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# Method of Calculating the Sampling Factor N=Fd/Fs for Analog Signal Conversion when the Maximum Amplitude Error Emax is Known and Vice Versa

Petre Tzvetanov Petrov

It is suggested the introduction of sampling factor (conversion factor, encoding factor) N=Fd/Fs=Fd/Fmax>0 where Fd is the sampling (encoding) frequency, Fmax is the maximal signal frequency to be reconstructed, Fs is the sinusoidal signal frequency to be reconstructed. It is proposed the following expressions for calculating the frequency of conversion Fd of an analog signal (in particular a sinusoidal signal (SS)) with an ideal converter (Analog to Digital Converter = ADC or Digital to Analog Converter = DAC): Fd = N*Fmax = N*Fs, N= 180(90-arcsin(1-Emax))>0 (usually N>=4), where N is the sampling factor which depends on the application. Emax is the maximum allowed amplitude error (0<=E<=1). For a cosinusoidal signal (CS) the following formulas are suggested N= 180/arccos (1-Emax) 1>=Emax = (1-cos (180/N))>=0. The discussed models for SS and CS are applicable for N>=2. For N<2 a different models should be used. The method shown in the paper has some advantages compared with the theorems of V.A. Kotelnikov (1933) and C. E. Shannon (1949). It is declared that from the practical point of view the sampling frequency Fd=4*Fs is the most important frequency because is guaranteeing the maximal amplitude conversion error Emax lower than –29.3% (-3dB).
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