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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 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.

Keywords−channel 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 ri−ˆxi+ 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

(ri−ˆxi+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+νi−ˆxi+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)

n−1(14)

and

β1(ˆ

x) = 6−(n+ 1)ζ0(ˆ

x)+2ζ1(ˆ

x)

n2−1,(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

(xi−ˆxi+ϕi(ˆ

x) + νi)2

≡

n

X

i=1

(xi−ˆxi+ϕi(ˆ

x)−ϕi(x) + νi)2

=

n

X

i=1

(xi−ˆxi−ϕi(x−ˆ

x) + νi)2.(16)

where the equivalence symbol ≡denotes that the expressions

on both sides of ≡yield xoafter the minimization (7). Let

e=x−ˆ

x, then we obtain

δ(r,ˆ

x)≡

n

X

i=1

(ei−ϕi(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

(ei−ϕi(e))νi+

n

X

i=1

(ei−ϕi(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(x→ˆ

x) = P r(δ(r,ˆ

x)< δ(r,x))

=Qd(x,ˆ

x)

2σ,(20)

where

Q(x) = 1

√2πZ∞

x

e−u2

2du, (21)

and the squared noise distance d2(x,ˆ

x)between xand ˆ

xis

d2(x,ˆ

x) =

n

X

i=1

(ei−ϕi(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 =

n2−1

16n, n = 3,5,7,9,

n(n2−4)

16(n2−1) , n = 2,4,6,8,10,

(n−1)(n−2)

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

n≥12, the loss is less than 1.5 dB, (for n≥18 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)≈12s−6nw

n3,

where we use the short-hand notation w=nζ0(e)and s=

nζ1(e). Then, after working out (22), we obtain

δ(0,e)≈dH+ 43nws −n2w2−3s2

n3,(27)

where dH=Pe2

idenotes the Hamming distance between x

and ˆ

x. For 1≤dH≤n−1, we ﬁnd that minimizing (27)

over wand sgives

min

s,w δ(0,e)≈dH−4n2d2

H−6nd3

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−(n−dH)(n−dH+

1)/2. The expression in (28) is at a minimum for dH= 1 and

dH=n−1, 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 1−4/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=n−2, and

s=n(n+ 1)/2−2. 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 ≈1−4

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)],

WERp≈nQ 1

2σr1−1

n!,(31)

and for Curve (1a) (Euclidean distance detection) by

WERe≈nQ 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 i−n+ 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(n2−1)

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

2n−1, 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 2n−1memory 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(q−1)/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| = 2n−1

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 2n−1computations 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 n≥12, 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.

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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.