In practical applications, adaptive notch filter is usually implemented on the fixed-point hardware. Therefore, the finite precision effects play an important role on the convergence performance and speed for the adaptive filter. It is difficult to maintain the accuracy of the small coefficient and to prevent the order of magnitude of weights from overflowing simultaneously, as the ratio of two weights in the steady state is very large. When the ratio of two weights in the steady state, , is close to one, the dynamic range of weight value in adaptive processing is fairly small [11]. Thus, the filter can be implemented on the fixed-point hardware with shorter word length, or the coefficients will have higher precision (less coefficient quantization noise) for given a word length.

Based on the concepts of signal space and phasor, the relationship of signal phasors for the above-mentioned two-weight adaptive filters is shown in Figure 3. Figure 3(a) illustrates that the combination of the signal bases (phasors), and , with the respective components in , is able to synthesize the signal phasor . Since the weight vector is only the function of system parameter , it is difficult to control the ratio of these two weights in steady state by the designer. Figure 4 shows that only some narrow regions in the -plane with specified values of satisfy the condition (i.e., ), where is a small value. If the FIR-type adaptive filter [11] is used, Figure 3(b) shows the relationship of the signal phasors , and , where the inserted delay holds. Figure 5 illustrates that the desired regions, in which the ratio of two taps satisfies (), in -plane have been rearranged. We can find that there are two solutions to achieve the requirement, . One solution is to translate the operation point along the vertical axis (-axis) by way of changing the sampling frequency. Therefore, the ratio of two weights for the optimal solution can be controlled by changing the sampling frequency to design the normalized frequency . That is, when the system parameter and the primary noise frequency are given, the designer can adjust the sampling rate to locate the operation point S in the desired region as shown in Figure 5. Another solution is that we can shift the operation point along the horizontal axis to locate the operation point S in the desired region by compensating the system phase .

If the multiple narrowband ANC systems are used, the same sampling frequency is suggested such that the synthesis noises for secondary source can therefore work concurrently. If the sampling rate has been fixed, Kuo et al. [11] suggested inserting a delay unit to control the quantity of weights. The inserted delay can compensate the system phase parameter . This system-phase compensation can move the operation point from *S* to () along the -axis, as shown in Figure 5. When the system phase has been compensated, the operation point in -plane can locate in the desired region which the ratio of two weights is close to one. Using the signal bases and , the ratio of two weights satisfies

The solution to (9) is , where *k* is any integer. The optimal delay *D* can be expressed as samples, where the operation denotes to take the nearest integer. These solutions confirm the results in [11] in which the solution is derived by transfer-function representation. Besides, since the relationship holds, there are four solutions for delay *D* ; these solutions are the possible operation points, , and , as shown in Figure 5. From the phasor point of view, the operation points and mean that the synthesis phasor *y* (*n*) is located in the acute angle formed by basis phasors and , as shown in Figure 3(c). Therefore, the range of weights value can be efficiently used. In addition, observing Figure 5, it can be found that the area of the desired regions varies with the normalized frequencies. It means that the performance will vary with the normalized frequency. This fact also confirms the experimental results in [11]. To solve the problem that the performance depends on the normalized frequency, another signal bases should be found for the two-tap adaptive filters.

In the desired signal space, the phasors and are linearly independent but not orthogonal. Based on the convergence comparison [19] according to the eigenvector and eigenvalue, the convergence speed of Kuo's FIR-type approach will be slow. To accelerate the convergence speed, the signal bases can be setup as orthogonal as possible. As shown in Figure 3(d), the near orthogonal bases and should be found to improve the performance. Based on this motivation, a new delay unit , is introduced as shown in Figure 6. The optimal weight vector of the proposed two-tap adaptive filter is therefore obtained as

such that the signal can be represented as a linear combination of and . That is,

Since the signal bases in the proposed two-tap adaptive filter can be controlled by the delays and , the signal bases can be setup as orthogonal as possible in order to accelerate the convergence speed and to compensate the system phase. Therefore, the delay should hold such that the signal phasor can be approximated as close as possible to . The ratio of two weights will be close to one when the system phase has been compensated by the delay . That is,

The solution to (12) is , . The optimal delays can therefore be found as samples. The desired regions in -plane for the proposed two-tap adaptive filter are similar to that of the desired regions shown in Figure 4. Theoretically, the desired regions do not depend on the normalized frequency in theory. To achieve a better performance for fixed-point implementation, the operation point in -plane can be shifted to the desired area along the horizontal axis (-axis) after the delay is inserted.