2.1 P-norm-like Affine Projection Algorithm (APPA)

Consider the data d(k) that is obtained from an unknown system

where $\boldsymbol{w_{o}}$ is an n-dimensional column vector that we expect to estimate, $v(k)$ accounts for a measurement noise with variance $\sigma_{v}^{2}$, and input vector is defined as

The output error vector, the desired output data vector, and the data matrix are defined as

where $\hat{\boldsymbol{w}}(k)$, which is the estimate of $\boldsymbol{w_{o}}$ at iteration k.

The proposed algorithm is derived by minimizing the p-norm-like of an error vector as follows:

where $|| \boldsymbol{x}(k)||_{p}=\sum_{i=1}^{n}|x(k)|^{p},\: 0\le p\le 1$ ^[9]. From (6), the proposed cost function is obtained as

To minimizing the cost function with respect to the weight vector $\hat{\boldsymbol{w}}(k+1)$, the cost function is differentiated as follows:

where $sgn(·)$ is the sign function, and

The update equation for the APPA is derived by the normalized gradient method ^[10] as follows:

where $\mu$ is a step size.

2.2 Variable Step-Size APPA

We define the weight-error vector as $\widetilde{\boldsymbol{w}}(k)= \boldsymbol{w_{o}}- \hat{\boldsymbol{w}}(k)$ and can rewritten in terms of $\widetilde{\boldsymbol{w}}(k)$ by subtracting $\boldsymbol{w_{o}}$ both side of (10) as follows:

The update recursion of mean-square deviation (MSD) is derived by taking the expectation after squaring both sides of (11) as follows:

where

If $g(k)<0$, MSD decreases monotonically. By minimizing the value of MSD$({k}+1)$ with respect to the step size $\mu$, the optimal step size $\mu^{*}(k)$ of APPA is derived by

where

Because the noise $v(k)$ is unknown, however, it is difficult to calculate the exact value of the optimal step size $\mu^{*}(k)$. Therefore, the proposed step size is obtained as

Since it is hard to determine the exact step size (11) directly, we propose to calculate $\mu(k+1)$ recursively by time-averaging as follows:

with a smoothing factor $\alpha(0\le\alpha <1)$. If $p=1$, this algorithm performs like a variable step-size APSA^[8].

Fig. 2. Illustration of the relationship between $\mu(k)$ and $g(k)$

Fig. 2. illustrates the relationship between $\mu(k)$ and $g(k)$. The optimal step size $\mu^{*}(k)$ leads to the largest decrease in the MSD because $g(k)$ has the smallest value when $\mu(k)=\mu^{*}(k)$. Because we cannot calculate $\mu^{*}(k)$, however, we use the step size (17) that is in the bracket whose size is related to the measurement noise. Therefore, the MSD of the proposed step-size algorithm decrease monotonically until $2\mu^{*}(k)$ is smaller than the bracket.

2.3 Simulation Results

Computer simulations in the channel identification are used to evaluate the performance of the proposed algorithm. In these simulations, the unknown channel is the acoustic impulse response of a room truncated to 512 taps ($n=512$) as shown in Fig. 3, and we assume that the adaptive filter and the unknown channel have same number of taps.

Fig. 3. Acoustic impulse response of a room.

The colored input signals are generated by filtering white Gaussian noise through a first-order system as follows:

The signal-to-noise ratio (SNR) and the normalized mean squared deviation (NMSD) are defined as

where $y(k)= \boldsymbol{u^{T}}(k)\boldsymbol{w_{o}}$. The measurement noise is added to the output y(k) such that the SNR is set to 30dB. The impulsive noise $\eta(k)$ is generated as $\eta(k)=\kappa(k)A(k)$, where $\kappa(k)$ is a Bernoulli process with probability of success $P[\kappa(k)=1]= Pr$, and $A(k)$ is a zero-mean Gaussian with power $\sigma^{2}_{A}= 1000\sigma^{2}_{y}$. Each adaptive filter is tested for $M=4$ and Pr that denotes the probability occurring the impulsive noise is set 0.5. The simulation results are obtained by ensemble averaging over 10 trials.

Fig. 4. MSD learning curves of APPAs at $(\mu =0.005)$ for colored input generated by G(z) and impulsive noises with $Pr = 0.5$

Fig. 5. MSD learning curves of VSS-APPAs for colored input generated by G(z) and impulsive noises with $Pr = 0.5$

Fig. 4 shows the NMSD learning curves for APPAs with fixed step size $(\mu =0.005)$. In high probability impulsive noise, the proposed APPAs with small $p$ has smaller misalignment errors than the APPA with $p=1$, which is APSA. Fig. 5 shows the NMSD learning curves of VSS-APPAs for $\alpha = 0.8$, and $\mu(0)=\sqrt{\dfrac{\sigma^{2}_{d}}{M\sigma^{2}_{u}}}$, where $\sigma^{2}_{d}$ are $\sigma^{2}_{u}$ the power of the observed output and input respectively. As can be seen, the proposed VSS-APPAs have smaller misalignment errors than VSS-APSA when $p$ is set to less than 1.