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Paper Memory truncation
and crosstalk cancellation
for efficient Viterbi detection
in FDMA systems
Alfred Mertins
Abstract — In this paper, the design of optimal receive filter
banks for frequency division multiple access (FDMA) over fre-
quency selective channels is investigated. A new design strat-
egy based on the principle of memory truncation, rather than
equalization, is presented. Through the receive filters, each
subchannel is truncated to a pre-defined length, and the fi-
nal data recovery is carried out via low complexity Viterbi
detectors. Both closed form designs and adaptive techniques
are discussed. Design examples are presented for high speed
transmission over copper wires. The examples show that mem-
ory truncation allows significant performance improvements
over the often used minimum mean squared error (MMSE)
equalization.
Keywords — transmultiplexer, memory truncation, Viterbi de-
tector, dispersive channel, filter banks.
1. Introduction
The performance of transmission systems based on dis-
crete multitone (DMT) modulation [1
3] or orthogonal
frequency division multiplex (OFDM) [4] degrades rapidly
when the length of the channel impulse response exceeds
the length of the guard interval, which is introduced to cope
with non-ideal channels. As a result of an insufficient guard
interval, intersymbol interference (ISI) and inter-channel in-
terference (ICI, crosstalk) will occur. To cope with longer
channel impulse responses one can increase the length of
the guard interval, but this will decrease the efficiency, as
less data symbols can be transmitted. Increasing both the
length of the guard interval and the number of subchannels
allows one to maintain a desired bandwidth efficiency, but
this strategy also has its limits. For example, the delay be-
tween transmitter and receiver may become unacceptably
high. Also, the hardware requirements increase with an in-
creasing number of subchannels. Finally, channels which
can be regarded as slowly time-varying when the number
of subchannels is low may turn into fast time-varying ones
if the number of subchannels and thus the lengths of the
transmit and receive filters are significantly increased.
In this paper, new methods for the design of optimal re-
ceive filter banks in multichannel transmission systems are
proposed. The techniques are presented for a multirate fil-
ter bank framework, which gives a common description of
a variety of transmission techniques [5]. The solutions ap-
ply to DMT [3, 6], OFDM [4], coded-OFDM [7], transmul-
tiplexers [8, 9], and other transmission techniques where the
transmit signal is created as a weighted linear combination
of basis sequences with the data symbols being the weights.
Even code division multiple access (CDMA) [10, 11] can
be seen as a multirate filter bank. Figure 1 shows the gen-
eral structure of the transmit/channel/receive model used in
this paper. Depending on the actual modulation technique
(DMT, OFDM, CDMA, etc.), the upsampling factor N,the
number of subchannels, M, and the impulse responses gk
(
n
)
and hk
(
n
)
are chosen.
Fig. 1. Multirate discrete-time transmitter/channel/receiver mo-
del: (a) transmitter; (b) channel; (c) receiver.
Classically, the system in Fig. 1 is used to convert time-
division multiplexed (TDM) data signals into frequency
29
Alfred Mertins
division multiplexed (FDM) signals and vice versa. The
synthesis filter bank in Fig. 1a then provides the TDM-to-
FDM conversion. The FDM signal is transmitted through
the channel, and the receive signal is finally fed into the
analysis filter bank in Fig. 1c which converts the FDM
signal back into a TDM one [8, 9]. Examples are DMT
and OFDM, which both use block wise DFT’s to do the
TDM-to-FDM and FDM-to-TDM conversions. For DMT
and OFDM the impulse responses of the filters in Fig. 1 are
the complex exponentials occurring in the DFT, with max-
imum filter length N. A better frequency selectivity can be
obtained if one uses longer filters, designed according to
multirate filter bank theory. In [12] a comparison between
cosine-modulated filter banks and DMT was given which
shows that filter banks offer greater potential than block
transforms.
Various solutions to the problem of reducing ISI through
channel equalization have been proposed [5, 10, 12
19].
Most of them are based on minimizing the mean squared
error (MSE) between the sent data and the equalizer output,
either using a general MSE or a zero forcing (ZF) concept.
Decision feedback equalizers (DFE) have been considered
in [17]. Minimum mean squared error and ZF solutions
with a joint design of receiver and transmitter have been
proposed in [5, 16, 18]. Such a joint design can be useful in
cases where communication takes place in both directions.
In this paper, we concentrate on the optimal receiver design,
thus addressing cases where the transmitter is fixed. The
methods proposed in this paper are extensions of the tech-
nique in [19] to the design of entire receive filter banks for
the critically sampled and oversampled cases (i.e N
M).
Furthermore, methods for adaptive receiver design are pre-
sented. The design criterion is based on the idea of memory
truncation [20, 21], where the receiver does not try to fully
equalize the channel and leaves a residual system in the
data path. In the optimum, the MSE between the equal-
izer output and a filtered version of the input data sequence
is minimized. The final data detection then takes place via
a Viterbi detector which needs to consider only the residual
impulse responses. The lengths of the residual filters can
be chosen arbitrarily and will typically be a few taps, thus
allowing the use of low-complexity Viterbi detectors. The
advantage of memory truncation over equalization is that
critical channel zeros (e.g. zeros close to or even on the
unit circle) need not be equalized, so that the problem of
noise amplification through the equalizer can be avoided.
Note that for DMT transmission, memory truncation has
also been proposed in a different form where the channel
memory is shortened to the length of the guard interval
prior to the DFT analysis in the receiver [22, 23]. In the
present paper, however, memory truncation is incorporated
as a property of the receive filters, and we can even treat
cases where no guard interval is introduced at all.
The paper is organized as follows. In Section 2 the input-
output relations for the multirate system in Fig. 1 are dis-
cussed. Section 3 addresses the design of optimal receive
filter banks. Methods for adaptive receiver design are pre-
sented in Section 4. Results are discussed in Section 5, and
finally, conclusions are given in Section 6.
Notation: The superscript Tdenotes transposition of a vec-
tor or matrix. The superscripts * and Hdenote complex
conjugation and conjugate transposition
r
r
rH
=[
r
r
r
]
T
,re-
spectively. I
I
Iis an identity matrix of appropriate size.
E
fg
denotes the expectation operation, and
δ
i
;
kis the Kro-
necker symbol.
2. Input-output relations
We consider the system in Fig. 1. The sequences dk
(
m
)
,
k
=
0
;
1
;::: ;
M
1are created through a series-to-parallel
conversion of a single data sequence d
(
m
)
in the form
dk
(
m
)=
d
(
mM
k
)
,k
=
0
;
1
;::: ;
M
1.Inotherwords,
they are polyphase components of the sequence d
(
n
)
.In
the next step, the data sequences dk
(
m
)
are upsampled by
a factor of Nand then fed into the Mrespective synthesis
filters with impulse responses gk
(
n
)
,k
=
0
;
1
;::: ;
M
1.
The sum of the filtered signals finally forms the transmit
signal
s
(
n
)=
M
1
∑
k
=
0
∞
∑
m
=
∞dk
(
m
)
gk
(
n
mN
)
:
(1)
Typically, the filters gk
(
n
)
are chosen to be frequency se-
lective, so that each data sequence dk
(
m
)
is transmitted in
a distinct frequency band.
To make certain that the input data can be recovered at least
theoretically from the transmit signal s
(
n
)
, the upsampling
factor Nmust be chosen such that N
M[9]. In many
practical systems N
>
Mis used, which means that the
transmitter introduces redundancy. This redundancy can
be utilized in the receiver for enhancing the performance
in the presence of frequency selective channels.
For the discussion in this paper, the transmission channel
is assumed to be time-invariant. However, since adaptive
methods for the receiver design are proposed the channel
may, in practice, even be slowly time varying with respect
to the filter lengths involved. Considering a time invariant
channel, the receive signal is given by
r
(
n
)=
∞
∑
m
=
∞c
(
m
)
s
(
n
m
)
+
η
(
n
)
;
(2)
where
η
(
n
)
is an additive, data independent noise process
and c
(
n
)
is the channel impulse response. The noise is
assumed to be zero mean and wide-sense stationary.
On the receiver side, the signal r
(
n
)
isfedintotheanalysis
filter bank, as shown in Fig. 1, and the filter output signals
are subsampled by a factor of Nto form the final output
signals
xk
(
m
)=
Lh
1
∑
n
=
0hk
(
n
)
r
(
mN
n
)
;
(3)
k
=
0
;
1
;::: ;
M
1
:
30
Memory truncation and crosstalk cancellation for efficient Viterbi detection in FDMA systems
In Eq. (3), Lhis the length of the receive filters. Combining
Eqs. (1), (2) and (3) we get the input-output relation
xk
(
m
)=
Lh
1
∑
n
=
0
∞
∑
µ
=
∞
M
1
∑
i
=
0
∞
∑
`
=
∞
hk
(
n
)
c
(
µ
)
di
(
`
)
gi
(
mN
n
µ
`
N
)+
Lh
1
∑
n
=
0hk
(
n
)
η
(
mN
n
)
;
(4)
k
=
0
;
1
;::: ;
M
1
:
Under ideal conditions where the analysis and synthesis fil-
ters of the transmission system form a perfect reconstruc-
tion (PR) filter bank and where the channel is noise free and
ideal (i.e.
η
(
n
)=
0
8
nand c
(
n
)=
δ
n
;
0) the transmit/receive
system allows us to recover the data dk
(
m
)
without error:
xk
(
m
)=
dk
(
m
m0
)
:
(5)
The term m0is the overall delay of the system. The PR
conditions for the filter bank itself are
Lh
1
∑
n
=
0hk
(
n
)
gi
(
mN
n
)=
δ
i
;
k
δ
m
;
m0(6)
with i
;
k
=
0
;::: ;
M
1. A practical problem is that even
transmitter/receiver systems satisfying Eq. (6) will be un-
able to perfectly recover the data if a non-ideal channel is
introduced. Thus, the channel should be taken into account
when designing the receive filter bank. Methods for this
will be discussed in the next section. Since the channel is
usually not known a priori in practice, adaptation rules will
be presented in Section 4.
3. Design of optimal receive filter banks
In this section, we derive methods for the design of optimal
receive filter banks. For this we define an error signal as
the difference between the receiver output signals xk
(
m
)
and filtered versions of the data sequences:
ek
(
m
)=
Lh
1
∑
j
=
0hk
(
j
)
r
(
mN
j
)+
Lp
1
∑
i
=
0pk
(
i
)
dk
(
m
m0
i
)
;
(7)
k
=
0
;::: ;
M
1
:
The optimality criteria for the design of the Mreceive filters
are the MSE’s given by
Qk
=
E
j
ek
(
m
)
j
2
;
k
=
0
;::: ;
M
1
;
(8)
which are to be minimized under the energy constraints
Lp
1
∑
n
=
0
j
pk
(
n
)
j
2
=
1
;
k
=
0
;
1
;::: ;
M
1
:
(9)
The constraints (9) are needed to avoid the trivial solution
hk
(
n
)=
0,pk
(
n
)=
0.
Note that the error measure (7) is different from the MSE
as defined for conventional MSE equalizers [10, 12
15].
The idea behind the proposed approach is to truncate the
channel memory and not to delete it completely. The im-
pulse responses pk
(
m
)
are to be understood as residual im-
pulse responses of arbitrarily chosen length Lp. Both the
optimal residual systems pk
(
m
)
and receive filters hk
(
n
)
need to be found through minimization of Eq. (8).
Because of the existence of residual systems pk
(
m
)
, mini-
mizing Eq. (8) does, in general, not result in an equalization
of the channel. Even if Qk
=
0there will be a remaining ISI
between Lpconsecutive data samples in each of the sub-
channels. The crosstalk between different channels i
6
=
k
will be reduced as much as possible with FIR filters of the
given length Lh.
With analysis filters designed through the minimization of
Eq. (8) the overall system can be modeled with little error
as a set of Mindependent channels with
xk
(
m
)=
"
Lp
1
∑
i
=
0pk
(
i
)
dk
(
m
m0
i
)
#
+
η
0
k
(
m
)
;
(10)
k
=
0
;::: ;
M
1
:
The modified noise processes
η
0
k
(
m
)
contains the filtered
and subsampled original noise and all modeling errors
made by simplifying the real system to the form (10).
To recover the data, the signals xk
(
m
)
,k
=
0
;::: ;
M
1
are fed into Mindependently operating Viterbi detectors
which have to consider the respective channels pk
(
m
)
,
k
=
0
;::: ;
M
1. Since the lengths of these channels are
chosen arbitrarily, one can choose lengths which result in
a manageable computational cost for the Viterbi detectors
while maintaining a low noise variance at the detector in-
puts. Clearly, the longer the systems pk
(
m
)
are, the smaller
the modeling errors in (10) and thus the smaller the vari-
ances E
fj
η
0
k
(
m
)
j
2
g
are. For Lp
=
1the Viterbi detectors
degenerate to simple threshold detectors, at the expense
of an increased noise variance compared to cases where
Lp
>
1.
Note that in the special case of Lp
=
1, Eq. (8) states a stan-
dard MSE criterion, and the optimized analysis filters hk
(
n
)
can be regarded as MMSE equalizers. Then the proposed
solution becomes equivalent to other known MMSE solu-
tions [10, 12
15].
To obtain a compact formulation of the objective function,
we now introduce the following vectors:
h
h
hk
=
hk
(
0
)
;::: ;
hk
(
Lh
1
)
T
;
(11)
e
r
r
r
(
m
)=
r
(
mN
)
;::: ;
r
(
mN
Lh
+
1
)
T
;
(12)
p
p
pk
=
pk
(
0
)
;::: ;
pk
(
Lp
1
)
T
;
(13)
d
d
dk
(
m
)=
dk
(
m
)
;::: ;
dk
(
m
Lp
+
1
)
T
:
(14)
31
Alfred Mertins
We ge t
ek
(
m
)=
e
r
r
rT
(
m
)
h
h
hk
d
d
dT
k
(
m
)
p
p
pk
:
(15)
Using this notation the cost functions (8) can finally be
written as
Qk
=
h
h
hH
kR
R
Rrrh
h
hk
h
h
hH
kR
R
R
(
k
)
rd p
p
pk
p
p
pH
kR
R
R
(
k
)
dr h
h
hk
+
p
p
pH
kR
R
R
(
k
)
dd p
p
pk(16)
with
R
R
Rrr
=
E
r
r
r
(
m
)
r
r
rT
(
m
)
;
R
R
R
(
k
)
rd
=
R
R
R
(
k
)
dr
H
=
E
r
r
r
(
m
)
d
d
dT
k
(
m
m0
)
;
R
R
R
(
k
)
dd
=
E
d
d
d
k
(
m
m0
)
d
d
dT
k
(
m
m0
)
:
For the sake of simplicity, let us assume that all data
sequences dk
(
m
)
are spectrally white and have the same
variance
σ
2
d. Then the autocorrelation matrices R
R
R
(
k
)
dd ,
k
=
0
;::: ;
M
1are diagonal with diagonal entries
σ
2
d,
R
R
R
(
k
)
dd
=
σ
2
dI
I
I
;
(17)
and Eq. (16) simplifies to
Qk
=
h
h
hH
kR
R
Rrrh
h
hk
h
h
hH
kR
R
R
(
k
)
rd p
p
pk
p
p
pH
kR
R
R
(
k
)
dr h
h
hk
+
σ
2
dp
p
pH
kp
p
pk
:
(18)
We now consider the minimization of Eq. (18) with respect
to p
p
pkand h
h
hkunder the energy constraints (9). To derive the
optimal filters we first derive the optimal vector h
h
hkgiven
a fixed residual system p
p
pk.From
∂
Qk
=
∂
h
h
hk
=
0with Qkas
in Eq. (18), we get
h
h
h
(
opt
)
k
=
R
R
R
1
rr R
R
R
(
k
)
rd p
p
pk
:
(19)
Substituting h
h
h
(
opt
)
kinto Eq. (18) results in
Qk
=
p
p
pH
k
R
R
R
(
k
)
rd
HR
R
R
1
rr R
R
R
(
k
)
rd p
p
pk
+
σ
2
dp
p
pH
kp
p
pk
;
(20)
which now is to be minimized with respect to p
p
pkunder the
constraint (9). This yields the eigenvalue problems
σ
2
dI
I
I
R
R
R
(
k
)
rd
HR
R
R
1
rr R
R
R
(
k
)
rd
p
p
pk
=
λ
kp
p
pk
;
(21)
k
=
0
;
1
;::: ;
M
1
which are essentially similar to the one in [20] for the
single-channel case. The optimal vectors p
p
pkare the eigen-
vectors that belong to the respective smallest eigenvalues
λ
k,k
=
0
;::: ;
M
1.
The receive filters designed according to the method de-
scribed above minimize the error measures Qkunder the
energy constraint and thus maximize the signal-to-noise
ratios (SNR’s) at the filter outputs. Since the filter out-
put signals, together with the residual systems, are fed into
the Viterbi detectors, the algorithm maximizes the SNR’s
as seen by the Viterbi detectors.
4. Adaptive receiver design
The receiver design method presented in the previous sec-
tion may be difficult to implement under real-world condi-
tions where the required computational power and accuracy
are not available. Also, a real-world channel may be slowly
time varying, which causes problems for the receiver design
above. To avoid such problems we now derive adaptation
rules for the receiver design. For this we follow the strategy
for the single-channel case in [20]. During adaptation, we
assume that the data sequences dk
(
m
)
are known or have
been correctly estimated by the receiver, and we use the
received samples as noisy estimates of the required cor-
relation terms. We first look at the design of filter hk
(
n
)
(vector h
h
hk) according to the rule
h
h
h
(
µ
+
1
)
k
=
h
h
h
(
µ
)
k
γ
hek
(
µ
)
e
r
r
r
k
(
µ
)
;
(22)
where
µ
denotes the iteration step. The value ek
(
µ
)
is the
value of the error defined in Eq. (7) in the
µ
th iteration
step, and
e
r
r
rk
(
µ
)
is the receive vector in the
µ
th step. Finally,
γ
his a factor that controls the step size and convergence
speed. Thus, Eq. (22) is similar to the well-known LMS
adaptation rule for equalizer design.
A rule for adapting p
p
pkcan be stated as
q
q
q
(
µ
+
1
)
k
=
p
p
p
(
µ
)
k
+
γ
pek
(
µ
)
d
d
dk
(
µ
)
;
(23)
p
p
p
(
µ
+
1
)
k
=
q
q
q
(
µ
+
1
)
k
q
q
q
(
µ
+
1
)
k
:
(24)
The entire iteration is given by Eqs. (22), (23) and (24),
where the normalization step in Eq. (24) is needed to en-
sure that the energy condition (9) will be satisfied by the
final filter p
p
p
(
∞
)
k. Using the same arguments as in [20] one
can show that the iteration indeed converges to the MMSE
solution where the final vector p
p
p
(
∞
)
kis the eigenvector of
σ
2
dI
I
I
R
R
R
(
k
)
rd
HR
R
R
1
rr R
R
R
(
k
)
rd
that corresponds to the smallest
eigenvalue.
5. Results
To demonstrate the performance of the proposed algo-
rithms, we consider data transmission over telephone lines
in an ADSL/VDSL related setting [24]. Figure 2 shows the
channel impulse response considered in this example. It is
assumed that the channel noise is comprised of near and
far end crosstalk as well as white Gaussian noise, result-
ing in the total power spectral density depicted in Fig. 3.
We consider the use of a cosine-modulated filter bank for
creating the transmit signal, which is an interesting alter-
native to blockwise DFT’s as in DMT. In [12, 25] it was
shown that such filter bank based systems offer greater po-
tential than blockwise DFT’s because of their longer im-
pulse responses and better frequency selectivity. However,
32
Memory truncation and crosstalk cancellation for efficient Viterbi detection in FDMA systems
they need equalization on the receiver side. In the present
example, the transmit signal is synthesized via a 16-band
cosine-modulated filter bank with extended lapped trans-
form (ELT) prototype [26]. As in [12, 25] pulse amplitude
modulation is used to create a real-valued transmit signal.
Fig. 2. Transmission channel impulse response.
Fig. 3. Noise power spectral density.
Figure 4 shows the signal-to-noise ratios within the dif-
ferent bands at the equalizer output for several configura-
tions. In all cases the lengths of the receive filters are
chosen as Lh
=
128. We first look at the results depicted
in Fig. 4a. In this case, all bands are loaded with the same
input power
σ
2
d. This means that the transmission sys-
tem is critically sampled and that no redundancy (e.g. in
form of a guard interval) is introduced. The comparison
of the three curves in Fig. 4a shows that, especially for
the low-frequency channels, memory truncation (Lp
>
1)
significantly outperforms MMSE equalization (Lp
=
1).
A significantly better performance of all methods under
consideration is obtained if the first frequency band re-
mains unloaded. Results are depicted in Fig. 4b. This
strategy has been proposed in [25] as a possibility to in-
troduce redundancy. Leaving out a particular band has two
effects. Firstly, the system becomes oversampled, which
means that the transmitter introduces redundancy in form
Fig. 4. Signal-to-noise ratios at detector input using a 16-band
cosine-modulated filter bank as transmit filters: (a) all bands are
loaded; (b) only bands 1
15 are loaded.
of excess bandwidth. Secondly, the receive filters do not
need to suppress crosstalk from the dropped channel and
have more freedom to equalize their own data paths. As the
results in Fig. 4b show, almost all channels gain from the
fact that the first band has been left out. When comparing
the three curves in Fig. 4b, we see that memory truncation
still results in a noticeable improvement over MMSE equal-
ization for a number of bands. The performance difference
between Lh
=
2and Lh
=
3, however, is only marginal in
Fig. 4b.
From the above examples we see that a receiver based on
memory truncation receive filters and low-cost Viterbi de-
tectors can yield a significant improvement over MMSE
equalization and threshold detection. In general, the amount
of SNR improvement of memory truncation over MMSE
equalization depends on the channel in question. Signifi-
cant improvements can be expected whenever it is difficult
to equalize a channel because of extreme frequency selec-
tivity.
To demonstrate the adaptive approach, we consider the
receive filter design for the first channel in the above
setting. The residual filter length is chosen as Lp
=
3.
At the beginning of the iteration, the receive filter h1
(
n
)
was set to zero, and the residual system p1
(
n
)
was set
33
Alfred Mertins
Fig. 5. Receiver adaptation: (a) error signal e1
(
µ
)
; (b) residual
filter coefficients p
(
µ
)
1
(
m
)
.
to
f
p
(
0
)
1
(
m
)
g
=
f
1
;
0
;
0
g
. The step sizes were chosen as
γ
h
=
0
:
0015 and
γ
p
=
0
:
33. Figure 5a depicts the error
signal e1
(
µ
)
, and it can be seen that the error rapidly
decreases during the first few hundred iterations. Fig-
ure 5b shows the adaptation of the three residual coeffi-
cients. For comparison, the closed-form solution yields
f
p
(
opt
)
1
(
m
)
g
=
f
0
:
382
;
0
:
822
;
0
:
421
g
. These values are ap-
proached after some hundred iterations.
6. Conclusions
In this paper, optimal receive filter banks for FDM transmis-
sion systems have been presented. The receive filters are
designed in such a way that the overall subchannel impulse
responses become truncated to predefined lengths. Using
an example of high-speed transmission over copper wires
it was shown that the SNR can be significantly improved
over MMSE equalizer banks. In general, the amount of
improvement clearly depends on the channel in question,
and there may be cases where MMSE approaches work
equally well. The design methods presented are applicable
to all transmultiplexing systems where the transmit signal
is formed as a linear combination of transmit filter im-
pulse responses with the data symbols being the weights
(e.g. DMT, OFDM, CDMA). Extensions of the proposed
methods to a joint transmitter/receiver design are under in-
vestigation.
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Alfred Mertins received his Dipl.-Ing. degree from the
University of Paderborn, Germany, in 1984, the Dr.-Ing.
degree in electrical engineering and the Dr.-Ing. habil. de-
gree in telecommunications from the Hamburg University
of Technology, Germany, in 1991 and 1994, respectively.
From 1986 to 1991 he was with the Hamburg University of
Technology, Germany, from 1991 to 1995 with the Micro-
electronics Applications Center Hamburg, Germany, from
1996 to 1997 with the University of Kiel, Germany, and
from 1997 to 1998 with the University of Western Aus-
tralia. Since September 1998 he has been with the Uni-
versity of Wollongong as a Senior Lecturer. His research
interests include image and video processing, wavelets and
filter banks, and digital communications.
e-mail: mertins@uow.edu.au
University of Wollongong
School of Electrical, Computer,
and Telecommunications Engineering
Northfields Avenue
Wollongong, NSW 2522, Australia
35
