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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

x∈Sis deﬁned by

ρx(i) = 1

n

n−i

j=1

xjxj+i,0≤i≤n−1.(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|

x∈S

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

n−1

i=1

ρ(i) cos(iω),(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

n−1

i=1

ρ(i)−

n−1

i=1

i2ρ(i)ω2+1

12

n−1

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)(2K−2)(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

il−1xi.(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 x∈SK. 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 ≤n−1,

0, i ≥n,

(8)

where

φK(i) = β(iK−1+a0iK−2+· ·· +aK−2)(i−n).(9)

The coefﬁcients βand ai,0≤i≤K−2, of the polynomial

φK(i)are real numbers to be determined below. The factor

i−nis applied since we have the condition ρ(n) = 0 for

a block code whose codewords are drawn independently. The

Kunknown parameters, βand ai,0≤i≤K−2, are found

by satisfying the Kconditions (5). Thus from (4) we obtain

1+2

n−1

i=1

ρK(i) = 0,(10)

and n−1

i=1

i2kρK(i)=0, k = 1, . . . , K −1.(11)

The above Kconditions deﬁne a system of linear equations

in the unknown βand ai,0≤i≤K−2, 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)!

n−1

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(n−1)(i−n),(14)

H1(ω) = n

n−11−sin nω

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)(i−n).(17)

The parameters βand a0are found by solving

1 + 2β

n−1

i=1

(i+a0)(i−n) = 0

and n−1

i=1

i2(i+a0)(i−n) = 0.

3

We obtain

a0=−3n2−2

5n

and

β=−15

(n−1)(n−2)(4n+ 3).

After working out (13) for K= 2, we obtain

χ2=n(n+ 1)(n+ 2)(4n2−1)

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)(i−n),(19)

where

a0=−n(19n2−31)

4(4n2−1) ,

a1=5(n2−1)(n2−2)

4(4n2−1) ,

and

β=12(4n2−1)

n(4n+ 5)(n2−1)(n−2)(n−3).

For n > 2, we can simply verify that a2

0−4a1>0so that

i2+a0i+a1has two real roots, denoted by β0,1, so that

φ3(i) = β(i−β0)(i−β1)(i−n),1≤i≤n−1.(20)

For large n, we simply have

β0≈19 −√41

32 n, β1≈19 + √41

32 n, n ≫1.

After working out (13) for K= 3, we obtain

χ3=−2

6!

n−1

i=1

i6ρ3(i)(21)

=(n−1)n(n+ 1)(n+ 2)(n+ 3)(4n2−1)

151200(4n+ 5)

and

χ3∼n6

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 −1’s. 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 n≤32 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.

REFERENCES

[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, pp. 1182-1184, Dec.

1972.

[3] 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.

[4] 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, pp. 477-482, Nov. 2017.

[5] 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.

[6] 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.

[7] 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.

[8] V. Skachek, T. Etzion, and R. M. Roth, “Efﬁcient Encoding Algorithm

for Third-order Spectral-Null Codes,” IEEE Trans. Inform. Theory, vol.

IT-44, pp. 846-851, March 1998.

[9] 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.

[10] R. M. Roth, P. H. Siegel, and A. Vardy, “Higher-Order Spectral-Null

Codes: Constructions and Bounds,” IEEE Trans. Inform. Theory, vol.

IT-40, pp. 1826-1840, Nov. 1994.

[11] K. A. S. Immink, Codes for Mass Data Storage Systems, Second Edi-

tion, ISBN 90-74249-27-2, Shannon Foundation Publishers, Eindhoven,

Netherlands, 2004.

[12] 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.

[13] R. Karabed and P. H. Siegel, “Matched Spectral-Null Codes for Partial-

Response Channels,” IEEE Trans. Inform. Theory, vol. IT-37, no. 3, pt.

II, pp. 818-855, May 1991.

[14] G. Freiman and S. Litsyn, “Asymptotically Exact Bounds on the Size

of High-Order Spectral-Null Codes,” IEEE Trans. Inform. Theory, vol.

IT-45, no. 6, pp. 1798-1807, Sept. 1999.

[15] K. A. S. Immink, “Spectrum Shaping with Binary DC2-constrained

Channel Codes,” Philips J. Res., vol. 40, pp. 40-53, 1985.