Sign Subband Adaptive Filter with Selection of Number of Subbands
Jae Jin Jeong
1
, Seung Hun Kim
1
, Gyogwon Koo
1
and Sang Woo Kim
1,2
1
Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH),
Pohang, Kyungbuk 790-784, South Korea
2
Department of Creative IT Excellence Engineering and Future IT Innovation Laboratory, Pohang University of Science
and Technology (POSTECH), Pohang, Kyungbuk 790-784, South Korea
Keywords:
Adaptive Filter, Impulsive Noise, Sign Algorithm, Mean-Square Deviation.
Abstract:
The sign subband adaptive filter (SSAF) algorithm is introduced to reduce performance degradation of least-
mean-square-type algorithms due to a correlated input signal or an impulsive noise environments. However,
this algorithmh has huge computational complexity when the length of the unknown system is large. In
this paper, we focus on reduce computational complexity of the conventional SSAF algorithm and propose
an SSAF algorithm which selects number of subbands according to convergence state. The specific bands
which contributes to decrease the mean-square deviation are used to update the adaptive filter. Thus, the
proposed algorithm reduces the computational complexity compared to the conventional SSAF algorithm.
The selection mehtod is derived by analysing the mean-square deviation. Through the computer simulation,
simulation results are presented that demonstrate the fast convergence rate of the proposed algorithm and save
the computational cost.
1 INTRODUCTION
Adaptive filter algorithm has many applications such
as channel equalization, echo cancellation, and sys-
tem identification (Sayed, 2003; Lee et al., 2009).
The least-mean-square (LMS) and normalized least-
mean-square algorithm (NLMS), which are derived
by minimizing the L
2
-norm of the error function, are
widely used in this area because of its simplicity and
robustness against background noise. However, these
algorithms exhibit slow convergence rate when an in-
put signal is correlated or a measured signal contains
impulsive noise.
To overcome each problem, two categorizations
are presented. First, the normalized subband adaptive
filter (NSAF) algorithm was developed to improve the
performance of the NLMS algorithm for highly cor-
related input signals (Lee and Gan, 2004). By tak-
ing a pre-whitening operation on the input signal, the
NSAF algorithm achieves fast convergence rate. Sec-
ond, the sign algorithm was developed to improve the
performance of the NLMS algorithm for impulsive
noise environments, because it is obtained by mini-
mizing the L
1
-norm of the error function (Mathews
and Cho, 1987).
Combining the advantages of these two tech-
niques, in (Ni and Li, 2010), the sign subband
adaptive filter (SSAF) algorithm was introduced,
i.e., the SSAF algorithm is derived by taking the
pre-whitening process and minimizing the L
1
-norm.
Therefore, the SSAF algorithm can have good perfor-
mance in correlated input signal and impulsive noise
environments.
For subband-type algorithms, the correlated input
signal is close to the white signal in each band when
the number of subbands is high (Lee et al., 2009).
However,this leads to a huge computationalcomplex-
ity when the length of the unknown system is long
(Kim et al., 2010). Therefore, the SSAF algorithm
also has the complexity problem when the algorithm
is applied the long-tap unknown system.
In this paper, we focus on reducing computational
complexity of the conventional SSAF algorithm and
propose an SSAF algorithm with selection of num-
ber of subbands. For every iteration, the only specific
bands which contributes to decrease the mean-square
deviation (MSD) are used to update the adaptive filter
coefficient. That is, the proposed algorithm implies
smaller number of subbands, so it reduces compu-
tational complexity of the conventional SSAF algo-
rithm. In addition, the proposed algorithm achieves
a fast convergence rate than the conventional SSAF
algorithm in impulsive-noise environments. Through
the computer simulation, simulation results are pre-
407
Jeong J., Kim S., Koo G. and Kim S..
Sign Subband Adaptive Filter with Selection of Number of Subbands.
DOI: 10.5220/0005537604070411
In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 407-411
ISBN: 978-989-758-122-9
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

sented that demonstrate the fast convergence rate of
the proposed algorithm.
This paper is organized as follows. In Section 2,
the SSAF algorithm is reviewed. Section 3 presents
the proposed algorithm. Section 4 deals with the sim-
ulation results which compare the proposed algorithm
with the SSAF algorithm for system identification.
Finally, conclusions are given in Section 5.
2 SIGN SUBBAND ADAPTIVE
FILTER
The output signal d(n) of the system is obtained as
d(n) = w
T
opt
u(n) + η(n), (1)
where w
opt
= [w
0
,w
1
,...,w
M− 1
]
T
denotes the opti-
mal weight vector, which is predicted by an adaptive
filter; superscript T is the vector transpose; u(n) =
[u(n),u(n −1),...,u(n −M + 1)]
T
denotes the input
signal vector; η(n) is a noise that consists of a back-
ground and an impulsivenoise; n is time index; and M
is the length of the optimal weight vector. The prob-
ability density function of the noise is expressed as
(Rey Vega et al., 2008)
p
η(n)
(η(n)) =pN (0,(K + 1)σ
2
b
) + (1− p)N (0,σ
2
b
),
(2)
where p is the probability of impulsive noise, K is
the magnitude of impulsive noise, and σ
2
b
is the noise
variance without the impulsive noise.
The structure of the SSAF is shown in Figure
1. The subband signals d
i
(n) and u
i
(n) are obtained
by filtering d(n), and u(n) via analysis filters H
i
(z)
for i = 0,1, . . . ,N −1, respectively, where u
i
(k) =
[u
i
(kN),u
i
(kN −1),...,u
i
(kN −M+1)]
T
and N is the
number of subbands. The decimated desired signal
d
i,D
(k) and output signal y
i,D
(k) are obtained by crit-
ically decimating d
i
(n) and y
i
(n), respectively, where
y
i,D
(k) = u
T
i
(k)w(k) and subscript D means the dec-
imated signal. n is index of original sequences and k
is index of decimated sequences. G
i
(z) is synthesis
filter for i = 0,1, . . . , N −1.
The weight update equation for the conventional
SSAF algorithm is (Ni and Li, 2010)
ˆ
w(k+ 1) =
ˆ
w(k) + µ
U(k)sign(e
D
(k))
q
∑
N−1
i=0
||u
i
(k)||
2
, (3)
where µ is a step size, || · || denotes the L
2
-norm,
sign(·) denotes the sign function,
U(k) = [u
0
(k),u
1
(k),... ,u
N−1
(k)], (4)
e
D
(k) = [e
0,D
(k),e
1,D
(k),... , e
N−1,D
(k)]
T
, (5)
e
i,D
= d
i,D
(k) −y
i,D
(k), (6)
Figure 1: Structure of the SSAF.
and d
i,D
(k) = d
i
(kN) denotes the decimated desired
signal.
3 PROPOSED ALGORITHM
3.1 Proposed Algorithm
The proposed algorithm is determined by maximiz-
ing the decrease in the MSD. The MSD is defined as
MSD(k) , E{
˜
w
T
(k)
˜
w(k)}, where
˜
w(k) = w
opt
(k) −
ˆ
w(k) is the weight error vector, and E{·} is the expec-
tation of random variables. By subtracting (3) from
w
opt
(k), the equation is expressed in terms of
˜
w(k) as
follows:
˜
w(k+ 1) =
˜
w(k) −µ
U(k)sign(e
D
(k))
q
∑
N−1
i=0
||u
i
(k)||
2
. (7)
By taking squared L
2
-norm and expectation, the MSD
can be obtained as
MSD(k+ 1) = MSD(k)
−2µE



˜
w
T
(k)U(k)sign(e
D
(k))
q
∑
N−1
i=0
||u
i
(k)||
2



+ µ
2
. (8)
For a sufficiently long length of the weight vector
(Rey Vega et al., 2008; Bershad et al., 2014), we ob-
tain
E



˜
w
T
(k)U(k)sign(e
D
(k))
q
∑
N−1
i=0
||u
i
(k)||
2



≈ γE{
˜
w
T
(k)U(k)sign(e
D
(k))}, (9)
where
γ , E



1
q
∑
N−1
i=0
||u
i
(k)||
2



. (10)
ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics
408

Substituting (9) into (8) yields
MSD(k+ 1) = MSD(k)
−2µγ
N−1
∑
i=0
E{e
i,a
(k)sign(e
i,D
(k))}+ µ
2
, (11)
where e
i,a
(k) = u
T
i
(k)
˜
w(k). The second term on the
right-hand side of (11) is calculated by assuming
that e
i,D
(k) and e
i,a
(k) are jointly Gaussian and have
zero mean (Sayed, 2003). By using Price’s theorem
(Sayed, 2003), we get
E{e
i,a
(k)sign(e
i,D
(k))} = a
i
σ
2
e
i,a
(k), (12)
where σ
2
e
i,a
(k) , E{e
2
i,a
(k)} is the undisturbed error
variance of the ith subband, and
a
i
=
r
2
π



1− p
q
σ
2
e
i,a
(k) + σ
2
b
i,D
+
p
q
σ
2
e
i,a
(k) + (K + 1)σ
2
b
i,D



. (13)
By assuming that the input signal and noise are mu-
tually independent (Shin and Sayed, 2004), the i−th
subband error variance is written as
σ
2
e
i,D
(k) = σ
2
e
i,a
(k)+ (1− p)σ
2
b
i,D
+ p(K + 1)σ
2
b
i,D
. (14)
Combining (12) and (14) into (11), the resulting equa-
tion is expressed in terms of σ
2
e
i,D
(k) as follows:
MSD(k+ 1) = MSD(k)
−2µγ
N−1
∑
i=0
a
i

σ
2
e
i,D
(k) −ασ
2
b
i,D

+ µ
2
, (15)
where α = (1+ pK).
In (15), the MSD tends to decrease when σ
2
e
i,D
(k)
is larger than ασ
2
b
i,D
. On the other hand, the
MSD increases when σ
2
e
i,D
(k) is smaller than ασ
2
b
i,D
.
The proposed algorithm selects subbands satisfying
σ
2
e
i,D
(k) > ασ
2
b
i,D
at every iteration for the largest de-
crease in the MSD. Consequently, the number of se-
lected subbands, which is to update the weight vector,
is less than or equal to that of the conventional SSAF
algorithm.
3.2 Practical Consideration and
Computational Computation
In practical application, we can not obtain the exact
expected values, so we assume that the expected value
is approximately the same as an instantaneous value.
E{e
2
i,D
(k)} ≈ e
2
i,D
(k). (16)
Table 1: Proposed Algorithm Summary.
Initialization :
ˆ
w(0) = [0,0,...,0]
T
Parameters : α ≥ 1
Update :
for i = 0,1,...,N −1 do
S
L(k)
= [ ]
If |e
i,D
(k)| >
√
ασ
b
i,D
s
l
is selected, S
L(k)
= [S
L(k)
,s
l
]
end
If L(k) 6= 0
ˆ
w(k+ 1) =
ˆ
w(k) + µ
∑
L(k)
l=0
u
s
l
(k)sign(e
s
l
,D
(k))
q
∑
L(k)
l=0
||u
s
l
(k)||
2
end
end for
The noise variance σ
2
b
can be easily estimated dur-
ing silences (Yousef and Sayed, 2001; Benesty et al.,
2006).
Let S
L(k)
= [s
1
,s
2
,··· ,s
L(k)
] means a subset with
L(k) members of the set 0, 1,··· ,N −1, where s
l
denotes the index of the chose subbands, and L(k)
means the number of selected subbands at iteration
k. Finally, the update equation of the proposed algo-
rithm is expressed as
ˆ
w(k+ 1)
=



ˆ
w(k) + µ
∑
L(k)
l=0
u
s
l
(k)sign(e
s
l
,D
(k))
q
∑
L(k)
l=0
||u
s
l
(k)||
2
L(k) 6= 0
ˆ
w(k) L(k) = 0
,
(17)
where |e
s
l
,D
(k)| >
√
ασ
b
s
l
,D
(l = 1, 2, . . . , L(k)). The
proposed algorithm is summarized in Table 1. Table
2 shows the computational cost of the conventional
SSAF and the proposed algorithm.
4 SIMULATION RESULTS
The performance of the proposed algorithm is com-
pared to the conventional SSAF algorithm via com-
SignSubbandAdaptiveFilterwithSelectionofNumberofSubbands
409

Table 2: Computational Complexity.
SSAF Proposed algorithm
Multiplication 3M + 3NL (1+ 2N(k))M + 3NL
Division 1 N(k)
Comparison
- 1
(1/T
S
, N(k) = L(k)/N)
0 2 4 6 8 10
x 10
5
−40
−30
−20
−10
0
Number of iterations
NMSD (dB)
(a) SSAF (µ = 0.005) (Ni and Li, 2010)
(b) Proposed algorithm (µ = 0.005, α = 1)
(c) Proposed algorithm (µ = 0.005, α = 2)
(d) SSAF (µ = 0.001) (Ni and Li, 2010)
(e) Proposed algorithm (µ = 0.001, α = 1)
(f) Proposed algorithm (µ = 0.001, α = 2)
(a)
(b)
(c)
(e)
(f)
(d)
Figure 2: NMSD learning curve for the conventional SSAF
(Ni and Li, 2010) and proposed algorithm with various step
sizes.
0 2 4 6 8 10
x 10
5
0
2
4
6
8
Number of iterations
Number of subbands
Proposed algorithm (µ = 0.005, α = 1)
Proposed algorithm (µ = 0.005, α = 2)
Proposed algorithm (µ = 0.001, α = 1)
Proposed algorithm (µ = 0.001, α = 2)
Figure 3: Average number of selected subbands in the pro-
posed algorithm.
puter simulation. The adaptive filter has the same
length as the optimal weight vector with 512 or 1024
taps. The input signal is generated by passing a zero-
mean white Gaussian random sequence through
G(z) =
1
1−0.9z
−1
. (18)
The background noise is added to the system output
with a signal-to-noise ratio (SNR) = 30 dB. Further-
more, an impulsive noise is also added to the system
output with p = 0.01 and K = 10
5
. In order to com-
pare the performance, we use the normalized MSD
(NMSD), which is defined as ||w
opt
−
ˆ
w(k)||
2
/||w
opt
||
2
and calculated by ensemble averaging over 50 inde-
pendent trials. We assume that the background noise
variance σ
2
b
is known (Yousef and Sayed, 2001; Ben-
esty et al., 2006), so i− th subband noise variance is
obtained as σ
2
b
i,D
= σ
2
b
/N (Yin and Mehr, 2011). In
the simulations, the number of subbands (N = 8) are
used. The length of the prototype filter is 64.
0 2 4 6 8 10
x 10
5
−40
−30
−20
−10
0
10
Number of iterations
NMSD (dB)
(a) SSAF (µ = 0.005) (Ni and Li, 2010)
(b) Proposed algorithm (µ = 0.005, α = 1)
(c) Proposed algorithm (µ = 0.005, α = 2)
(d) SSAF (µ = 0.001) (Ni and Li, 2010)
(e) Proposed algorithm (µ = 0.001, α = 1)
(f) Proposed algorithm (µ = 0.001, α = 2)
(b)
(f)
(e)
(c)
(d)
(a)
Figure 4: NMSD learning curve for the conventional SSAF
(Ni and Li, 2010) and proposed algorithm with various step
sizes.
0 2 4 6 8 10
x 10
5
0
2
4
6
8
Number of iterations
Number of subbands
Proposed algorithm (µ = 0.005, α = 1)
Proposed algorithm (µ = 0.005, α = 2)
Proposed algorithm (µ = 0.001, α = 1)
Proposed algorithm (µ = 0.001, α = 2)
Figure 5: Average number of selected subbands in the pro-
posed algorithm.
Figure 2 shows the normalized MSD learning
curve for the conventional SSAF (Ni and Li, 2010)
and the proposed algorithm for M = 1024, various
step sizes (µ = 0.005 and µ = 0.001), and values of
α (α = 1 and α = 2). As can be seen, the proposed
algorithm leads to a fast convergence rate when step
size is small. The proposed algorithm has a fast con-
vergence rate but high steady-state MSD if α is large
because the number of subbands quickly decreases.
Figure 3 shows the average number of selected sub-
ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics
410

bands. In this result, the proposed algorithm has a
lower computation complexity than the conventional
SSAF algorithm, because the number of used subb-
bands is decreased.
Tracking performance is an important issue in
adaptive filter (Benesty et al., 2006). The unknown
system is changed to −w
opt
at 5 ×10
5
to evaluate
its tacking performance (Zou et al., 2000). Figure
4 shows the NMSD learning curve of the conven-
tional SSAF (Ni and Li, 2010) and proposed algo-
rithm for M = 1024, various step sizes (µ = 0.005 and
µ = 0.001), and values of α (α = 1 and α = 2). As can
be seen, the proposed algorithm has fast convergence
rate after the system change. That is the proposed
algorithm properly tracks the changed system coeffi-
cient. AS can be seen from Figure 5, the average num-
ber of selected subbands is increased when the system
changed, but it is decreased again as the iteration in-
creases. Therefore, the proposed algorithm efficiently
reduces the computational cost even for system track-
ing scenario.
In practical application, we can not exactly know
the values of p and K. Therefore, it is difficult to se-
lect α. However, the user chooses α ≥ 1, because p
and K are always positive values.
5 CONCLUSIONS
In this paper, we have proposed a new SSAF algo-
rithm with a low computational complexity. By an-
alyzing the MSD, the proposed algorithm selects the
number of subbands at each iteration. In conclusion,
the proposed algorithm was derived by maximizing
the decrease in the MSD at every iteration. Conse-
quentially, the proposed algorithm reduces the com-
putational complexity compared to the conventional
SSAF algorithm. In addition, the simulation results
show the proposed algorithm achieves a fast conver-
gence rate in impulsive-noise environments.
ACKNOWLEDGEMENTS
This research was supported by the MSIP(Ministry
of Science, ICT and Future Planning), Korea, under
the ICT Consilience Creative Program (IITP-2015-
R0346-15-1007) supervised by the IITP(Institute for
Information & communications Technology Promo-
tion) and by the Basic Science Research Program
through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education (NRF-
2013R1A1A2058975).
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