This paper proposes an affine projection algorithm (APA) that determines its projection order using a pseudo-fractional method. The pseudo-fractional method adjusts the projection order by comparing the averages of the accumulated squared errors. The method relaxes the constraint of the conventional APA that the projection order must be integral, and it includes both the integral projection order and the fractional projection order. Simulation results show that the proposed algorithm achieves a faster convergence rate and a smaller steady-state estimation error than the existing algorithms.

#### The Transactions of the Korean Institute of Electrical Engineers

**ISO Journal Title**Trans. Korean. Inst. Elect. Eng.

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## 1. 서 론

The affine projection algorithm (APA) has a fast convergence rate for highly correlated input data compared to the normalized least-mean-squares (NLMS) algorithm, because it employs multiple input vectors rather than only one [1-3]. However, the APA has the disadvantages of high computational complexity and a large steady-state estimation error. A high projection order leads to fast convergence but a large estimation error. Meanwhile, a low projection order leads to slow convergence but a small estimation error.

Therefore, it is worth considering the adjustment of the projec- tion order to produce a fast convergence rate and a small steady- state estimation error.

Recently, several papers have been published that deal with the study of the projection order to improve the performance of APAs. Among these works, representative algorithms include an APA with dynamic selection of input vectors (DS-APA), an APA with selective regressors (SR-APA) and an APA with evolving order (E-APA) [4-6]. Although these algorithms show faster con- vertgence and smaller estimation errors than the conventional APA, there is still room for improvement in terms of the con- ver-gence rate and steady-state estimation error.

This paper proposes a new APA that controls the projection order by using a pseudo-fractional
method based on the concept of pseudo-fractional projection order in order to achieve
a fast convergence rate and a small steady-state estimation error, motivated by the
concept of the pseudo-fractional tap-length ^{(6)}. The pseudo-fractional method employs both the integral projection order and the
fractional projection order by relaxing the constraint of the conventional APA that
the projection order must be integral.

Using this method, the projection order for the proposed algorithm is increased or decreased by comparing the averages of the accumulated errors. Moreover, the proposed pseudo-fractional method makes the convergence rate and steady-state estimation error of the proposed algorithm faster and smaller, respectively, than those of the conventional APA, DS-APA, SR-APA, and E-APA.

## 2. 본 론

### 2.1 Conventional Affine Projection Algorithm

Consider reference data $d_{i}$ obtained from an unknown system,

$$d_{i}= u_{i}^{T} w +v_{i}$$

where $ w$ is the n-dimensional column vector of the unknown system that is to be
estimated, $v_{i}$ accounts for measurement noise, which has variance $\sigma_{v}^{2}$,
and $ u_{i}$ denotes an n-dimensional column input vector, $ u_{i}=[u_{i}u_{i-1}\cdots
u_{i-n+1}]^{T}$. The update equ- ation of the conventional APA can be summarized as
^{(3)}:

$$\hat w_{i+1}=\hat w_{i}+\mu U_{i}( U_{i}^{T} U_{i})^{-1} e_{i}$$

where $ e_{i}= d_{i}- U_{i}^{T}\hat w_{i}$, $\hat w_{i}$ is an estimate of $ w$ at iteration $i$, $\mu$ is the step-size parameter, $M$ is the projection order defined as the number of the current input vector used for the update, and

$$ \begin{aligned} U_{i} &=\left[u_{i} u_{i-1} \cdots u_{i-M+1}\right], \\ d_{i} &=\left[d_{i} d_{i-1} \cdots d_{i-M+1}\right]^{T}. \end{aligned} $$

Consider reference data $d_{i}$ obtained from an unknown system,

$$d_{i}= u_{i}^{T} w +v_{i}$$

where $ w$ is the n-dimensional column vector of the unknown system that is to be
estimated, $v_{i}$ accounts for measurement noise, which has variance $\sigma_{v}^{2}$,
and $ u_{i}$ denotes an n-dimensional column input vector, $ u_{i}=[u_{i}u_{i-1}\cdots
u_{i-n+1}]^{T}$. The update equation of the conventional APA can be summarized as
^{(3)}:

$$\hat w_{i+1}=\hat w_{i}+\mu U_{i}( U_{i}^{T} U_{i})^{-1} e_{i}$$

where $ e_{i}= d_{i}- U_{i}^{T}\hat w_{i}$, $\hat w_{i}$ is an estimate of $ w$ at iteration $i$, $\mu$ is the step-size parameter, $M$ is the projection order defined as the number of the current input vector used for the update, and

$$ \begin{aligned} \boldsymbol{U}_{i} &=\left[\boldsymbol{u}_{i} \boldsymbol{u}_{i-1} \cdots \boldsymbol{u}_{i-M+1}\right], \\ \boldsymbol{d}_{i} &=\left[d_{i} d_{i-1} \cdots d_{i-M+1}\right]^{T}. \end{aligned} $$

### 2.2 Affine Projection Algorithm wih Pseudo-Fractional Projection Order

There is a constraint that the projection order for the existing APAs must always be integral. If the projection order includes not only the integral part but also a non-integral part, then the algorithm will achieve better performance than the conventional APA. With the above motivation, we propose a novel APA using a pseudo-fractional method derived from the concept of pseudo-fractional projection order. The pseudo-fractional method includes both the integral projection order and the fractional projection order by relaxing the constraint for the projection order. The integral projection order is the integral part of the fractional projection order when the difference between the integral and fractional projection orders becomes greater than a predeter- mined value. This method adjusts the projection orders dynamically to improve the performance of the proposed algorithm in terms of its convergence rate and steady-state estimation error. Moreover, the leaky factor is applied in the adaptation rule of the fractional projection order in the proposed method.

According to this adaptation rule, the integral projection order remains unchanged until the change in the fractional projection order has accumulated to some extent.

To be specific, we define $P_{i}$ as the pseudo-fractional projection order, which can take positive integral values and construct the following adaptation rule:

$$ P_{i+1}=\left\{\begin{array}{l}{\left(P_{i}-\alpha\right)-\gamma\left(A A S E_{M_{i}}(i)-A A S E_{M_{i}-1}(i)\right), \text { if } M_{i} \geq 2} \\ {\left(P_{i}-\alpha\right)-\gamma\left(A A S E_{M_{i}+1}(i)-A A S E_{M_{i}}(i)\right), \text { otherwise }}\end{array}\right\} $$

where both $\alpha$ and $\gamma$ are small positive numbers, $\alpha$ is a leaky factor that satisfies $\alpha\ll\gamma$, $M_{i}$ is the integral projection order at time instant $i$, and the average of the accumulated squared error (AASE) is defined as

$$AASE_{M}(i)=\dfrac{\sum_{N=0}^{M-1}e_{N}^{2}(i)}{M}$$

Then, the integral projection order $M_{i}$ is determined according to

$$ M_{i}=\left\{\begin{array}{l}{\max \left[\min \left[\left\lfloor P_{i-1}\right\rfloor, M_{\max }\right], 1\right], \text { if }\left|M_{i-1}-P_{i-1}\right| \geq \delta} \\ {M_{i-1}, \text { otherwise }}\end{array}\right\} $$

where the $⌊\cdots ⌋$ operator rounds to the nearest integer and $\delta$ is the threshold parameter.

It is to be noted that $M_{i}$ is updated to satisfy $1\le M_{i}\le M_{\max}$, where $M_{\max}$ is the maximum projection order. In this paper, the threshold parameter $\delta$ is set to 1.

The update equation of the proposed APA is given as follows:

$$ \hat{\boldsymbol{w}}_{i+1}=\hat{\boldsymbol{w}}_{i}+\mu U_{i, M}\left(U_{i, M}^{T} U_{i, M}\right)^{-1} \boldsymbol{e}_{i, M} $$ where $$ \begin{array}{l}{U_{i, M}=\left[u_{i} u_{i-1} \cdots u_{i-M+1}\right]} \\ {e_{i, M}=\left[e_{0}(i) e_{1}(i) \cdots e_{M-1}(i)\right]^{T}}\end{array} $$

and $M_{i}$ is determined by the adaptation rule for the fractional projection order.

## 3. 실험 결과

We illustrate the performance of the proposed algorithm using channel estimation.
The channel of the unknown system is gener- ated by a moving average model with 16
taps (n=16). We assume that the adaptive filter and the unknown channel have the same
number of taps and that the noise variance $\sigma_{v}^{2}$ is known a priori, since
it can be estimated during silences in many practical applications ^{(7)}. The input signal $u_{i}$ is generated by filtering a white, zero-mean Gaussian random
sequence through the following system:

$$G_{1}(z)=\dfrac{1}{1-0.9z^{-1}},\: G_{2}(z)=\dfrac{1+0.6z^{-1}}{1+z^{-1}+0.21z^{-2}}$$

The measurement noise $v_{i}$ is added to $y_{i}$ with a signal-to- noise ratio (SNR) of 30dB, where the SNR is defined by $10\log_{10}(E[y_{i}^{2}]/E[v_{i}^{2}])$ and $y_{i}= u_{i}^{T} w$. Both $P_{0}$ and $M_{0}$ are set to $M_{\max}$, which is the initial projection order of the proposed APA. The mean squared deviation (MSD), i.e., $E\left\|w-\hat{w}_{i}\right\|^{2}$, is calculated to indicate the performance of the proposed algorithm. The simulation results are obtained through ensemble averaging over 100 independent trials, and the input signals are generated by $G_{1}(z)$ and $G_{2}(z)$. Furthermore, to check the tracking perfor- mance of the proposed algorithm, these simulations change the coefficients of the unknown filter taps abruptly at time $i=5000$. The proposed algorithm is applied with $M_{\max}=8$, $\mu =0.1$, and $\gamma =1-\alpha$.

Fig. 1 The MSD of the conventional APA ^{(3)}, DS-APA ^{(4)}, SR-APA ^{(5)}, E-APA ^{(6)}, and the proposed algorithm (the input signal is generated by $G_{1}(z)$, $n=16$,
$SNR=30 d B$).

## 4. 결 론

In this paper, we have proposed an APA with the pseudo- fractional projection order, which determines its projection order by using the pseudo-fractional method. The pseudo-fractional method not only relaxes the constraint that the projection order must be integral, but also adjusts the projection order dynamically by using the proposed adaptation rule for the fractional projection order. The proposed adaptation rule determines the current pro- jection order by comparing the averages of the accumulated squared errors. The channel estimation simulation results proved that the proposed algorithm achieves faster convergence and has a smaller steady-state estimation error than the existing algorithms.

### References

## 저자소개

JinWoo Yoo received his BS, MS, Ph.D. in electrical engineering from Pohang University of Science and Technology (POSTECH) in 2009, 2011, 2015, respectively. He was a senior engineer at Samsung Electronics from 2015 to 2019. He is currently an assistant professor in the department of automotive engi- neering at Kookmin University. His current research interests are signal/image proces- sing and autonomous driving.