M -ary optical data storage: fixed versus variable
length modulation
Estuardo Licona and Steven W. McLaughlin
School of Electrical and Computer Engineering, Georgia Institute of Technology
Atlanta, GA 30332 USA
estuardo@ece.gatech.edu, swm@ece.gatech.edu
Abstract— In this paper we consider theoretical and practical aspects
of an M-ary optical recording channel. Specifically we address the perfor-
mance tradeoffs between conventional pulse amplitude modulation (PAM)
and M-ary runlength limited (RLL) coding in an optical disc system with
compact disc parameters. Using a linear-time invariant channel model with
white and nonwhite noise components, we see that for signal-to-noise ratios
found in typical CD systems, coded PAM outperforms M-ary RLL codes
for a range of storage densities. In addition, this work also shows that for
systems that require moderate-to-small values of M, M-ary RLL codes can
provide significant advantages.
I. I NTRODUCTION
Conventional optical recording systems like the compact disc
(CD) and digital versatile disc (DVD) use binary signaling in
conjunction with runlength limited codes. This modulation is
essentially pulse width modulation (PWM) where information
is conveyed in the length of the marks on the disc and the spac-
ing between them. The dominant source of random errors in
these systems is caused by jitter, producing variations in the
pulse widths of the recorded waveforms. Increasing storage
density using binary signaling (without changes to the optical
or mechanical parameters of the system) requires more infor-
mation in the pulse widths which is very difficult because of
the inherent jitter limitations. Additive amplitude noise in the
signal may not be the major source of noise. This has led
to research and implementations of systems storing nonbinary
(M-ary) signals using fixed-length, pulse amplitude modulation
(PAM) in an attempt to store more information per unit area.
Various papers addressing fixed-length multilevel marks have
appeared, using M = 8 - 12 levels [1–4]. The first multilevel
CD-R and CD-RW compatible system achieves 2.5 bits/0.6-
micron mark compared to a conventional CD system that stores
1.4 bits/0.833-micron mark. The write-once and re-writable op-
tical discs are highly nonlinear and the process of storing M-ary
signals requires a nonlinear, but straightforward, precompensa-
tion algorithm. This precompensation effectively linearizes the
channel so conventional coding and signal processing for a lin-
ear bandlimited channel can be employed [3]. Recently [5] has
shown that it is possible to write variable length marks, opening
the possibility of using M-ary runlength limited coding.
In this paper we expand a linear, time-invariant (LTI) model
of the M-ary optical recording channel with CD parameters
[6] and show maximum likelihood sequence detection (MLSD)
performs using the model. The model is generic enough to in-
clude both fixed and variable length marks, so we can include
runlength limited (RLL) modulation with M levels. We com-
pare multilevel storage with fixed length marks (with and with-
out coding) with M-ary RLL methods and theoretical limits.
✲
✍✌
✎☞
❅
✲
h
c
(t )
✲
✍✌
✎☞
✲
✻ ✻
˜ n(t )
p
T
(t - kT )
p
T
(t )
-
T
2
T
2
a
k
x(t ) y(t )
h
r
(t )
r(t )
❍ ✲
r(kT )
Fig. 1. Optical Recording Channel model
This paper is organized as follows. Section II presents the
channel model and the sufficient statistics discrete-time channel
model. Section III expands the model for the RLL constrained
channel. Section IV describes the runlength limited modula-
tion scheme. Section V shows the performance of maximum
likelihood sequence detection (MLSD) on the channel model.
Finally, Section VI provides a summary of the results.
II. A MODEL FOR THE MULTI -LEVEL OPTICAL
RECORDING CHANNEL
In Fig. 1 the optical recording channel is modeled as having a
discrete-time, discrete-valued multi-level input, and a discrete-
time, continuous-valued output. It is assumed to be linear, with
additive noise. Its output is
r(t )=
∑
k
a
k
p
T
(t - kT )
∗ h
c
(t ) ∗ h
r
(t )+ ˜ n(t ), (1)
where T is the duration of one channel symbol. The model
consists of a linear filter h
c
(t ), a write pulse p
T
(t ), a noise signal
˜ n(t ) and a receive filter h
r
(t ).
The input to the model is an M-ary sequence {a
k
} a
k
∈
{±1, ±3,..., ±(M - 1)}, which gets multiplied by pulses p
T
(t )
with a duration of T seconds. The frequency response of the
channel is the modulation transfer funcion (MTF)
H
c
( f
n
)=
2
π
arccos | f
n
|-| f
n
|
1 - f
2
n
2
(2)
where f
n
= f / f
c
is a normalized frequency variable, such that
f
n
∈ [-1, 1]. The frequency f
c
is the critical frequency of the
optical recording channel given by f
c
= 2NA/λ , where NA is the
numerical aperture of the focusing lens and λ is the wavelength
of the laser.
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