We obtain the functional defining the price and quality of sample readings of the generalized velocities. It is shown that the optimal sampling frequency, in the sense of minimizing the functional quality and price depends on the sampling of the upper cutoff frequency of the analog signal of the order of the generalized velocities measured by the generalized coordinates, the frequency properties of the analog input filter and a maximum sampling rate for analog-digital converter (ADC). An example of calculating the frequency quantization for two-tier ADC with an input RC filter.
Deep Dive into Optimization of frequency quantization.
We obtain the functional defining the price and quality of sample readings of the generalized velocities. It is shown that the optimal sampling frequency, in the sense of minimizing the functional quality and price depends on the sampling of the upper cutoff frequency of the analog signal of the order of the generalized velocities measured by the generalized coordinates, the frequency properties of the analog input filter and a maximum sampling rate for analog-digital converter (ADC). An example of calculating the frequency quantization for two-tier ADC with an input RC filter.
The well-known sampling theorem specifies the sampling frequency of and is in many publications [1]. It is applicable only to send the generalized coordinate systems, for example, the deviation of the analog signal from the zero level in the absence of noise as the substitution frequencies. Sampling frequency is chosen from , where -cut-off frequency range of the analog signal. Sampling frequency of is called the minimum sampling rate of the zero-order generalized speed.
Let an almost periodic signal is the Fourier , .
Sampling theorem is proved only for signals that satisfy all harmonic amplitudes , if the frequency of .
(
It is known [2] that the simultaneous assignment of all the coordinates and velocities completely defines the state of one-dimensional and multidimensional systems. In this case it is assumed that enough to know information about the generalized coordinate and generalized velocity of the first derivative. If the generalized coordinates and generalized velocities are described by continuously differentiable functions, higher derivatives are the first derivative for any value of the current time. The situation changes if we are given discrete samples of the generalized coordinates at time intervals . To find the approximate value of the first derivative on the interval requires at least one additional count in the middle of the interval. The minimum sampling frequency for estimating the derivative of the th order associated with a minimum frequency of the zero order by the obvious relation , where order derivative, 0
. It follows that knowledge of an approximate estimate of the first derivative is a necessary but not sufficient condition for an approximate estimate of higher order derivatives.
In this regard, the task of sampling the generalized coordinates and generalized velocities -th order one-dimensional case.
The quality (accuracy) estimates for the derivatives of the generalized coordinates with increasing frequency quantization, but the fee increases (price) as the volume of information being processed for the same duration of implementation. Necessary to determine the optimum sampling frequency , where the price-quality rate reaches a minimum value for the evaluation of the derivative -th order.
In reality, the original signal with a limited range of is given as a finite function. It is the product of the implementation model signal (1) defined on the whole line, and finite window function , which differs from zero only on the observation interval signal. The spectrum of the product of two functions is the convolution of the spectra of the original signal and finite window function [3]. The convolution of the spectra of the product of these functions is defined on the whole infinite frequency axis. Therefore, the condition (2) does not hold in the real world well-known theorem of quantization.
Analog low-pass filter mounted in front of an analog-digital converter (ADC) to approximate the condition of quantization of a theorem (2). The real low-pass filter may have limited the rate of decrease of the amplitude-frequency response. The cutoff frequency low-pass filter is chosen for a given level of suppression of high-frequency part of the spectrum, where -filter bandwidth at half power. So instead of frequency quantization , or in the ADC is sampling frequency , which substitute for the suppression of frequencies chosen from the condition After the suppression substitution frequencies continued use of the frequency quantization resulting in redundancy of information. This shortcoming is eliminated by the digital output low pass filter in the form of a frequency divider with division ratio equal .
The condition of the limited spectrum for finite processes is approximately satisfied by the input analog low-pass filter. With the limited frequency spectrum, for example, F there is a boundary harmonic component
of non-zero amplitude A , for example,
which defines the boundary harmonic component of the generalized coordinates. It is believed that the amplitude of harmonics with frequencies > are so small that their influence can be neglected. In determining the quality (accuracy) estimate of the approximate values derived from their exact values will use the mean-square numerical characteristics of the species .
(
If you put in this functional , then the value of the functional will not depend on the initial phase. Therefore, the value of the initial phase we assume to be zero in order to simplify the expressions obtained.
Boundary harmonic component of the generalized velocity -th order coordinate ) (t y is the time derivative of . With digital signal processing instead of an infinitely small dt quantity may be taken only a finite quantity t , the maximum value which is determined by the sampling theorem for the frequency of , and the minimum value is limited to the maximum sampling frequency (speed) applied to the ADC. With these constraints, the value t can be wr
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