Content uploaded by Kees Schouhamer Immink
Author content
All content in this area was uploaded by Kees Schouhamer Immink on Apr 11, 2019
Content may be subject to copyright.
Computation of the spectrum of dc2-balanced codes
Kees A. Schouhamer Immink, Fellow, IEEE, and Kui Cai, Senior Member, IEEE
Abstract—We apply the central limit theorem for deriving ap-
proximations to the auto-correlation function and power density
function (spectrum) of second-order spectral null (dc2-balanced)
codes. We show that the auto-correlation function of dc2-balanced
codes can be accurately approximated by a cubic function. We
show that the difference between the approximated and exact
spectrum is less than 0.03 dB for codeword length n= 256.
I. INTRODUCTION
Spectral null, or dc-balanced, codes have been applied in
cable transmission [1, 2, 3], magnetic recording [4, 5] and
optical recording systems [6, 7, 8]. Spectral null codes have re-
cently been advocated in visible light communications (VLC)
systems, where light intensity of solid-state light sources,
mostly LEDs, are varied [9]. It is desirable that the intensity
variation of the light is invisible to the users, that is, annoying
flicker should be mitigated [10]. This requirement implies that
the spectrum of the modulated signal should not contain low-
frequency components. Light sources are usually connected
to the AC power grid, and therefore generate interference
components at 50, 60 Hz, or the higher harmonics. Rejection
of these interfering components can easily be accomplished
by high-pass filtering, but in order not to degrade the wanted
communication signal by this filtering, low-frequency compo-
nents should be absent in the modulated signal. Three types
of dc-balanced codes, the Manchester code (bi-phase), a 4B6B
code, and an 8B10B code, have been adopted in VLC standard
IEEE 802.15.7-2011 [11] for flicker mitigation and dimming
control [9, 12].
Higher-order spectral null codes, such as dc2-balanced
codes, offer a greater rejection of the low-frequency compo-
nents than regular dc-balanced codes [13]. Most of the prior art
literature has focused on efficient constructions of higher-order
spectral null codes, see for example [14, 15, 16, 17, 18, 19, 20].
Spectral properties of higher-order spectral null codes have
been computed for relatively small values of the codeword
length nby enumerating all codewords [13]. For larger values
of n, Immink and Cai [21] have postulated expressions for
approximating the auto-correlation function and spectrum of
higher-order spectral null codes.
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.
This work is supported by Singapore Ministry of Education Academic
Research Fund Tier 2 MOE2016-T2-2-054, and Singapore Agency of Science
and Technology (A*Star) PSF research grant
The main contribution of this paper is the derivation of
improved approximations to the auto-correlation function and
spectrum of dc2-balanced codes by applying the central limit
theorem. The paper is organized as follows. Section II com-
mences with prior art background on dc2-balanced codes. In
Section III, we derive an approximation to the auto-correlation
function and spectrum of dc2-balanced codes for asymptot-
ically large values of the codeword length nby counting
dc2-balanced codewords using the central limit theorem. The
approximations found are compared with the exact values for
n < 256. Further approximations for asymptotically large n
will be discussed in Section III-C. In Section IV, we appraise
the spectral performance of dc2-balanced codes. Section V
shows our conclusions.
II. BACKGROU ND O N DC2-B AL AN CE D BLO CK CODES
Let the n-bit codeword x= (x1, x2, . . . , xn)over the binary
symbol alphabet Q={0,1}, be a member of a codebook
S. The encoder emits codewords from Srandomly and in-
dependently (i.i.d.). The auto-correlation function, ρ(i), of a
sequence of codeword symbols is given by [22, 23, 24]
ρ(i) = 1
n|S|
x∈S
n−i
j=1
x′
jx′
j+i,0≤i≤n−1,(1)
where |S|denotes the cardinality of Sand x′
i= 2xi−1,
x′
i∈ {−1,1}, is the bipolar representation of xi. If both x
and its inverse ¯
xare members of S, then the power spectral
density (psd), in short spectrum, versus frequency ωof the
emitted symbol sequence is
H(ω) = 1 + 2
n−1
i=1
ρ(i) cos(iω).(2)
A regular ‘full-set’ dc-balanced block code comprises all
possible codewords that have equal numbers of 0’s and 1’s
(neven). Franklin and Pierce [2] showed that the spectrum
of a full-set dc-balanced block code has a null at the zero
frequency, that is, H(0) = 0. Dc2-balanced spectral null codes
are dc-balanced codes that satisfy a second condition, namely
H(0) = H(2)(0) = 0,(3)
where H(2)(0) denotes the second derivative of H(ω)at ω=
0. Note that the above frequency domain conditions imply, see
(2), that
n−1
i=1
ρ(i) = −1
2and
n−1
i=1
i2ρ(i)=0.(4)
A codeword, x, is dc2-balanced if it satisfies [13, 25]
n
i=1
xi=n
2and
n
i=1
ixi=n(n+ 1)
4.(5)
A block code comprising a full set of dc2-balanced codewords,
denoted by S2, is defined by
S2=x∈ Qn:xi=n
2;ixi=n(n+ 1)
4.(6)
The set S2is empty if nmod 4 ̸= 0 [13]. Let x∈S2then
its reverse xr= (xn, . . . , x1)∈S2, since for a x∈S2
n
i=1
ixi=
n
i=1
(n+ 1 −i)xi=n(n+ 1)
4.(7)
A useful metric of the low-frequency spectral content, denoted
by χ, called Low Frequency Spectral Weight (LFSW) [18], is
the first non-zero coefficient of the Taylor expansion of (2),
that is,
H(ω)∼χω4, ω ≪1.(8)
We derive from (2) that
χ=1
12
n−1
i=1
i4ρ(i).(9)
The number of dc2-balanced codewords, denoted by Ndc2=
|S2|, can be approximated for asymptotically large n,
by [19, 26]
Ndc2∼4√3
πn22n, n mod 4 = 0, n ≫1.(10)
In the range n < 256 we have found experimentally that a
better approximation is found by applying a small correction
term, namely
Ndc2∼4√3
πn22n1−1.211
n, n mod 4 = 0, n ≫1.
(11)
We consider here the spectral properties of full-set block
codes, that is, S2denotes the set of all possible words, x, that
satisfy condition (5). Finding an expression of the spectral
properties of a full-set S2for large values of nis an open
problem as the computation requires the evaluation of (1) for
each x∈S2[15]. In the next section, our main contribution,
we address an alternative method, which is based on statistical
analysis, which gives a simple and good approximation to the
spectrum.
III. AU TO-CORRELATION FUNCTION
Let xbe a codeword in S2, and let i0and i1,i0̸=i1,
1≤i0, i1≤n, be two (different) index positions in the code-
word x. Define the average correlation, denoted by r(i0, i1),
between the symbols at positions i0and i1averaged over all
codewords x∈S2by
r(i0, i1) = 1
Ndc2
x∈S2
(2xi0−1)(2xi1−1).(12)
Note that the bipolar variables x′
i0= 2xi0−1and x′
i1=
2xi1−1∈ {−1,1}, so that x′
i0x′
i1= 1 for xi0=xi1and
x′
i0x′
i1=−1for xi0̸=xi1. Then,
r(i0, i1) = Ndc2(xi0=xi1)−Ndc2(xi0̸=xi1)
Ndc2
,(13)
where Ndc2(A)denotes the number of dc2-balanced code-
words xthat satisfy condition A. Since, by definition,
Ndc2=Ndc2(xi0=xi1) + Ndc2(xi0̸=xi1),
and as both xand its inverse ¯
xare members of S2,
Ndc2(xi0=xi1= 0) = Ndc2(xi0=xi1= 1),
we have
r(i0, i1) = 2Ndc2(xi0=xi1)
Ndc2−1
=4Ndc2(xi0=xi1= 1)
Ndc2−1.(14)
Applying (1) and (13), we find the auto-correlation function
ρ(i) = 1
n
n−i
j=1
r(j, j +i),0≤i≤n−1.(15)
By using the central limit theorem, we compute below an
approximation to the number of dc2-balanced codewords that
have a ‘1’ at positions i0and i1,Ndc2(xi0=xi1= 1), for
asymptotically large values of n. Then, using (15), we are able
to derive an approximation to the auto-correlation function
ρ(i).
A. Counting of codewords using the central limit theorem
The number of dc2-balanced codewords, x,Ndc2(xi0=
xi1= 1), that is required for computing the auto-correlation
function using (15) and (14), can be computed using generat-
ing functions. For very large n, however, this rapidly becomes
an impractically cumbersome exercise, and an efficient alter-
native method is considered a desideratum.
To that end, we exploit the central limit theorem by regard-
ing the integer variables xi∈ {0,1}as i.i.d. binary random
variables whose numerical outcomes ‘0’ or ‘1’ are equally
likely. We define the stochastic variables cand pby
c=x1+x2+···+xn(16)
and
p=x1+ 2x2+· ·· +nxn,(17)
where xi0=xi1= 1,i0, i1∈ {1, . . . , n}.
The central limit theorem [27, Chapter 8], states that
for asymptotically large nthe distribution of the stochastic
variables cand p, which are obtained by summing a large
number, n, of independent stochastic variables, approaches a
two-dimensional Gaussian probability distribution.
Define E[.], the expected value operator for all possible
codewords in S2. Let the parameters µc=E[c]and µp=E[p]
denote the average of cand p, and let σ2
c=E[(c−µc)2]
and σ2
p=E[(p−µp)2]denote the variance of cand p. The
parameter rdenotes the linear correlation coefficient between
the random variables cand p. Then the probability density
function of the bi-variate Gaussian distribution, denoted by
G(c, p), is given by
G(c, p) = 1
2πϕ1
e−ϕ(c,p),(18)
where
ϕ2
1=σ2
cσ2
p(1 −r2),(19)
ϕ(c, p) = 1
2(1 −r2)f(c, p),(20)
and
f(c, p) = c−µc
σc2
+p−µp
σp2
−2r(c−µc)(p−µp)
σcσp
.
We have
E[xi] = E[x2
i] = 1
2,and E[xixj] = 1
4, i ̸=i0, i1,(21)
E[xi] = E[x2
i] = 1, i =i0, i1, E[xixj] = 1
2, i, j =i0, i1,
and E[xi0xi1]=1. We may find after a routine computation
using (21) that
µc=En
i=1
xi=
n
i=1
E[xi] = n−2
2+ 2
and similarly
µp=En
i=1
ixi=
n
i=1
E[ixi] = n(n+ 1)
4+i0+i1
2.
The variances σ2
c,σ2
p, and the correlation coefficient rcan be
found without too much difficulty:
σ2
c=En
i=1
(xi−µc)2=n−2
4,(22)
σ2
p=En
i=1
(ixi−µp)2
=n(n+ 1)(2n+ 1)
24 −i2
0+i2
1
4,(23)
and
r2=E[n
i=1(xi−µc).n
i=1(ixi−µp)]
σ2
cσ2
p
=3
2(n−2)
(n2+n−2(i0+i1))2
2n3+ 3n2+n−6(i2
0+i2
1).(24)
The total number of n-sequences with xi0=xi1= 1, equals
2n−2, so that for asymptotically large n, the number of n-
sequences versus cand p, denoted by N(c, p;xi0=xi1= 1),
can be approximated by
N(c, p;xi0=xi1= 1) ∼2n−2G(c, p) = 2n
8πϕ1
e−ϕ(c,p).
(25)
A dc2-balanced codeword satisfies, by definition, the con-
ditions, see (5), c=n/2and p=n(n+ 1)/4. Then,
Ndc2(xi0=xi1= 1) is found after substituting c=n/2
and p=n(n+ 1)/4into (25). We find
Ndc2(xi0=xi1= 1) ∼2n
8πϕ1
e−ϕ2,(26)
where, see (19) and (20),
ϕ2=ϕc=n
2, p =n(n+ 1)
4
=1
2(1 −r2)1
σc2
+i0+i1
2σp2
−r(i0+i1)
σcσp
=4σ2
p+σ2
c(i0+i1)2−4rσcσp(i0+i1)
8ϕ1
.(27)
After combining (10), (14), and (26), we obtain
r(i0, i1) = n2
√192 ϕ1
e−ϕ2−1.(28)
In order to reduce the clerical work and offer more insight,
we define the four (real) variables
γ= 12n[(i0−n−1)i0+ (i1−n−1)i1],
δ= (i0−i1)2,
r1=−1
n4(8n3+ 13n2+ 4n+γ−12δ),
and
r2=1
8n3[12n2+ 4n+γ−6(n+ 2)δ].
With some effort we find the expressions
ϕ2
1=n4
192(1 + r1)(29)
and
ϕ2=8
n
1 + r2
1 + r1
.(30)
We finally obtain
r(i0, i1) = n2
√192 ϕ1
e−ϕ2−1
=1
√1 + r1
e−8
n
1+r2
1+r1−1,(31)
where we can easily verify, since xand xr∈S2, see (7), that
r(i0, i1) = r(n+ 1 −i0, n + 1 −i1).(32)
The auto-correlation function, ρ(i), is found using (15). In the
next subsection, we show results of computations.
B. Results of computations
By invoking (15) and (31) we are now able to compute
an estimate of the auto-correlation function ρ(i). Figure 1
shows results of computations for n= 32, 64, and 128. As a
comparison we plotted the exact auto-correlation function of
a full set of dc2-balanced sequences, denoted by ˆρ(i), which
was computed using an enumeration technique and generating
functions [13].
The accuracy of the approximate auto-correlation function,
ρ(i), cannot easily be determined from Figure 1 for the larger
values of n. Figure 2, Curve ‘without correction’, shows
0 0.2 0.4 0.6 0.8 1
i/n
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
Auto-correlation function
n=32
n=64
n=128
exact
exact
Fig. 1. Auto-correlation functions, ρ(i)(estimate, using (15) and (31))
and ˆρ(i)(exact, full set), versus i/n for n= 32,64 and 128.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
i/n
10-8
10-7
10-6
10-5
10-4
10-3
Difference
n=128
with correction
n=256
n=128
n=256 without correction
without correction
Fig. 2. Difference, with and without correction, between estimated
and exact auto-correlation functions, |ˆρ(i)−ρ(i)|, versus i/n for
n= 128 and 256.
|ˆρ(i)−ρ(i)|, the difference between the two auto-correlation
functions versus i/n for the selected n= 128 and 256. We
notice that the difference between the two functions decreases
with increasing iand n. For n= 256 in the whole range the
difference, |ˆρ(i)−ρ(i)|, is less than 10−4.
Although the difference, |ˆρ(i)−ρ(i)|, is relatively small,
especially for larger n, see Figure 2, the ‘checks’, see (4),
ρ(i) = −1/2and i2ρ(i) = 0, which accumulate the
small error differences, are not necessarily satisfied. We have
observed that with increasing nthat ρ(i)+1/2is converging
to zero (as it should), while i2ρ(i)is not. As a result,
the spectra, computed using ρ(i)do not satisfy the spectral
conditions (3).
We propose to add a small correction term to ρ(i)so that
both ‘checks’, ρ(i) = −1
2and i2ρ(i) = 0, are satisfied.
We add to ρ(i)the correction term a+bi, where the (real)
parameters, aand b, are chosen such that i(ρ(i)+ a+bi) =
−1
2and ii2(ρ(i) + a+bi) = 0. Define
a0=ρ(i) + 1
2
and
a1=i2ρ(i),
then we find two linear equations with two unknowns, aand
b, namely
n−1
i=1
(ρ(i) + a+bi) = a0−1
2+na +b
n−1
i=1
i=−1
2
and
n−1
i=1
i2(ρ(i) + a+bi) = a1+a
n−1
i=1
i2+b
n−1
i=1
i3= 0.
After solving the above system, where we substitute the well-
known expressions for ik,k= 1,2,3, we obtain
a=−3n(n−1)a0−2a1
n(n−1)(n−2) (33)
and
b= 2n(2n−1)a0−6a1
n2(n−1)(n−2) .(34)
For example, for n= 128, we find that a0=−0.0156 and
a1=−22.21. So that a= 0.0003063 and b=−0.0000029.
The result of the correction can be seen in Figure 2, curves
‘with correction’, for n= 128 and 256. We notice in the range
i/n < 0.6a significant improvement in the accuracy of the
estimate of the auto-correlation function.
C. Further approximations for asymptotically large n
With (31) we can straightforwardly compute the auto-
correlation function ρ(i)and spectrum H(ω). In this section,
we attempt to approximate ρ(i)for asymptotically large n,
which might offer more insight in the trade-offs between
redundancy and spectral properties. We apply to (31) the well-
known series approximations
1
√1 + x= 1 −x
2+3
8x2−5
16x3+···
and 1
1 + x= 1 −x+x2−x3+· ·· .
We have experimented with the various options available
for trading accuracy versus simplicity of the expression, and
propose
r(i0, i1)∼ −8
n(1 + r2)−r1
2, n ≫1.(35)
Using (15), we obtain
ρ(i)∼n−i
2n512i2+ 4ni2+ 4n2i−4n3+n2+ 4n.(36)
Then, after deleting the smallest terms, we obtain the simple
cubic function
ρ(i)∼2
n4(n−i)(i2+in −n2),(37)
which can be rewritten as
ρ(i)∼2
n4(n−i)(i−c0n)(i−c1n),(38)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
i/n
10-8
10-7
10-6
10-5
10-4
10-3
Difference
(b)
(a)
Fig. 3. Deviation between exact and estimated auto-correlation func-
tion a) |ρ′(i)−ˆρ(i)|and b) |ρa(i)−ˆρ(i)|versus i/n for n= 256,
where ρa(i)is defined in (41).
where c0,1= (−1∓√5)/2. The checks (4) for the above ρ(i)
yield
a0=1
2+ρ(i) = 1
n−1
2n2
and
a1=i2ρ(i) = −1
6+1
6n2.
In order to satisfy both checks (4), we add to ρ(i)the
correction term a+bi, and define
ρ′(i) = 2
n4(n−i)(i−c0n)(i−c1n) + a+bi, (39)
where after using (33) and (34), we obtain
a=−6n2−n+ 2
2(n−2)n3∼ − 3
n2
and
b=4n3−2n2+n−2
n4(n−1)(n−2) ∼4
n3.
Note that aand bare relatively small terms in (39) for
asymptotically large n. Figure 3 shows the difference between
exact and estimated auto-correlation function |ρ′(i)−ˆρ(i)|
versus i/n for n= 256. As a final proof of the pudding,
we compare the (exact) spectrum of full set codewords versus
the spectrum, denoted by H′(ω), which is computed using
the above approximated auto-correlation function ρ′(i). The
difference, H′(ω)/ˆ
H(ω)(dB), between the spectrum, H′(ω),
computed using ρ′(i), and the exact spectrum, ˆ
H(ω), of full set
codewords, which was computed using generating functions,
is plotted in Figure 4. We may observe that the difference
between the two spectra is very small, less than 0.05 dB for
n= 128 and less than 0.03 dB for n= 256.
The LFSW metric, χ′is, using (9),
χ′=1
12
n−1
i=1
i4ρ′(i)∼n4
720 1 + 4
n.(40)
Table I shows χ′for selected values of n, where as a
comparison we have listed the LFSW of full set dc2-balanced
codes, denoted by ˆχ. We may notice that for n= 256 the
difference between χand ˆχis less than half a percent.
10-4 10-3 10-2 10-1 100
Frequency f (log)
-0.01
0
0.01
0.02
0.03
0.04
0.05
Deviation (dB)
n=256
n=128
Fig. 4. Deviation between exact and estimated spectrum H′(ω)/ˆ
H(ω)
(dB) versus ωfor n= 128 and 256.
10-4 10-3 10-2 10-1 100
Frequency f (log)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Deviation (dB)
(b)
(a)
Fig. 5. Deviation between exact and estimated spectrum a)
H′(ω)/ˆ
H(ω)(dB) and b) Ha(ω)/ˆ
H(ω)(dB) versus ωfor n= 256.
D. Comparison with prior art
In [21], it is postulated that the auto-correlation of dc2-
balanced spectral null codes, denoted by ρa(i), can be mod-
elled by the simple parabola’s equation
ρa(i) = β(i+α)(i−n),(41)
where the (real) parameters αand βare given by
α=−3n2−2
5n
TABLE I
LFSW χ′AN D ˆχVE RSU S n.
n χ′ˆχ
32 1629.48 1576.72
64 24723.13 24250.79
128 384339.75 380367.61
256 6057889.79 6025352.62
and
β=−15
(n−1)(n−2)(4n+ 3).
It has been shown in [21] that the parabola’s equation (41)
is an accurate approximation to the exact correlation func-
tion of full-set dc2-balanced spectral null codes. Figure 3,
Curve (b), shows the difference between exact and estimated
auto-correlation function |ρa(i)−ˆρ(i)|versus i/n for n= 256.
We notice that the newly developed ρ′(i), Curve (a), is almost
an order more accurate than (41) presented in the prior art.
Figure 5, Curve (b), shows that the quotient of the exact
spectrum and the one based on prior art (41), ˆ
H(ω)/Ha(ω), is
for n= 256 less than 0.7 dB, and also here we notice that the
newly developed theory is more than an order more accurate.
IV. APPRAISAL OF SPECTRAL PERFORMANCE
A system designer is usually confronted with a restricted
redundancy budget, so that with a given redundancy the
designer searches for a balanced code that offers the best
rejection of low-frequency components. In this section, we
compare the spectral performance of regular dc-balanced codes
with that of dc2-balanced codes. We start with a summary of
properties of dc-balanced codes.
A. Codes with a first-order spectral null
Let the codeword length of a regular full-set dc-balanced
code be denoted by n1,n1even. Each codeword has an equal
number of 0’s and 1’s, so that the number of available dc-
balanced codewords, denoted by Ndc, is simply [28]
Ndc =n1
n1/2∼1
π
2n1
2n1, n1≫1.(42)
The auto-correlation function, ρ1(i), and the spectrum, H1(ω),
of dc-balanced codes is [2]
ρ1(i) = 1
n1(n1−1)(i−n1)(43)
and
H1(ω) = n1
n1−11−sin n1ω
2
n1sin ω
22.(44)
At the very low-frequency end, we have [18]
H1(ω)∼χ1ω2, ω ≪1,(45)
where
χ1=n1(n1+ 1)
12 .(46)
B. Performance comparison
We compare the spectral content of dc-balanced versus that
of dc2-balanced codes, where we assume that both types of
codes have the same redundancy. Let Rand R1denote the
maximum information rate of a dc2-balanced code or dc-
balanced of length nand n1, respectively, then we have, using
(10),
R=1
nlog2Ndc2=1
nlog2
4√3
πn22n
= 1 −1
nlog2
πn2
4√3(47)
TABLE II
COD E LE NGT H nAND n1FOR R=R1.
R=R1n1n
0.90 28 132
0.92 38 172
0.94 54 248
0.96 90 408
0.98 210 932
10−4 10−3 10−2
−40
−35
−30
−25
−20
−15
−10
−5
0
5
Spectrum (dB)
Frequency f (log)
(a) (c)
(b)
Fig. 6. Spectra of dc-balanced and dc2-balanced codes with the same
redundancy versus frequency for a) R=R1= 0.98, b) R=R1=
0.94, and c) R=R1= 0.90, see also Table II. The points of
intersection are around −20 dB.
and, using (42),
R1= 1 −1
2n1
log2
π
2n1.(48)
Table II shows a few examples of the codeword lengths n
and n1for which dc-balanced and dc2-balanced codes have
equal redundancy, respectively, that is, R=R1. In the range
shown in Table II, the codeword length nof a dc2-balanced
code is approximately a factor of 4.5 larger than the codeword
length n1of a dc-balanced code for achieving the same rate
R=R1. Figure 6 shows three examples of spectrum pairs of
dc-balanced and dc2-balanced codes with the same redundancy
versus frequency for a) R=R1= 0.98, b) R=R1= 0.94,
and c) R=R1= 0.90, see also Table II. We may notice the
points of intersection of the spectra of dc-balanced and dc2-
balanced codes. A further perusal of the diagram reveals that
the points of intersection are at around −20 dB, which implies
that dc2-balanced codes are to be preferred when a low-
frequency spectral suppression is required better than around
−20 dB. Additional computations show that this ‘20 dB rule’
applies to all codes with a rate larger than 0.75.
V. CONCLUSIONS
By applying the central limit theorem, we have derived
an approximate expression for the auto-correlation function
and spectrum of full-set dc2-balanced codes for asymptotically
large values of the codeword length n. We have shown that
the auto-correlation function of dc2-balanced codes can be
accurately approximated by a simple cubic function. We have
compared the approximate spectrum with the exact spectrum
of full set dc2-balanced codes. We have shown that the
difference between the approximated and exact spectrum is
less than 0.03 dB for n= 256.
REF ER EN CE S
[1] K. W. Cattermole, “Principles of Digital Line Coding,” Int. Journal of
Electronics, vol. 55, pp. 3-33, July 1983.
[2] J. N. Franklin and J. R. Pierce, “Spectra and Efficiency of Binary Codes
without DC,” IEEE Trans. Commun., vol. COM-20, no. 6, pp. 1182-
1184, Dec. 1972.
[3] C. Cao and I. Fair, “Minimal Sets for Capacity-Approaching Variable-
Length Constrained Sequence Codes,” IEEE Trans. Commun., Accepted
Sept. 2018.
[4] Y. Ng, K. Cai, K. S. Chan, M. R. Elidrissi, M. Y. Lin, Z. Yuan, C .L.
Ong, and S. Ang, “Signal Processing for Dedicated Servo Recording
System,” IEEE Trans. Magn., vol. 51, no. 10, Oct. 2015.
[5] Z. M. Yuan, J. Shi, C. L. Ong, P. S. Alexopoulos, C. Du, A. Kong, S.
Ang, B. Santoso, S. H. Leong, K. S. Chan, Y. Ng, K. Cai, J. Tsai, H. Ng,
and H. K. Tan, “Dedicated Servo Recording System and Performance
Evaluation,” IEEE Transactions on Magnetics, vol. MAG-51, no. 4,
2015.
[6] K. Cai, K. A. S. Immink, M. Zhang, and R. Zhao, “Design of Spectrum
Shaping Codes for High-Density Data Storage,” Trans. on Consumer
Electronics, vol. CE-63, no. 4, pp. 477-482, Nov. 2017.
[7] K. A. S. Immink, “Spectral Null Codes,” IEEE Trans. Magn., vol. MAG-
26, no. 2, pp. 1130-1135, March 1990.
[8] K. Cai, A. Kong, and Z. Yuan, “Systems and methods for decoding using
run-length limited (RLL) codes,” US Patent 9,852,757, Dec. 2017.
[9] A. R. Ndjiongue, H. C. Ferreira, and T. M. N. Ngatched, Visible Light
Communications (VLC) Technology, Wiley Encyclopedia of Electrical
and Electronics Engineering, 2015.
[10] M. Oh, “A Flicker Mitigation Modulation Scheme for Visible Light
Communications”, 2013 15th International Conference on Advanced
Communications Technology (ICACT), PyeongChang, South Korea,
Jan. 2013.
[11] S. Rajagopal, R. D. Roberts, and S-K. Lim, “IEEE 802.15.7 Visible
Light Communication: Modulation Schemes and Dimming Support,”
IEEE Communications Magazine, vol. 50, no. 3, pp. 72-82, March 2012.
[12] Z. Wang, Q. Wang, W. Huang, and Z. Xu, Visible Light Communications:
Modulation and Signal Processing, Wiley-IEEE Press, Jan 2018.
[13] K. A. S. Immink and G. F. M. Beenker, “Binary Transmission Codes
with Higher Order Spectral Zeros at Zero Frequency,” IEEE Trans.
Inform. Theory, vol. IT-33, no. 3, pp. 452-454, May 1987.
[14] L. G. Tallini and B. Bose, “On Efficient High-Order Spectral-Null
Codes,” IEEE Trans. Inform. Theory, vol. IT-45, no. 7, pp. 2594-2601,
Nov. 1999.
[15] Y. Xin and I. J. Fair, “Algorithms to Enumerate Codewords for DC2-
Constrained Channels,” IEEE Trans. Inform. Theory, vol. IT-47, no. 7,
pp. 3020-3025, Nov. 2001.
[16] C. N. Yang, “Efficient Encoding Algorithm for Second-Order Spectral-
Null Codes Using Cyclic Bit Shift,” IEEE Transactions on Computers,
vol. 57, no. 7, pp. 876-888, July 2008.
[17] V. Skachek, T. Etzion, and R. M. Roth, “Efficient Encoding Algorithm
for Third-order Spectral-Null Codes,” IEEE Trans. Inform. Theory, vol.
IT-44, no. 2, pp. 846-851, March 1998.
[18] Y. Xin and I. J. Fair, “A Performance Metric for Codes with a High-
Order Spectral Null at Zero Frequency,” IEEE Trans. Inform. Theory,
vol. IT-50, no. 2, pp. 385-394, Feb. 2004.
[19] R. M. Roth, P. H. Siegel, and A. Vardy, “Higher-Order Spectral-Null
Codes: Constructions and Bounds,” IEEE Trans. Inform. Theory, vol.
IT-40, no. 6, pp. 1826-1840, Nov. 1994.
[20] L. G. Tallini, D. Pelusi, R. Mascella, L. Pezza, S. Elmougy, and B.
Bose, “Efficient Non-Recursive Design of Second-Order Spectral-Null
Codes,” IEEE Transactions on Inform. Theory,, vol. IT-62, no. 6, pp.
3084-3102, June 2016.
[21] K. A. S. Immink and K. Cai, “Estimated Spectra of Higher-Order
Spectral Null Codes,” IEEE Commun. Letters, vol. 23, no. 1, pp. 20-23,
Jan. 2019.
[22] B.S. Bosik, “The Spectral Density of a Coded Digital Signal,” Bell Syst.
Tech. J., vol. 51, pp. 921-932, April 1972.
[23] G. L. Pierobon, “Codes for Zero Spectral Density at Zero Frequency,”
IEEE Trans. Inform. Theory, vol. IT-30, no. 2, pp. 435-439, March 1984.
[24] J. Justesen, “Information Rates and Power Spectra of Digital Codes,”
IEEE Trans. Inform. Theory, vol. IT-28, no. 3, pp. 457-472, May 1982.
[25] C. M. Monti and G. L. Pierobon, “Codes with a Multiple Spectral Null
at Zero Frequency,” IEEE Trans. Inform. Theory, vol. IT-35, no. 2, pp.
463-472, March 1989.
[26] H. Prodinger, “On the Number of Partitions of {1,...,n}into Two
Sets of Equal Cardinalities and Equal Sums,” Canad. Math. Bull., vol.
25, no. 2, pp. 238-241, 1982.
[27] P. Flajolet and R. Sedgewick, Analytic Combinatorics, ISBN 978-0-521-
89806-5, Cambridge University Press, 2009.
[28] K. A. S. Immink and J. H. Weber, “Very Efficient Balanced Codes,”
IEEE Journal on Selected Areas of Communications, vol. 28, no. 2, pp.
188-192, 2010.
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 in-
novative start-up focused on novel signal processing
for DNA-based storage, where he currently holds
the position of president. Immink designed coding
techniques of digital video, audio, and data 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.
