# Variable Tap-Length Algorithm with Mixed Parameter
## 2 Variable Tap-Length Algorithm
### Mean Square Deviation Gradient  
Next, the mean square deviation (MSD) gradient is used to discuss the effect of different tap-lengthson the algorithm. For a more intuitive comparison, we analyzed the MSD gradient first. Define \(\mathbf{p}\) as follows:  
\[\mathbf{p}(n)=\mathbf{W}_{opt}-\mathbf{W}(n) \tag{4}\]  
Taking \(\mathrm{W}_{\mathrm{opt}}\) to subtract Equation (3), Equation (3) becomes as follows:  
\[\mathbf{p}(n)=\mathbf{p}(n-1)-\mu e(n)\mathbf{X} \tag{5}\]  
Using \(t(n)\) to replace \(E\{\parallel\mathbf{p}(n-1)\parallel^{2}\}-E\{\parallel\mathbf{p}(n) \parallel^{2}\}\), the equation becomes as follows:  
\[t(n)=E\{\parallel\mathbf{p}(n-1)\parallel^{2}\}-E\{\parallel\mathbf{p}(n) \parallel^{2}\}=E\{\parallel\mathbf{p}(n-1)\parallel^{2}\}-E\{\parallel \mathbf{p}(n-1)-\mu\mathbf{X}e(n)\parallel^{2}\} \tag{6}\]  
Equation (6) can be simplified as follows:  
\[t(n)=2\mu E\{e(n)^{2}\}-2\mu\sigma_{v}^{2}-M\mu^{2}\sigma_{x}^{2}E\{e(n)^{2}\} \tag{7}\]  
where \(\sigma_{v}^{2}\) is the variance of the noise and \(\sigma_{x}^{2}\) is the variance of the input. The MSD gradient is related to tap-length and noise signal according to Equation (7). Next, an experiment was used to compare the MSD gradient of different tap-lengths.  
Figure 1 plots the MSD gradient of different tap-lengths (8, 12, 16, and 20) in an identification system. This system was set up with the best tap-length of 16. The reference input was obtained by passing white Gaussian noise through a transfer system of \(\mathrm{H}(\mathrm{z})=0.35+\mathrm{z}^{-1}+\mathrm{z}^{-2}\). Transfer system H(z) was referenced from [PERSON] and [PERSON] [7]. The length 16 of the identification system was adopted from the white Gaussian random sequence with zero-mean and unity variance. The signal-to-noise ratio (SNR) of 10 dB was obtained by adding white Gaussian interference noise. Different tap-lengths had different MSD gradient performance. In the first half of the iteration, the red line (M = 16) had the largest \(t(n)\). In the latter half, although the black line (M = 8) had a fluctuating and negative MSD gradient, part of the black line was larger than the red line. This is because under-modeling appears in the black line. As its tap-length is smaller than the optimum value, the difference between its coefficient and the optimum is very large. Large and fluctuant MSD results in a large MSD gradient that is incorrect. Thus, the closer to optimum the tap-length is, the larger the amplitude of MSD gradient that will be obtained. For achieving the fastest convergence rate and the best tracking capability, the tap-length should be close to the optimum. In this case, MSD will undergo the largest decrease at each iteration and also guarantees the achievement of the lowest possible steady-state MSE [14]. Furthermore, a variable tap-length potentially gains significant computational savings. For an unknown environment, an efficient variable tap-length algorithm provides better results. The fractional variable tap-length algorithm is a favorable strategy, as explained below.  
Figure 1: Mean square deviation (MSD) gradient of different tap-lengths.