Optimization of frequency quantization. VN Tibabishev. Keywords: optimization, sampling frequency, the substitution frequencies.
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1 UDC Otimizatin f frequency quantizatin VN Tibabishev Asvt51@nard.ru We btain the functinal defining the rice and quality f samle readings f the generalized velcities. It is shwn that the timal samling frequency, in the sense f minimizing the functinal quality and rice deends n the samling f the uer cutff frequency f the analg signal f the rder f the generalized velcities measured by the generalized crdinates, the frequency rerties f the analg inut filter and a maximum samling rate fr analg-digital cnverter (ADC). An examle f calculating the frequency quantizatin fr tw-tier ADC with an inut RC filter. Math.OC, 62D05, 42A38 Keywrds: timizatin, samling frequency, the substitutin frequencies. 1. Intrductin The well-knwn samling therem secifies the samling frequency f and is in many ublicatins [1]. It is alicable nly t send the generalized crdinate systems, fr examle, the deviatin f the analg signal frm the zer level in the absence f nise as the substitutin frequencies. Samling frequency is chsen frm, where - cut-ff frequency range f the analg signal. Samling frequency f is called the minimum samling rate f the zer-rder generalized seed. Let an almst eridic signal is the Furier,. (1) Samling therem is rved nly fr signals that satisfy all harmnic amlitudes, if the frequency f. (2) It is knwn [2] that the simultaneus assignment f all the crdinates and velcities cmletely defines the state f ne-dimensinal and multidimensinal systems. In this case it is assumed that enugh t knw infrmatin abut the generalized crdinate and generalized velcity f the first derivative. If the generalized crdinates and generalized velcities are described by cntinuusly differentiable functins, higher derivatives are the first derivative fr any value f the current time. The situatin changes if we are given discrete samles f the generalized crdinates at time intervals. T find the arximate value f the first derivative n the interval requires at least ne additinal cunt in the
2 middle f the interval. The minimum samling frequency fr estimating the derivative f the th rder assciated with a minimum frequency f the zer rder by the bvius relatin, where rder derivative, 0. It fllws that knwledge f an arximate estimate f the first derivative is a necessary but nt sufficient cnditin fr an arximate estimate f higher rder derivatives. In this regard, the task f samling the generalized crdinates and generalized velcities -th rder ne-dimensinal case. The quality (accuracy) estimates fr the derivatives f the generalized crdinates with increasing frequency quantizatin, but the fee increases (rice) as the vlume f infrmatin being rcessed fr the same duratin f imlementatin. Necessary t determine the timum samling frequency, where the rice-quality rate reaches a minimum value fr the evaluatin f the derivative - th rder. In reality, the riginal signal with a limited range f is given as a finite functin. It is the rduct f the imlementatin mdel signal (1) defined n the whle line, and finite windw functin, which differs frm zer nly n the bservatin interval signal. The sectrum f the rduct f tw functins is the cnvlutin f the sectra f the riginal signal and finite windw functin [3]. The cnvlutin f the sectra f the rduct f these functins is defined n the whle infinite frequency axis. Therefre, the cnditin (2) des nt hld in the real wrld well-knwn therem f quantizatin. Analg lw-ass filter munted in frnt f an analg-digital cnverter (ADC) t arximate the cnditin f quantizatin f a therem (2). The real lw-ass filter may have limited the rate f decrease f the amlitude-frequency resnse. The cutff frequency lw-ass filter is chsen fr a given level f suressin f high-frequency art f the sectrum, where - filter bandwidth at half wer. S instead f frequency quantizatin, r in the ADC is samling frequency, which substitute fr the suressin f frequencies chsen frm the cnditin the frequency quantizatin After the suressin substitutin frequencies cntinued use f resulting in redundancy f infrmatin. This shrtcming is eliminated by the digital utut lw ass filter in the frm f a frequency divider with divisin rati equal. 2. Determining the timal frequency f quantizatin f the generalized velcity
3 The cnditin f the limited sectrum fr finite rcesses is arximately satisfied by the inut analg lw-ass filter. With the limited frequency sectrum, fr examle, F there is a bundary harmnic cmnent (t) with the highest incidence 2F f nn-zer amlitude A, fr examle, y t) A cs( t ) which defines the bundary harmnic cmnent f the ( generalized crdinates. It is believed that the amlitude f harmnics with frequencies > are s small that their influence can be neglected. In determining the quality (accuracy) estimate f the arximate values derived frm their exact values will use the mean-square numerical characteristics f the secies. (3) If yu ut in this functinal, then the value f the functinal will nt deend n the initial hase. Therefre, the value f the initial hase we assume t be zer in rder t simlify the exressins btained. Bundary harmnic cmnent f the generalized velcity - th rder crdinate y(t) is the time derivative f. With digital signal rcessing instead f an infinitely small dt quantity may be taken nly a finite quantity t, the maximum value which is determined by the samling therem fr the frequency f, and the minimum value is limited t the maximum samling frequency (seed) alied t the ADC. With these cnstraints, the value t can be written as, where is nt necessarily a sitive integer which can take values in the range where t 1/ 2F N where N, which can take values in the interval 1, [ ] m N, where - the maximum allwable value. An arximate estimate f the harmnic cmnent f the generalized velcity -th rder is given by v ( ( t, t) y( t, t) /( t) ( k ) k ) k ( k ) where y( t, t) - the finite difference - th rder cntinuus functin n the entire line. Knwn cnnectin with the finite difference derivatives f the crresnding rder [4], y We chse α such that wuld,. At the same time
4 . Deviatin f the arximate estimate f (t, frm its exact value f ( k) ( k) is given by R( ) v ( ) y ( t). As an abslute measure f quality assessment f the bundary f the harmnic cmnent f the generalized velcities using the functinal frm (3). We define the functinal significance fr the exact derivative f harmnic cmnent f the bundary at a frequency - th rder We find a numerical characteristic f the relative errr f estimate f the bundary f the harmnic cmnent f the derivative - th rder., (4),,. With the increasing N imrtance f decreasing the value f the functinal J ( k, ), but it increases the number f cunts er unit time, which acts as a fee 1 N (rices) t imrve the quality f measurement. If at a frequency quantizatin samling therem the number f cunts er unit time G 2F, with increasing values f the number f cunts er unit time N 1increases t a maximum value =. As a functinal measure takes int accunt bth the quality f the bundary f the generalized harmnic cmnent f seed and cst (rice) fr the quality f measurements can be taken frm the functinal J( N) r( k, N) J ( N) 2, (5) where J ( ) - the functinal defining the cst (rice) fr measuring the quality 2 N f the bundary f the harmnic cmnent f the generalized velcities. Functinal r( k, N) [0,1 ] is the relative numerical characteristic. Therefre, the functinal J ( ) must als be defined as the relative numerical characteristic 2 N J ( N) [0.1] 2. As such a functinal can be, fr examle, the functinal frm 1
5 J 2NF / F. (6) ( N) G / G 2 1 m s At the same time /. (7) Number N fr which the functinal J ( N ) attains its minimum value determines the timal samling frequency F 2N F fr estimating the bundary f the harmnic cmnent f the generalized velcities in terms f minimum errr fr the minimum samling rate at which we can neglect the nise substitutin frequencies. Fund functinality is a transcendental equatin. This circumstance des nt allw t btain an exressin fr the arameter N in an exlicit frm, in which the functinal J(N) attains its minimum value. In this regard, evaluatin f timal samling frequency is a numerical methd. An examle. Let the sectrum f the surce signal frequency is limited = 2 khz. The task is t determine the timum samling frequency fr estimating the first rder derivative in the sense f minimizing the functinal "rice - quality" (7). The minimum samling frequency is кhz fr estimating the generalized first-rder rate. Cnsider the rder f slving the rblem fr tw-link inut RC = T lw-ass filter. Fr a frequency f 2 khz determined by the filter time T n the half-wer s. Cutff frequency bandwidth f the cnvlutin f the sectra is determined by the amlitude frequency characteristic f tw-link RC-lw ass filter n the selected level, fr examle, 0.01 signal wer. It is 4,56 khz. Assume that the maximum frequency cnversin in the ADC F_m (seed) is 500 khz. Numerical methd is established that the minimum f functinal (7) is reached fr = 5. Hence, the timal samling frequency = 5 * 2 * 4.56 = 45.6 khz. Standard errr f change in the generalized first-rder rate is 4.8%. The frequency f the utut ulses fr ADC evaluatin f the generalized first-rder rate is 2 * 4.56 = 9.12 khz. Cefficient Decimatin is 5. When these data t estimate the generalized crdinates must be taken in the timal samling frequency f the ADC is equal t khz samling frequency instead f samling therem 4 khz. The cefficient f decimatin is 8. This is a fee fr the suressin f the substitutin frequencies. If yu need infrmatin abut the generalized crdinate and generalized velcity f the first rder, then received the tw frequencies f quantizatin is chsen the largest value, and the tw values f the cefficients f decimatin selects the smallest value. Fr mre infrmatin n the quantizatin f the signals in the slutin f rblems f identificatin f the dynamics f multidimensinal cntrl bjects can be fund at htt://asvt51.nard.ru/ The authr exresses his gratitude t Assciate Prfessr O.M. Pklenk fr valuable cmments. Cnclusin Frm the knwn samling therem samling rate determined by the inequality under tw cnditins. First, it is estimated nly a generalized crdinate,
6 and, secndly, it is believed that there is n substitute fr digitizing frequency. In many cases it is necessary t digitize the generalized velcities f finite rders and eliminate the henmenn f substitutin frequencies. It is established that there exists an timal samling frequency in the sense f minimizing the functinal, which determines the rice and quality measurement. This frequency deends n the uer cutff frequency, f the signal the rder f the derivative (generalized velcity), the frequency rerties f the inut analg lw-ass filter and the maximum frequency reference t the ADC. Is functinal, which determines the rice and quality measurements. An examle is given fr calculating the timum frequency quantizatin fr tw-link RC-lw ass filter. References 1. Middletn D. An intrductin t statistical cmmunicatin thery. New Yrk, McGraw-Hill, LD Landau, EM Lifshitz A mechanic. - Mscw: Fizmatgiz, Otnes RK, Enchsn L. Alied time series analysis. Vlume I basic techniques, New Yrk-Chichester-Brisbane-Trnt, Higher Mathematics. Secial glavy. / Ed. PI Chinaeva. - Kiev: Highest Schl, /10/2011 V. Tibabishev
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