Consider a periodic signal f (t)

(1)

Where: k is the harmonic order; A_k and _k are amplitude and initial-phase of the k-th harmonic; f₁ is the frequency of f (t).

According to the sampling frequency f_S and the number of sampling points within a cycle N, in the range of [t ₀, t ₀ + W × T_S] uniformly-spaced sampling W+1 times to obtain the sampled sequence f(i), i = 0 to W. Where W is determined by the integral method, W = n N when using Trapezoidal integration methods; n is the number of iterations.

Appling Eq. (2) and (3) to calculate the real part a_k, the imaginary part b_k, A_k and φ_k of the k-th harmonic, this is QSDFT [12X.Z. Dai, and R. Gretsch, "Quasi-synchronous sampling algorithm and its applications", IEEE Trans. Instrum. Meas., vol. 43, no. 2, pp. 204-209, 1994.
[http://dx.doi.org/10.1109/19.293421] ].

(2)

(3)

Where: i is the start sampling point, normally i = 0; γ_j is the weighting coefficient, determined by the integral method, n and is the sum of all weighting coefficients, and Q = (N+1)ⁿ when W = nN.

The signal used for harmonic analysis comes from rectangular window truncation of the infinite length signal which duration is T = NT_S, T_S = 1/f_s and that leads to the spectral leakage of Fourier transform.

As to a sinusoidal signal f (t) = sin (100 πt) and N = 128, its DFT value can be retrieved from frequency domain at fixed sampling frequency , as is shown in Fig. (1). The analog frequency corresponding to 2π/N is f_s/N. Fig. (1) illustrates that the discrete amplitude spectrum is no longer a single peak, instead countless peaks of varying sizes distribute in the entire frequency axis. In other words, the energy of frequency spectrum is leaked and no longer concentrated, which is commonly named long-range leakage. In the process of harmonic analysis, the leaked energy from each harmonic will affect each other, and cause analysis error [1F.J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform", Proc. IEEE, vol. 66, no. 1, pp. 51-83, 1978.
[http://dx.doi.org/10.1109/PROC.1978.10837] ].

For QSDFT, the main lobe of the discrete spectral is widened and the side lobe will be suppressed under the convergence of quasi-synchronous sampling. Fig. (1). shows the discrete amplitude spectrum of QSDFT, where the number of iterations n = 8, the main lobe is converged and the mutual influence of leaked energy from each harmonic is trivial. From this perspective, the quasi-synchronous sampling can be treated as a good window function and it can inhibit long-range leakage significantly, so sometimes it is also referred to as quasi-synchronous window [19H.F. Qiu, and H. Zhou, "Study on the approachs of electrical harmonic analysis for asynchronous sampling", Chinese J. Relay, vol. 36, no. 1, pp. 57-62, 2008.].

Fig. (1) The discrete spectrum of DFT and QSDFT.

In practical measurement, considering the influence of multi-factors - precision of sampling clock, integer multiple error of sampling clock, frequency drift of signals - it is difficult to implement exact whole-cycle synchronous sampling. The above factors lead to the short-range leakage of discrete spectrum, of which the causes can mainly be attributed to the frequency drift of signals. In theory, based on the convergence characteristics of multiple iterations, QSDFT can suppress the short-range leakage caused by frequency drift. However, in practical applications, the inhibitory effect on the short-range leakage is not significant.

Fig. (2) shows the QSDFT harmonic analysis error (number of iterations n=5) of the waveforms generated from Eq. (4).

(4)

Where: A_2k-1 and ϕ_2k-1 are arbitrary; f₁ = 45 to 55Hz; Amplitude Relative Error = (Analyzed - Theoretical) / Theoretical* 100%; Initial-phase Absolute Error = (Analyzed - Theoretical).

As is in Fig. (2), high analytical precision of the amplitude of QSDFT occurs only in the range of 50Hz ± 1%, and the maximum relative error of the amplitude nearly reach 100% when in the range from 45 to 55Hz. It suggests that initial-phase from QSDFT is untrustworthy except when it is close to original frequency. For that reason, QSDFT can be applied only to such occasions with small signal frequency drift and low amplitude precision requirements, and it is not suitable for initial-phase analysis.

Definition: The rate of signal frequency deviation , is the degree of the signal frequency drift, which is defined as follows:

(5)

If the signal frequency does not drift, µ = 1; and if the signal frequency drifts, μ ≠ 1.

Assume that sampling frequency f_S and N is fixed, the sampled sequence f_i acquired from Eq. (1) is as follows:

(6)

Fig. (2) The harmonic analysis errors of QSDFT (n=5).

The frequency spectrum function of x-th harmonic is:

(7)

If the signal frequency has deviated µ,the sampled sequence f^* (i) is:

(8)

Thus, the corresponding frequency spectrum function of x-th harmonic is:

(9)

Compare Eq. (7) and Eq. (9) can be obtained:

(10)

Eq. (10) means, if signal frequency has deviated µtimes, then the discrete spectrum of QSDFT is stretched μ times along the frequency axis, and the discrete spectrum peak of the k-th harmonic will appear in the position of kµf₁ from kf₁ (Fig. 3).

Fig. (3) The discrete amplitude spectrum of QSDFT with the signal frequency drift.

Fig. (4) The variable picket fence.

Eq. (10) shows that the signal frequency drift will let the spectral peak deviated from the ideal position. When the signal frequency drift occurs, there will be great error if the sampling position in frequency domain still is 2π/N (the corresponding analog frequency f_s/N). For example, when f₁ = 55HZ, µ = 1.1, and the fundamental peak position is Nf₁/f_s = 1.1. However, in accordance with frequency domain sampling position 2π/N, the fundamental information obtained from Nf₁/f_s = 1 is inaccurate. If μ can be accurately estimated, and we can adjust frequency domain sampling position to µ2π/N (the corresponding analog frequency µf_S/N), the amplitude and initial-phase information of the fundamental and the k-th harmonic will be obtained accurately (Fig. 4).

So, the improved equation is:

(11)

Variable Picket Fence (VFP) is a modified algorithm in which the frequency domain sampling position is not fixed, instead the position changes with the signal frequency drift. So VPF can accurately estimate the peak position of each harmonic, thereby obtain high precision results of the amplitude and initial-phase.

Suppose , and n is a large number, and the following equations can be deduced from [20X. Z. Dai, and T. Y. Tang, "Quasi-synchronous sampling algorithm and its applications-Part 3: High accurate measurement of frequency, frequency deviation, and phase angle difference in power systems", in Proceeding. IEEE IMTC/93,, pp. 726-729, .]:

(12)

Where: M is the maximum harmonic order,

(13)

(14)

The fundamental initial-phase φ₁(i) analyzed by QSDFT from the i-th sampling point, is:

(15)

As to two adjacent sampling points, the fundamental initial-phase difference:

(16)

Eq. (16) means, although the fundamental initial-phase acquired by QSDFT is totally incorrect when the signal frequency drifts, the fundamental phase difference between adjacent sampling points is fixed, and it is only interrelated with the signal frequency f₁ and the sampling frequency f_S.

For engineering applications, the fundamental initial-phase φ₁(0) and φ₁(1) of the two adjacent sampling points can be calculated from two starting points i = 0 and i = 1 with Eq. (2), so µ is

(17)

According to the Eq. (11) and (17), the analysis process of VFP consists of five steps:

1. Sampling W + 2 data. Standard QSDFT algorithm requires sampling W + 1 points, but due to the needs of two starting points i = 0 and i = 1, adding one more sampling point for VFP is necessary.
2. Calculating Fⁿ_a1 (0) and Fⁿ_b1 from i = 0 by Eq. (2).
3. Calculating Fⁿ_a1 (1) and Fⁿ_a1 from i = 1 by Eq. (2).
4. Calculating µ by Eq. (17).
5. Calculating the amplitude and the initial-phase of each harmonic by Eq. (11).

If it is necessary, we can calculate the frequency of each harmonic according to μ and the sampling frequency f_S.

The analysis accuracy of VPF is directly dependent on the precision of μ. The μ relative error from VPF is given in Table (1). The simulation results show the high accuracy of μ is available.

Table 1
The relative error(%) of µ.

Due to pages limitation, only the simulation error figures of the amplitude and initial-phase from VPF are given. Fig. (5) shows the relative error of the amplitude. If n = 5, the maximum amplitude error equals to 0.0059632%; if n = 8, the maximum amplitude error equals to -0.0000015%.

Fig. (6) illustrates the simulation absolute error of the initial-phase from VPF. If n = 5, maximum initial-phase error equals to -0.3083002°; if n = 8, the maximum initial-phase error equals to 0.0002375°.

The following conclusions can be drawn from the simulation results:

1. The amplitude and initial-phase obtained from VPF have a high precision.
2. The number of iterations n has great impact on the analysis accuracy. With the increase of n, the analysis precision increases for almost one order of magnitude. The higher n selected by the system, the more accurate analysis results will be achieved.
3. With the drift of the signal frequency, the analysis errors of μ, amplitude and initial-phase increases, and VPF has higher analysis accuracy within 47 to 53Hz.
4. The analysis accuracy of the high-order harmonic is lower than the low-order harmonics, but it is not evident in the range from 47 to 53Hz.

The main source of the VPF analysis error is the tiny leak of the discrete spectrum caused by non-integer-period sampling, but this shortcoming can be compensated by increasing the number of iterations.

Fig. (5) The amplitude errors of VPF.

Fig. (6) The initial-phase errors of VPF.