(r,x),(18)
or, after using (17),
2
n
X
i=1
(eiϕi(e))νi+
n
X
i=1
(eiϕi(e))2<0.(19)
The noise samples, νi, are assumed to be white and drawn
from N(0, σ2), so that the pairwise error probability P r(x
ˆ
x)equals
P r(xˆ
x) = P r(δ(r,ˆ
x)< δ(r,x))
=Qd(x,ˆ
x)
2σ,(20)
where
Q(x) = 1
2πZ
x
eu2
2du, (21)
and the squared noise distance d2(x,ˆ
x)between xand ˆ
xis
d2(x,ˆ
x) =
n
X
i=1
(eiϕi(e))2=δ(0,e).(22)
The union bound offers a useful tool to approximate the
average word error rate (WER). The WER is upperbounded
by [6]
WER X
d
KdQd(x,ˆ
x)
2σ,(23)
where Kdis the average number of neighbors at distance d=
d(x,ˆ
x). The minimum noise distance between any pair of
distinct codewords, denoted by dmin, is defined by
dmin = min
x,ˆ
x∈S,x6=ˆ
x
d(x,ˆ
x).(24)
The union bound estimate of the word error rate is
WER Ndmin Qdmin
2σ, σ 1,(25)
where the average number of neighbors at minimum noise
distance dmin is denoted by Ndmin .
1) Analysis of minimum noise distance dmin:For relatively
small values of nwe can, using exhaustive search (24), find
the worst case error vector e. Substituting the found einto (22)
yields the expression (26) for various values of n,n < 30 (the
maximum word length, n, of our search), so that
d2
min =
n21
16n, n = 3,5,7,9,
n(n24)
16(n21) , n = 2,4,6,8,10,
(n1)(n2)
n(n+1) ,11 n < 30.
(26)
Figure 2 displays the minimum noise distance,
20 log(dmin)(dB) versus codeword length n, where the
results are obtained using (26). As a reference we plotted the
minimum noise distance of the prior art method that offers
constant offset immunity, based on the distance measure
(3). We notice for small values of na significant loss in
the receiver’s noise margin with respect to conventional
Euclidean distance detection due to the decrease in dmin. For
n12, the loss is less than 1.5 dB, (for n18 the loss
is less than 1 dB). Note that the method advocated by Bu
et al. [10] that aims to solve the same problem has a 3 dB
noise penalty, irrespective of the codeword length.
For large n, the minimum distance computation is amenable
for analysis. We approximate β0(e)and β1(e)by, see (14) and
(15),
β0(e)4nw 6s
n2and β1(e)12s6nw
n3,
where we use the short-hand notation w=0(e)and s=
1(e). Then, after working out (22), we obtain
δ(0,e)dH+ 43nws n2w23s2
n3,(27)
where dH=Pe2
idenotes the Hamming distance between x
and ˆ
x. For 1dHn1, we find that minimizing (27)
over wand sgives
min
s,w δ(0,e)dH4n2d2
H6nd3
H+ 3d4
H
n3, n 1,(28)
where the minimum is achieved at the maximum values for w
and s, i.e., w=dHand s=n(n+ 1)/2(ndH)(ndH+
1)/2. The expression in (28) is at a minimum for dH= 1 and
dH=n1, which shows that a large Hamming distance does
not necessarily lead to a large noise distance δ(0,e). Observe
that this minimum is approximately 14/n. For the remaining
case dH=n, we find from (27) that mins,w δ(0,e)4,
achieved when wand sare maximum, i.e., w=n2, and
s=n(n+ 1)/22. Note that the choice w=dH=n
and s=n(n+ 1)/2, corresponding to e=1, would lead to
δ(0,e)=0, but this undesirable case is avoided by excluding
the all-zero vector from the code S. In conclusion, we obtain
d2
min 14
n, n 1.(29)
For detection based on (3), which refers to a constant offset,
the minimum squared noise distance equals
d2
min = 1 1
n.(30)
The minimum distance of the new method is smaller than
that of the prior art, which accounts for the immunity against
varying offset that we created.
4 6 8 10 12 14 16 18 20
n
-7
-6
-5
-4
-3
-2
-1
0
prior art
new
Fig. 2. Minimum noise distance, 20 log(dmin)(dB), versus codeword
length nfor the new scheme and the prior art using (3) as a reference.
2) Analysis of Ndmin :Let 1be the all-1 word, and
x1= (0,...,0,1), and x2= (1,0,...,0). For n > 11,
each codeword x S \ {1,x1,x2}, has two words ˆ
xat
minimum noise distance d(x,ˆ
x) = dmin that differ at the
first or last position. The words x1= (0,...,0,1) and
x2= (1,0,...,0) also have two neighbors at minimum
noise distance, namely ˆ
x=1and ˆ
x= (1,0,...,0,1). The
all-1 word, x=1, has four neighbors at minimum noise
distance, namely (1,0,...,0),(0,1,...,1),(0,...,0,1), and
(1,...,1,0), so that the average number of neighbors at
minimum noise distance is Ndmin = 2 + 2/|S| ≈ 2.
C. Results
We have conducted simulations and computations to eval-
uate the error performance of the new detection method. We
started by comparing the word error rate of the new detection
scheme as computed by two methods and by simulations.
Results are shown in Figure 3, which displays the WER
versus SNR = 20 log(σ)(dB) for the union bound (23),
union bound estimate (25), and computer simulations. The
error performance is independent of mismatch terms b0and b1.
We note that the simulations agree favorably with the union
bound (23); the difference between union bound and union
bound estimate is large in the range of small SNR.
We also appraised the error performance of various detec-
tion schemes. Figure 1 shows results of computations (union
bound), where we compare the word error rate of various
scenarios of offset and detection schemes.
Figure 4 displays the word error rate versus SNR for
three detection methods, namely Curve 1) Euclidean distance
detection, Curve 2) modified Pearson distance detection us-
ing distance measure (3), and Curve 3) the new detection
method. Results are shown for the ideal noisy channel and
the mismatched channel, b0= 0, b1= 0.025. Without offset
mismatch, both Euclidean and modified Pearson distance
detection perform better than the new method. The situation
changes when there is a varying offset. Then, both Euclidean
and modified Pearson distance detection perform less than
the new detection method. The error performance of the new
17 17.5 18 18.5 19 19.5 20
SNR
10-5
10-4
10-3
10-2
WER
(a)
(b)
Fig. 3. Word error rate, WER versus SNR = 20 log(σ)(dB) of
the new detection method computed using (a) union bound (23), (b)
union bound estimate (25), and by simulations. The points marked
with ‘*’ result from simulations. Note that Curve (a) is the same as
Curve (b) in Figure 1. For all curves, n= 12.
detection method is independent of the offset terms b0and
b1. If the channel is ideal b0=b1= 0, the new detection
method loses error performance with respect to the prior art
detection methods. If, however, the channel is mismatched
b0, b16= 0, we notice, see Curve (3), that the error performance
is unaffected, while the alternative schemes lose performance.
For all curves, we have n= 12. The performance curves for
the ideal channel, b0=b1= 0, have been computed for
Curve (2a) (Pearson distance detection) using, see [6, eqn.
(28)],
WERpnQ 1
2σr11
n!,(31)
and for Curve (1a) (Euclidean distance detection) by
WERenQ 1
2σ,(32)
where WERpand WERedenote the word error rate of
Pearson and Euclidean distance detection, respectively. We
used computer simulations for the mismatched case shown by
Curves (1b) and (2b). Curve (3) showing the new method’s
error performance was computed using union bound (23) and
confirmed by simulations.
D. Relationship with mass-centered codewords
In [9], an alternative approach has been disclosed for ob-
taining immunity against varying offset by using a constrained
code. The constrained set, Sm, of mass-centered codewords is
defined by [9]
Sm={x∈ {0,1}n: Ω(x)=0},(33)
where
Ω(x)=2
n
X
i=1 in+ 1
2xi.(34)
17 17.5 18 18.5 19 19.5 20
SNR
10-6
10-5
10-4
10-3
10-2
10-1
WER
(2a)
(1b)
(2b)
(1a)
(3)
Fig. 4. Word error rate, WER versus SNR = 20 log(σ)(dB) of (1a)
Euclidean detection, no offset; (1b) Euclidean detection, b0= 0, b1=
0.025; (2a) Pearson detection, no offset; (2b) Pearson detection, b0=
0, b1= 0.025; (3) new detection method, with and without offset.
For all curves, n= 12.
We simply rewrite (34) as, see (13) and (15),
Ω(x)=2
n
X
i=1
ixi(n+ 1)
n
X
i=1
xi
=n(2ζ1(x)(n+ 1)ζ0(x))
=n(n21)
6β1(x).(35)
For mass-centered codewords, where Ω(x)=0, we may write
ζ1(x) = n+ 1
2ζ0(x),
so that, using (14) and (15),
β0(x) = ζ0(x) = xand β1(x)=0.(36)
Clearly, by selecting a set of mass-centered codewords, we are
able to significantly simplify the detection routine as the term
β1(x)iis absent in (5), at the cost of extra code redundancy.
We also note that as β0(x) = x, the distance measure δ(r,ˆ
x)
changes into the prior art δ0(r,ˆ
x), see (3). The rate of a mass-
centered code is not attractive for many applications, see, for
example, Table 1 of [9], which shows the size of Smversus
n.
IV. RECEIVER IMPLEMENTATION COMPLEXITY
The complexity of the encoder and detector/decoder can
be partitioned into three major blocks, namely a) encoding
arbitrary user data into a codeword in the Pearson code Sand
vice versa, b) computation or storage of the coefficients β0(ˆ
x)
and β1(ˆ
x), and c) evaluation of the distance measure (7) for
all codewords in S.
A. Encoder/decoder complexity
Systematic methods for designing Pearson codes that effi-
ciently translate (arbitrary) source data into n-bit codewords
in the codebook S={0,1}n\ {0}and vice versa have
been presented in [11, 12, 13]. The rate of the encoder,
R= 1 1
2n1, presented in [13] is close to the maximum
possible, and the complexity of the encoder and decoder scales
linearly with n.
B. Coefficient storage
The receiver requires the coefficients β0(ˆ
x)and β1(ˆ
x)
for computing (5). The coefficients β0(ˆ
x)and β1(ˆ
x)can be
calculated ‘on the fly’ for each codeword ˆ
x∈ S, but in order
to save computation time, they are preferably pre-calculated
and stored in a memory. The coefficients storage requires at
first sight 2n1memory cells. We may, however, save on the
coefficients storage as illustrated by the following observation.
We partition the codebook set Sin distinct subsets of
codewords that have equal zeroth- and first-order moments.
Let m0and m1be two positive integers. The codeword subset
Sm0,m1with prescribed moments m0and m1is defined by
Sm0,m1=(x∈ S :
n
X
i=1
xi=m0
n
X
i=1
ixi=m1).(37)
Let N(n)denote the number of distinct nonempty subsets
Sm0,m1.
Theorem 1: The number of distinct nonempty subsets
Sm0,m1is upper bounded by
N(n)n(n2+ 5)
6.(38)
Proof: Let q=Pn
i=1 xi=m0denote the number of 1’s in x
(weight), then
m1=
n
X
i=1
ixi∈ {q1, q1+ 1, q1+ 2, . . . , q2},
where q1=q(q+ 1)/2(all q1’s at the beginning of x) and
q2=nq q(q1)/2(all q1’s at the end of x). Then the
number of distinct values of m1for a given m0equals q2
q1+ 1 = nq q2+ 1. The number of distinct pairs (m0, m1)
is n
X
q=1
(nq q2+ 1) = n(n2+ 5)
6,
which proves the theorem.
We conclude that the number of distinct coefficient pairs
β0(ˆ
x)and β1(ˆ
x)equals N(n), so that the decoder storage
complexity grows polynomially, n3, with the codeword
length n. If the size of the subset Sm0,m1is small, this does
not provide great solace, but for larger subset sizes it may
offer an attractive saving in coefficient storage.
C. Time complexity
For evaluating (7), the decoder requires |S| = 2n1
computations of δ(r,ˆ
x)plus comparisons, which makes the
new method unattractive for very large n. It is shown in [6] that
the (time) complexity of the prior art method based on (3) can
be reduced to ncomputations and comparisons using Slepian’s
method [14]. This significant reduction in time-complexity is
possible as Sconsists of npermutation codes. We cannot
apply Slepian’s method here as the subset Sm0,m1is not a
simple permutation code. Although the storage requirements
of the precalculated coefficients can be reduced as shown
above, the evaluation of (7) requires 2n1computations and
comparisons per decoded codeword.
V. CONCLUSIONS
We have presented a new detection method for noisy
channels with an offset of monotonically increasing time-
varying magnitude. The error performance of the new detec-
tion method is independent of both unknown base offset and
offset’s slew rate. The rate of the requisite constrained code
is very high as only one codeword has to be barred. Com-
puter simulations have been conducted to appraise the error
performance of the new detection method. The simulations
compare favorably with theory based on the union bound.
For large signal-to-noise ratios, the new method requires, for
a codeword length n12, less than 1.5 dB more noise
margin than Euclidean distance detection or less than 1 dB
loss with respect to Pearson detection. The relationship with
constrained codes based on mass-centered codewords and the
new detection method has been discussed.
REFERENCES
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Dc-free RLL codes for Optical Recording,” IEEE Transactions on
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[3] B. Peleato, R. Agarwal, J. M. Cioffi, M. Qin, and P. H. Siegel,
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CAS), Florence, Italy, pp. 1-5, 2018, doi: 10.1109/ISCAS.2018.8351740.
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[7] K. A. S. Immink, “Coding Schemes for Multi-Level Channels with
Unknown Gain and/or Offset Using Balance and Energy constraints,
pp. 709-713, IEEE International Symposium on Information Theory,
(ISIT), Istanbul, July 2013, doi: 10.1109/ISIT.2013.6620318.
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Detection Based on Pearson Distance Detection,” IEEE Transactions
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[10] R. Bu and J. H. Weber, “Minimum Pearson Distance Detection Using a
Difference Operator in the Presence of Unknown Varying Offset,IEEE
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doi: 10.1109/LCOMM.2019.2917677.
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Sequence Codes,” IEEE Access, doi: 10.1109/ACCESS.2021.3065067,
2021.
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Kees A. Schouhamer Immink (M’81-SM’86-F’90) received
his PhD degree from the Eindhoven University of Technol-
ogy. 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 currently focused on novel signal
processing for DNA-based storage, 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 Sci-
ences, the Royal Holland Society of Sciences and Humanities,
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.
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 Telecom-
munications) degrees from Delft University of Technology,
Delft, The Netherlands, in 1985, 1989, and 1996, respectively.
Since 1985 he has been with the Delft University of
Technology. Currently, he is an associate professor at the De-
partment of Applied Mathematics. He was the chairman of the
Werkgemeenschap voor Informatie- en Communicatietheorie
from 2006 until 2021. He is the secretary of the IEEE Benelux
Chapter on Information Theory since 2008. He was a visiting
researcher at the University of California (Davis, CA, USA),
the Tokyo Institute of Technology (Japan), the University of
Johannesburg (South Africa), EPFL (Switzerland), and SUTD
(Singapore). His main research interests are in the area of
channel coding.
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