Journal of Signal and Information Processing, 20 11 , 2, 26 - 32
doi:10.4236/jsip.2011.21004 Published Online February 2011 (http://www.SciRP.org/journal/jsip)
A Reduced Complexity Quasi-1D Viterbi Detector
R. N. Panda1, N. Panda2, S. P. Panigrahi2, R. R. Mohanty2, M. Sahu2, H. Sahu2, A. Majhi2, U. Mishra2
Electrical Engineering; 1NIT; 2KIST, Bhubaneswar, India
Received December 10th, 2010; revised January 11th, 2011; accepted February 18th, 2011
ABSTRACT
This paper develops a reduced complexity quasi-1D detector for optical storage devices and digital communication
system. Superior performance of the proposed detector is evidenced by simulation results.
Keywords: Channel equalization, Viterbi Detector
1. Introduction
Recent literature is rich enough for improvements in
multi-user detection system like that of Digital Commu-
nication or Optical Storage system. Such improvement
with use of turbo encoding/decoding algorithms [1] for
digital communication, non-coherent Ultra Wide band
(UWB) detector for in the context of distributed wireless
sensor networks [2]. However, in this paper, we focus on
Optical storage systems.
The perpetual push for higher track density necessi-
tates the two-dimensional optical storage (Two-DOS)
systems to have large number of tr acks in a single group.
In the current stage, the number of tracks is chosen to be
11 within the group [3]. The complexity of two-dimen-
sional (2D) Viterbi detector (VD) grows exponentially
with both the target length
g
Nand number of tracks r
N
in a single group. Hen ce, truncating the channel memory
by means of pre-filtering techniques does not sufficiently
reduce the complexity of 2D VD for the current Two-
DOS system. For example, though we have shortened the
channel memory by setting 3
g
N, it is by far imprac-
tical because the number of states for the full-edged 2D
VD will reach 222 for 11
r
N. For this reason, in this
paper, we develop a quasi-one-dimensional (quasi-1D)
VD, which exploits the cross-track decisions as the
feedback to facilitate the implementation of reduced-
complexity 2D Viterbi-like detectors for systems with
large number of tracks per group.
2. Background
2.1. Decision Feedback Equalization
Decision feedback equalization is a nonlinear detection
technique that is quite popular in digital communication
systems [4,5]. Figure 1 shows the block diagram of a
discrete time decision feedback equalizer (DFE). In the
figure, hk is the discrete-time channel symbol response,


n  is the additive white Gaussian noise (AWGN)
with variance 2

, and k w and k
f
represent the taps
of the forward and feedback equalizer, respectively. The
forward equalizer shapes the channel into a prescribed
target k
g
, which is constrained to be causal and the first
tap 0
g
is constrained to be one. Feedback equalizer has
a strictly causal impulse response k
f
that should match
k
g
for all 1k in order to cancel the causal inter-
symbol interference (ISI), i.e. the ISI due to the symbols
that have already been detected. By removing the causal
ISI, the DFE uses the threshold comparator to make the
bit decision based on the input of the slicer. Though the
DFE is the optimum detector that has no detection delay
[6], its performance lags behind that of the VD because
of the following two main reasons.
• Error propagation: Any decision errors at the out-
put of the slicer will cause a corrupted estimation
of the causal ISI, which is to be generated by the
feedback equalizer. The result is that a single error
causes the detector to be less tolerant of the noise
for a number of future decisions. This phenomenon
is referred to as th e error propagation and degrades
the performance of the detector.
• Energy reduction: Even in the absence of error
propagation, the DFE is still sub-optimum com-
pared to the VD in terms of performance. This is
because in the decision process, the DFE subtracts
the causal ISI and thus ignores the signal energy
embedded in this causal ISI component. In other
words, some signal energy that is beneficial for the
optimum detection is neglected. The adverse effect

A Reduced Complexity Quasi-1D Viterbi Detector 27
θ(n)
Figure 1. Block diagram of a discrete-time decision feedback equalizer.
on the detection performance is referred to as the
energy reduction. To minimize the energy reduction
effect due to neglecting the energy of causal ISI, the
target is designed to have minimum-phase charac-
teristics, i.e. the energy of the target is optimally
concentrated near the time origin.
2.2. Fixed-Delay Tree Serch
Unlike the DFE that makes the bit decision instantly, the
fixed-delay tree search (FDTS) detection technique makes
the bit decision after a delay of D [7]. In this technique,
the bit decision is based on a sequence of D + 1 input
samples before the detector and uses the maximum-like-
lihood (ML) decision rule for the bit decision with a de-
lay of D. The ML decision exploits partly or all of the
signal energy embedded in the causal ISI components,
and thus reduces the energy reduction effect compared to
the DFE. The choice of parameter D is limited by the
compromise between performance and complexity. If
D+1 is smaller than the target length Ng, the FDTS is
referred to as the fixed delay tree search with decision
feedback (FDTS/DF) [8]. In fact, the FDTS can be con-
sidered as a generalization of the DFE since the FDTS is
essentially equivalent to the DFE when D = 0.
Similar to the DFE, the FDTS first uses the forward
equalizer to shape the channel into a known target.
Then, the noiseless input of the detector is


dn


1
01
g
N
i
iga n


, where i
g
represent the coefficients
of the target whose length is Ng, and a(n) is the channel
input bit at time index n. The FDTS uses a fixed-depth
ML decision rule implemented as a tree search algorithm.
The tree representation with depth D = 2 is shown in
Figure 2 for illustration. Each branch corresponds to one
input bit at a particular time. A sequence of branches
through the tree diagram is referred to as a path. Each
possible path corresponds to one input sequence and vice
versa. At time index n, the tree diagram consists of D + 1
bits. Thus, at each time index, the trellis contains 2D + 1
paths that represent all the possible 2D + 1 input se-
quences. Detection based on the smallest Euclidian dis-
tance between the detector input z(n) and the desired
noiseless detector input d(n) is optimum in the ML sense
when the noise component of the detector input is white
and Gaussian.
Thus, similar to the trellis diagram that corresponds to
the VD, the Euclidian distance
 
2
zn dn


is de-
fined as the branch metric for each branch, and the sum-
mation of the branch metrics associated with each path is
called the path metric. Since the FDTS performs ML
detection based on a sequence of samples, it chooses the
path whose path metric is minimum as the most likely
transmitted sequence and releases the first bit associated
with this path as the detected bit. More specifically, the
FDTS operates recursively as follows [8]:
• Initial condition: At the end of

th
1n step, the
tree structure has a depth of 1D. Each path re-
tains the path metric obtained from the previous it-
eration.
• Path extension: At the nth step, the tree structure is
extended such that the depth is increased to D.
The new input sample


zn is used to compute the
branch metric
 
2
z
ndn



 for each extended
Figure 2. Tree representation with depth D = 2 for the un-
coded binary channel input data.

A Reduced Complexity Quasi-1D Viterbi Detector 28
branch. To compute

dn for each extended
branch, the possible channel input bit sequence
consists of the preceding 1
g
N input bits lying
on the path leading to that extended branch. If the
channel memory 1
g
N exceeds the detection
delay D, the already detected bits are also used to
compute

dn.
• Path selection: After computing all the path metrics
for the extended paths, the first bit of the path that
has the smallest path metric is selected and released
as the detected bit. Then, half of the total paths that
are incompatible with the detected bit are discarded.
As a result, the tree structure that remains has a
depth of 1D.
As time progresses, the root node moves along the ML
path and a fixed-size identical tree structure is main-
tained at each time index. Therefore, the complexity of
the FDTS is kept constant for each time index. Similar to
the VD, the ML decision rule makes the FDTS unduly
complicated if D is large. An efficient and simple re-
alization of the FDTS for systems using run length- lim-
ited (RLL) (1; k) codes can be found in [9,10].
2.3. Sequence Detection with Local Feedback
Many detection techniques with sequence feedback, such
as the DFE and FDTS/DF, use the detected bits as the
input of the feedback equalizer, resulting in the error
propagation problem. Nevertheless, this problem can be
reduced by resorting to local feedback [11,12]. The local
feedback is based on the trellis structure, and uses the
path memory associated with the current state instead of
the past decisions to estimate the causal ISI. The local
feedback guarantees that the branch metric of the correct
path is the ML metric, as long as it is discarded in favor
of some incorrect path [11]. As a result, it improves the
performance of those detectors with sequence feedback
at the price of requiring a large memory to store paths
associated with each state.
3. Quasi-1D Viterbi Detector
3.1. Complexity of 2D VD
2D PR equalization to shape the 2D channel into a
known 2D target with contro lled ISI and intertrack inter-
ference (ITI). These controlled ISI and ITI are left to be
handled by the 2D VD. The noiseless input of the 2D VD
is given by
 
1
01
g
N
i
i
dn gan



, where, i
g
is the
target matrix whose length is Ng, and a(n) is the chann el
input vector at time index n. As indicated earlier, the
complexity of 2D VD grows exponentially with both the
target length Ng and number of tracks r
N in a single
group. For a better understanding, the trellis structure for
the case of target length 3
g
N

and number of tracks
per group 2
r
N

is shown in Figure 3. In this figure,
the ‘+’ and ‘

’ represent the bits ‘+ 1’ and ‘1’, r espec-
tively. The trellis is assumed to start at the node S0, and
then becomes steady at instant 3n (i.e.
g
nN

).
Here, the labels of states represent the channel memory
and number of tracks per groups associated with the
paths that pass through these states. At time index n, each
state consists of


1
rg
NN

bits. Thus, at each time
index, the trellis contains

1
2rg
NN states. At time index
n, each branch specifies the channel memory associated
with the state that the branch originates from and the
possible channel input vector _a(n). Therefore, each
branch corresponds to one possible noiseless detector
input
 
1
01
g
N
i
i
dn gan




. For the binary channel
input bit, each state possesses 2r
N incoming and
2r
Noutgoing branches and thus there are totally
2rg
N
Nincoming and 2rg
N
Nof outgoing branches for each
time index of the trellis.
In Figure 3, it is clear that even in this simple 2D case,
the trellis of 2D case is much more complicated than the
one-dimensional (1D) case though the target length is the
same. Thus, the practical implementation of the 2D Vi-
terbi-like detector for large Nr also requires the signifi-
cant reduction of the complexity arising from the


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




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
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








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
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
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
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
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
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

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

Figure 3. Trellis structure for a channel with Ng = 3 and Nr
= 2.

A Reduced Complexity Quasi-1D Viterbi Detector 29
cross-track direction. In [13], a technique using the
Viterbi detector track-by-track, as well as the decision
feedback to estimate the ITI between tracks was pro-
posed. We call this detector the DFE-VD. It uses a set of
sub-2D VDs, each corresponding to one track. In the bit
decision process for a given track, the known bits just
above (or below) the current track are used as the feed-
back to calculate par t of the ITI. Th ese known bits can be
previously detected bits, or can be zeros if the upper (or
lower) track is the guard-band.
The branch metric is then computed by subtracting the
effect of these known bits. However, in this track-by-
track technique, the ITI from either only the upper
track(s) or only the lower track(s) estimated, and the re-
maining ITI estimations are still dependent on the trellis
states. As a result, the number of states should be larger
than that of 1D VD with the same target length. More-
over, this redundant complexity will not benefit per-
formance much since the detector makes the detection
based still only on the input samples from the current
single track. An improved detector is the stripe-wise Vi-
terbi detector (SWVD) [3,14]. This detector consists of a
set of sub-2D VDs, each dealing with one stripe that
consists of a limited number of tracks. The number of
stripes is equal to that of tracks in a single group. The
preliminary decisions from one sub-2D VD is used for
estimating the ITI in the next sub-2D VD, which is
shifted up (or down) by one track. This procedure is con-
tinued for all the stripes and the full procedure from bot-
tom to top (or top to bottom) of the group is considered
to be one iteration. Note that at least two iterations are
required in order to estimate the ITI from both upper and
lower tracks. Unlike the DFE-VD that resorts to the trel-
lis states to estimate the ITI from the lower (or upper)
track(s), the SWVD uses the preliminary decisions from
the previous iteration to estimate the ITI from the lower
(or upper) track(s). This additional decision feedback not
only reduces the complexity but also improves the per-
formance compared with the DFE-VD since its decisions
exploit the input information from both upper and lower
track(s) as well as that from current. However, the use of
iterations increases complexity as well as latency. Our
new proposal, whereas, is a non-iterative reduced-com-
plexity detector that is applicable to any 2D system.
3.2. Causal ITI Target
In this subsection, we introduce the causal ITI target as a
starting point for the development of our reduced-com-
plexity 2D Viterbi-like detectors. Conventionally, the
causal and anticausal ISI are referred to as the ISI from
the past and future bit decisions, respectively [6]. Simi-
larly, we refer to the causal and anticausal ITI as the ITI
resulting from the lower and upper tracks, respectively.
The concept of causal ITI was first used in the multi-
channel DFE [15]. Similar as shown in Figure 1, this
multi-channel DFE consists of a multi-channel forward
filter, a multi-channel feedback filter, and a decision
block. The multi-channel forward filter is designed to
constrain the channel to be causal ISI and ITI. The multi-
channel feedback filter is designed to remove the causal
ISI based on the previous bit decisions. The causal ITI is
left to be handled by the decision block. Motivated by
this, we propose the causal ITI target such that the 2D
target matrices are constrained to be the right triangular
matrices. It should be noted that this target is the basis
for the development of our reduced-complexity 2D Vi-
terbi-like detectors. As a starting point for our develop-
ment, we first examine the suitability of the causal ITI
target in Two-DOS. Figure 4 shows the performance of
full-edged 2D VD for four different targets when
5
r
N

and target length 3
g
N. In the figure, the di-
agonal elements of G0 in the causal ITI target are con-
strained to be 1s to avoid trivial solutions of the target
and equalizer. We use a fixed 2D target with elements


12 and 2D monic constrained target, which are rea-
sonable targets described in the last chapter for Two
DOS, as reference targets. Note that we impose a sym-
metry constraint, which constrains all the tracks within
the same group to suffer the same amount of ITI, in the
design of the 2D monic constrained target. In other
words, after the finite length equalizer, all the tracks
within the same group ideally suffer the same amount of
ITI. However, due to the presence of guard-bands serv-
ing as boundaries of the group, before the finite length
equalizer, not all the tracks suffer the same amount of ITI.
In addition, the 2D monic constrained target only allows
ITI from adjacent tracks. Therefore, the symmetry con-
straint will burden the design of finite length equalizer
and result in residual ISI and ITI.
Figure 4. BER performance for different target constraints.

A Reduced Complexity Quasi-1D Viterbi Detector 30
Note that the causal ITI target does not have this
symmetry constraint, and allows ITI not only from the
adjacent tracks. Therefore, compared with the 2D monic
constrained target, the causal ITI target burdens th e finite
length equalizer less and is expected to achieve better
performance. From Figure 4, it is shown that the causal
ITI target outperforms all the targets at every SNR. This
result indicates that it is reasonable to use the causal ITI
target for Two-DOS. More importantly, based on this
target, we propose some reduced-complexity 2D Viter-
bi-like detectors that are quite different from DFEVD
and SWVD since the latter two detectors suffer ITI from
both lower and upper tracks.
3.3. Principle of Quasi-1D VD
Since the causal ITI target contains ITI only from the
lower tracks, the bits in the upper tracks will not affect
the desired output. Based on this idea, a set of 1D VDs
are used to detect the bits, each deals with one track.
More specifically, as shown in Figure 5, the first 1D VD
that deals with the lowest track is processed with no de-
lay and the bits are detected after a delay D. The second
1D VD that deals with the second lowest track is proc-
essed with the delay D in order to use the detected bits
from the lowest track to estimate all the ITI in the second
lowest track. The third 1D VD that deals with the third
lowest track is processed with a delay D after the second
1D VD, and the detected bits from the lowest two tracks
are used to estimate the ITI in the third lowest track. This
procedure continues for all the tracks. Since the bits de-
tection does not need to consider the interference from
the upper tracks, this detector is distinct from the
DFE-VD and SWVD. Compared with the DFE-VD, this
detector has less computational complexity since fewer
states are needed for bit detection. More importantly, the
quasi-1D VD has better BER performance since it uses
all, while DFE-VD uses part, of the input information
that is needed in the cross-track direction. As illustrated
in Figure 6, the quasi-1D VD outperforms the DFE-VD
significantly no matter what target is chosen for the
DFE-VD. Compared with the SWVD, as mentioned pre-
viously, it has much lower complexity since it has no
iterative procedures.
Link with QR Detector
Our quasi-1D VD is developed for the Two-DOS sys-
tem, which is a multiple-input multiple-output system
having a large temporal span of the channel. Obviously,
this quasi-1D VD is applicable to multiple-input multi-
ple-output systems having an arbitrary temporal span of
the channel. In many wireless communication systems,
the multiple-input multiple-output channel is assumed to
be at-fading [16,17], i.e. the temporal span 1
h
N

. In
such systems, the channel is characterized by a matrix,
Figure 5. Principle of the quasi-1D VD. The solid lines rep-
resent the input and output of sub-VDs, the dashed lines
represent the feedback coming from the output of the pre-
vious sub-VDs.
Figure 6. Performance comparison of different detection
techniques.
instead of a sequence of matrices in the Two-DOS sys-
tem. Let 1
N and 2 N represent the number of transmit
and receive antennas, respectively, in multiple-input
multiple-output wireless communication systems. Then,
the channel output vector at a given time is given by
zHa

(1)
Where, z and a are the (21N) channel output vec-
tor, and (11N

) channel input vector, respectively, H is
the (21 NN

) at-fading channel matrix. For the sake of
simplicity, the time index is ignored here. Then, QR de-
composition of the channel matrix yields QRH

,
where Q is an (21 NN

) ortho-normal matrix con-
structed to make the (11 NN

) matrix
R
right triangu-
lar [19]. Pre-multiplying the channel output vector z
with
H
Q, the resulting vector ˆ z is given by
ˆH zQzRa (2)
Note that if the noise in z is additive white Gaussian
noise (AWGN), the noise in ˆ z remains AWGN since
H
QQ is an (11 NN

) identity matrix. Comparing R
with the causal ITI target discussed in the previous sub-
section, we find that R can be seen as a special case of

A Reduced Complexity Quasi-1D Viterbi Detector 31
causal ITI targets. Then, like the quasi-1D VD, the first
element from the bottom of the channel input vector a is
first detected. The detected element is used to estimate
interferences for the detection of the second element
from the bottom of a. This procedure continues until all
the elements in a are detected.
This detector is commonly referred to as the QR de-
tector and has been investigated in multiple-input multi-
ple-output at-fading channels [19-21]. The QR detector is
also applicable in multiple-input multiple-output or-
thogonal frequency division multiplexing (MIMO-OFDM)
systems [20,22], since the channel at each sub-carrier of
MIMO-OFDM systems is considered as a multiple-input
multiple-output at-fading channel. Note that our pro-
posed quasi-1D VD is suitable for any multiple-input
multiple-output channel with arbitrary positive h
N,
while the QR detector is only applicable for multi-
ple-input multiple-output at-fading channel, i.e. 1
h
N

.
Therefore, the QR detector is considered as a special case
of our proposed quasi-1D VD.
4. Performance of 1D VD
As shown in Figure 5, though the quasi-1D has much
lower complexity than the DFE-VD and SWVD, it
causes significant detraction from optimality. We con-
sider three factors that affect the performance of quasi-
1D VD: target length, error propagation and energy re-
duction. In Figure 7, “L4” and “L5” represent that the
lengths of targets are four and five, respectively. Other-
wise, the length of target is three. “No EP” means detec-
tors without suffering error propagation. In simulation,
“No EP” is achieved by use of correct input bits to esti-
mate ITI. The length of the equalizer is 31 in all the
simulations. As illustrated in Figure 7, the BER per-
formance is not significantly improved by increasing the
target length. Further investigation shows that all the
elements in target matrices 3
g
and 4
g
approach zero,
therefore confirming that there is no need to increase the
channel memory beyond two. Figure 7 also shows that
the error propagation degrades performance by 1 dB for
BER is 4
10. Thus, the energy reduction should be the
dominant factor that degrades the performa nce.
5. Conclusions
In this paper, we have first briefly reviewed prior work
on the detectors with sequence feedback. Then, by con-
straining the target with causal ITI, we have developed a
quasi-1D VD, which uses a computationally efficient
technique whose complexity, grows only linearly with
the number of tracks. This is a significant complexity
reduction compared to the conventional 2D VD whose
complexity grows exponentially with the number of
tracks. We have shown that the quasi-1D VD improves
Figure 7. BER performance of quasi-1D VD with different
target lengths.
over the DFE-VD and SWVD in terms of complexity.
Further, we have shown that the widely known QR de-
tector is a special case of our proposed quasi-1D VD.
However, we have found that the quasi-1D VD still
causes significant detraction from optimality in the
Two-DOS system. Therefore, effective compensation
techniques are needed to ensure reliable data recovery.
To achieve this goal, we have investigated the factors
that might degrade the performance. Our simulation re-
sults implied that the energy reduction is the dominant
factor that degrades the performance of the quasi-1D VD.
Therefore, in the next chapter, we develop some effective
techniques to reduce the effect of this energy reduction
problem. In addition, the effect of error propagation still
needs to be minimized since it degrades the performance
by roughly 1 dB when BER is 4 10. However, increas-
ing the target length beyond three is of no practical value
for the Two-DOS system since it hardly improves the
performance while introducing excessive complexity.
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