【正文】 Lowfrequency electrical metrology1. IntroductionIt is fundamental to take into account measurement errors when applying sampling techniques to highaccuracy digital measurements [1–3]. Reconstruction ofdeterministic,nonbandlimited, real signals from sampled values also benefit from a careful study of measurement errors [4–11].In this paper, we apply qualitative and quantitative analysis of some of error sources in sampling theory to asynchronous digital sampling measurements. We verify the significance of each error source in simulations and in laboratory measurements, and also the efficiency of errorreduction strategies. Although there are many papers that address sampling error analysis, our objective here is to study the most significant of those errors sources for electrical highaccuracy measurements.It is a mathematical idealization to assume that a signal function has limited bandwidth with finite energy and infinite duration. In practice, the signal to be sampled by an analogtodigital converter (ADC) is of limited time duration and often possesses a much wider frequency bandwidth than that of the converter. These limitations are responsible for aliasing error, one of the most significant error sources in digital sampling. Integration is another significant error source, for most ADCs average the input signal during a time interval. In addition to aliasing and integration, we also consider here two other error sources: quantization and jitter. This paper is organized as follows: in the following section, we present an introduction to sampling analysis and aliasing error. In the third section, we present an analysis of integration error in sampling systems and study a known method to pensate for it. These calculations are then applied to asynchronous data acquisition of periodical signals. In Section 4 we describe quantization and jitter errors. In Section 5 we present the conclusions.2. Sampling analysis and aliasing errorThe sampling theorem [1] states that a signal function, , defined over the field of real numbers R and square integrable over that field, with bandwidth limited to an interval radians per second, can be pletely reconstructed from its sampled values f（n），naswhere=Ts seconds is the sampling period and .Finiteduration signals do not have limited bandwidth,as stated by the uncertainty principle [12]. Therefore the signal function f(t) takes the formR[f(t)] is the aliasing error, whose norm is limited by [1] where is the Fourier transform of ，which shall be in the space of continuous real absolutely integrable functions， [1]: (5)2. 1 Aliasing error: theoretical calculationsFor signals with high harmonic distortion, the harmonic ponents’ energy can be significant even if the operational bandwidth is much larger than the fundamental frequency, Vo Hz, causing considerable aliasing error. For many reasons, evaluation of (3) is of great importance in error analysis for high accuracy measurements. However,as we will be able to verify in the examples that will follow,for some important signals (3) is a very loose upper bound. Signal 1: rectified sinusoidal waveformAs a first example, we discuss below the solution of (3) for the rectified sinusoidal waveform with period seconds. For this signal, all conditions for the signal function,f(t), are satisfied and (3) applies directly.Let denote the rectified sinusoidal waveform, , which can be rewritten, using its Fourier series expansion, asThe Fourier transform of is where radians per second.For a bandwidth of we have [13]where denotes the smaller integer not less than x and denotes the larger integer not greater than x. For M = 25, the upper bound for the aliasing error can becalculated as which indicates that even for a bandwidth 25 times larger than the fundamental frequency, , aliasing error contaminates harmonic measurement for most practical purposes. However, increasing the sampling rate even further may not be a viable solution: High sampling rates may cause high quantization errors due to internal limitations of the ADC. On the other hand, small sampling rates may cause very high aliasing error, eventually requiring the use of antialiasing filters, which may distort the original signal. Signal 2: sawtooth waveformFor a second example, we discuss the solution of (3) for the sawtooth waveform,which can be rewritten, using its Fourier series expansion,as （12）The Fourier transform of is （13）The sawtooth is also a wellknow waveform. However, unlike the rectified sinusoidal waveform, for this particular signal, (3) cannot be applied directly because and. The function is not continuous in time and its Fourier transform is not absolutely integrable.However, far from being an obstacle, the sawtooth waveform is an interesting signal for study. Due to practical limitations, signal generators construct approximate sawtooth signals, which are continuous in time and limited in bandwidth. The generator output signal can be approximated by a highly asymmetric triangular waveform such aswhere and seconds is the period of the signal. Parameter controls the degree of asymmetry: l=2renders a symmetric triangular waveform, whereas eads to the sawtooth waveform. Fig. 1 illustrates one period of .Eq. (14) can be rewritten in terms of its Fourier series expansion asAlthough is a good representation of the approximate sawtooth signal, programmable signal generators have limited bandwidth. For a more exact representation of the approximate sawtooth signal used in practical measurementscenarios, one may wish to limit in (15), where is the generator bandwidth.Fig. 1. Sawtooth waveform as an asymmetric triangular waveform.The Fourier transform of is given byThe signal represented by is continuous and absolutely integrable, therefore (3) can be evaluated. For a bandwidth we ha