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A new realization of 2-D adaptive separable-denominator state-space filters based on DLMS algorithm suitable for parallel processing
ARTICLE IN PRESS
Journal of the Franklin Institute 341 (2004) 431–442
A new realization of 2-D adaptive separabledenominator state-space filters based on DLMS algorithm suitable for parallel processing$ Naoya Ishizakia, Mitsuji Muneyasub,*, Takao Hinamotoa a
Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi, Hiroshima 739-8527, Japan b Faculty of Engineering, Kansai University, 3-3-35 Yamate-cho, Suita 564-8680, Japan
Abstract In the application of 2-D adaptive filters, a large amount of data must be processed and real-time processing is often required. In this paper, a new parallel-form realization of 2-D adaptive separable-denominator state-space filters suitable for high-speed processing is proposed. First, the 2-D local state-space model suitable for parallel processing is introduced. Next, the adaptive algorithm for this model is developed. This algorithm is based on the delayed least mean square (DLMS) method. In addition, the computation time required for the update of the coefficients is investigated. Finally, the proposed technique is applied to the design of 2-D digital filters in the spatial domain. r 2004 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: 2-D adaptive filter; State-space model; Parallel processing; High-speed processing
1. Introduction An adaptive filter has been applied to various fields of signal processing, such as echo canceller, system identification and adaptive equalizer. Nowadays, it also has been extended to process multi-dimensional signals for the purpose of the $
The results in this paper were partially presented at the 1998 IEEE International Workshop on Intelligent Signal Processing and Communication Systems. *Corresponding author. Tel.: +81-824-24-7682; fax: +81-824-22-7195. E-mail address: [email protected] (M. Muneyasu). 0016-0032/$30.00 r 2004 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2004.04.002
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noise reduction of image data, multi-dimensional system identification and multi-dimensional digital filter design, which results that multi-dimensional adaptive filters have been proposed and their researches are getting more popular [8,10–12]. One concern, however, is that a multi-dimensional adaptive filter not only has to process a large amount of data, like an image, but sometimes has to perform realtime processing to handle such large data. Therefore, high-speed processing should be required for high-dimensional adaptive filters. One of the ideas for that is to apply the parallel processing structure of adaptive filters. For 2-D digital filters, several methods suitable for the parallel processing have been proposed [1,2]. Besides, we have some studies on 2-D adaptive filters suitable for the parallel processing where 2-D digital filter for parallel processing is applied to the structure of 2-D adaptive filters [3–5] and also extended to 3-D case [6]. This paper proposes a new parallel-form realization of a 2-D adaptive filter based on the Roesser local state-space model with separable denominator. The DLMS algorithm [7] is adopted as an adaptive algorithm in this realization. The class of separable denominator is one of the most important classes, since almost frequency specification can be implemented. The effective parallel form realization can also be achieved. However, such realizations by the general local state-space model are difficult because of its complicated feedback structure. At first, the parallel-form realization of a 2-D digital filter with separable denominator proposed by Dabbagh and Alexander [1] is brought up for consideration. Then, a new adaptive algorithm is introduced by the combination of the DLMS algorithm and the adaptive algorithm for a 2-D state-space digital filter with separable denominator [8]. The computational time required for the filtering and the coefficient update is also evaluated to prove that the use of the DLMS algorithm leads a faster algorithm than the algorithms reported in [3,4]. Finally, the effectiveness of the proposed technique is confirmed by applying it to 2-D digital filter design in the spatial domain.
2. Parallel implementation of 2-D state-space filters with separable denominator In this section, the parallel-form implementation of 2-D state-space filter with separable denominator reported in [5] is introduced to be prepared for deriving the corresponding adaptive algorithm [9]. Consider the following irreducible 2-D separable-denominator transfer function:
Hðz1 ; z2 Þ ¼
Nðz1 ; z2 Þ ; D1 ðz1 ; z2 ÞD2 ðz1 ; z2 Þ
ð1Þ
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where Nðz1 ; z2 Þ ¼
M X N X i¼0
D1 ðz1 Þ ¼
M X
bij zi1 zj2 ;
j¼0
ai zi1 ;
i¼0
D2 ðz2 Þ ¼
N X
bj zj2 ;
j¼0
aM ¼ bN ¼ 1: The 2-D separable denominator transfer function expressed in Eq. (1) can be represented by PM PN iM j z2 1 i¼0 j¼0 bij z1 PM Hðz1 ; z2 Þ ¼ PN j i i¼0 ai z1 j¼0 bj z2 ¼ H1 ðz1 ; z2 Þ H2 ðz1 Þ:
ð2Þ
Therefore, Hðz1 ; z2 Þ can be described by the cascade form of H1 ðz1 ; z2 Þ and H2 ðz1 Þ: H2 ðz1 Þ is rewritten as H2 ðz1 Þ ¼ PM
zM ¼ PM 1 i i¼0 ai z1
1
ai ziM 1 PM1 i¼0 ai zi1 ¼ 1 þ PM : i i¼0 ai z1 i¼0
ð3Þ
Then, by employing a controllable canonical form, the implementation of the statespace model of H2 ðz1 Þ is shown as xh ðn1 þ 1; n2 Þ ¼ A1 xh ðn1 ; n2 Þ þ b1 wðn1 ; n2 Þ; yðn1 ; n2 Þ ¼ cT1 xh ðn1 ; n2 Þ þ d1 wðn1 ; n2 Þ; where 2
0 ? 61 6 A1 ¼ 6 4 &
0 0
0 2 3 1 607 6 7 b1 ¼ 6 7 ; 4^5
1
0
3 a0 a1 7 7 7; ^ 5 aM1
ð4Þ
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cT1 ¼ g0
g1
?gM1
¼ aM1
aM2
2
6 6 ?a0 6 4
1
aM1
?
1
&
a0
31
7 7 7 ; & aM1 5 1 ^
d1 ¼ 1: In this model, xh ðn1 ; n2 Þ is an M 1 horizontal state vector, wðn1 ; n2 Þ is a scalar input, and yðn1 ; n2 Þ is a scalar output. Likewise, H1 ðz1 ; z2 Þ can be rewritten as PN1 j j¼0 lj ðz1 Þz2 H1 ðz1 ; z2 Þ ¼ lN ðz1 Þ þ PN ; ð5Þ j j¼0 bj z2 where M X
lN ðz1 Þ ¼
biN ziM ; 1
i¼0
lj ðz1 Þ ¼
M X
bij ziM bj 1
i¼0
M X
biN ziM 1
ð j ¼ 0; y; N 1Þ:
i¼0
From Eq. (5) and an observable canonical form, the state-space model of Eq. (5) is given by xv ðz1 ; n2 þ 1Þ ¼ A4 xv ðz1 ; n2 Þ þ kðz1 Þuðz1 ; n2 Þ; wðz1 ; n2 Þ ¼ cT2 xv ðz1 ; n2 Þ þ lN ðz1 Þuðz1 ; n2 Þ; where
2
0 61 6 A4 ¼ 6 4
? &
0 0
ð6Þ
3 b0 b1 7 7 7; ^ 5
1 bN1 3 l0 ðz1 Þ 6 7 kðz1 Þ ¼ 4 ^ 5; lN1 ðz1 Þ
cT2 ¼ 0 ? 0 1 : 0 2
xv ðz1 ; n2 Þ is an N 1 vertical state vector, uðz1 ; n2 Þ is a scalar input, and wðz1 ; n2 Þ is a scalar output with z-transformation in terms of z1 : Thus, defined the inverse z-transformation of lj ðz1 Þuðz1 ; n2 Þ as Z11 ½lj ðz1 Þuðz1 ; n2 Þ ¼ fj ðn1 ; n2 Þ ð j ¼ 0; y; NÞ:
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Eq. (7) can be apparently described as xv ðn1 ; n2 þ 1Þ ¼ A4 xv ðn1 ; n2 Þ þ f ðn1 ; n2 Þ; wðn1 ; n2 Þ ¼ cT2 xv ðn1 ; n2 Þ þ fN ðn1 ; n2 Þ;
ð7Þ
where 2 6 f ðn1 ; n2 Þ ¼ 4
f0 ðn1 ; n2 Þ
3
7 ^ 5: fN1 ðn1 ; n2 Þ
ð8Þ
From Eqs. (7) and (8), the input f ðn1 ; n2 Þ is considered as a bank of 1-D FIR filters with a transfer function, lk ðz1 ; z2 Þ: Finally, arranging Eqs. (4) and (7) by removing wðn1 ; n2 Þ gives us the following desired result: " # # " A1 A2 xh ðn1 ; n2 Þ xh ðn1 þ 1; n2 Þ ¼ þ f ðn1 ; n2 Þ; 0 A4 xv ðn1 ; n2 þ 1Þ xv ðn1 ; n2 Þ " # T T xh ðn1 ; n2 Þ ð9Þ yðn1 ; n2 Þ ¼ c1 c2 þ fN ðn1 ; n2 Þ: xv ðn1 ; n2 Þ Note that A2 ¼ b1 cT2 and f ðn1 ; n2 Þ is redefined as 3 2 fN ðn1 ; n2 Þ 7 6 0 7 6 7 6 7 6 ^ 7 6 7 6 0 f ðn1 ; n2 Þ ¼ 6 7 7 6 6 f0 ðn1 ; n2 Þ 7 7 6 7 6 ^ 5 4 fN1 ðn1 ; n2 Þ
ð10Þ
in which each element is described as fj ðn1 ; n2 Þ ¼
M X
bij uðn1 i; n2 Þ
ð j ¼ 0; y; NÞ:
ð11Þ
i¼0
This model is, consequently, identified with the combination of 2 filters. One is a filter bank, f ðn1 ; n2 Þ; and other is an IIR filter with multiple inputs from a filter bank output and one output. Also, the parallel processing enables this model to calculate a filtering result with the time required for 1 multiplication and 1 addition except for an initial delay [1].
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3. Adaptive algorithm and its evaluation In this section, an adaptive algorithm derived from the DLMS algorithm and the corresponding computational time is estimated in order to show the effect of the parallel processing. 3.1. Adaptive algorithm Defined an error signal eðn1 ; n2 Þ as eðn1 ; n2 Þ ¼ rðn1 ; n2 Þ yðn1 ; n2 Þ;
ð12Þ
where rðn1 ; n2 Þ is a reference signal, filter coefficients are updated so that the mean square error signal E½e2 ðn1 ; n2 Þ can be minimized during adaptation process with a gradient signal. Assuming the data size to be N1 N2 ; k ¼ l1 l2 where l1 ¼ N1 n2 þ n1 and l2 are an iteration counter, an equation of updating coefficients, pðkÞ; is given by qE½e2 ðn1 ; n2 Þ ; pðk þ 1Þ ¼ pðkÞ meðn1 ; n2 Þ qpðkÞ
ð13Þ
where m is a step-size parameter which controls convergence property of the algorithm. The DLMS, however, not only uses an instantaneous value of error signal instead of the mean square error signal for approximation, but also replaces the current value of error signal with a past value which is supposed to be already calculated so that we can reduce the computational time. Then, from Eq. (13), we obtain qyðn1 ; n2 Þ pðk þ 1Þ ¼ pðkÞ meðk l1 ðl2 qÞÞ ; qpðkÞ
ð14Þ
where q means a certain amount of delay. Therefore, Eq. (14) uses the error signal which is calculated before q iterations. With this equation, we can calculate the error signal separately from updating the coefficients, resulting in parallel processing between error signal calculation and coefficient updates. From Eq. (14), we can derive the updating formula for each system coefficient in the following: a1i ðk þ 1Þ ¼ a1i ðkÞ þ 2meðk l1 ðl2 qÞÞa1i ðn1 ; n2 Þ; a4i ðk þ 1Þ ¼ a4i ðkÞ þ 2meðk l1 ðl2 qÞÞa4i ðn1 ; n2 Þ; biN ðk þ 1Þ ¼ biN ðkÞ þ 2meðk l1 ðl2 qÞÞfb1i1 ðn1 ; n2 Þ þ uðn1 þ i M; n2 Þg; bij ðk þ 1Þ ¼ bij ðkÞ þ 2meðk l1 ðl2 qÞÞb2ij ðn1 ; n2 Þ; c1i ðk þ 1Þ ¼ c1i ðkÞ þ 2meðk l1 ðl2 qÞÞxhi ðn1 ; n2 Þ;
ð15Þ
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where i and j are the index of the elements of each matrix. In addition, a1i ; a4i ; b1i1 ; b2ij can be calculated by a1 ðn1 þ 1; n2 Þ ¼ AT1 a1 ðn1 ; n2 Þ þ c1 xhM ðn1 ; n2 Þ; a2 ðn1 þ 1; n2 Þ ¼ AT1 a2 ðn1 ; n2 Þ þ c1 xvN ðn1 ; n2 Þ; a4 ðn1 ; n2 þ 1Þ ¼ AT2 a2 ðn1 ; n2 Þ þ AT4 a4 ðn1 ; n2 Þ þ c2 xvN ðn1 ; n2 Þ; b1i ðn1 þ 1; n2 Þ ¼ b1i ðn1 ; n2 ÞA1 þ cT1 uðn1 þ i M; n2 Þ; b2i ðn1 ; n2 þ 1Þ ¼ b1i ðn1 ; n2 ÞA2 þ b2i ðn1 ; n2 ÞA4 þ cT2 uðn1 þ i M; n2 Þ ði ¼ 0; y; MÞ;
ð16Þ
where a1 ðn1 ; n2 Þ ¼ ½a11 ðn1 ; n2 Þ; y; a1M ðn1 ; n2 Þ T ; a2 ðn1 ; n2 Þ ¼ ½a21 ðn1 ; n2 Þ; y; a2M ðn1 ; n2 Þ T ; a4 ðn1 ; n2 Þ ¼ ½a41 ðn1 ; n2 Þ; y; a4N ðn1 ; n2 Þ T ; b1i ðn1 ; n2 Þ ¼ ½b1i1 ðn1 ; n2 Þ; y; b1iM ðn1 ; n2 Þ ; b2i ðn1 ; n2 Þ ¼ ½b2i0 ðn1 ; n2 Þ; y; b2iN1 ðn1 ; n2 Þ : As a result, applying DLMS algorithm allows us to separate the error signal calculation and coefficients updating to realize the 2-D adaptive state-space filter suitable for parallel processing. 3.2. Evaluation In this section, the processing time of the proposed algorithm is estimated. Applying the DLMS algorithm allows us to calculate the entire system as a pipeline structure because error signal calculation and coefficients updating are separately calculated during adaptive process. First we consider the minimum time required for the calculation of the output, yðn1 ; n2 Þ; of the adaptive filter with parallel processing. In this case, we have to consider the time required for the calculation of the filter bank, the state variables, and the output portion separately. From Eq. (11), fj ðn1 ; n2 Þ can be calculated by M þ 1 multiplications and M additions. Executing M þ 1 multiplications in parallel at the same time and preparing enough adders for M additions to be executed in parallel, fj ðn1 ; n2 Þ can be calculated within the time for one multiplication plus Jlog2 Mn additions where J n indicates the minimum integer larger than : With regard to the state vector, each element of both xv ðn1 ; n2 þ 1Þ and xh ðn1 þ 1; n2 Þ can be calculated at most within the time for one multiplication and one addition. In the calculation of yðn1 ; n2 Þ ¼ cT1 xh ðn1 ; n2 Þ þ cT2 xv ðn1 ; n2 Þ þ fN ðn1 ; n2 Þ; the third term is supposed to have been already calculated above and the second term has no calculation because of cT2 ¼ ½0 ? 0 1 : Therefore, yðn1 ; n2 Þ can be
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calculated with M multiplications and M additions. Executing M multiplications in parallel and preparing enough adder for M additions to be executed in parallel, such as filter bank portion, we can complete yðn1 ; n2 Þ within one multiplication plus Jlog2 Mn additions. In the end, the minimum calculation time Ty for this adaptive filter output results in Ty ¼ 2tp þ 2Jlog2 Mnta ;
ð17Þ
where ta ðtp Þ indicates the time for one addition (multiplication). Next, the time for the calculation of gradients is evaluated. From Eq. (16), the only Mth element of a1 ðn1 þ 1; n2 Þ; a2 ðn1 þ 1; n2 Þ and b11 ðn1 ; n2 Þ has M þ 1 multiplications and M additions, respectively. The other elements need only one multiplication and one addition. On the other hand, the Nth element of a4 ðn1 ; n2 þ 1Þ and b21 ðn1 ; n2 Þ have N multiplications and N additions and the other elements need no calculation. Therefore, executing the multiplications in parallel and preparing enough adders for parallel processing, the gradients can be calculated within the time for one multiplication and Jlog2 Mn: After all, the minimum time Tg for gradients ends up Tg ¼ tp þ Jlog2 Mnta :
ð18Þ
Obviously Tg oTy holds, then the calculation of gradients can be completed within the time for the filter output calculation. Finally, we evaluate the time for updating coefficients. From Eqs. (12) and (14), the coefficients can be updated within the time Tc with two multiplications and two additions, i.e. Tc ¼ 2tp þ 2ta :
ð19Þ
However, one of the multiplications for updating formula contains an error signal which is supposed to have been already calculated in the past, so we can complete the multiplication without waiting for the calculation results of gradients. Executing the multiplication and the calculation of the gradients at the same time allows us to reduce one multiplication from Tc : So the time from starting out the gradients calculation to the completion of updating coefficients, Tgc ; results in Tgc ¼ tp þ 2ta þ Tg ¼ 2tp þ ðJlog2 Mn þ 2Þta :
ð20Þ
For example, consider the use of q ¼ 1; the relation of Ty XTc can be obviously held. Therefore, the parallel processing with updating coefficients and error signal calculation is feasible and the minimum time Tmin to be required for updating the coefficients for each input is given as Tmin ¼ Ty ¼ 2tp þ 2Jlog2 Mnta
ð21Þ
For the comparison, we consider another implementation where an adaptive filter is comprised of single multiplier and single adder derived from the following
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difference equation: yðn1 ; n2 Þ ¼
M X N X i¼0
bij uðn1 i; n2 jÞ þ
j¼0
M N X X
li mj yðn1 i; n2 jÞ:
i¼0 j¼0 iþja0
The time required for calculating of the output is [6] Ty ¼ tp f2ðM þ 1ÞðN þ 1Þ 1g þ ta f2ðM þ 1ÞðN þ 1Þ 2g and the time for updating the coefficients Tc ¼ tp f3MN þ 2M þ 2N þ 1g þ ta f3MN þ 2M þ 2N þ 1g: Consequently, the minimum time Tmin required from input processing to updating the coefficients takes Tmin ¼ tp f5MN þ 4M þ 4Ng þ ta f5MN þ 4M þ 4N þ 1g:
ð22Þ
Comparing Eq. (21) to (22), the method we proposed is more suitable in terms of high-speed processing than that implemented with difference equations. Furthermore, when we compare our algorithm to another implementation for parallel processing addressed in Ref. [3] as Tmin ¼ 4tp þ ð4 þ Jlog2 ðM þ 1Þn þ Jlog2 NnÞta
ð23Þ
and in Ref. [4] as Tmin ¼ 4tp þ 2ðJlog2 Mn þ 1Þta :
ð24Þ
The advantage of our algorithm in terms of the minimum calculation time Tmin is obvious. In addition, another evaluation approach is to count the number of multipliers and adders required for implementation. To calculate the filter output yðn1 ; n2 Þ; we need ðM þ 1ÞðN þ 1Þ multipliers and MðN þ 1Þ adders for filter bank, M þ N multipliers and M þ N adders for state-space vectors. So Eq. (10) means that ðM þ 1ÞðN þ 1Þ þ ðM þ NÞ þ M multipliers and MðN þ 1Þ þ ðM þ NÞ þ M adders for the filter output yðn1 ; n2 Þ are required. Regarding the calculation of gradient vectors, we need 2M multipliers and 2M 1 adders for a1 ðn1 þ 1; n2 Þ and a2 ðn1 þ 1; n2 Þ; N multipliers, and N adders for a4 ðn1 ; n2 þ 1Þ; 2MðM þ 1Þ multipliers and ð2M 1ÞðM þ 1Þ adders for b1i ðn1 þ 1; n2 Þ and NðM þ 1Þ multipliers, and ð2M 1ÞðM þ 1Þ adders for b2i ðn1 ; n2 þ 1Þ: Totally, 2M 2 þ 7M þ MN þ 2N þ 1 multipliers and 2M 2 þ 7M þ MN þ 2N 2 adders for the calculation of gradient vectors are required. Regarding the coefficients updates, we need only 1 multiplier and 1 adder for each coefficient, since Eq. (15) indicates that we have 2 multiplications and 2 additions which cannot be processed in parallel. In the end, we need M þ N þ ðM þ 1Þ þ NðM þ 1Þ þ M multipliers and M þ N þ ðM þ 1Þ þ NðM þ 1Þ þ M adders for coefficient updates. Table 1 shows us the number of multipliers and adders for some filter orders.
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Table 1 The number of multipliers and adders M¼N
2
3
4
5
Multiplier Adder
61 55
105 98
159 151
223 214
4. Simulation To demonstrate the effectiveness of the proposed algorithm, we have carried out the simulation applied to 2-D digital filter design in the spatial domain. The unit sample response of the Gaussian filter is given by the following equation: rðn1 ; n2 Þ ¼ 0:256322 exp ½0:103203fðn1 4Þ2 þ ðn2 4Þ2 g
ðn1 ; n2 ÞASh ;
where Sh ¼ fðn1 ; n2 Þj0pn1 p10; 0pn2 p10g: The order of the filter was chosen as M ¼ N ¼ 3; and the input signal was assumed to be 1 ðn1 ¼ n2 ¼ 0Þ; uðn1 ; n2 Þ ¼ 0 ðotherwiseÞ: The step-size parameter was chosen to be m ¼ 0:0007 and the data frðn1 ; n2 Þjðn1 ; n2 ÞASh g was used repeatedly during the adaptation. The delay q was set to 1. The filter coefficients with separable denominator was obtained at k ¼ 121; 000; 000 as 2 3 0:0 0:0 0:256657 6 7 A1 ¼ 4 1:0 0:0 0:403207 5; 2
0:0 1:0
0:436897
0:0 0:0 6 A4 ¼ 4 1:0 0:0
3 0:271496 7 1:144697 5;
0:0 1:0 0:005606 6 0:013204 6 B1 ¼ 6 4 0:015765
1:757631 3 0:0 0:0 0:0 0:0 7 7 7; 0:0 0:0 5
0:009470 0:007669 6 0:018052 6 B2 ¼ 6 4 0:021553
0:0 0:0 0:001899 0:004453
2
2
0:012950
0:005315 0:003201
3 0:011712 0:027575 7 7 7 0:032922 5 0:019780
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c1 ¼ 0:389255
1:397843
441
1:209952 :
Moreover, the deviation of the resulting filter is evaluated by hP i1=2 N 1 PN 2 2 n1 ¼0 n2 ¼0 ðrðn1 ; n2 Þ yðn1 ; n2 ÞÞ e¼ 100; hP i1=2 N 1 PN 2 2 rðn ; n Þ 1 2 n1 ¼0 n2 ¼0 where N1 ¼ N2 ¼ 10: The resulting filter obtained at k ¼ 121; 000; 000 had e2 ¼ 2:69% and the initial error was e2 ¼ 92:87%: The 2-D digital filters designed by our proposed method have the advantage of its filter structure suitable for parallel processing. Finally, the numerical example shows the effect of the proposed adaptive algorithm.
5. Conclusion The new parallel-form realization of 2-D adaptive filters, based on the Roesser local state-space model with separable denominator, has been proposed. The DLMS method has been applied to implement the adaptive algorithm and proved out to reduce the minimum computation time for each input. The proposed adaptive filter has been developed by combining the parallel-form realization with the adaptive algorithm for 2-D separable-denominator digital filters. Also, the computational time required for updating the coefficients has been investigated and shown that two multiplications and Jlog2 Mn þ 2 additions were required in each iteration. Finally, the effectiveness of the proposed algorithm has been demonstrated by applying it to 2-D digital filter design in the spatial domain.
References [1] M.Y. Dabbagh, W.E. Alexander, Multiprocessor implementation of 2-D denominator-separable digital filters for real-time processing, IEEE Trans. Acoust. Speech Signal Process. ASSP-37 (1989) 872–881. [2] T. Hinamoto, M. Muneyasu, Realization of 2-D separable-denominator digital filters for real-time and parallel processing, Trans. IEICE, J74-A 1 (1991) 39–46 (in Japanese). [3] M. Muneyasu, T. Hinamoto, Parallel-from realization of 2-D adaptive separable-denominator statespace filters, Trans. IEICE J77-A 6 (1994) 844–853 (in Japanese). [4] M. Muneyasu, N. Ishizaki, T. Hinamoto, A realization of 2-D adaptive separable-denominator statespace filters for parallel processing, Proceedings of the JTC-CSCC’95, July 1995, pp. 9–12. [5] M. Muneyasu, E. Uemoto, T. Hinamoto, A 2-D cascade-parallel adaptive IIR filter using backpropagation method, J. Franklin Institute 333(B) (4) (1996) 453–465. [6] M. Muneyasu, N. Ishizaki, T. Hinamoto, A realization of 3-D adaptive separable-denominator statespace filters for parallel processing, Proceedings of the ISCAS’97, Vol. 4, June 1997, pp. 2300–2303. [7] G. Long, F. Ling, J.G. Proakis, The LMS algorithm with delayed coefficient adaptation, IEEE Trans. Acoust. Speech Signal Process. ASSP-37 9 (1989) 1397–1405. [8] M. Muneyasu, T. Hinamoto, Two-dimensional adaptive state-space filters using LMS method, Proceedings of the ISCAS’94, Vol. 2, 1993, pp. 65–68.
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[9] M. Muneyasu, N. Ishizaki, T. Hinamoto, A new parallel-form realization of 2-D adaptive separable-denominator state-space filters using delayed LMS algorithm, IEEE International Workshop on Intelligent Signal Processing and Communication Systems, ISPACS’98, Vol. 2, Nov. 1998, pp. 777–781. [10] M.M. Hadhoud, D.W. Thomas, The two-dimensional adaptive LMS(TDLMS) algorithm, IEEE Trans. Circuits Syst. CAS-35 5 (1988) 485–494. [11] T. Hinamoto, A. Doi, M. Muneyasu, 2-D adaptive state-space filters based on the Fornasini– Marchesini second model, IEEE Trans. Circuits Syst. II 44 (8) (1997) 667–670. [12] M. Muneyasu, T. Hinamoto, A new 2-D adaptive filter using affine projection algorithm, Proceedings of the ISCAS’98, MPA1-7, June 1998.
Journal of the Franklin Institute 341 (2004) 431–442
A new realization of 2-D adaptive separabledenominator state-space filters based on DLMS algorithm suitable for parallel processing$ Naoya Ishizakia, Mitsuji Muneyasub,*, Takao Hinamotoa a
Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi, Hiroshima 739-8527, Japan b Faculty of Engineering, Kansai University, 3-3-35 Yamate-cho, Suita 564-8680, Japan
Abstract In the application of 2-D adaptive filters, a large amount of data must be processed and real-time processing is often required. In this paper, a new parallel-form realization of 2-D adaptive separable-denominator state-space filters suitable for high-speed processing is proposed. First, the 2-D local state-space model suitable for parallel processing is introduced. Next, the adaptive algorithm for this model is developed. This algorithm is based on the delayed least mean square (DLMS) method. In addition, the computation time required for the update of the coefficients is investigated. Finally, the proposed technique is applied to the design of 2-D digital filters in the spatial domain. r 2004 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: 2-D adaptive filter; State-space model; Parallel processing; High-speed processing
1. Introduction An adaptive filter has been applied to various fields of signal processing, such as echo canceller, system identification and adaptive equalizer. Nowadays, it also has been extended to process multi-dimensional signals for the purpose of the $
The results in this paper were partially presented at the 1998 IEEE International Workshop on Intelligent Signal Processing and Communication Systems. *Corresponding author. Tel.: +81-824-24-7682; fax: +81-824-22-7195. E-mail address: [email protected] (M. Muneyasu). 0016-0032/$30.00 r 2004 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2004.04.002
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noise reduction of image data, multi-dimensional system identification and multi-dimensional digital filter design, which results that multi-dimensional adaptive filters have been proposed and their researches are getting more popular [8,10–12]. One concern, however, is that a multi-dimensional adaptive filter not only has to process a large amount of data, like an image, but sometimes has to perform realtime processing to handle such large data. Therefore, high-speed processing should be required for high-dimensional adaptive filters. One of the ideas for that is to apply the parallel processing structure of adaptive filters. For 2-D digital filters, several methods suitable for the parallel processing have been proposed [1,2]. Besides, we have some studies on 2-D adaptive filters suitable for the parallel processing where 2-D digital filter for parallel processing is applied to the structure of 2-D adaptive filters [3–5] and also extended to 3-D case [6]. This paper proposes a new parallel-form realization of a 2-D adaptive filter based on the Roesser local state-space model with separable denominator. The DLMS algorithm [7] is adopted as an adaptive algorithm in this realization. The class of separable denominator is one of the most important classes, since almost frequency specification can be implemented. The effective parallel form realization can also be achieved. However, such realizations by the general local state-space model are difficult because of its complicated feedback structure. At first, the parallel-form realization of a 2-D digital filter with separable denominator proposed by Dabbagh and Alexander [1] is brought up for consideration. Then, a new adaptive algorithm is introduced by the combination of the DLMS algorithm and the adaptive algorithm for a 2-D state-space digital filter with separable denominator [8]. The computational time required for the filtering and the coefficient update is also evaluated to prove that the use of the DLMS algorithm leads a faster algorithm than the algorithms reported in [3,4]. Finally, the effectiveness of the proposed technique is confirmed by applying it to 2-D digital filter design in the spatial domain.
2. Parallel implementation of 2-D state-space filters with separable denominator In this section, the parallel-form implementation of 2-D state-space filter with separable denominator reported in [5] is introduced to be prepared for deriving the corresponding adaptive algorithm [9]. Consider the following irreducible 2-D separable-denominator transfer function:
Hðz1 ; z2 Þ ¼
Nðz1 ; z2 Þ ; D1 ðz1 ; z2 ÞD2 ðz1 ; z2 Þ
ð1Þ
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where Nðz1 ; z2 Þ ¼
M X N X i¼0
D1 ðz1 Þ ¼
M X
bij zi1 zj2 ;
j¼0
ai zi1 ;
i¼0
D2 ðz2 Þ ¼
N X
bj zj2 ;
j¼0
aM ¼ bN ¼ 1: The 2-D separable denominator transfer function expressed in Eq. (1) can be represented by PM PN iM j z2 1 i¼0 j¼0 bij z1 PM Hðz1 ; z2 Þ ¼ PN j i i¼0 ai z1 j¼0 bj z2 ¼ H1 ðz1 ; z2 Þ H2 ðz1 Þ:
ð2Þ
Therefore, Hðz1 ; z2 Þ can be described by the cascade form of H1 ðz1 ; z2 Þ and H2 ðz1 Þ: H2 ðz1 Þ is rewritten as H2 ðz1 Þ ¼ PM
zM ¼ PM 1 i i¼0 ai z1
1
ai ziM 1 PM1 i¼0 ai zi1 ¼ 1 þ PM : i i¼0 ai z1 i¼0
ð3Þ
Then, by employing a controllable canonical form, the implementation of the statespace model of H2 ðz1 Þ is shown as xh ðn1 þ 1; n2 Þ ¼ A1 xh ðn1 ; n2 Þ þ b1 wðn1 ; n2 Þ; yðn1 ; n2 Þ ¼ cT1 xh ðn1 ; n2 Þ þ d1 wðn1 ; n2 Þ; where 2
0 ? 61 6 A1 ¼ 6 4 &
0 0
0 2 3 1 607 6 7 b1 ¼ 6 7 ; 4^5
1
0
3 a0 a1 7 7 7; ^ 5 aM1
ð4Þ
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cT1 ¼ g0
g1
?gM1
¼ aM1
aM2
2
6 6 ?a0 6 4
1
aM1
?
1
&
a0
31
7 7 7 ; & aM1 5 1 ^
d1 ¼ 1: In this model, xh ðn1 ; n2 Þ is an M 1 horizontal state vector, wðn1 ; n2 Þ is a scalar input, and yðn1 ; n2 Þ is a scalar output. Likewise, H1 ðz1 ; z2 Þ can be rewritten as PN1 j j¼0 lj ðz1 Þz2 H1 ðz1 ; z2 Þ ¼ lN ðz1 Þ þ PN ; ð5Þ j j¼0 bj z2 where M X
lN ðz1 Þ ¼
biN ziM ; 1
i¼0
lj ðz1 Þ ¼
M X
bij ziM bj 1
i¼0
M X
biN ziM 1
ð j ¼ 0; y; N 1Þ:
i¼0
From Eq. (5) and an observable canonical form, the state-space model of Eq. (5) is given by xv ðz1 ; n2 þ 1Þ ¼ A4 xv ðz1 ; n2 Þ þ kðz1 Þuðz1 ; n2 Þ; wðz1 ; n2 Þ ¼ cT2 xv ðz1 ; n2 Þ þ lN ðz1 Þuðz1 ; n2 Þ; where
2
0 61 6 A4 ¼ 6 4
? &
0 0
ð6Þ
3 b0 b1 7 7 7; ^ 5
1 bN1 3 l0 ðz1 Þ 6 7 kðz1 Þ ¼ 4 ^ 5; lN1 ðz1 Þ
cT2 ¼ 0 ? 0 1 : 0 2
xv ðz1 ; n2 Þ is an N 1 vertical state vector, uðz1 ; n2 Þ is a scalar input, and wðz1 ; n2 Þ is a scalar output with z-transformation in terms of z1 : Thus, defined the inverse z-transformation of lj ðz1 Þuðz1 ; n2 Þ as Z11 ½lj ðz1 Þuðz1 ; n2 Þ ¼ fj ðn1 ; n2 Þ ð j ¼ 0; y; NÞ:
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Eq. (7) can be apparently described as xv ðn1 ; n2 þ 1Þ ¼ A4 xv ðn1 ; n2 Þ þ f ðn1 ; n2 Þ; wðn1 ; n2 Þ ¼ cT2 xv ðn1 ; n2 Þ þ fN ðn1 ; n2 Þ;
ð7Þ
where 2 6 f ðn1 ; n2 Þ ¼ 4
f0 ðn1 ; n2 Þ
3
7 ^ 5: fN1 ðn1 ; n2 Þ
ð8Þ
From Eqs. (7) and (8), the input f ðn1 ; n2 Þ is considered as a bank of 1-D FIR filters with a transfer function, lk ðz1 ; z2 Þ: Finally, arranging Eqs. (4) and (7) by removing wðn1 ; n2 Þ gives us the following desired result: " # # " A1 A2 xh ðn1 ; n2 Þ xh ðn1 þ 1; n2 Þ ¼ þ f ðn1 ; n2 Þ; 0 A4 xv ðn1 ; n2 þ 1Þ xv ðn1 ; n2 Þ " # T T xh ðn1 ; n2 Þ ð9Þ yðn1 ; n2 Þ ¼ c1 c2 þ fN ðn1 ; n2 Þ: xv ðn1 ; n2 Þ Note that A2 ¼ b1 cT2 and f ðn1 ; n2 Þ is redefined as 3 2 fN ðn1 ; n2 Þ 7 6 0 7 6 7 6 7 6 ^ 7 6 7 6 0 f ðn1 ; n2 Þ ¼ 6 7 7 6 6 f0 ðn1 ; n2 Þ 7 7 6 7 6 ^ 5 4 fN1 ðn1 ; n2 Þ
ð10Þ
in which each element is described as fj ðn1 ; n2 Þ ¼
M X
bij uðn1 i; n2 Þ
ð j ¼ 0; y; NÞ:
ð11Þ
i¼0
This model is, consequently, identified with the combination of 2 filters. One is a filter bank, f ðn1 ; n2 Þ; and other is an IIR filter with multiple inputs from a filter bank output and one output. Also, the parallel processing enables this model to calculate a filtering result with the time required for 1 multiplication and 1 addition except for an initial delay [1].
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3. Adaptive algorithm and its evaluation In this section, an adaptive algorithm derived from the DLMS algorithm and the corresponding computational time is estimated in order to show the effect of the parallel processing. 3.1. Adaptive algorithm Defined an error signal eðn1 ; n2 Þ as eðn1 ; n2 Þ ¼ rðn1 ; n2 Þ yðn1 ; n2 Þ;
ð12Þ
where rðn1 ; n2 Þ is a reference signal, filter coefficients are updated so that the mean square error signal E½e2 ðn1 ; n2 Þ can be minimized during adaptation process with a gradient signal. Assuming the data size to be N1 N2 ; k ¼ l1 l2 where l1 ¼ N1 n2 þ n1 and l2 are an iteration counter, an equation of updating coefficients, pðkÞ; is given by qE½e2 ðn1 ; n2 Þ ; pðk þ 1Þ ¼ pðkÞ meðn1 ; n2 Þ qpðkÞ
ð13Þ
where m is a step-size parameter which controls convergence property of the algorithm. The DLMS, however, not only uses an instantaneous value of error signal instead of the mean square error signal for approximation, but also replaces the current value of error signal with a past value which is supposed to be already calculated so that we can reduce the computational time. Then, from Eq. (13), we obtain qyðn1 ; n2 Þ pðk þ 1Þ ¼ pðkÞ meðk l1 ðl2 qÞÞ ; qpðkÞ
ð14Þ
where q means a certain amount of delay. Therefore, Eq. (14) uses the error signal which is calculated before q iterations. With this equation, we can calculate the error signal separately from updating the coefficients, resulting in parallel processing between error signal calculation and coefficient updates. From Eq. (14), we can derive the updating formula for each system coefficient in the following: a1i ðk þ 1Þ ¼ a1i ðkÞ þ 2meðk l1 ðl2 qÞÞa1i ðn1 ; n2 Þ; a4i ðk þ 1Þ ¼ a4i ðkÞ þ 2meðk l1 ðl2 qÞÞa4i ðn1 ; n2 Þ; biN ðk þ 1Þ ¼ biN ðkÞ þ 2meðk l1 ðl2 qÞÞfb1i1 ðn1 ; n2 Þ þ uðn1 þ i M; n2 Þg; bij ðk þ 1Þ ¼ bij ðkÞ þ 2meðk l1 ðl2 qÞÞb2ij ðn1 ; n2 Þ; c1i ðk þ 1Þ ¼ c1i ðkÞ þ 2meðk l1 ðl2 qÞÞxhi ðn1 ; n2 Þ;
ð15Þ
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where i and j are the index of the elements of each matrix. In addition, a1i ; a4i ; b1i1 ; b2ij can be calculated by a1 ðn1 þ 1; n2 Þ ¼ AT1 a1 ðn1 ; n2 Þ þ c1 xhM ðn1 ; n2 Þ; a2 ðn1 þ 1; n2 Þ ¼ AT1 a2 ðn1 ; n2 Þ þ c1 xvN ðn1 ; n2 Þ; a4 ðn1 ; n2 þ 1Þ ¼ AT2 a2 ðn1 ; n2 Þ þ AT4 a4 ðn1 ; n2 Þ þ c2 xvN ðn1 ; n2 Þ; b1i ðn1 þ 1; n2 Þ ¼ b1i ðn1 ; n2 ÞA1 þ cT1 uðn1 þ i M; n2 Þ; b2i ðn1 ; n2 þ 1Þ ¼ b1i ðn1 ; n2 ÞA2 þ b2i ðn1 ; n2 ÞA4 þ cT2 uðn1 þ i M; n2 Þ ði ¼ 0; y; MÞ;
ð16Þ
where a1 ðn1 ; n2 Þ ¼ ½a11 ðn1 ; n2 Þ; y; a1M ðn1 ; n2 Þ T ; a2 ðn1 ; n2 Þ ¼ ½a21 ðn1 ; n2 Þ; y; a2M ðn1 ; n2 Þ T ; a4 ðn1 ; n2 Þ ¼ ½a41 ðn1 ; n2 Þ; y; a4N ðn1 ; n2 Þ T ; b1i ðn1 ; n2 Þ ¼ ½b1i1 ðn1 ; n2 Þ; y; b1iM ðn1 ; n2 Þ ; b2i ðn1 ; n2 Þ ¼ ½b2i0 ðn1 ; n2 Þ; y; b2iN1 ðn1 ; n2 Þ : As a result, applying DLMS algorithm allows us to separate the error signal calculation and coefficients updating to realize the 2-D adaptive state-space filter suitable for parallel processing. 3.2. Evaluation In this section, the processing time of the proposed algorithm is estimated. Applying the DLMS algorithm allows us to calculate the entire system as a pipeline structure because error signal calculation and coefficients updating are separately calculated during adaptive process. First we consider the minimum time required for the calculation of the output, yðn1 ; n2 Þ; of the adaptive filter with parallel processing. In this case, we have to consider the time required for the calculation of the filter bank, the state variables, and the output portion separately. From Eq. (11), fj ðn1 ; n2 Þ can be calculated by M þ 1 multiplications and M additions. Executing M þ 1 multiplications in parallel at the same time and preparing enough adders for M additions to be executed in parallel, fj ðn1 ; n2 Þ can be calculated within the time for one multiplication plus Jlog2 Mn additions where J n indicates the minimum integer larger than : With regard to the state vector, each element of both xv ðn1 ; n2 þ 1Þ and xh ðn1 þ 1; n2 Þ can be calculated at most within the time for one multiplication and one addition. In the calculation of yðn1 ; n2 Þ ¼ cT1 xh ðn1 ; n2 Þ þ cT2 xv ðn1 ; n2 Þ þ fN ðn1 ; n2 Þ; the third term is supposed to have been already calculated above and the second term has no calculation because of cT2 ¼ ½0 ? 0 1 : Therefore, yðn1 ; n2 Þ can be
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calculated with M multiplications and M additions. Executing M multiplications in parallel and preparing enough adder for M additions to be executed in parallel, such as filter bank portion, we can complete yðn1 ; n2 Þ within one multiplication plus Jlog2 Mn additions. In the end, the minimum calculation time Ty for this adaptive filter output results in Ty ¼ 2tp þ 2Jlog2 Mnta ;
ð17Þ
where ta ðtp Þ indicates the time for one addition (multiplication). Next, the time for the calculation of gradients is evaluated. From Eq. (16), the only Mth element of a1 ðn1 þ 1; n2 Þ; a2 ðn1 þ 1; n2 Þ and b11 ðn1 ; n2 Þ has M þ 1 multiplications and M additions, respectively. The other elements need only one multiplication and one addition. On the other hand, the Nth element of a4 ðn1 ; n2 þ 1Þ and b21 ðn1 ; n2 Þ have N multiplications and N additions and the other elements need no calculation. Therefore, executing the multiplications in parallel and preparing enough adders for parallel processing, the gradients can be calculated within the time for one multiplication and Jlog2 Mn: After all, the minimum time Tg for gradients ends up Tg ¼ tp þ Jlog2 Mnta :
ð18Þ
Obviously Tg oTy holds, then the calculation of gradients can be completed within the time for the filter output calculation. Finally, we evaluate the time for updating coefficients. From Eqs. (12) and (14), the coefficients can be updated within the time Tc with two multiplications and two additions, i.e. Tc ¼ 2tp þ 2ta :
ð19Þ
However, one of the multiplications for updating formula contains an error signal which is supposed to have been already calculated in the past, so we can complete the multiplication without waiting for the calculation results of gradients. Executing the multiplication and the calculation of the gradients at the same time allows us to reduce one multiplication from Tc : So the time from starting out the gradients calculation to the completion of updating coefficients, Tgc ; results in Tgc ¼ tp þ 2ta þ Tg ¼ 2tp þ ðJlog2 Mn þ 2Þta :
ð20Þ
For example, consider the use of q ¼ 1; the relation of Ty XTc can be obviously held. Therefore, the parallel processing with updating coefficients and error signal calculation is feasible and the minimum time Tmin to be required for updating the coefficients for each input is given as Tmin ¼ Ty ¼ 2tp þ 2Jlog2 Mnta
ð21Þ
For the comparison, we consider another implementation where an adaptive filter is comprised of single multiplier and single adder derived from the following
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difference equation: yðn1 ; n2 Þ ¼
M X N X i¼0
bij uðn1 i; n2 jÞ þ
j¼0
M N X X
li mj yðn1 i; n2 jÞ:
i¼0 j¼0 iþja0
The time required for calculating of the output is [6] Ty ¼ tp f2ðM þ 1ÞðN þ 1Þ 1g þ ta f2ðM þ 1ÞðN þ 1Þ 2g and the time for updating the coefficients Tc ¼ tp f3MN þ 2M þ 2N þ 1g þ ta f3MN þ 2M þ 2N þ 1g: Consequently, the minimum time Tmin required from input processing to updating the coefficients takes Tmin ¼ tp f5MN þ 4M þ 4Ng þ ta f5MN þ 4M þ 4N þ 1g:
ð22Þ
Comparing Eq. (21) to (22), the method we proposed is more suitable in terms of high-speed processing than that implemented with difference equations. Furthermore, when we compare our algorithm to another implementation for parallel processing addressed in Ref. [3] as Tmin ¼ 4tp þ ð4 þ Jlog2 ðM þ 1Þn þ Jlog2 NnÞta
ð23Þ
and in Ref. [4] as Tmin ¼ 4tp þ 2ðJlog2 Mn þ 1Þta :
ð24Þ
The advantage of our algorithm in terms of the minimum calculation time Tmin is obvious. In addition, another evaluation approach is to count the number of multipliers and adders required for implementation. To calculate the filter output yðn1 ; n2 Þ; we need ðM þ 1ÞðN þ 1Þ multipliers and MðN þ 1Þ adders for filter bank, M þ N multipliers and M þ N adders for state-space vectors. So Eq. (10) means that ðM þ 1ÞðN þ 1Þ þ ðM þ NÞ þ M multipliers and MðN þ 1Þ þ ðM þ NÞ þ M adders for the filter output yðn1 ; n2 Þ are required. Regarding the calculation of gradient vectors, we need 2M multipliers and 2M 1 adders for a1 ðn1 þ 1; n2 Þ and a2 ðn1 þ 1; n2 Þ; N multipliers, and N adders for a4 ðn1 ; n2 þ 1Þ; 2MðM þ 1Þ multipliers and ð2M 1ÞðM þ 1Þ adders for b1i ðn1 þ 1; n2 Þ and NðM þ 1Þ multipliers, and ð2M 1ÞðM þ 1Þ adders for b2i ðn1 ; n2 þ 1Þ: Totally, 2M 2 þ 7M þ MN þ 2N þ 1 multipliers and 2M 2 þ 7M þ MN þ 2N 2 adders for the calculation of gradient vectors are required. Regarding the coefficients updates, we need only 1 multiplier and 1 adder for each coefficient, since Eq. (15) indicates that we have 2 multiplications and 2 additions which cannot be processed in parallel. In the end, we need M þ N þ ðM þ 1Þ þ NðM þ 1Þ þ M multipliers and M þ N þ ðM þ 1Þ þ NðM þ 1Þ þ M adders for coefficient updates. Table 1 shows us the number of multipliers and adders for some filter orders.
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Table 1 The number of multipliers and adders M¼N
2
3
4
5
Multiplier Adder
61 55
105 98
159 151
223 214
4. Simulation To demonstrate the effectiveness of the proposed algorithm, we have carried out the simulation applied to 2-D digital filter design in the spatial domain. The unit sample response of the Gaussian filter is given by the following equation: rðn1 ; n2 Þ ¼ 0:256322 exp ½0:103203fðn1 4Þ2 þ ðn2 4Þ2 g
ðn1 ; n2 ÞASh ;
where Sh ¼ fðn1 ; n2 Þj0pn1 p10; 0pn2 p10g: The order of the filter was chosen as M ¼ N ¼ 3; and the input signal was assumed to be 1 ðn1 ¼ n2 ¼ 0Þ; uðn1 ; n2 Þ ¼ 0 ðotherwiseÞ: The step-size parameter was chosen to be m ¼ 0:0007 and the data frðn1 ; n2 Þjðn1 ; n2 ÞASh g was used repeatedly during the adaptation. The delay q was set to 1. The filter coefficients with separable denominator was obtained at k ¼ 121; 000; 000 as 2 3 0:0 0:0 0:256657 6 7 A1 ¼ 4 1:0 0:0 0:403207 5; 2
0:0 1:0
0:436897
0:0 0:0 6 A4 ¼ 4 1:0 0:0
3 0:271496 7 1:144697 5;
0:0 1:0 0:005606 6 0:013204 6 B1 ¼ 6 4 0:015765
1:757631 3 0:0 0:0 0:0 0:0 7 7 7; 0:0 0:0 5
0:009470 0:007669 6 0:018052 6 B2 ¼ 6 4 0:021553
0:0 0:0 0:001899 0:004453
2
2
0:012950
0:005315 0:003201
3 0:011712 0:027575 7 7 7 0:032922 5 0:019780
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c1 ¼ 0:389255
1:397843
441
1:209952 :
Moreover, the deviation of the resulting filter is evaluated by hP i1=2 N 1 PN 2 2 n1 ¼0 n2 ¼0 ðrðn1 ; n2 Þ yðn1 ; n2 ÞÞ e¼ 100; hP i1=2 N 1 PN 2 2 rðn ; n Þ 1 2 n1 ¼0 n2 ¼0 where N1 ¼ N2 ¼ 10: The resulting filter obtained at k ¼ 121; 000; 000 had e2 ¼ 2:69% and the initial error was e2 ¼ 92:87%: The 2-D digital filters designed by our proposed method have the advantage of its filter structure suitable for parallel processing. Finally, the numerical example shows the effect of the proposed adaptive algorithm.
5. Conclusion The new parallel-form realization of 2-D adaptive filters, based on the Roesser local state-space model with separable denominator, has been proposed. The DLMS method has been applied to implement the adaptive algorithm and proved out to reduce the minimum computation time for each input. The proposed adaptive filter has been developed by combining the parallel-form realization with the adaptive algorithm for 2-D separable-denominator digital filters. Also, the computational time required for updating the coefficients has been investigated and shown that two multiplications and Jlog2 Mn þ 2 additions were required in each iteration. Finally, the effectiveness of the proposed algorithm has been demonstrated by applying it to 2-D digital filter design in the spatial domain.
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