Advances in Environmental Biology

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1 AESI Jornals Advances in Environmental Biology ISS EISS Jornal home page: The Generic Mathematical Model of Freency Sampling Digital Filters with Shiftable Phase-Freency Response Characteristic Victor Vasilievich Ivanov, Victor ikolaevich Bdilov, Vladimir Ivanovich Volovach, Maxim Victorovich Shakrskiy Volga region state niversity of service, Rssia, , Togliatti, Gagarin Street, 4 A R T I C L E I F O Article history: Received 25 Jne 2014 Received in revised form 8 Jly 2014 Accepted 10 Agst May 2014 Available online 30 Agst 2014 Keywords: Digital filters, freency sampling method, phase-freency response characteristic, elementary digital filter, finite-implse response. A B S T R A C T The article contains mathematical models for the digital filters of freency sampling with shiftable phase-freency response characteristic and the generic mathematical model which takes into accont all ways of shifting the phase-freency response characteristic. The development of filters with shifted phase-freency response characteristic is cased by the development of sensitive self-oscillating systems. The filters themselves are sed in data processing systems. The research condcted made it possible to develop a class of narrow-band filters with opportnity to both shift the phase-freency response characteristic and change the instantaneos phase of otpt signal in real time AESI Pblisher All rights reserved. To Cite This Article: Victor Vasilievich Ivanov, Victor ikolaevich Bdilov, Vladimir Ivanovich Volovach, Maxim Victo-rovich Shakrskiy, The Generic Mathematical Model of Freency Sampling Digital Filters with Shiftable Phase-Freency Response Characteristic. Adv. Environ. Biol., 8(13), , 2014 ITRODCTIO The development of digital measring 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 sed in measring devices, it is necessary to have filters with adjstable phase-freency response characteristic [3, 4, 5]. The research condcted showed the possibility to synthesize digital filters with shiftable phase-freency response characteristic on the basis on both freency sampling method and the classical algorithm of time-domain convoltion. Main part: Every algorithm is implemented in terms of its mathematical formlation [6]. Let s consider varios ways of shifting the phase-freency response characteristic for digital FIR-filters on the basis of freency sampling method and the sliding discrete complex Forier transformation. We will get mathematical models for digital filters withot the shift of the phase-freency response characteristic [7] and digital filters with the shift of the phase-freency response characteristic. Let s consider the work of elementary digital filter (EDF) on the basis of the sliding discrete complex Forier transformation [8, 9]. To do this, we will get an expression which allows s to find the vale of otpt sample by the well-known sliding inpt sampling. The otpt sample is formed at every interval of discretization by the last sample of Forier inversion [10, 11]: 1 2 j( 1) 1 X e. (1) 0 where X is samples of complex spectrm for sliding sampling, is the array of inpt samples; is the nmber of F-series spectral component. Corresponding Athor: Victor Vasilievich Ivanov, Volga region state niversity of service, Rssia, , Togliatti, Ga garin Street, 4

2 310 Victor Vasilievich Ivanov et al,2014 The freency sampling method is comptationally efficient with small vales as compared with the 2 j( 1) time-domain convoltion method [8, 9]. In this case, we can think factor e to be eal to nity. Comptational investigations showed that this approximation leads to a slight shift of PFRC [7] and can be considered its error [6]. However the reslt of addition in (1) will no longer be valid. Let s find vales of otpt samples for each EDF. According to the freency sampling method, we find a sample for each otpt signal of EDF as a reslt of filtration of certain spectral components from inpt sliding sampling spectrm, ths the nmber of EDF will coincide with the nmber of spectral component in what follows. Constant spectral component corresponds to a zero EDF. The corresponding resltant vales of addition will have a complex character. If we bear this in mind and take into accont the symmetry of spectrm concerning zero freency, the expression for EDF otpt samples will look like this: ex, n 2 Re X, n. (2) So, EDF can be synthesized withot Forier inversion. In expression (2), we se the real component of otpt vector. The imaginary component of the vector allows s to shift the resltant PFRC throgh angle of 2 radian. This can be implemented in a nmber of applications. So, the otpt sample can be presented the following way: ex, n 2 ImX, n. (3) The inpt sampling shifts one sample at each step of discretization. When the inpt sampling shifts one sample, the first component of the sm vale is deleted according to the direct Forier transformation which has a zero angle; and one new component is added. All other components will remain nchanged, only their serial nmbers vary per nit that is eal to sm vector rotation throgh angle of 2 j n. So, the vale of the next vector can be fond on the basis of the vale of the previos vector [12, 13]: 2 j 2 j 1 n 1 n X, n X, e e, (4) where X, is the previos sample of the vector, n1 is the oldest inpt sample which is deleted, n is the crrent inpt sample which is added. In brackets (4), the first component of the vector is sbtracted from its previos vale in brackets. The reslt rotates throgh angle 2 j n de to the shift of serial nmbers per nit. Then a new crrent inpt sample is added. If we remove brackets in expression (4), we get expressions for otpt samples of EDF in the following form: ex ex 2 j 2 j 2 j n 1 n, n 2 Re X, e e e ; (5) 2 j 2 j 2 j n 1 n, n 2 Im X, e e e. (6) The obtained expressions (5) and (6) are the mathematical model for EDF on the basis of sliding discrete complex Forier transformation with the featre of shifting the PFRC to 2. Let s consider a strctre chart which realizes digital filters according to the obtained mathematical models (Fig. 1). At the strctre chart, block 1 is the memory block containing the inpt sampling. It works by the FIFO principle (First in, first ot) and otpts the oldest sample (the first incoming one). In block 2, we can see the addition of inpt 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, fond at the previos interval of discretization. In block 5, we see the complex mltiplication of the real and imaginary components of the signal sent to the inpt of block 5 by nit 2 j vector which can be described in expression e. The real and imaginary component vales of this expression are sent to block 5 from memory block 4. From the otpt of block 5, the signal is sent to the otpt of the EDF and to blocks 3 and 5 to be sed at the next level of discretization.

3 311 Victor Vasilievich Ivanov et al,2014 Fig. 1: The strctre chart of EDF. Let s consider the mathematical model for the zero EDF. The zero EDF corresponds to the constant component of Forier transformation. For the sliding window, the constant component is fond as an average vale 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-freency response characteristic. Sppose all inpt samples, presented in direct complex Forier transformation by vectors in the complex plane, are additionally rotated throgh angle. The expression we got is the mathematical model of EDF with phase shift at the inpt. At the same time, the PFRC of EDF is shifted too. Term at the inpt is sed becase the extra phase shift is added to inpt 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: 2 j 2 j j n n 1, n X, e e. (7) In fact, vales expression for X, n, n and X, n are eal. That is why, hereinafter, if we find vale, n, we get an too. The difference between them consists only in the fact that we mean a complex otpt vale by the first one and vales sed at the next level of discretization X, by the second one. Figre 2 shows the strctre chart of EDF corresponding to expression (7). Fig. 2: The strctre chart of EDF with inpt phase shift. At the chart, block 1 is the memory block containing the inpt sampling. It works by the FIFO principle (First in, first ot) and otpts the oldest sample (the first incoming one). In block 2, we can see the addition of inpt sample and the signal from block 1. In block 4, we see the complex mltiplication of the signal sent from 2 j j block 2 by vale e. The real and imaginary components of this vale are sent to block 4 from memory block 3. The reslt of mltiplication, the real and imaginary components respectively, are sent to the first inpts of smmarizing blocks 7 and 8. In block 5, we see the complex mltiplication of vale X,, the real and imaginary components of which are sent from the previos discretization interval, by vale e 2 j, the real and

4 312 Victor Vasilievich Ivanov et al,2014 imaginary components of which are sent to block 6 from memory block 5. The mltiplication reslt, the real and imaginary components respectively, are sent to the second inpts of smmarizing blocks 7 and 8. In blocks 7 and 8, the real and imaginary components of vale X, are formed. They are sent to the otpt of EDF and n the inpt of block 6 at the next discretization interval. Let s consider the method of phase shifting at the expense of rotating inpt samples throgh an additional angle. Besides, we will consider another method of controlling the PFRC if otpt signal. The otpt signal is formed by the real or imaginary component of spectral sample vector which is stored in the memory of EDF. The modls of vector determines the amplitde of otpt sample. The angle determines the crrent phase of the signal. It is inadmissible to rotate this vector throgh a fixed angle in the strctre of EDF, becase this will lead to the dysfnction of the EDF. However, it is possible in the chain of signal transmission to filter otpt. To do this, one shold send the real and imaginary components of EDF spectral component vector to the otpt of each EDF and rotate this vector by means of mltiplication with nit vector. The angle of nit vector is determined by the shift angle of the PFRC. In case when only one component of complex otpt signal is sent to the otpt of the DF, two mltiplications and one addition will be needed. It shold be noted that the vector is rotated otside the strctre of EDF and does not inflence 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 throgh a predetermined angle (in any direction). The advantage of this method consists in simple implementation, comptational efficiency and the opportnity to change the crrent phase of otpt signal in real time thorogh any angle withot transitional process. ow 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 nchanged. Only the phase of otpt signal changes. Ths the mathematical model will look like this: 2 j 2 j 2 j n1 n j n X n e e e,, 1 e. (8) In case when angle is the same for all EDF in the strctre of DF, we can perform rotation for the resltant vector. This will redce the total nmber of mathematical operations. Let s consider a strctre chart corresponding to expression (8). The strctre chart is shown in Figre 3. Its difference from the strctre chart withot phase shift (see Fig. 1) consists in the fact that complex otpt signal is sent from block 5 to block 7 j where the additional mltiplication by vale e takes place. The vales of its real and imaginary components are sent from memory block 6 to block 7. The complex otpt signal is sent to the otpt of EDF. Fig. 3: The strctre chart of EDF with otpt phase shift. Expressions (5) and (6) are the mathematical models of EDF withot PFRC shifting. Expression (7) is the mathematical model of EDF with inpt shifting of PFRC. Expression (8) is the mathematical model with otpt shifting of PFRC. On the basis of these models we can imagine the generic mathematical model of EDF which allows s to take into accont both methods of PFRC shifting. To do this, we se to designate the rotation angle of inpt samples. This angle is formed by inpt control. Then we se to designate the rotation angle of the resltant signal formed by otpt control. The resltant mathematical model will look like this: 2 j 2 j n j n n 1 1 j n X e e, e. (9)

5 313 Victor Vasilievich Ivanov et al,2014 Expression (9) is the generic mathematical model of elementary digital FIR-filter based on the sliding discrete complex Forier transformation. Besides, the complex otpt vale which allows s to se two otpt projections real and imaginary is meant by n. This makes it possible to work with two otpt vales which give the relative PFRC shift of 2 radian. If we create DF based on the sperposition of EDF sing the freency sampling method, we need to distribte the obtained mathematical models to the array of EDF. At the same time, the amplitde coefficients of EDF otpt signals are taken into accont to get the comptational efficiency and given freency band. In case when the amplitde coefficient of EDF otpt signal is eal 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 continos band pass at the expense of mltiplying neven EDFs by mins nit. In compliance with the above we can write down the generic mathematical model of digital FIR-filter based on the sliding discrete complex Forier transformation: n m X n e e 2 j 2 j j n j 1, n 1 e. (10) The otpt signal of DF can be obtained by the isolation of the real or imaginary component from expression (10) Conclsion: In the generic mathematical model, the PFRC is shifted throgh angle 2 by means of isolation of the real or imaginary component of complex otpt signal. The gradal shifting of PFRC is achieved by means of rotating inpt samples throgh an additional angle in their vector addition. Besides, the gradal shifting of PFRC is achieved by the additional rotation of otpt signal vector. Findings: The athors proposed an approximation which makes it possible to exclde Forier 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 reslt if the sperposition of EDF. ACKOWLEDGEMETS The paper is created within the framework of the fndamental research financed by the Ministry of Edcation and Science of the Rssian 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 sb-department Information and electronic services of the Volga Region State niversity of Service. REFERECES [1] Bdilov, V.., V.I. Volovach, M.V. Shakrskiy and S.V. Eliseeva, Atomated Measrement of Digital Video Cameras Exposre Time. In the Proceedings of 2013 IEEE East-West Design & Test Symposim (EWDTS 2013), pp: [2] Artyshenko, V.M. and V.I. Volovach, Statistical Characteristics of Envelope Otliers Dration of non-gassian Information Processes. Proceedings of IEEE East-West Design & Test Symposim (EWDTS 2013). Rostov-on-Don, Rssia. Kharkov: KRE, pp: [3] Shakrskiy, M.V., V.K. Shakrskiy and V.V. Ivanov, Digital converter of freency deviation based on three freency generator. In the Proceedings of 2013 IEEE East-West Design & Test Symposim (EWDTS 2013), pp: [4] Prokopenko,.., A.I. Serebryakov and P.S. Bdyakov, Perspective high-freency correction in differential and broadband amplifiers. Proceedings of Papers - 5th Eropean Conference on Circits and Systems for Commnications, ECCSC'10, pp: [5] Prokopenko,.., D.. Konev and A.I. Serebriakov, Two channel AC amplifier with antiphase control of transfer factor. KpbiMKo 2009 CriMiCo th International Crimean Conference Microwave and Telecommnication Technology, Conference Proceedings, pp:

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