Abstract and Figures

We investigate a new approach for designing spectral shaping block codes with a target spectrum, H_t(f), that has been specified at a plurality of frequencies. We analyze the probability density function of the spectral power density function of uncoded n-symbol bipolar code words. We present estimates of the redundancy and the spectrum of spectral shaping codes with specified target spectral densities H_t(f_i) at frequencies f_i. Constructions of low-redundancy codes with suppressed low-frequency content are presented that compare favorably with conventional dc-balanced codes currently used in data transmission and data storage devices with applications in consumer electronics.
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Spectral Shaping Codes
Kees A. Schouhamer Immink, Life Fellow, IEEE, and Kui Cai, Senior Member, IEEE
Abstract—We investigate a new approach for designing spec-
tral shaping block codes with a target spectrum, Ht(f), that
has been specified at a plurality of frequencies. We analyze
the probability density function of the spectral power density
function of uncoded n-symbol bipolar codewords. We present
estimates of the redundancy and the spectrum of spectral
shaping codes with specified target spectral densities Ht(fi)
at frequencies fi. Constructions of low-redundancy codes with
suppressed low-frequency content are presented that compare
favorably with conventional dc-balanced codes currently used in
data transmission and data storage devices with applications in
consumer electronics.
Keywordsspectral shaping block codes, spectral power
density function, dc-free codes, spectral notch codes.
I. INTRODUCTION
Spectral shaping codes, specifically codes with spectral
nulls, have been applied extensively in digital audio and
video magnetic tape as well as the optical disc recording
systems [1, 2, 3], which are main constituents of consumer
electronics products. In recent years, codes with spectral nulls
at non-zero frequencies have also been proposed for the dedi-
cated servo recording systems [4, 5, 6] in data storage systems
for digital consumer electronics products that are designed for
the ultra-mobile hard-disk drive (HDD) for tablets.
Our paper aims at the system designer who specifies the tar-
get spectrum, denoted by Ht(f), where fis the frequency, at
m,m1,spectral points fi,Ht(fi) = di,1im, where
di,di0, is the desired power spectral density. To realize
the proposed coding scheme, a major technical issue is how to
minimize the coding redundancy introduced into the encoded
sequence while satisfying the desired spectrum. We have
developed theoretical tools for computing the redundancy of
spectral shaping codes and for estimating the spectrum of the
encoded sequence. Our designed low-frequency suppression
codes outperform the prior art dc-balanced codes with lower
redundancy and better spectral shaping. The redundancy of
spectral shaping codes currently used in consumer electronics
devices can be as high as 25 % in the 8B10B code [7] or
10-15 % in the EFM and EFMPlus codes [8]. A lower code
redundancy is welcome as it has an immediate effect on the
device’s capacity, leading to a higher playing time and higher
data throughput.
We start in Section II with relevant background and defini-
tions of spectra and auto-correlation functions of block coded
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),
Science, Mathematics and Technology Cluster, 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 MOE2019-T2-2-123.
signals. In Section III, we analyze the probability density dis-
tribution of the spectral density function and auto-correlation
function of n-bit bipolar sequences. In Sections IV and V,
we present constructive approaches for designing memoryless
block codes having a desired spectrum. In Section V-A, we
briefly review relevant attributes of prior art spectral null
codes, and compare their spectral performance with newly
developed low-frequency rejecting codes. Estimates of the
spectrum are presented in Section VI. Section VII shows our
conclusions.
II. BACKGROU ND ,DEFINITIONS
Let the n-bit codeword x= (x1, x2, . . . , xn)over the
bipolar symbol alphabet Q={−1,1}be a member of a
codebook S ⊆ Qn. The code’s redundancy, denoted by r,
is defined by
r=nlog2|S|,(1)
where |S| denotes the cardinality of S. The encoder emits
codewords from Srandomly and independently (i.i.d.).
We define the aperiodic auto-correlation function,ρx=
(ρx,1, ρx,2, . . . , ρx,n1), of the codeword xby
ρx,i =1
n
ni
X
j=1
xjxj+i,1in1.(2)
It is immediate that ρx=ρx, where x= (x1,x2,...,
xn), denotes the inverse of x, and ρx=ρxr, where xr=
(xn, xn1, . . . , x1)is the mirror version of x. The spectrum
of the codeword xis defined by
Hx(f) = 1 + 2
n1
X
i=1
ρx,i cos(i2πf ),(3)
where fdenotes the frequency. Alternatively, we may write [9]
Hx(f) = 1
n
n
X
i=1
xieji2πf
2
,(4)
where j = 1. The auto-correlation function, ρ=
(ρ1, . . . , ρn1), of the cyclo-stationary process formed by a
sequence of symbols of codewords drawn at random from Sis
the average of the auto-correlation functions of the codewords
xin S, that is [10]
ρi=1
|S| X
x∈S
ρx,i,1in1.(5)
If both xand its inverse, x, are members of S, and the
words are sent with equal probabilities, then the spectrum of
the emitted symbol sequence is a continuous function [11] and
given by the average of the individual spectra, Hx(f), or
H(f) = 1 + 2
n1
X
i=1
ρicos(i2πf ) = 1
|S| X
x∈S
Hx(f).(6)
III. PROBABILITY DENSITY FUNCTION OF THE SPECTRAL
CONTENT
As our aim is to construct a code set Swhose codewords
have specified spectral content at specified frequencies, we
must, in order to compute the size |S| and redundancy r,
estimate the number of codewords x∈ Qnwith a given power
spectral density at given frequencies.
If we see the generation of a word x∈ Qnas a random
event, then, as (6) assigns to each xa value to Hx(f)
THx(f), we conclude that for a specified frequency f, the
spectral density Hx(f)is a random variable with a discrete
probability density function that depends on both fand n.
Let the probability density function of Hx(f)be denoted by
P r(Hx(f) = ζ).
Clearly, the number of distinct values of Hx(f)for all x
Qn, denoted by |THx(f)|, is at most equal to the number of
distinct non-periodic auto-correlation functions ρx. Since the
codewords xrand x∈ Qnhave the same auto-correlation
function as x, we expect that the number of distinct auto-
correlation functions is around 2n2. An exact number was
formulated by Whitehead [12], who enumerated the number
of distinct non-periodic auto-correlation functions, ρx, of n-bit
bipolar sequences. For specific frequencies, f=k/n,kis an
integer, |THx(f=k/n)|is much less than 2n2. For example,
for the null (f= 0) and the Nyquist (f= 1/2) frequency we
may easily verify that |THx(0)|=|THx(1/2)|=d(n+ 1)/2e.
In general, it is hard to write down such an expression, let
alone to find the probability density function P r(Hx(f) =
ζ), and in most cases we must therefore rely on computer
searches.
Although, in general, we do not have exact knowledge of
the probability density function, P r(Hx(f) = ζ), we are able
to formulate simple expressions for its main parameters. In the
next subsections, we analyze the mean and variance of Hx(f)
for all fand n, and the correlation coefficient between the
probability density functions Hx(f=f0)and Hx(f=f1),
where f0and f1are two distinct frequencies 0f0, f10.5.
A. Principal parameters of the probability density function
P r(Hx(f) = ζ)
Let E[a(x)] denote the expected value of the variable a(x)
of all possible x∈ Qn. For symmetry reasons, it is immediate
that, see (2), that E[ρx,i]=0,1in1.Using (3) and
(6), we infer that E[Hx(f)] = 1, which is the well-known
result that the mean of the spectral content is independent of
the frequency f(white noise). Below we study the variance
of the spectral power density, which, as we show, does depend
on the frequency.
The (unnormalized) auto-correlation coefficient, x,i, is
the sum of the nibipolar variables x1xi+1,x2xi+2, . . .,
xnixn, see (2), so that x,i ∈ {−n+i, n+i+2, . . . , ni}.
For j∈ {−n+i, n+i+ 2, . . . , n i}we have
P r ρx,i =j
n= 2inni
ni+j
2.(7)
For 1i, j n1, we obtain from (2) and the above that
E[ρx,iρx,j ] = ni
n2, i =j,
0, i 6=j. (8)
The variance of the spectral power density Hx(f), denoted by
σ2
f, is found using (3) and (8), namely
σ2
f=E[(Hx(f)1)2]
= 4
n1
X
i=1
Eh(ρx,i cos(i2πf ))2i
=4
n2
n1
X
i=1
(ni) cos2(i2πf )
=2
n2
n1
X
i=1
(ni){1 + cos(i4πf )}(9)
=n1
n+2
n2
n1
X
i=1
(ni) cos(i4πf ).(10)
(11)
After substituting the trigonometric identity
n1
X
i=1
(ni) cos(i2πf ) = n
2+n2
2F(f),(12)
where
F(f) = sin πnf
nsin πf 2
,(13)
we obtain
σ2
f= 1 2
n+F(2f).(14)
In case fis a multiple of half the codeword frequency, f=
i/(2n),1in1, we simply find F(f=i/(2n)) = 0,
so that
σ2
f=i/(2n)= 1 2
n.(15)
For f= 0 and f= 1/2, we obtain
σ2
f=0 =σ2
f=1/2= 2 11
n.(16)
Figure 1 depicts σ2
fversus ffor n= 16 and 32. Note that for
the major range of f, we have |F(2f)|< n2, so that, except
for fnear 0 or 1/2, we have
σ2
f12
n.
The covariance of the spectral power density at frequency
f0and f1, denoted by C(f0, f1), is defined by
C(f0, f1) = E[(Hx(f0)1)(Hx(f1)1)].(17)
We find, using (8) and (12), that
C(f0, f1)=4
n1
X
i=1
E[ρ2
x,i] cos(i2πf0) cos(i2πf1)
=2
n2
n1
X
i=1
(ni){cos(2πi(f0f1)) +
+ cos(2πi(f0+f1))}
=2
n2G(f0, f1),(18)
0 0.05 0.1 0.15 0.2 0.25
Frequency f
0.8
1
1.2
1.4
1.6
1.8
2
2
f
n=16
n=32
Fig. 1. Variance of the spectral power density Hx(f),σ2
f, versus
frequency, f, for a word length n= 16 and 32.
0 0.1 0.2 0.3 0.4 0.5
Frequency f
-0.2
0
0.2
0.4
0.6
0.8
1
H(f0,f1)
n=16
n=32
Fig. 2. Pearson correlation coefficient, ρHx(1/4,f1), between the
spectral power densities Hx(1/4) and Hx(f1)versus f1for a word
length n= 16 and n= 32.
where
G(f0, f1) = n+n2
2{F(f0f1) + F(f0+f1)}.(19)
The Pearson correlation coefficient [13] between the spectral
power densities at frequency f0and f1, denoted by ρHx(f0,f1),
is given by
ρHx(f0,f1)=C(f0, f1)
σf0σf1
.(20)
Figure 2 shows an illustrative example of the correlation
coefficient ρHx(1/4,f1)versus f1for n= 16 and n= 32.
We infer that if the difference between the two frequencies,
f0and f1, is small, or |f0f1|<1/n, that the correlation
coefficient is close to unity. For a larger frequency difference,
we have ρHx(f0,f1)≈ −2/n,|f0f1|  1/n.
The principal parameters of the distribution P r(Hx(f) = ζ)
have been analyzed above. A general expression for the
distribution, however, for all fand ncould not be found.
For our analysis of the redundancy of spectral shaping codes,
it is mandatory to have a simple and reliable model for
P r(Hx(f) = ζ)as it describes the number of sequences of
given spectral density.
B. Exponential distribution function
Rice [14] found that for asymptotically large nthe probabil-
ity density function, P r(Hx(f) = ζ), is independent of fand
given by the (continuous) exponential distribution function (a
χ2-distribution with two degrees of freedom [13])
P r(Hx(f) = ζ) = λeλζ , ζ 0,
0, ζ < 0,(21)
where λis a positive real constant, often called the rate
parameter. Rice’s findings are based on (4), which can be
written as
Hx(f) = 1
n
n
X
i=1
xisin(i2πf )!2
+ n
X
i=1
xicos(i2πf )!2
.
(22)
For asymptotically large n, the distribution of the sum of the
(co)sine’s converges to a Gaussian distribution according to
the Central Limit Theorem [13]. The sum of the squares of
two stochastic variables with a Gaussian distribution has a
χ2-distribution. Clearly, the χ2-distribution used is an approx-
imation for asymptotically large n, so that a validation of the
accuracy is needed for smaller n.
According to the (assumed) χ2- or exponential distribution
of the spectral density (21), we obtain for the mean, µ0, and
variance, σ2
0,
µ0=1
λand σ2
0=1
λ2.(23)
Note that both mean, µ0, and variance, σ2
0, are independent
of f. We derive, however, in Section III that the mean and
variance of the spectral power density are
E[Hx(f)] = 1
and
EHx(f)1)2=σ2
f= 1 2
n+F(2f),
so that, clearly, see (23), the assumption on the expo-
nential distribution is false for finite nas E[Hx(f)]26=
E(Hx(f)1)2. In the next subsection, we report on ex-
perimental evaluations of P r(Hx(f) = ζ)for finite n.
C. Experimental studies regarding the spectral content distri-
bution
A sound way to verify the (in)validity of the χ2- or
exponential distribution, P r(Hx(f) = ζ), for finite n, is the
measurement of the distribution. For various values of foand
n, we have studied myriad histograms of Hx(f=fo), where
we count the number of observations of Hx(fo)that fall into
each of the Nbin disjoint bins after randomly generating a
large number of codewords x. The ith bin ranges from ibin
to (i+ 1)bin,0iNbin , where bin, called bin width,
is a positive real parameter conveniently chosen to cover
the Hx(fo)histogram (horizontal) axis with Nbin bins. In
our experiments, we generate Non-bit codewords, compute
Hx(fo)for each codeword, and increase the jth bin, where
j=bHx(fo)/binc. Let the number of observations in the ith
012345678
H(fo)
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
histogram w
Fig. 3. Histogram of wiversus Hx(fo),n= 64,fo= 7/n, and
bin = 0.03, based on 107observations. The straight line shown is
found using linear regression analysis. The regression estimate equals
ˆ
λ= 1.0021 for the histogram shown.
bin be denoted by Ni,PNi=No, then, we normalize Ni
and define wiby
wi= ln Ni
binNo
,0iNbin,(24)
so that an exponential curve turns into a straight line.
Figure 3 shows a plot of the histogram wiversus Hx(fo)
for bin = 0.03,n= 100 and fo= 7/n. Using standard
linear regression analysis [13], we obtain the least squares
estimate, denoted by ˆ
λ, of the rate parameter λ, using the
pairs (i+1
2bin, wi)for the Nbin bins. An example of the
straight regression line obtained is shown in Figure 3. The
regression estimate equals ˆ
λ= 1.0021. We have evaluated the
distribution at various frequencies and codeword lengths, and
found that λis sufficiently close to unity for our engineering
applications. In the next section, we study the focal point of
our paper, the feasibility of spectral shaping by a judicious
choice of the codebook S.
IV. SEL EC TI NG T HE C OD EW OR DS
Let the target spectrum, denoted by Ht(f), be specified at
mspectral points, Ht(fi) = di,1im. We compute the
spectrum, Hx(f), of each possible n-bit word x∈ Qn, and
select those x’s for which each mspecified Hx(fi)s lie within
a judiciously chosen acceptance interval [p1, p2]i,1im.
In case Qnis too large for generating all the n-bit words, we
randomly draw words from Qn.
A. Selection criterion
Let the acceptance interval length be independent of di, then
S={x:|diHx(fi)|< δ, 1im},(25)
where the (real) parameter, δ,0< δ 1, called tolerance
level, is a designer’s choice. We assume that the differences
between the spectral points, fi, are at least 1/n, so that the
dependence between the Hx(fi)’s can be neglected. Then,
0 0.5 1 1.5 2
d
2.5
3
3.5
4
4.5
5
5.5
redundancy
measured
computed
Fig. 4. Redundancy ˆr1(d), computed using (27), and redundancy,
r1(d), measured using exhaustive search for a single spectral point,
m= 1, at f= 9/n versus d1=d, for n= 32 and δ= 0.1.
following (21), where for convenience we choose λ= 1, the
number of accepted codewords, |ˆ
S|, can be estimated by
|ˆ
S| = 2nZd1+δ
x=max(0,d1δ)
exdx ·· ·Zdm+δ
x=max(0,dmδ)
exdx,
(26)
so that, for di> δ, the estimated redundancy, denoted by ˆr1,
see (1), equals
ˆr1=nlog2|ˆ
S| =mlog2(eδeδ)+log2(e)
m
X
i=1
di,(27)
which shows a simple linear relationship between the target
spectral content, di, and redundancy ˆr1. Figure 4 displays
for m= 1 the estimated, ˆr1, using (27), and the measured
redundancy, r1, using selection criterion (25), versus d=d1
for n= 32,f1= 9/n, and δ= 0.1. We notice a nice agree-
ment between the two curves, less than a percent difference,
which supports Rice’s exponential distribution approximation
presented in Subsection III-B.
It is unsatisfactory, see (27) and Figure 4, that with selection
criterion (25) for the target power spectral density di= 1 a
redundancy is required of around 3.7 bits (δ= 0.1), as, clearly,
di= 1 is the average power density of uncoded sequences, and
redundancy is thus a waste. Secondly, the redundancy depends
on an arbitrarily chosen value of the parameter δ. Therefore,
we propose an alternative criterion without such flaws.
B. Alternative selection criterion
We present an alternative selection criterion, where the
acceptance interval depends on the desired spectral power
density, di, in such a way that both the average spectral density
in the interval [p1, p2]equals the target value di, that is,
I1(p1, p2) = Rp2
x=p1xexdx
Rp2
x=p1exdx =di,(28)
and the redundancy
I2(p1, p2) = log2Zp2
x=p1
exdx (29)
is minimized. Minimizing the redundancy I2(p1, p2)is accom-
plished by choosing p1= 0 or p2=. Then we have either
0I1(0, p2)1or 1I1(p1,)1 + p1. In case di1,
we select the acceptance interval p1,], such that
I1(p1= ˆp1,) = eˆp1(1 + ˆp1)
eˆp1=di,
or simply
ˆp1=di1.(30)
In case di<1, we choose the acceptance interval [0,ˆp2], such
that ˆp2satisfies the equation
I1(0, p2= ˆp2) = 1eˆp2(1 + ˆp2)
1eˆp2=di.(31)
We may find ˆp2by numerically solving (31). In summary, we
accept a word xif its spectral contents Hx(fi),i= 1, . . . , m,
are within the interval
[p1, p2]i=[di1,], di1,
[0,ˆp2], di<1,(32)
where ˆp2satisfies (31). Clearly, for di= 1, we have
[p1, p2] = [0,]and may accept all available words, so that
the redundancy is nil. The estimated redundancy, denoted by
ˆr2(di), for a single spectral point at the spectral level diequals,
see (29),
ˆr2(di) = log2eˆp1= (di1) log2e, di1,
log2(1 eˆp2), di<1.(33)
The accumulated redundancy for mspectral points at level di,
1,im, assuming independence, denoted by ˆr2, equals
ˆr2=
m
X
i=1
ˆr2(di).(34)
Clearly, criterion (32) leans heavily on the validity of the
exponential distribution premise (21), and a further validation
by measurement is necessary. Figure 5 shows results for a
single spectral point, m= 1, the redundancy ˆr2, computed
using (33) and r2, measured, versus spectral density d=d1
with the parameters n= 32 and f1= 9/n. The diagram
shows a good agreement between the computed and measured
redundancy.
Our designed spectrum shaping codes can be easily im-
plemented on various consumer electronics devices, such as
magnetic tape and other data storage products such as ultra-
mobile HDDs [3, 4, 5, 6]. The above described code designs
and constructions are all done off-line, and once the codes are
designed, the encoder and decoder can be implemented based
on simple look-up tables. Therefore, our codes are suitable for
efficient hardware implementations in practical devices.
V. SPECTRUM SHAPING
As a first illustration of the sought for spectral shaping
effect, Figure 6 shows the power density function, H(f),
versus frequency ffor two cases. The first plot, Curve (a)
shows the case f1= 1.5/n,d1= 1.5and Curve (b) shows
the case f1= 2.5/n,d1= 0.1. For both spectra, we have
n= 32 and m= 1. Curve (a) shows a peak d1= 1.5at
f1= 1.5/n, while Curve (b) shows a notch of depth d= 0.1
0 0.5 1 1.5 2
d
0
0.5
1
1.5
2
2.5
redundancy
computed
measured
Fig. 5. Computed redundancy, ˆr2(d), using (33), and measured
redundancy, r2(d), versus dfor n= 32 and f1= 9/n.
0 0.05 0.1 0.15
Frequency f
0
0.5
1
1.5
Spectrum H(f)
(b)
(a)
Fig. 6. Power density function, H(f), versus frequency fof two
cases: (a) f1= 1.5/n,d1= 1.5and the second case (b) f1= 2.5/n,
d1= 0.1,n= 32,m= 1.
at f1= 2.5/n. The measured redundancy equals r2= 0.72
for the first case and r2= 2.4for the second case.
As a second illustration, Figure 7, Curve ’measured’, shows,
for n= 32,m= 3 spectral points at f1= 2/n,f2= 3/n, and
f3= 4/n. The target spectral density of the mspectral points
is fixed, di= 0.1,1i3, so that the target spectrum
is shaped as the response of a band stop filter. The second
curve, Curve ’computed’, shows a computed spectrum using
the theory developed in Section VI, and we conclude that
the measured spectrum is in concordance with the computed
spectrum. The estimated redundancy ˆr2, see (34), equals 7.24,
and the measured redundancy equals r2= 7.63, which is close
to the estimated one.
The ringing artifacts seen in the above diagram are due
to the well-known Gibbs phenomenon. The ringing can be
reduced by taking a larger value of mand closer spaced
spectral points fi. Figure 8 shows for the same parameters
as above a band stop with m= 5 spectral points at f1= 2/n,
f2= 2.5/n,f3= 3/n,f4= 3.5/n, and f5= 4/n,di= 0.1,
i= 1,...,5. The choice for a smoother spectrum has a
0 0.1 0.2 0.3 0.4 0.5
Frequency f
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Spectrum H(f)
measured
computed
Fig. 7. Power density function, H(f), versus frequency fof (label:
measured) selected codewords of length n= 32 with spectral points
at fi= 2/n, 3/n, 4/n,di= 0.1,i= 1,2,3, and (label: computed)
spectrum for the same parameters computed using (44) and (54).
0 0.1 0.2 0.3 0.4 0.5
Frequency f
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Spectrum H(f)
(a)
(b)
Fig. 8. Power density function, H(f), versus frequency fof (a)
selected codewords of length n= 32 with m= 5 spectral points at
fi= 2/n, 2.5/n, 3/n, 3.5/n, 4/n,di= 0.1,i= 1,...,5, and (b)
spectrum for the same parameters computed using (44).
bearing on the code redundancy as the measured redundance
is r2= 9.2, which is slightly larger than in the previous case
m= 3.
A. Spectral null codes
Spectral null codes have been applied in a myriad practical
communication [15] and data storage systems [2, 3, 4, 5, 6]. In
this section, we analyze the difference in spectral performance
and redundancy of the newly developed spectral shaping codes
and prior art dc-balanced codes having a null at dc, f= 0.
1) Conventional balanced codes, null at f= 0:Conven-
tional dc-balanced block codes have codewords with equal
numbers of 1’s and -1’s, so that, clearly, nis even. Efficient
methods for encoding and decoding dc-balanced codes have
been developed by Knuth [16]. The spectrum of a conventional
dc-balanced code is given by [17]
Hdc(f) = n
n1{1F(f)}.(35)
The redundancy of an n-bit dc-balanced code using Knuth’s
implementation equals [16]
rdc log2n, n 1.(36)
0 0.05 0.1 0.15
Frequency f
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Spectrum H(f)
m=4
m=2
Fig. 9. Power density function, H(f), versus frequency fof encoded
sequences for d= 0.1,m= 2,m= 4 spectral points and word
length n= 32, both computed and measured.
In the next section, we study an alternative to dc-balanced
codes, namely spectral shaping codes that suppress the spectral
power density in the low-frequency (LF) range.
B. New codes with low-frequency suppression
In order to suppress LF spectral components, we specify m
spectral power points by
Htfi=2i1
2n=d, i = 1, . . . , m, (37)
where the target spectral content dis sufficiently small for
the application at hand. As an illustration of the spectral
performance of the new spectral shaping codes, we have
plotted in Figure 9, for m= 2 and m= 4 spectral points, the
measured spectrum, H(f), versus frequency ffor d= 0.1and
n= 32. The spectra are computed using (54) and measured
after a search using codeword selection criterion (32). The
estimated redundancy of the new codes with mspectral points,
all at the same spectral level d, denoted by ˆrn, equals, see (34),
rn=mˆr2(d). For m= 2 and m= 4, respectively we find
rn= 4.83 and rn= 9.66. The measured redundancy is slightly
higher, namely 4.98 and 10.32, respectively.
C. Performance evaluation of codes with suppressed low-
frequency content
The spectral performance of dc-balanced codes is mea-
sured by the width of the spectral notch, called the cut-off
frequency [18, 19]. The cut-off frequency, denoted by fc, is
defined by [19]
H(fc) = 1
2.(38)
For conventional dc-balanced codes we may numerically solve
(38) using (35). Using a Taylor series approximation of (35),
we obtain the useful approximation [20]
fcadc
n, n 1,(39)
0 0.05 0.1 0.15
Frequency f
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Spectrum H(f)
d=0.2
d=0.1
Fig. 10. Spectra of the new LF-suppressing codes with m= 4 spectral
points at level d= 0.1or d= 0.2for a word length n= 32 and
number of spectral points m= 4.
0 0.02 0.04 0.06 0.08 0.1
Frequency f
0
0.2
0.4
0.6
0.8
1
1.2
Spectrum H(f)
dc-balenced
new
Fig. 11. Low-frequency spectra of a conventional dc-balanced code,
n= 32, and a new code, n= 32,m= 2, and d= 0.1
computed using (35) and (54). The redundancy of both schemes is
approximately equal.
where
adc =p632
π0.422.(40)
We have evaluated the spectral notch width fcusing definition
(38) of the newly developed codes with low-frequency sup-
pression, whose spectrum is given by (54). The notch width,
fc, depends on the number of spectral points m, the choice of
the frequencies fi, and the spectral depth d. We found that the
spectral width hardly depends on the the notch depth d. As an
example, we plotted the spectra, see Figure 10, of a spectral
notch depth at d= 0.1and at d= 0.2(both for n= 32
and m= 4). We found that the notch width is for d < 0.2
independent of the notch depth d, and approximately given by
fcm1/2 + adc
n.(41)
Figure 11 compares the low-frequency spectra of conventional
dc-balanced code, n= 32, and new code n= 32,m= 2, and
d= 0.1. The spectral notch of the new code is a factor of 4.5
wider than that of the conventional dc-balanced code, where
it should be appreciated that the redundancy of both schemes
is approximately equal, rdc = 5 ˆr2= 4.98.
VI. ES TI MATE D SP EC TR A
In this section, we analyze the spectrum of encoded se-
quences defined by a single or multiple prescribed spectral
points fiat spectral depth di.
A. Estimate of the spectrum with multiple spectral points
The estimated auto-correlation function corresponding with
a spectrum with multiple, m, spectral points, f=fj,1j
mis [3]
ˆρi=
m
X
j=1
γj(ni) cos(2πifj),1in1,(42)
where γjare mreal constants to be determined below. The
corresponding power density function, H(f), is given by, see
(6),
H(f) = 1 + 2
n1
X
i=1
ˆρicos(i2πf)
= 1 + 2
m
X
j=1
γj
n1
X
i=1
(ni) cos(2πifj) cos(2πif )
= 1 +
m
X
j=1
γj
n1
X
i=1
(ni){cos(2πi(ffj)) +
+ cos(2πi(f+fj))}.(43)
After applying (19), we obtain
H(f) = 1 +
m
X
j=1
γjG(fj, f ).(44)
Since H(fj) = dj, we obtain a linear system of equations in
the munknown variables, γi,i= 1, . . . , m,
m
X
j=1
γjG(fj, fi) = di1.(45)
Define the column vectors
γ= [γ1, . . . , γm]Tand d= [d11, . . . , dm1]T,(46)
and the m×mmatrix Aby its elements Ai,j =G(fi, fj),1
i, j, m. Then for (45) we obtain the short-hand notation
Aγ=d.(47)
The matrix Ais symmetric as G(fi, fj) = G(fj, fi)for
all indices iand j. The solution of the above linear system
is straightforwardly accomplished using numerical methods.
Some special cases are amenable to analysis.
B. Case fi=ki/n and fi= (ki1/2)/n
Let the m,2m<n, specified spectral points be rational,
fi=ki/n, where the integer kiVm⊆ {1,...,dn/2e − 1},
(that is, excluding the zero, f= 0, and Nyquist frequency,
f= 1/2) then in (19) we have F(fi)=0. Alternatively, if the
m,2m<n, spectral,points are given by fi= (2ki1)/(2n),
kiVmthen, clearly, in (19) we have F(fi1fi2)=0,
i16=i2. In both cases we obtain
A=a1I+a2U,(48)
where Iand Udenote the identity and all-one matrix, respec-
tively, and
a1=n2
2and a2=n. (49)
We have
γ=A1d,(50)
where
A1=c1I+c2U,(51)
where
c1=1
a1
and c2=a2
a1(a1+ma2).(52)
Let di=dfor i= 1, . . . , m, then, after substituting (49), we
obtain
γi=γ=d1
a1+ma2
=2(d1)
n(n2m),1im. (53)
After working out (44) and (53), we obtain the LF-suppressed,
‘brick-wall’ shaped, spectrum Hbw(f), shown in Figures 9
and 11:
Hbw(f) = 1 + γ
m
X
i=1
G(fi, f ).(54)
The spectrum Hbw(f)agrees nicely with the measured one,
as can be seen in Figure 9.
VII. CONCLUSIONS
We have presented a new approach for designing spectral
shaping codes. We analyzed the probability distribution of the
spectral power density and auto-correlation coefficients of un-
coded bipolar sequences. Based on a simple exponential prob-
ability distribution assumption of the spectral power density,
we have given estimates of the redundancy of spectral shaping
codes. We have investigated codes with a wide frequency range
of suppressed power density. The frequency range is flexible
and can be better suited by the designer. Constructions of low-
redundancy codes with suppressed low-frequency content have
been presented that compare favorably with conventional dc-
balanced codes currently used in data transmission and data
storage devices applied in consumer electronics.
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Kees A. Schouhamer Immink (M’81-SM’86-F’90)
founded Turing Machines Inc., an innovative start-
up focused on novel signal processing for DNA-
based storage, where he currently holds the position
of president. Among the accolades received are a
personal Emmy award in 2004, the 2017 IEEE
Medal of Honor, the 1999 AES Gold Medal, the
IEEE Masaru Ibuka Consumer Electronics Award,
the 2004 SMPTE Progress Medal, and the 2015 IET
Faraday Medal. He was inducted into the Consumer
Electronics Hall of Fame and the (US) National
Academy of Engineering.
Kui Cai (M’07-SM’10) 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 Techni-
cal University of Eindhoven, The Netherlands, and
National University of Singapore. Currently, she is
an Associate Professor with Singapore University of
Technology and Design (SUTD). She received 2008
IEEE Communications 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.
... The 8B10B code has many embodiments [2,9], and is widely used in gigabit telecommunication systems and data storage media. Combinations of RLL and balanced codes can be found in data storage, energy harvesting, and communications codes [10,11,12,13]. ...
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