1. Introduction

Several kinds of methods to design quadrature filters are found in the literature and references therein [1–6]. However, most of them give an algorithm designed for a specific phase step [1–8]. In a previous work, a theory to design tunable filters was presented [9]. Then, in this paper another method to obtain tunable filters is introduced. Like in the previous work [9], the quadrature conditions of the filter and errors like the phase shift detuning error and bias modulation are represented as geometrical conditions to be satisfied by the filter. Thus, from these conditions and the desired errors to minimize, a set of two-frame filters for each condition is obtained. Hence, trough the convolution of this set of individual filters we recover the desired filter as an algebraic problem. Finally, the method is applied to obtain novel results for tunable four and five-frame algorithms.

The estimated phase of a quadrature filter order M is given by [1–9],

It should be noticed that $\overrightarrow{I}={\left[\begin{array}{cccc}{I}_1& I_2& \mathrm{...}& I_M\end{array}\right]}^T$ is the column vector of frames, and N and D are the desired numerator and denominator row vectors; then, for a symmetrical filter the corresponding temporal impulse response $h\left(t\right)$ is given by [6–9]

In other words, m gives the order of insensitivity to the phase shift error, and $H^m\left(\omega \right)$ is the m^th derivate of $H\left(\omega \right)$ with regard to ω. In the same way, the condition to be satisfied by a filter which is insensitive to the m^th order bias variation error $H\left(\omega \right)$ is [9],

2. Convolution algorithm

Assuming that the Fourier transform of a filter can be factorized in two functions such as $H(\omega )=H_1(\omega )H_2(\omega )$, where $H_1(\omega )$ and $H_2(\omega )$ are the Fourier transforms of $h_1\left(t\right)=D_1\overrightarrow{\delta }+i{N}_1\overrightarrow{\delta }$ and $h_2\left(t\right)=D_2\overrightarrow{\delta }+i{N}_2\overrightarrow{\delta }$ which are an n and m order filters respectively, the individual estimated phases ${\phi }_1$ and ${\phi }_2$ are given by

Hence, the desired filter $h\left(t\right)$ is obtained from the expression $h\left(t\right)=h_1\left(t\right)\ast h_2\left(t\right)$, where ∗ denotes the temporal discrete convolution, and $h\left(t\right)$ becomes,

In other words, as mentioned before, a new (n + m-1) frame filter from two individual filters is obtained. Likewise, the design of a tunable quadrature filter is seen as an algebraic problem without the use of Fourier formalisms. The convolution properties allow this case to be extended for three or more filters.

3. Design of tunable filters

The design of a quadrature filter order M implies that only M-1 parameters are free, two of which are the quadrature conditions and the other M-3 are used to compensate some errors.

3.1 The tunable two-frame algorithm

The general form of a symmetric two-frame algorithm is [9],

As in the previous work [9], the Fourier transform of Eq. (9) for a given α step is,

Tuning this filter to cut off the frequency $\omega =\alpha$, the condition $H(\omega =\alpha ,\alpha )=0$ is satisfied. Hence, the solution of Eq. (10) is given by the ratio $b_1/a_1=\cos \left(\alpha /2\right)/\sin \left(\alpha /2\right)$. And from Eq. (9) the corresponding tunable two-frame filter and the ratio N/D are respectively,

Furthermore, this two-frame filter has the following Fourier transform

Therefore, from Eq. (3) the Fourier transform and the corresponding algorithm for a filter insensitive to the 0, 1, 2 … and m orders phase shift detuning error are respectively

Then, from Eqs. (4) and (16) the filter insensitive to the m order bias modulation error is,

In the same way, other possible conditions working over the frequency $\omega =\pi$ are,

3.2 The tunable three-frame algorithm

The Fourier transform that meets the two quadrature conditions H(0) = 0 and H(α) = 0 is

This function is the product of the two filters that cuts off individually both frequencies ω = 0 and ω = α. Then, in terms of the corresponding two-frame filters, and through the convolution algorithm Eq. (8), the desired N/D becomes

3.3 The tunable four-frame algorithm

Like Eq. (1), the general form for the estimated phase of a four-frame algorithm is

From Eq. (13), the Fourier transform that cuts off the frequencies $\omega =0$, $\omega =\alpha$ and $\omega =\beta$ is

That is, two conditions correspond to the necessary quadrature conditions, and $H(\omega ,\beta )=0$ is the particular condition that cuts off the frequency β. Then, by applying the corresponding two-frame filters, the result is

From Eq. (8) the last two terms become,

This is the general four-frame filter that cuts off both frequencies, the α step and an arbitrary frequency β. Thus, by using the geometrical condition $\beta =\pi$, a new four-frame algorithm is obtained. Therefore, as reported in [5] we named it “tunable four-frame algorithm in X”,

3.4 The tunable five-frame algorithm

For the α step and the two arbitrary frequencies β and γ, the Fourier transform of the filter is,

From Eqs. (7) and (8) the desired phase is obtained through the following expression

In Table 1 , particular cases of several five-frame algorithms are shown. Case 1 meets the conditions $H\left(0\right)=H^{\prime }\left(0\right)=H^{″}\left(0\right)=H\left(\alpha \right)=0$, therefore from Eq. (32) $H\left(\omega \right)$ becomes,

Table 1. Several particular five-frame temporal phase shifting algorithms

View Table

In case 2, $H\left(0\right)=H\left(\alpha \right)=H^{\prime }\left(0\right)=H\left(\pi \right)=0$ and $H\left(\omega \right)$ becomes,

In case 3, $H\left(0\right)=H\left(\alpha \right)=H^{\prime }\left(\alpha \right)=H\left(\pi \right)=0$ and $H\left(\omega \right)$ gives

In case 4, $H\left(0\right)=H\left(\alpha \right)=H\left(\pi \right)=H^{\prime }\left(\pi \right)=0$ and $H\left(\omega \right)$ results

In case 5, the filter meets the conditions $H\left(0\right)=H\left(\alpha \right)=H^{\prime }\left(0\right)=H^{\prime }\left(\alpha \right)=0$ and gives

This is a tunable five-frame algorithm which is simultaneously insensitive to linear bias modulation and linear phase shift detuning errors. It comprises the results reported in [6]. To conclude, in case 6 the tunable filter insensitive to second order phase shift detuning error is,

Finally, from [7,8], the detuning error of the function $\sin \left[\left(\omega -\alpha \right)/2\right]$ gives $\mathrm{\Delta }\phi =-\tan \left(\mathrm{\Delta }/2\right)$ for the angle Δ. Therefore, for ${\sin }^q\left[\left(\omega -\alpha \right)/2\right]$ the detuning error is $\mathrm{\Delta }\phi ={\left(-1\right)}^q{\tan }^q\left(\mathrm{\Delta }/2\right)$. Thus, cases 3 and 5 have q = 2, while case 6 has q = 3, and the others cases have the value q = 1.

4. Conclusions

A new method to design quadrature filters is presented. The design problem is reduced to an algebraic problem that through the convolution of a set of two-frame filters gives the desired phase without the use of Fourier’s formalisms. Each individual condition of the filter corresponds with a specific two-frame filter. Therefore, several new tunable four and five-frame symmetrical algorithms are reported. This method is easily evaluated numerically.