# The generic mathematical model of frequency sampling digital filters with shiftable phase-frequency response characteristic.

INTRODUCTION

The development of digital measuring devices is connected with the task to synthesize digital filters for the given parameters [1, 2]. At the same time, when digital phase converters are used in measuring devices, it is necessary to have filters with adjustable phase-frequency response characteristic [3, 4, 5]. The research conducted showed the possibility to synthesize digital filters with shiftable phase-frequency response characteristic on the basis on both frequency sampling method and the classical algorithm of time-domain convolution.

Main part:

Every algorithm is implemented in terms of its mathematical formulation . Let us consider various ways of shifting the phase-frequency response characteristic for digital FIR-filters on the basis of frequency sampling method and the sliding discrete complex Fourier transformation. We will get mathematical models for digital filters without the shift of the phase-frequency response characteristic  and digital filters with the shift of the phase-frequency response characteristic.

Let us consider the work of elementary digital filter (EDF) on the basis of the sliding discrete complex Fourier transformation [8, 9]. To do this, we will get an expression which allows us to find the value of output sample by the well-known sliding input sampling.

The output sample is formed at every interval of discretization by the last sample of Fourier inversion [10, 11]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [X.bar] is samples of complex spectrum for sliding sampling, u is the array of input samples; q is the] number of F-series spectral component.

The frequency sampling method is computationally efficient with small values N as compared with the time-domain convolution method [8, 9]. In this case, we can think factor [e.sup.2j[pi](N-1)q/N] to be equal to unity. Computational investigations showed that this approximation leads to a slight shift of PFRC  and can be considered its error . However the result of addition in (1) will no longer be valid.

Let us find values of output samples for each EDF. According to the frequency sampling method, we find a sample for each output signal of EDF as a result of filtration of certain spectral components from input sliding sampling spectrum, thus the number of EDF will coincide with the number of spectral component in what follows. Constant spectral component corresponds to a zero EDF. The corresponding resultant values of addition will have a complex character. If we bear this in mind and take into account the symmetry of spectrum concerning zero frequency, the expression for EDF output samples will look like this:

[u.sub.ex q,n] = 2 x (Re ([X.bar].sub.q,n]). (2)

So, EDF can be synthesized without Fourier inversion.

In expression (2), we use the real component of output vector. The imaginary component of the vector allows us to shift the resultant PFRC through angle of -[pi]/2 radian. This can be implemented in a number of applications. So, the output sample can be presented the following way:

[u.sub.ex q,n] = 2 x Im ([X.bar].sub.q,n). (3)

The input sampling shifts one sample at each step of discretization. When the input sampling shifts one sample, the first component of the sum value is deleted according to the direct Fourier transformation which has a zero angle; and one new component is added. All other components will remain unchanged, only their serial numbers vary per unit that is equal to sum vector rotation through angle of 2 j[pi]nq/N. So, the value of the next vector can be found on the basis of the value of the previous vector [12, 13]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](4)

where [[X.bar].sub.q,n-1] is the previous sample of the vector, [u.sub.n-N-1] is the oldest input sample which is deleted, [u.sub.n] is the current input sample which is added.

In brackets (4), the first component of the vector is subtracted from its previous value in brackets. The result rotates through angle 2j[pi]nq/N due to the shift of serial numbers per unit. Then a new current input sample is added.

If we remove brackets in expression (4), we get expressions for output samples of EDF in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

The obtained expressions (5) and (6) are the mathematical model for EDF on the basis of sliding discrete complex Fourier transformation with the feature of shifting the PFRC to [pi]/2. Let us consider a structure chart which realizes digital filters according to the obtained mathematical models (Fig. 1).

At the structure chart, block 1 is the memory block containing the input sampling. It works by the "FIFO" principle (First in, first out) and outputs the oldest sample (the first incoming one). In block 2, we can see the addition of input sample and the signal from block 1. In block 3, we see the addition of the signal from block 2 and the real component of [[X.bar].sub.q,n-1] found at the previous interval of discretization. In block 5, we see the complex multiplication of the real and imaginary components of the signal sent to the input of block 5 by unit vector which can be described in expression [e.sup.2j[pi]q/N]. The real and imaginary component values of this expression are sent to block 5 from memory block 4. From the output of block 5, the signal is sent to the output of the EDF and to blocks 3 and 5 to be used at the next level of discretization.

Let us consider the mathematical model for the zero EDF. The zero EDF corresponds to the constant component of Fourier transformation. For the sliding window, the constant component is found as an average value for all samples. This corresponds to the well-known mathematical model of digital integrator.

We get the expression for the mathematical model of digital filter with shiftable phase-frequency response characteristic. Suppose all input samples, presented in direct complex Fourier transformation by vectors in the complex plane, are additionally rotated through angle [PSI].

The expression we got is the mathematical model of EDF with phase shift at the input. At the same time, the PFRC of EDF is shifted too. Term "at the input" is used because the extra phase shift is added to input samples. In order to show the possibility of changing angle for each EDF independently, we will now write down the generic mathematical model of EDF:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

In fact, values [[U.bar].sub.q,n] and [[X.bar].sub.q,n] are equal. That is why, hereinafter, if we find value [[bar.U].sub.qn], we get an expression for [[X.bar].sub.q,n] too. The difference between them consists only in the fact that we mean a complex output value by the first one and values used at the next level of discretization [[X.bar].sub.q,n-1] by the second one. Figure 2 shows the structure chart of EDF corresponding to expression (7).

At the chart, block 1 is the memory block containing the input sampling. It works by the "FIFO" principle (First in, first out) and outputs the oldest sample (the first incoming one). In block 2, we can see the addition of input sample and the signal from block 1. In block 4, we see the complex multiplication of the signal sent from block 2 by value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The real and imaginary components of this value are sent to block 4 from memory block 3. The result of multiplication, the real and imaginary components respectively, are sent to the first inputs of summarizing blocks 7 and 8. In block 5, we see the complex multiplication of value [[X.bar].sub.q,n-1], the real and imaginary components of which are sent from the previous discretization interval, by value [e.sup.2j[pi]q/N], the real and imaginary components of which are sent to block 6 from memory block 5. The multiplication result, the real and imaginary components respectively, are sent to the second inputs of summarizing blocks 7 and 8. In blocks 7 and 8, the real and imaginary components of value [[X.bar].sub.qn] are formed. They are sent to the output of EDF and the input of block 6 at the next discretization interval.

Let us consider the method of phase shifting at the expense of rotating input samples through an additional angle. Besides, we will consider another method of controlling the PFRC if output signal. The output signal is formed by the real or imaginary component of spectral sample vector which is stored in the memory of EDF. The modulus of vector determines the amplitude of output sample. The angle determines the current phase of the signal. It is inadmissible to rotate this vector through a fixed angle in the structure of EDF, because this will lead to the dysfunction of the EDF. However, it is possible in the chain of signal transmission to filter output. To do this, one should send the real and imaginary components of EDF spectral component vector to the output of each EDF and rotate this vector by means of multiplication with unit vector. The angle of unit vector is determined by the shift angle of the PFRC. In case when only one component of complex output signal is sent to the output of the DF, two multiplications and one addition will be needed. It should be noted that the vector is rotated outside the structure of EDF and does not influence the work of the EDF. This makes for the absence of transitional process in changing the rotation angle and the absence of PFRC distortions.

In fact, this method of shifting PFRC is an integration of EDF and device which performs the phase shift through a predetermined angle (in any direction). The advantage of this method consists in simple implementation, computational efficiency and the opportunity to change the current phase of output signal in real time thorough any angle without transitional process.

Now we will get the mathematical model of EDF on the basis of the method of phase shifting described above.

The work of every EDF remains unchanged. Only the phase of output signal changes. Thus the mathematical model will look like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

In case when angle is the same for all EDF in the structure of DF, we can perform rotation for the resultant vector. This will reduce the total number of mathematical operations. Let us consider a structure chart corresponding to expression (8). The structure chart is shown in Figure 3. Its difference from the structure chart without phase shift (see Fig. 1) consists in the fact that complex output signal is sent from block 5 to block 7 where the additional multiplication by value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] takes place. The values of its real and imaginary components are sent from memory block 6 to block 7. The complex output signal is sent to the output of EDF.

Expressions (5) and (6) are the mathematical models of EDF without PFRC shifting. Expression (7) is the mathematical model of EDF with input shifting of PFRC. Expression (8) is the mathematical model with output shifting of PFRC. On the basis of these models we can imagine the generic mathematical model of EDF which allows us to take into account both methods of PFRC shifting. To do this, we use [PSI]' to designate the rotation angle of input samples. This angle is formed by input control. Then we use [PSI]" to designate the rotation angle of the resultant signal formed by output control. The resultant mathematical model will look like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Expression (9) is the generic mathematical model of elementary digital FIR-filter based on the sliding discrete complex Fourier transformation. Besides, the complex output value which allows us to use two output projections--real and imaginary--is meant by [[U.bar].sub.q] (n). This makes it possible to work with two output values which give the relative PFRC shift of [pi]/2 radian.

If we create DF based on the superposition of EDF using the frequency sampling method, we need to distribute the obtained mathematical models to the array of EDF. At the same time, the amplitude coefficients of EDF output signals are taken into account to get the computational efficiency and given frequency band. In case when the amplitude coefficient of EDF output signal is equal to zero, the EDF is practically absent. Moreover, it is necessary to provide the coincidence for PFRC of certain EDF in order to form a continuous band pass at the expense of multiplying uneven EDFs by minus unit.

In compliance with the above we can write down the generic mathematical model of digital FIR-filter based on the sliding discrete complex Fourier transformation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

The output signal of DF can be obtained by the isolation of the real or imaginary component from expression (10)

Conclusion:

In the generic mathematical model, the PFRC is shifted through angle [pi]/2 by means of isolation of the real or imaginary component of complex output signal.

The gradual shifting of PFRC is achieved by means of rotating input samples through an additional angle in their vector addition. Besides, the gradual shifting of PFRC is achieved by the additional rotation of output signal vector.

Findings:

The authors proposed an approximation which makes it possible to exclude Fourier inversion from the algorithm of EDF.

Besides, they proposed methods for shifting the PFRC of EDF and obtained the generic mathematical model for digital FIR-filter as a result if the superposition of EDF.

ACKNOWLEDGEMENTS

The paper is created within the framework of the fundamental research financed by the Ministry of Education and Science of the Russian Federation (Governmental task for 2014, code 226) "The development and investigation of methods for data processing in contactless systems of motion estimation and observable object control" at sub-department "Information and electronic services" of the Volga Region State University of Service.

ARTICLE INFO

Article history:

Received in revised form 8 July 2014

Accepted 10 August May 2014

Available online 30 August 2014

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Victor Vasilievich Ivanov, Victor Nikolaevich Budilov, Vladimir Ivanovich Volovach, Maxim Victorovich Shakurskiy

Volga region state university of service, Russia, 445677, Togliatti, Gagarin Street, 4

Corresponding Author: Victor Vasilievich Ivanov, Volga region state university of service, Russia, 445677, Togliatti, Gagarin Street, 4
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