【正文】 gital 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, ft, defined over the field of real numbers R and square integrable over that field, with bandwidth limited to an interval Ω，Ω radians per second, can be pletely reconstructed from its sampled values f（nπ/Ω），n∈Z， aswhere π/Ω=Ts seconds is the sampling period and sincx=sinxx .Finiteduration signals do not have limited bandwidth,as stated by the uncertainty principle [12]. Therefore the signal function f(t) takes the formRΩ[f(t)] is the aliasing error, whose norm is limited by [1] wherefω∈L1R is the Fourier transform of ft ，which shall be in the space of continuous real absolutely integrable functions，ft∈CR∩L1R [1]:ft=12π∞∞Ftejωtdω (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 12Vo seconds. For this signal, all conditions for the signal function,f(t), are satisfied and (3) applies directly.Let f1t denote the rectified sinusoidal waveform, f1t=sinω0t, which can be rewritten, using its Fourier series expansion, asThe Fourier transform of f1t is where ω0=2πV0 radians per second.For a bandwidth of Ω=2πMVo，M?N we have [13]where x denotes the smaller integer not less than x and xdenotes the larger integer not greater than x. For M = 25, the upper bound for the aliasing error can becalculated as RΩf1t≤，which indicates that even for a bandwidth 25 times larger than the fundamental frequency, Vo, 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,asf2t=2πk=1∞1k1sinω0ktk （12）The Fourier transform of f2t isF2ω=k=1∞21k1jkδωkω0δω+kω0 （13）The sawtooth is also a wellknow waveform. However, unlike the rectified sinusoidal waveform, for this particular signal, (3) cannot be applied directly becausef2t?CR∩L1R andF2ω?L1R. 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 m∈N and To=1Vo seconds is the period of the signal. Parameter l1controls the degree of asymmetry: l=2renders a symmetric triangular waveform, whereas l→∞ eads to the sawtooth waveform. Fig. 1 illustrates one period of f3t.Eq. (14) can be rewritten in terms of its Fourier series expansion asAlthough f3t 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 k≤VmaxV0 in (15), where Vmaxis the generator bandwidth.Fig. 1. Sawtooth waveform as an asymmetric triangular waveform.The Fourier transform of f3t is given byThe signal represented by f3t is continuous and absolutely integrable, therefore (3) can be evaluated. For a bandwidth Ω=2πMνo，M∈N， we haveThis function is highly dependent on parameter l, which itself depends on the signal generator. For a signal generator with bandwidth equal to 50 MHz, the typical rising time for a sawtooth waveform with 1 V of amplitude is in the order of magnitude of 20 nanoseconds. If υ0= 50 Hz, then l=5105. Eq. (19) gives RΩf3t≤ V for M = 50,which is an upper bound unrealistically high (too pessimistic) for the total aliasing error. Aliasing error–simulation resultsIn order to illustrate better the importance of a careful samplingerror analysis, we simulated the signal f3t represented by (14) and its version with limited bandwidth for M = 50, denoted here byf3，Mt, for values of t=nπ/Ω 。n∈N. In both cases, we used υ0 = 50 Hz and Ts=200μs.We define here F3kω0 and F3，Mkω0 the discretetime Fourier transforms of f3nπΩ and f3，MnπΩ，respectively, where the latter was constructed with bandwidth limited to M harmonic ponents and therefore is free from aliasing error. The difference E3kω0=F3kω0F3，Mkω0is the