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Abstract

We present an estimate of the power density function (spectrum) of binary K-th order spectral null codes. We work out the auto-correlation model in detail for second-and third-order spectral null codes. We compare the auto-correlation functions and spectra predicted by the model with those generated by full-set K-th order spectral null block codes.
1
Estimated Spectra of Higher-Order Spectral Null
Codes
Kees A. Schouhamer Immink, Fellow, IEEE, and Kui Cai, Senior Member, IEEE
Abstract—We present an estimate of the power density function
(spectrum) of binary K-th order spectral null codes. We work out
the auto-correlation model in detail for second- and third-order
spectral null codes. We compare the auto-correlation functions
and spectra predicted by the model with those generated by full-
set K-th order spectral null block codes.
I. INTRODUCTION
Traditionally, spectral shaping codes with a null at the
zero frequency, also called dc-balanced codes, have been
employed to counter the effects of the channel’s low-frequency
cut-off [1], [2]. Spectral shaping codes have been applied
extensively in magnetic recording products such as Digital
Video Recording (DV) and optical recording systems, such
as Compact Disc, DVD, Blu-ray disc [3], [4], where spectral
null codes are applied for avoiding the interaction between the
written data and the mechanical servo systems that are used for
tracking the data. Constructions of spectral null block codes
have received widespread attention in the literature, see for
example [5], [6], [7], [8], [9], [10].
Higher-order spectral null codes, introduced by Immink [11]
and Immink and Beenker [12], exhibit a spectral null at the
zero frequency, ω= 0, and also the derivatives of the power
density function or spectrum H(ω)at ω= 0 are zero. Higher-
order spectral zeros will result in a substantial decrease of
the power at low frequencies for a fixed code redundancy
as compared with the conventional designs on the bounded
digital sum concept. Higher-order spectral null block codes
also show attractive distance properties, which have been
studied in [12], [13]. A challenging combinatorics problem of
estimating the redundancy of codes that satisfy higher-order
spectral null constraints have been solved by, for example,
Freiman and Litsyn [14], and Roth, Siegel, and Vardy [10].
The computation of the spectrum of such codes is a prereq-
uisite for the evaluation of such code for practical application.
Spectral properties of higher-order spectral null codes have
been published for small values of the codeword length n,
but for larger values of nthe spectra computations become
prohibitively complex since each codeword in the codebook
has to be evaluated [9]. As a consequence, the shape of the
spectrum at the low-frequency end, is not known for asymp-
totically large values of n. The computation of the spectrum
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 Agency of Science and Technology
(A*Star) PSF research grant and Singapore Ministry of Education Academic
Research Fund Tier 2 MOE2016-T2-2-054.
of a code serves as a theoretical framework for analyzing
a spectrum shaping code, and hence the proposed spectrum
computation for large nwill provide valuable guidelines for
the design and analysis of less redundant spectrum shaping
codes. It has been a desideratum for a long time to bring
forward less computationally intensive methods for computing
the spectra for large values of n.
In this paper, we introduce a simple model for the auto-
correlation function of higher-order spectral null codes. The
model presented is able to estimate the spectra of extremely
large codebooks, where the prior art exhaustive method is
futile. We start, in Section II, with a description of the
properties of higher-order spectral null codes. Then, in Sec-
tion III, we propose a simple model for characterizing the
auto-correlation function of full-set higher-order spectral null
codes. In Section IV, we compare the new estimate of the auto-
correlation function and spectrum with those of full-set higher
order spectral null codes. Section V concludes the paper.
II. SP EC TR AL NU LL BL OC K COD ES
We consider a communication codebook, S, of chosen
bipolar n-bit codewords x= (x1, x2, . . . , xn)over the bipolar
alphabet Q={−1,1}, where n, the length of x, is a positive
integer. The auto-correlation function, ρx(i)of the codeword
xSis defined by
ρx(i) = 1
n
ni
j=1
xjxj+i,0in1.(1)
It is assumed that the encoder translates source words into
codewords taken from S, which, in turn, are concatenated
to form an infinite sequence of symbols. An infinitely long
sequence of symbols generated by cascading randomly and
independently (i.i.d.) generated codewords from the codebook
Sis a cyclo-stationary process of period n[11]. The auto-
correlation function, ρ(i), of a plurality of codewords, which
are randomly drawn from the codebook, S, to form an infinite
sequence of symbols equals the average of the auto-correlation
functions of the individual codewords. Or
ρ(i) = 1
|S|
xS
ρx(i),(2)
where |S|denotes the cardinality of S. Clearly ρ(0) = 1, and
since the codewords are assumed to be taken randomly we
have ρ(i) = 0, i n. The power spectral density (psd) versus
frequency ω, in short spectrum, of the infinite sequence is
given by
H(ω) = 1 + 2
n1
i=1
ρ(i) cos(),(3)
2
where it is assumed that the codewords are equiprobable
and that both xand xare members of S, so that the
spectrum is continuous and spectral lines are absent [11].
We are specifically interested in the spectrum near the zero
frequency. The Taylor expansion of the spectrum H(ω)at the
low-frequency end, ω= 0, is
H(ω) = 1+2
n1
i=1
ρ(i)
n1
i=1
i2ρ(i)ω2+1
12
n1
i=1
i4ρ(i)ω4−··· .
(4)
Spectral null codes of order Kare characterized by the
property that
H(0) = H(0)(2)(0) = · ·· =H(0)(2K2)(0) = 0,(5)
where H(i)(0) denotes the i-th derivative of H(ω)at ω= 0.
There are various ways of implementing higher-order spectral
null codes. Immink and Beenker [12] defined a block code
comprising a set of K-th order balanced codewords, SK, by
SK={x∈ Qn:w1(x) = w2(x) = ·· · =wK(x) = 0},
(6)
where the codeword moments,wl(x),l= 1, . . . , K, of the
vector xare defined by
wl(x) =
n
i=1
il1xi.(7)
The spectrum of a block code that takes codewords from
SKexhibits a K-th order spectral null [12]. For a practical
code, the number of codewords is a power of two, and the
code designer must select the ‘best’ codewords available, and
usually, the designer evaluates a spectral performance metric
of candidate codewords [9].
We consider the spectral properties of full-set block codes,
that is, SKdenotes the set of all possible words, x, that satisfy
condition (6). For the full-set case K= 1, simple expressions
for the auto-correlation function and spectrum were found
by Franklin and Pierce [2]. For K > 1, finding a simple
expression of the spectral properties of a full-set SKis an
open problem. With prior art methods, the computation of the
spectrum requires the evaluation of (1) for each xSK. As
the size of SKgrows exponentially with increasing n[10],
the computation of the spectrum of SKbecomes prohibitive
for large values of n. Using the theory, Theorem 2, developed
in [9], we may slightly reduce the computational effort, but its
application also requires the evaluation of each codeword in
SK. The modification of the enumeration method shown for
K= 2 [15] is prohibitive for large nand K.
In the next section, we formulate an estimate of the spectral
properties of SKthat does not require the brute force evalua-
tion (3).
III. SPE CT RA O F HIGHER-OR DE R SPECTRAL NULL CODES
We propose to model the auto-correlation function, ρK(i),
of a K-th order spectral null code as the K-th order polyno-
mial
ρK(i) =
1, i = 0,
φK(i),0< i n1,
0, i n,
(8)
where
φK(i) = β(iK1+a0iK2+· ·· +aK2)(in).(9)
The coefficients βand ai,0iK2, of the polynomial
φK(i)are real numbers to be determined below. The factor
inis applied since we have the condition ρ(n) = 0 for
a block code whose codewords are drawn independently. The
Kunknown parameters, βand ai,0iK2, are found
by satisfying the Kconditions (5). Thus from (4) we obtain
1+2
n1
i=1
ρK(i) = 0,(10)
and n1
i=1
i2kρK(i)=0, k = 1, . . . , K 1.(11)
The above Kconditions define a system of linear equations
in the unknown βand ai,0iK2, parameters.
A useful metric of the low-frequency spectral content,
denoted by χK, is the first non-zero coefficient of the Taylor
expansion (4), that is,
HK(ω)χKω2K, ω 1.(12)
The metric, χK, has been coined Low Frequency Spectral
Weight (LFSW) by Xin and Fair [9]. We have
χK= (1)K2
(2K)!
n1
i=1
i2KρK(i).(13)
In the next subsections, we take a look at the spectral
properties for K= 1,2and 3.
A. First-order Spectral Null, K= 1
We simply obtain from (9)
φ1(i) = 1
n(n1)(in),(14)
H1(ω) = n
n11sin
2
nsin ω
22,(15)
and
χ1=n(n+ 1)
12 ,(16)
which agrees with results in [9].
B. Second-order Spectral Null, K= 2
We have from (9)
φ2(i) = β(i+a0)(in).(17)
The parameters βand a0are found by solving
1 + 2β
n1
i=1
(i+a0)(in) = 0
and n1
i=1
i2(i+a0)(in) = 0.
3
We obtain
a0=3n22
5n
and
β=15
(n1)(n2)(4n+ 3).
After working out (13) for K= 2, we obtain
χ2=n(n+ 1)(n+ 2)(4n21)
840(4n+ 3) n4
840, n 1.(18)
C. Third-order Spectral Null, K= 3
For the case K= 3, we derive from (9)
φ3(i) = β(i2+a0i+a1)(in),(19)
where
a0=n(19n231)
4(4n21) ,
a1=5(n21)(n22)
4(4n21) ,
and
β=12(4n21)
n(4n+ 5)(n21)(n2)(n3).
For n > 2, we can simply verify that a2
04a1>0so that
i2+a0i+a1has two real roots, denoted by β0,1, so that
φ3(i) = β(iβ0)(iβ1)(in),1in1.(20)
For large n, we simply have
β019 41
32 n, β119 + 41
32 n, n 1.
After working out (13) for K= 3, we obtain
χ3=2
6!
n1
i=1
i6ρ3(i)(21)
=(n1)n(n+ 1)(n+ 2)(n+ 3)(4n21)
151200(4n+ 5)
and
χ3n6
151200, n 1.(22)
In the next section, we compute the spectra of higher-order
spectral null codes and compare them to those of block codes.
IV. COM PU TATIO N AN D CO MPARISON OF SP EC TR A
For K= 1, the codewords of a balanced code, S1, have
a fixed number of 1’s and 1s. The code size is |S1|=
n/2
n. We find from the literature [2] that the auto-correlation
function of our model and that of a zero-disparity code are
perfectly in line, and further study is not required.
Using the theory developed, we have conducted computer
experiments for various values of nand K > 1. The spectrum
and auto-correlation function, denoted by ˆ
HK(ω)and ˆρK(i),
of the full-set block codes, SK, have been computed by
enumerating all allowed codewords and applying (1) and (3).
Figure 1 shows the auto-correlation function, ρ2(i), of the
model (17), and ˆρ2(i), of the actual block codes, S2, for
0 0.2 0.4 0.6 0.8 1
i/n
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
Correlation function
n=16
n=128
n=32
n=64
model
model
model
model
Fig. 1. Auto-correlation function ρ2(i)(model) and ˆρ2(i)(full set)
versus i/n for n= 16,32,64 and 128.
10-3 10-2 10-1
Frequency f (log)
-40
-35
-30
-25
-20
-15
-10
-5
0
5
Spectrum (dB)
n=64
n=32
n=128
model
n=16
model
model
model
Fig. 2. Power density function H2(f)and ˆ
H2(f)versus frequency
f=ω/2πfor n= 16,32,64, and 128.
n= 16,32,64, and 128. For n32 we applied an exhaustive
search for computing (2) of S2. For n > 32, we applied a mod-
ified enumeration method introduced in [15]. For n= 16, we
notice a raggedness of the correlation function, ˆρ2(i), which all
but disappears for the larger values of nshown. Figure 2 shows
the corresponding power density functions, ˆ
H2(ω)and H2(ω),
for the same values of n. The spectrum of the model and the
actual block codes show a good agreement. The raggedness of
the auto-correlation function for n= 16 is not visible in the
corresponding smooth spectrum. Figure 3 shows the quotient
10 log10 ˆ
H2(ω)/H2(ω)(dB) versus frequency f=ω/2πfor
K= 2 and n= 16,32,64 and 128. The maximum deviation
is less than 0.7 dB for n= 16 and 0.3 dB for n= 128, and
overall diminishing with increasing n.
Figure 4 shows the auto-correlation function, ρ3(i)(model)
and ˆρ3(i)(full-set S3) for n= 24 and n= 40. We begin to run
out of steam for values of the codeword length nexceeding
forty or so as the searching of the allowed codewords is
prohibitively complex. The calculation took around nine hours
for n= 40 using an everyday PC, and for the next larger
case, n= 44, we may expect weeks of computation time. The
4
10-2 10-1
Frequency f (log)
-1
-0.5
0
0.5
Deviation (dB)
n=32
n=64
n=128
n=16
Fig. 3. Quotient of the power density functions,
10 log10 ˆ
H2(ω)/H2(ω)(dB), versus frequency f=ω/2πfor
n= 16,32, 64, and 164.
0 0.2 0.4 0.6 0.8 1
i/n
-0.15
-0.1
-0.05
0
0.05
Correlation function
n=40
n=24
Fig. 4. Auto-correlation functions, ρ3(i)(smooth) and ˆρ3(i)(ragged),
versus i/n for n= 24 and n= 40.
raggedness of the auto-correlation function, ˆρ3(i), of the block
codes is quite pronounced. The corresponding spectra, ˆ
H3(ω),
shown in Figure 5, however, is quite smooth, and similar to
the spectrum, H3(ω), predicted by the model.
V. CONCLUSIONS
We have proposed a simple model of the auto-correlation of
full-set K-th order spectral null codes. We have evaluated the
auto-correlation functions and spectra predicted by the model
and compared them with those generated by full-size higher-
order spectral null block codes. For second-order spectral null
codes, K= 2, we have noticed for larger values of the
codeword length na good agreement between the spectra and
the auto-correlation function of the model and the actual full-
size block code. For K= 3, the auto-correlation function
of the full-set block codes is ragged for all values of nthat
we have evaluated. The corresponding spectra in the range
investigated are in good agreement with the spectra predicted
by the model.
10-3 10-2 10-1
Frequency f (log)
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Spectrum (dB)
n=40
n=24
model
Fig. 5. Power density function H3(f)and ˆ
H3(f)versus frequency
f=ω/2πfor n= 24 and n= 40.
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