Intermittently Updated Simplified Proportionate Affine Projection Algorithm

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Intermittently Updated Simplified Proportionate Affine Projection Algorithm
Felix Albu, Henri Coanda, Dinu Coltuc, Marius Rotaru
Dept. of Electronics
Valahia University of Targoviste
Targoviste, Romania
E-mails: {felix.albu, coanda, coltuc}@valahia.ro; marius.rotaru@gmail.com
Abstract—In this paper, an intermittent update interval for
filter coefficients and a simplified output error vector
computation is proposed for a proportionate affine projection
algorithm. It is shown that the proposed algorithm has good
convergence performance and much smaller computation
complexity than other proportionate-type APAs. Also, the
accuracy of its implementation using the logarithmic number
system was investigated. We demonstrated the performance of
the proposed algorithm for echo cancellation and adaptive
feedback cancellation applications.
Keywords-Proportionate-type algorithms; adaptive filters;
affine projection algorithm; logarithmic number system.
I. INTRODUCTION
There are many adaptive algorithms proposed for
adaptive systems [1][2]. The most used algorithms are: the
Normalized Least Mean Square (NLMS) algorithm, the
Affine Projection Algorithm (APA) [3], and fast versions of
APA for various applications like echo cancellation, hearing
aids and active noise control (e.g., [4]–[9]). It is known that
in echo cancellation systems, the echo paths are often sparse
[1]. An intuitive idea for this case is to exploit the sparseness
of the echo path by updating filter coefficients independently
and proportionally to their estimated magnitude. One of the
first such algorithms was proposed by Duttweiler [10], and it
was called the Proportionate Normalized Least-Mean-Square
(PNLMS) algorithm. Several proportionate algorithms were
designed (e.g., [11],
μ-law
PAPA [12], Improved PAPA
(IPAPA) [13], Memory IPAPA (MIPAPA) [14],
μ-law
MIPAPA (MMIPAPA) [15], and Approximated
MIPAPA (AMIPAPA) [16]). The latter algorithm is still too
complex, and an approximation for the output error
computation of AMIPAPA was proposed in [17]. It was
termed Simplified AMIPAPA (SAMIPAPA) and the
complexity reduction come at a price of a reduced
performance by several dB, especially when using speech
signals and sparse echo paths. In [18], an algorithm that uses
a combination of recursive filtering, dichotomous coordinate
descent iterations and an approximation of a matrix in order
to further reduce its numerical complexity in terms of
multiplications was also proposed.
Therefore, a new proportionate algorithm with little
performance degradation that incorporates an approximation
of the output error and an intermittent update of filter
coefficients depending on a computed threshold [19][20] is
proposed in this paper. The algorithm proposed by Albu et
al. in [20] used an intermittent update on an affine projection
algorithm. It is shown that the threshold derived for the
affine projection algorithm by Shin, Sayed & Song in [21] it
is good enough for the proposed proportionate APA. The
new algorithm is termed Intermittently Updated SAMIPAPA
(IUSAMIPAPA). IUSAMIPAPA distinguishes from the
algorithm proposed by Albu et al. in [20], called
Intermittently Updated APA (IU-APA), because it is a
proportionate-type algorithm and uses other steady-state
MSE estimation formula. Also, the update formula of [20] is
related linearly to the logarithm of the estimated variance of
the filter output error. IUSAMIPAPA is different from the
algorithm proposed by Albu in [18] because it does not
include DCD iterations and uses other approximation. The
algorithm proposed in Albu and Kwan [22] is a sign
algorithm without an intermittent weights update unlike the
proportionate algorithm presented in this paper.
The paper is organized as follows. Section 2 presents a
short overview of the proportionate-type algorithms for echo
cancellation. In Section 3, SAMIPAPA is derived and the
proposed intermittently updated SAMIPAPA is
investigated. In Section 4, the proposed algorithm is
compared with AMIPAPA and SAMIPAPA in the context
of echo cancellation and adaptive feedback cancellation.
Also, the accuracy of its simulation using the logarithmic
number system is verified. Finally, the conclusions are
given in Section 5.
II. PROPORTIONATE-TYPE ALGORITHMS
In an echo cancellation system, we consider the far-end
signal x(n), and the reference signal d(n), where n is the time
index. The adaptive FIR filter is given by the coefficients
vector
       
0 1 1
ˆ ˆ ˆ
ˆ, ,..., T
L
n h n h n h n



h
, where L is the
length of the adaptive filter and superscript T denotes
transposition. The error signal is given by [1]
       
ˆ1
T
e n d n n n  hx
(1)
where x(n) = [x(n), x(n–1),…, x(n–L+1)]T is a vector
containing the L most recent samples of the input signal. If p
is the projection order, the error signal vector is given by

       
ˆ1
T
n n n n  e d X h
, (2)
where
       
, 1 , , 1n n n n p

   

X x x x
is the input
signal matrix,
       
, 1 , , 1 T
n d n d n d n p

   

d
is
the reference signal vector, and
       
, 1 , , 1 T
n e n e n e n p

   

e
is the error vector.
The coefficients of the proportionate-type affine
projection algorithms (PAPA) are updated as follows [18]
       
       
1
ˆˆ
11
1 ,
T
p
n n n n
n n n n



    



h h G X
I X G X e
, (3)
where G(n – 1) is an L x L diagonal matrix, δ is a
regularization constant, μ is the normalized step-size
parameter, and Ip is the p x p identity matrix. In the case of
the improved PAPA (IPAPA) [13], the diagonal elements of
G(n – 1), denoted by gl(n – 1), are evaluated as
     
 
1
0
ˆ1
1
1 1 ,
ˆ
221
l
lLi
i
hn
gn Lhn





    

(4)
where
11

  
,
01lL  
and ξ is a small positive
constant. Let us denote [14]
     
       
1
1 1 1 ,
n n n
n n n n p

    


P G X
g x g x
(5)
where g(n – 1) is a vector containing the diagonal
elements of G(n – 1) and the operator denotes the
Hadamard product [14].
 
nP
is approximated with
         
' 1 ... 1 ,n n n n p n p    


P g x g x
(6)
where g(n – k) are the vectors containing the diagonal
elements of the matrixes G(n – k), with k = 1, 2, …, p [14].
We have
       
1
' 1 ' 1 ,n n n n


  

P g x P
(7)
where the matrix
 
       
1
'1
2 1 ... 1 ,
n
n n n p n p


    

P
g x g x
(8)
contains the first p – 1 columns of
 
'1nP
. The
MIPAPA equations are written as in [16]:
     
'
1T
p
n n n

S I X P
(9)
         
'1
1
ˆˆ
1n n n n n


  h h P S e
(10)
The coefficients of the approximated MIPAPA
(AMIPAPA) are given by [16]
         
'1
2
ˆˆ
1n n n n n


  h h P S e
(11)
where,
 
2nS
, is updated by changing both its first row
and column with
   
:,1
'
TnnXP
and adding

to the first
element.
 
:,1
'nP
denotes the first column of
 
'nP
and is
given by
   
1nngx
. The bottom-right
   
11pp  
submatrix of
 
2nS
is replaced with the top-left
   
11pp  
submatrix of
 
21nS
[16].
III. INTERMITTENTLY UPDATED SIMPLIFIED AMIPAPA
Firstly, an important numerical complexity reduction is
obtained if
       
ˆ1
T
n n n n  e d X h
is approximated
as in the original fast affine projection algorithm [4]
       
; 1 1 ,
T
T
n e n n


  

ee
(12)
where
 
1ne
represents the first
1p
elements of
 
1ne
. The algorithm proposed in [17] used (12) instead
of (2) and was called simplified AMIPAPA (SAMIPAPA).
The numerical co mplexity of the following algorithms in
terms of multiplications is presented in equations (13)-(15)
(
m
P
=O(p3) [23] indicates the numerical complexity in terms
of multiplications):
 
MIPAPA 41 m
C L p p P  
(13)
 
AMIPAPA 32 m
C L p p P  
(14)
 
SAMIPAPA 2 3 2 .
m
C L p p P  
(15)
It can be noticed that the complexity of SAMIPAPA is
roughly half of that of MIPAPA for typical echo cancellation
systems where
Lp
. However, the complexity can be
further reduced using the intermittently updated procedure
proposed in [19]. Thus, the update equation of (11) can be
replaced by

         
 
'1
2
ˆ1 , if mod 0
ˆ
ˆ1 otherwise
n
n n n n n i
nn


  




h P S e
hh
(16)
where
n
i
is the computed update interval at time n.
Starting with an initial update interval of 1,
n
i
is given by
 
 
 
2
1
1
max 1, 1 , if
min 1, otherwise
n
nnM
i e n
iii








(17)
where
M
i
is the maximum update interval and

is the
threshold [19] computed as in (18)
22,
2vv
p





(18)
where
2
v

is estimated during silences [24]. The
numerical savings are important because (11) requires
m
Lp P
multiplications and the filter can have hundreds of
coefficients in echo cancellation systems. The update of the
filter coefficients from (16) is performed only when
mod 0
n
ni
and not at every iteration like in (11). The new
algorithm is termed Intermittently Updated SAMIPAPA
(IUSAMIPAPA). The algorithm can have a periodic update
if the update interval is fixed to
1
n
i
.
IV. SIMULATION RESULTS
Most of the simulations were performed in the context of
echo cancellation, where the input signal is either white
Gaussian noise or speech. The first impulse response from
ITU-T G168 Recommendation [25] is padded with zeros in
order to have 512 coefficients. A white Gaussian noise with
a SNR = 30 dB is added at the output of the echo path. The
performance measure used is the normalized misalignment
(in dB), defined as 20log10(||h – ĥ(n)||2/||h||2), where h
denotes the true impulse response of the echo path. In the
simulations with white noise, the performance curves are
averaged over 10 independent trials. The regularization
constant is δ = 0.01, p = 8 and α = 0. In a ll the simulations
where the input signal is a white signal, the step size of all
algorithms is 0.11.
Figure 1 shows the misalignment performance of the
periodic SAMIPAPA with fixed periodically updated filter
coefficients. It can be noticed that the larger the update
interval, the lower steady-state error and the slower the
convergence speed. Therefore, s imilar conclusions as those
of [18] and [19] are obtained and this indicates that a
variable updating interval for SAMIPAPA could lead to a
good compromise between fast convergence and low
steady-state error.
Figure 2 shows the misalignment curves for the
proposed IUSAMIPAPA (
8
M
i
), SAMIPAPA, and the
periodic SAMIPAPA with
8i
. An abrupt change of the
echo path after 25000 iterations by shifting the impulse
response to the right by 12 samples was introduced in order
to verify the tracking ability of the algorithms. It can be seen
that IUSAMIPAPA has roughly the same initial
convergence as SAMIPAPA and steady-state error of the
periodic SAMIPAPA. The update of the filter weights is
made on average only on a fifth of the number of iterations.
Overall, for the investigated case, IUSAMIPAPA obtains an
impressive 35% co mplexity reduction over SAMIPAPA in
terms of multiplications (SAMIPAPA has 9884
multiplications, while IUSAMIPAPA has 6495
multiplications).
Figure 3 shows the misalignment curves for
IUSAMIPAPA for different update intervals.
Figure 1. Misalignment of periodic SAMIPAPA for different update
intervals, white noise, p = 8, L = 512, SNR = 30 dB.
Figure 2. Misalignment of SAMIPAPA, periodic SAMIPAPA, i = 8,
and IUSAMIPAPA
8iM
. Other conditions are the same as in Figure 1.

Similar conclusions with those obtained in [18] and [19]
are obtained regarding the influence of
M
i
. It can be seen
that the time to reach steady-state increases with
M
i
value.
For the considered case, the percentage of updates is
about 15% for
8
M
i
, 9% for
16
M
i
, and 6% for
32
M
i
. The overall number of updates is reduced by
increasing
M
i
. The maximum update interval is set to the
projection order in the following simulations. An example
of co mputed
n
i
values and their histogram for the case
8
M
i
(Figure 3) is shown in Figure 4. It can be seen that
during the initial convergence, the updating intervals are
closer to 1, while they are closer to 8 in the steady-state
region.
Figure 3. Misalignment of IUSAMIPAPA with
8iM
,
16iM
and
32iM
respectively. Other conditions are the same as in Figure 1.
Figure 4. Computed update interval values (upper); and histogram of
computed
n
i
values (lower)
In Figure 5, the input signal is speech, with p = 8, the
output of the echo path is corrupted by independent white
Gaussian noise SNR = 30 dB and the echo path changes
after 0.5 seconds. The step-size for all algorithms is 0.2 for
the following simulation. It was shown in [16] that
MIPAPA has virtually identical performance with
AMIPAPA at a higher computational cost. Therefore, for
the following simulations, there is no need to plot the
misalignment curves of MIPAPA. Also, the superiority of
MIPAPA to APA for echo cancellation applications has
been proved in previous publications [14]-[16]. Figure 5
shows that the approximation used by SAMIPAPA and the
intermittent update of filter weights lead to slightly reduced
performance (1 to 3 d B for this example) in comparison
with AMIPAPA in case of a speech signal input. However,
IUSAMIPAPA offers a better performance/complexity
tradeoff than AMIPAPA, due to its reduced numerical
complexity by about 42% (7766 multiplications vs. 13460
multiplications).
Figure 5. Misalignment of the AMIPAPA, SAMIPAPA and
IUSAMIPAPA. Speech sequence, p = 8, L = 512, SNR = 30 dB, and echo
path changes at time 0.5s.
The same conclusions can be drawn for results using
colored noise as input signal, different filter lengths or
maximum projection orders.
In the next simulation, the performance of MMIPAPA
[15], AMIPAPA [16], SAMIPAPA [17] and IUSAMIPAPA
is investigated in the acoustic feedback context [26]. The
feedback path and the adaptive filter have 64 coefficients. A
delay of 60 samples and a constant gain of 30 d B in the
forward path were assumed. The sampling frequency was 16
kHz,
8M
,
0.1


, and
0.001


. The logarithmic
factor of MMIPAPA [15] was 100. It can be seen from
Figure 6, that most of the time, the performance of
IUSAMIPAPA is superior to that of MMIPAPA,
SAMIPAPA and AMIPAPA in case of a coloured input
signal. IUSAMIPAPA obtains a s maller misalignment than
the other algorithms, although has a slower convergence
speed at some moments in time.

Figure 6. Misalignment of MMIPAPA, AMIPAPA, SAMIPAPA, and
IUSAMIPAPA for an AFC application with coloured input signal,
8M
and
0.1


.
Figure 7 shows the same behaviour for a speech input
signal. The parameters of the algorithms are the same as
above example. It can be noticed that the performance of
MMIPAPA, AMIPAPA and SAMIPAPA is most of the
time similar. However, MMIPAPA has the highest
numerical co mplexity from all the investigated algorithms.
MMIP-APSA requires additional L logarithmic functions
and L additions per iteration in comparison with MIPAPA.
Figure 7. Misalignment of MMIPAPA, AMIPAPA, SAMIPAPA, and
IUSAMIPAPA for an AFC application with speech input signal,
8M
and
0.1


.
We’ve a lso investigated the performance of the
algorithm using 32-bit simulation using the logarithmic
number system (LNS) and compared with 32-bit floating
point results for the AFC example. The logarithmic number
system is an alternative to floating-point that offers the
potential to perform real multiplication, division and square-
root at fixed-point speed and, in the case of multiply and
divide, with no rounding error at all [27]. The logarithmic
addition and subtraction are performed with the speed and
accuracy equivalent to that of floating-point. The LNS
format compares favorably against its floating-point
counterpart, having greater range and slightly smaller
representation error [27]. Impressive speed-ups were
obtained over conventional floating point implementations
for a wide range of algorithms [28][29]. More details about
the logarithmic number system are available at
http://www.ncl.ac.uk/eece/elm.
We considered the AFC experiment results for both 32-
bit LNS and 32-bit floating point simulations. An accurate
standard for comparison of the outputs was obtained by
considering the corresponding double precision version
results. The corresponding sum of absolute errors was
computed for IUSAMIPAPA. The 32-bit LNS and 32-bit
floating-point simulations have almost identical results. This
confirmed similar conclusions obtained in the past for a
wide range of algorithms. However, the sum of absolute
errors of the 32 bit LNS implementation of IUSAMIPAPA
was about 10% smaller than that of the 32-bit floating point
implementation. Therefore, an LNS implementation could
benefit from an increased accuracy.
V. CONCLUSION AND FUTURE WORK
In this paper, a low complexity proportionate-type AP
algorithm was proposed. IUSAMIPAPA offers an excellent
convergence performance/numerical complexity compromise
in comparison with other proportionate AP algorithms. The
performance was verified on an echo cancellation and
adaptive feedback cancellation applications. Also, an
accuracy investigation of an LNS implementation was
performed. Future work will be focused on investigating the
performance of the proposed algorithm on AFC application
using two microphones in hearing devices [30] and compare
it variable projection order versions [31].
ACKNOWLEDGMENT
This work was supported by a grant of the Romanian
National Authority for Scientific Research, CNCS-
UEFISCDI project number PN-II-ID-PCE-2011-3-0097.
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Affine Projection Algorithm with Periodically Evolved
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[20] F. Albu, M. Rotaru, R. Arablouei, and K.
Dogancay,”Intermittently-updated affine projection
algorithm,” Proc. of ICASSP 2013, May 2013, pp. 585 – 589,
doi: 10.1109/ICASSP.2013.6637715.
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NLMS and affine projection algorithms,” IEEE Signal
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[22] F. Albu and H.K. Kwan, “New proportionate affine projection
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doi: 10.1109/ISCAS.2013.6571895.
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[25] Digital Network Echo Cancellers, ITU-T Rec. G.168, 2002.
[26] M. Rotaru, C. Stanciu, S. Ciochina, F. Albu, and H. Coanda,
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Active Feedback Cancellation using Xilinx System
Generator,” Proc. of ADAPTIVE 2013, May 2013, pp. 26-29.
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Implementations of the A Priori Error-Feedback LSL
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[30] C. R. C. Nakagawa, S. Nordholm, F. Albu, W.-Y. Yan,
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and transform domain processing", Proceedings of ICASSP
2014, pp. 3673-3677.
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The matlab code for the proposed algorithm can be obtained by
filling the request form from
http://falbu.50webs.com/List_of_publications_aec.htm
The reference for the paper is:
F. Albu, H. Coanda, D. Coltuc, & M. Rotaru, “Intermittently
Updated Simplified Proportionate Affine Projection Algorithm”, in
Proc. of ADAPTIVE 2014, Venice, Italy, pp. 42-47, ISBN: 978-1-
61208-341-4