Science in China Series E: Technological Sciences © 2009
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Hanning self-convolution window and its application to harmonic analysis WEN He†, TENG ZhaoSheng†, GUO SiYu, WANG JingXun, YANG BuMing, WANG Yi & CHEN Tao College of Electrical and Information Engineering, Hunan University, Changsha 410082, China
The Hanning self-convolution window (HSCW) is proposed in this paper. And the phase difference correction algorithm based on the discrete spectrum and the HSCW is given. The HSCW has a low peak side lobe level, a high side lobe roll-off rate, and a simple spectrum representation. Hence, leakage errors and harmonic interferences can be considerably reduced by weighting samples with the HSCW, the parameter estimation by the HSCW-based phase difference correction algorithm is free of solving high order equations, and the overall method can be easily implemented in embedded systems. Simulation and application results show that the HSCW-based phase difference correction algorithm can suppress the impacts of fundamental frequency fluctuation and white noise on harmonic parameter estimation, and the HSCW is advantageous over existing combined cosine windows in terms of harmonic analysis performance. hanning self-convolution window, FFT, phase difference, spectral leakage, frequency fluctuation
The accurate dynamic estimation of signal parameters has been a hot topic in the field of signal processing[1]. The estimation can provide an information basis for power measurment[2], fault diagnosis[3], electrical harmonic compensation[4], and orthogonal frequency-division multiplexing[5]. Compared with wavelet transforms, the fast Fourier transform (FFT) is more computationally efficient and easier for implementations in embedded systems such as DSP and ARM, and is by now the most widely used in various signal parameter estimation algorithms[6,7]. For dynamic signals, it is difficult to achieve strict synchronous sampling even if frequency tracking technologies are adopted[8,9]. When using FFT to estimate signal parameters with asynchronous sampling, estimation error due to the spectral leakage and picket fence effect introduced by the asynchronous sampling and signal cutoff is relatively large [10]. Various kinds of windows, e.g., the rectangular window[11], the Hanning window [12], the Hamming window[13], the Blackman window[14], the Blackman-Harris window[15], the RifeVincent window[16], the Nuttall window[17], the polyno-
mial windows[18], the flat-top window[19], and the rectangular convolution window[20], have been proposed and used in the windowed interpolation FFT algorithms, and they can in some degree suppress spectral leakage and increase the accuracy of the signal parameter estimation. The use of the FFT algorithms with dual-spectrum-line[7,12,14] or multi-spectrum-line[6,8] interpolation based on high order combined cosine windows in fundamental and harmonics parameter estimation involves, however, solving high order equations, which is compu― tationally expensive[21 23]. Different approaches have been proposed to solve the problem. Yang, Ding et al. gave a discrete phase difference correction algorithm suitable for all kinds of symmetrical windows, which can calculate signal parameters without relying on the expression of the window spectral function[24]. Yang et al. proposed a harmonic analysis method which includes frequency and phase estimating algorithm, finite impulse response comb filter, and a correction factor[25]. Zhu Received July 10, 2008; accepted September 4, 2008 doi: 10.1007/s11431-008-0356-6 † Corresponding authors (email:
[email protected];
[email protected]) Supported by the National Natural Science Foundation of China (Grant No. 60872128)
Sci China Ser E-Tech Sci | Feb. 2009 | vol. 52 | no. 2 | 467-476
presented a practical numerical method for exact calculation of harmonic/interharmonics using adaptive window width[26]. Gu et al. proposed a spectrum-estimation method known as “estimation of signal parameters via rotational invariance techniques” based on the sliding-window[27]. However, the existing combined cosine windows and rectangular convolution windows which are slow in asymptotic decay rate of side lobes[28,29] cannot perform satisfactory estimation of signal parameters[30,31], and the harmonic interference of multi-component signals still exists[32,33]. To satisfy the requirements for accurate dynamic complicated signal parameter estimation and its high speed implementation in embedded terminals, this paper proposes a new kind of window, the Hanning self-convolution window (HSCW). Its time-domain and frequency-domain functions are both deduced, the performances of the first to the fourth order HSCWs and a phase difference correction algorithm based on this window are studied. Calculations of signal parameters, including frequency, phase and amplitude, are deduced in this paper. The HSCW has a low peak side lobe level and a high side lobe roll-off rate. Hence, considerable leakage errors and harmonic interferences can be reduced by weighting samples with the HSCW. Furthermore, the phase difference correction algorithm based on the HSCW has advantages of computational efficiency and easy implementation in embedded systems. Simulation results show that the proposed method can achieve higher estimation accuracy with the presence of fundamental frequency fluctuation and white noise, and that the HSCW is advantageous over existing combined cosine windows in terms of harmonic analysis performance. The effectiveness and practicability of the HSCW have been verified by its application to the three-phase multifunctional harmonics ammeter, which can achieve a precision of grade 0.2 s for electrical power measurement.
1 Hanning self-convolution window 1.1 Continuous-time Hanning self-convolution window The Hanning window, also known as “raised cosine” window, is defined as ⎧ 1 ⎡1 1 ⎛ πt ⎞ ⎤ ⎪ ⎢ + cos ⎜ ⎟ ⎥ , wH (t ) = ⎨ T ⎣ 2 2 ⎝ T ⎠⎦ ⎪0, ⎩ 468
t ≤T,
(1)
where T represents the width of the Hanning window in the continuous-time domain. The proposed pth-order Hanning self-convolution window (HSCW) is defined as the p−1 self-convolutions of p instances of the Hanning window:
wHp (t ) = wH (t ) ∗ wH (t ) ∗" wH (t ),
(2)
p
where p is the number of the Hanning windows, henceforth called the order of the HSCW, or simply the window order, and pT is the length of the pth-order HSCW wHp . According to eq. (2), the continuous-time second order and third order HSCWs are respectively given as ⎧ ⎛ πt ⎞ ⎛ πt ⎞ ⎪2πt + πt cos ⎜ T ⎟ + πT cos ⎜ T ⎟ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎛ πt ⎞ + T sin ⎜ ⎟ + 2πT , − T ≤ t ≤ T , ⎪ 1 ⎪ ⎝T ⎠ wH2 ( t ) = 2 ⎨ 8T π ⎪ ⎛ πt ⎞ ⎛ πt ⎞ −2πt − πt cos ⎜ ⎟ + 3πT cos ⎜ ⎟ ⎪ ⎝T ⎠ ⎝T ⎠ ⎪ ⎛ πt ⎞ ⎪ − T sin ⎜ ⎟ + 6πT , T < t ≤ 3T , ⎪⎩ ⎝T ⎠ (3) 1 ⋅ wH3 ( t ) = 64T 3 π2 ⎧ 22 ⎛ πt ⎞ ⎛ πt ⎞ 2 2 2 ⎪4π t + 2π t cos ⎜ T ⎟ + 8π Tt + 7T πt sin ⎜ T ⎟ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎛ πt ⎞ ⎛ πt ⎞ + 2π2Tt cos ⎜ ⎟ + 4π 2T 2 + 8T 2 + 8T 2 cos ⎜ ⎟ ⎪ ⎝T ⎠ ⎝T ⎠ ⎪ ⎪ ⎛ πt ⎞ ⎛ πt ⎞ + 7 πT 2 sin ⎜ ⎟ + π2T 2 cos ⎜ ⎟ , − T ≤ t ≤ T, ⎪ T ⎝ ⎠ ⎝T ⎠ ⎪ ⎪ ⎛ πt ⎞ ⎛ πt ⎞ ⎪−8π2 t 2 − 2π2t 2 cos ⎜ ⎟ + 8π2Tt cos ⎜ ⎟ + 32π2Tt ⎝T ⎠ ⎝T ⎠ ⎪ ⎪⎪ ⎛ πt ⎞ ⎛ πt ⎞ − 14T πt sin ⎜ ⎟ − 16T 2 − 8π2T 2 − 16T 2 cos ⎜ ⎟ ⎨ ⎝T ⎠ ⎝T ⎠ ⎪ ⎪ ⎛ πt ⎞ ⎛ πt ⎞ + 28πT 2 sin ⎜ ⎟ − 2π2T 2 cos ⎜ ⎟ , T < t ≤ 3T , ⎪ ⎝T ⎠ ⎝T ⎠ ⎪ ⎪ ⎛ πt ⎞ ⎛ πt ⎞ ⎪4π2t 2 + π2 t 2 cos ⎜ ⎟ − 40π2Tt − 10π2Tt cos ⎜ ⎟ T ⎝ ⎠ ⎝T ⎠ ⎪ ⎪ ⎛ πt ⎞ ⎛ πt ⎞ ⎪ + 7T πt sin ⎜ ⎟ + 100π2T 2 + 8T 2 + 8T 2 cos ⎜ ⎟ ⎪ ⎝T ⎠ ⎝T ⎠ ⎪ ⎛ πt ⎞ ⎛ πt ⎞ ⎪ − 35πT 2 sin ⎜ ⎟ + 25π2T 2 cos ⎜ ⎟ , 3T < t ≤ 5T . ⎝T ⎠ ⎝T ⎠ ⎩⎪
t > T,
WEN He et al. Sci China Ser E-Tech Sci | Feb. 2009 | vol. 52 | no. 2 | 467-476
(4)
1.2 Discrete-time Hanning self-convolution window
By discretizing the time-continuous Hanning window, the discrete-time Hanning window of length, or width, M is obtained as ⎡ ⎛ 2πm ⎞ ⎤ wH (m) = 0.5 ⎢1 − cos ⎜ ⎟ ⎥ , m = 0, 1, 2," , M − 1. (5) ⎝ M ⎠⎦ ⎣ For the sake of implementation of FFT, M is typically set to 2i, where i is a natural number. From eq. (2) it can be seen that the length of the result sequence of the convolution wH (m) ∗ wH (m) is 2M−1.
Zero padding can be applied on the result sequence to get the second order HSCW with length of 2M. Similarly, for the pth-order HSCW resulted from p−1 discrete convolutions and padding with p−1 zeros, the pth-order HSCW with length of pM can be obtained as wHp (n) = wH (m) ∗ wH (m) ∗ " ∗ wH (m)
(6)
p
where N=pM represents the width, in samples, of the pth-order HSCW.
2 Frequency response of the Hanning self-convolution window 2.1 Frequency response function
The discrete time Fourier transform (DTFT) of the Hanning window can be expressed simply as WH (ω ) = 0.5WR (ω ) ⎡ ⎛ 2π ⎞ 2π ⎞ ⎤ ⎛ + 0.25 ⎢WR ⎜ ω − ⎟ + WR ⎜ ω + ⎟⎥ , (7) M⎠ M ⎠⎦ ⎝ ⎣ ⎝ where ω is the angular frequency, and WR(ω ) is the DTFT of a rectangular window, which can be written as ⎛ ωM ⎞ sin ⎜ ⎟ ⎝ 2 ⎠ exp ⎡ − j ( M − 1) ω ⎤ . WR (ω ) = (8) ⎢ ⎥ 2 ⎛ω ⎞ ⎣ ⎦ sin ⎜ ⎟ ⎝2⎠ According to the convolution theorem in frequency domain, the DTFT of the pth-order HSCW is ⎧ ⎡ ⎛ 2π ⎞ WHp (ω ) = ⎨0.5WR (ω ) + 0.25 ⎢WR ⎜ ω − ⎟ M⎠ ⎣ ⎝ ⎩ p
2π ⎞ ⎤ ⎫ ⎛ + WR ⎜ ω + (9) ⎟⎥ ⎬ . M ⎠⎦ ⎭ ⎝ 2k π , where k is the By substituting ω in eq. (9) with N index of the discrete spectral lines and k=0, 1, " N−1,
the DFT of the pth-order HSCW is given as ⎧ ⎛ 2k π ⎞ ⎡ ⎛ 2k π 2π ⎞ WHp (k ) = ⎨0.5WR ⎜ − ⎟ + ⎢0.25WR ⎜ ⎟ M⎠ ⎝ N ⎠ ⎣ ⎝ N ⎩ p
⎛ 2k π 2π ⎞ ⎤ ⎫ +WR ⎜ + ⎟ ⎬ . M ⎠ ⎥⎦ ⎭ ⎝ N
(10)
2.2 Characteristics of the HSCW major lobe
The width of the major lobe of a window function is defined as the distance between zero-crossing in window spectrum[15]. As shown in eq. (10), the DFT of the pth-order HSCW is a periodic function with a period of 2π, and is symmetric with respect to the origin. Therefore, the major lobe width of the pth-order HSCW is the doubled spectral distance between the origin and the nearest zero point on either side to the origin in the spectrum. From eq. (10), the following conditions must be satisfied to make WHp ( k ) zero: ⎧ πkM ⎪⎪ N = d π, ⎨ ⎪π ⎛ k ± 1 ⎞ ≠ d π, ⎜ ⎟ ⎩⎪ ⎝ N M ⎠
When k =
d = 0, ± 1, ± 2, ".
(11)
2dN , conditions in eq. (11) are satisfied, M
and thus WHp ( k ) = 0 . Since k ∈ [ 0, N − 1] , the nearest zero point to the right of the origin is k = 2 p, given 4 pπ . The major N lobe width of the pth-order HSCW is henceforth given by d=1, whose distance to the origin is
BW =
8 p π 8π = . N M
(12)
Referring to eq. (10) one can see that the major lobe width of an HSCW is equal to that of the original Hanning window used in the convolutions. If the length of the pth-order HSCW is fixed, the value of M is inversely proportional to the order of the HSCW since N=pM. The major lobe width of an HSCW with a fixed length of N is thus determined by the order of the HSCW. The higher the order of the HSCW, the wider the major lobe. 2.3 Characteristics of the HSCW side lobes
The first order to the fourth order HSCWs are constructed using a Hanning window with length M=64, whose magnitude frequency responses are derived from
WEN He et al. Sci China Ser E-Tech Sci | Feb. 2009 | vol. 52 | no. 2 | 467-476
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eq. (10) and shown in Figure 1. The side lobe roll-off rates are also given in the figure. It can been seen in Figure 1 that the peak side lobe level and side lobe rolloff rate of an HSCW are proportional to the order of the window. The side lobe performance of the HSCW can thus be rapidly elevated with the increase of its order. Figure 2 shows the magnitude frequency responses of the second and the fourth order HSCWs constructed from a Hanning window with length M=64, the Hanning window and Hamming window with length N=128, and the Blackman window and Blackman-Harris window with length N=256. The major lobe width of the HSCW is subject to the length of the Hanning window used. For the second order HSCW, its major lobe width is two times that of the Hanning window and the Hamming window of the same length, and for the fourth order HSCW, two times that of the Blackman window and the Blackman-Harris window of the same length. Besides, with the increase of the
window order, the side lobe performances, such as peak side lobe level and side lobe roll-off rate, of the HSCW improve rapidly and are superior to those of the combined cosine window of the same length. Comparisons of the major lobe width, peak side lobe level and side lobe roll-off rate between the first to the fourth order HSCWs and the various classical windows are listed in Table 1. From Figures 1 and 2 and Table 1, it can be seen that when the sequence length is fixed to N, with the increase of the window order, the peak side lobe level of the HSCW decreases and the side lobe roll-off rate increases, both rapidly. Especially at the frequencies satisfying the conditions in eq. (11), or multiples of the major lobe width, the frequency response of the HSCW is of the lowest peak side lobe level. At these particular frequencies, the average side lobe levels of the HSCW are lower than those of the classical windows with the same length, and thus the HSCW can suppress spectral leakage more effectively.
Figure 1 Magnitude frequency responses. (a) Magnitude frequency responses of the first and the second order HSCWs; (b) Magnitude frequency responses of the third and the fourth order HSCWs.
Figure 2 Magnitude frequency responses. (a) Magnitude frequency responses of the second order HSCW, Hanning window, and Hamming window; (b) Magnitude frequency responses of the fourth order HSCW, Blackman window, and Blackman-Harris window.
470
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Table 1 Comparisons of window characteristics Window of length N
Rectangular Triangle Hamming Blackman Blackman-Harris Nuttall Second order HSCW
Fourth order HSCW
Major lobe width
4π N
8π N
8π N
12π N
16π N
16π N
8pπ N
8pπ N
Peak side lobe level (dB)
−13
−27
−43
−59
−92
−83
−64
−125
Side lobe roll-off rate (dB/oct)
6
12
6
18
6
30
36
72
3 Phase difference correction algorithm based on HSCW Because of its advantageous side lobe performances, the HSCW can be used to suppress spectral leakage by weighting the signal. Based on the time- and frequency-domain characteristics of the HSCW, accurate signal parameter estimation can be achieved by applying the phase difference correction algorithm on signals weighted by the HSCW. The phase difference correction algorithm based on HSCW includes discrete signal truncating, HSCW weighting, discrete phase difference correcting, and signal parameter estimating. The steps are as follows. (a) Discrete signal truncating. Let the discrete signal length N=2i, where i is a natural number, and is 9 or 10 for simulations in the paper, corresponding to N values of 512 and 1024. Also let L be the time domain shift of N the signal, 0 < L < . Truncate the sampled discrete 2 signal into the sequences S1 consisting of samples 1 to N and S2 of samples L+1~L+N. (b) HSCW weighting. Use the HSCW to weight S1 and S2, resulting in new sequences NS1 and NS2, respectively. (c) Discrete phase difference correcting. Perform DFT on NS1 and NS2, whose lengths are both N, and carry out correction according to the discrete phase difference to obtain the frequency offset correction. (d) Signal parameter estimating. Use the frequency response of the HSCW and the frequency offset correction to estimate the frequency, amplitude and phase of the fundamental and the harmonics. Take the analysis of a time-domain signal with multicomponent harmonic as an example. The time-domain representation of the signal is H
{
}
x ( t ) = ∑ Ah exp ⎡⎣ j ( 2πhf 0 t + ϕh ) ⎤⎦ , h =1
(13)
where H is the number of harmonics, h is the order of the harmonic, f0 is the fundamental frequency, and Ah
and ϕ h are respectively the amplitude and phase of the
hth harmonic. Sampling the signal x(t) with a sampling rate of fs, one gets the following discrete sequence. H ⎧ ⎡ ⎛ 2πhf 0 n ⎞ ⎤ ⎫⎪ ⎪ (14) + ϕh ⎟ ⎥ ⎬ . x ( n ) = ∑ ⎨ Ah exp ⎢ j ⎜ h =1 ⎩ ⎠ ⎦⎥ ⎭⎪ ⎪ ⎣⎢ ⎝ f s Truncate x(n) as described in Step (a) to obtain the sequences x1(n) and x2(n) with length of N, and weight
the sequences by using the HSCW WHp ( n ) . The spectrum of x1 ( n ) obtained by applying DFT is H
X ( k ) = ∑ ⎡⎣ Ah exp ( jϕ h )WHp ( Δkh ) ⎤⎦ ,
(15)
h =1
⎛ f N f N⎞ where Δkh = kh − hk0 ⎜ kh = h , k0 = 0 ⎟ is the fs fs ⎠ ⎝ difference between the actual frequency and the spectral line of the hth harmonic in the spectrum. For the hth harmonic, if synchronous sampling is employed, k0 is then an integer, and the spectral line corresponding to the hth harmonic is at kh = hk0 . If, however, asynchronous sampling is used, k0 is not an integer, and because of the picket fence effect, the peak corresponding to the hth harmonic appears at the spectral line kh = [ hk0 ]floor , where [ hk0 ]floor is the integer part of hk0. There is thus a deviation Δf h between the actual frequency f h = hf 0 of the hth harmonic and the frequency
k f f h′ = h s denoted by the spectral line kh N Δk f Δf h = f h − f h ' = h s . (16) N By utilizing the parameters of the spectral line kh and its adjacent spectral lines along with the frequency response of the HSCW, parameters of the h th harmonic can be solved. Let f h denote the frequency of the spectral line kh . The phase of the spectral line is
φh = ϕh + πΔkh .
(17)
The previous procedure can also be applied to the se-
WEN He et al. Sci China Ser E-Tech Sci | Feb. 2009 | vol. 52 | no. 2 | 467-476
471
quence x2 ( n ) , and the phase of the kh -th spectral line in its spectrum is
φh′ = ϕh + πΔkh −
2πhf 0 L . fs N
(18)
From eqs. (17) and (18), the phase difference Δφh = φh′ − φh between the kh-th spectral lines of the sequences x1(n) and x2(n) can be simplified as 2πhf 0 L Δφh = − , fs N
(19)
which further gives the frequency deviation of the hth harmonic as Δφ f N (20) Δf h = − h s − f h′ , 2πL and the error between the actual frequency and the frequency given by discrete spectral line of the h th harmonic is then obtained from eqs. (16) and (20) as Δf N Δφ N 2 f h′N Δkh = h = − h − . (21) fs 2πL fs Thus, the frequency, the amplitude, and the phase of the hth harmonic are respectively Δφ f N f h = Δf h +f h′ = − h s , (22) 2πL X ( kh )
Ah = WHp
⎛ 2πΔkh ⎞ ⎜ N ⎟ ⎝ ⎠
,
(23)
⎡ ⎛ 2πΔkh ⎞ ⎤ ϕh = arg ⎡⎣ X ( kh ) ⎤⎦ − arg ⎢WHp ⎜ ⎟⎥ . ⎝ N ⎠⎦ ⎣
simulation, as it has been used in many works for verifying the soundness of algorithms. The noiseless signal is
⎛ 2πf3 n ⎞ ⎛ 2πf1n ⎞ x ( n ) = A0 + A1 sin⎜ + ϕ1 ⎟ + A3 sin ⎜ + ϕ3 ⎟ , (25) ⎝ N ⎠ ⎝ N ⎠ where A0=0.2 V, A1=6 V, f1=20.2 Hz, ϕ1=0.1 rad, A3=1 V, f3=60.6 Hz, and ϕ3 = 0 rad/s. The second order and fourth order HSCWs are utilized to weight the signal and the parameters are calculated through phase difference correction algorithm. The Hanning window constructing the HSCWs is of a length of M=128. Table 2 shows the results given by the various windows and phase difference correction algorithms. Absolute errors are also given in the Table 2. Compared with classical windows such as the Hanning window, the Hamming window, the Blackman window, and the Blackman-Harris window, the HSCW gives the lowest peak side lobe level and the highest side lobe roll-off rate at the frequency of the multiples of the major lobe width. This shows that the HSCW can effectively suppress the spectral leakage. It can also be seen in Table 2 that when the sequence length decreases, the accuracy of harmonics parameter estimation given by HSCW-based phase difference correction algorithm is 1―2 magnitude higher than those given by classical windows. 4.2 Complex signal harmonics parameter estimation
(24)
4 Simulation results 4.1 Comparison with classical windows
The signal model given in [11] and [12] is adopted in the Table 2 Comparisons of simulation results given by different windows Rectangular Hanning Blackman window[11] window[12] window N 2048 2048 512
When applying DTFT to truncated signals, harmonics components can interfere with each other due to the spectral leakage and thus lead to large parameter estimation errors. Weak components can very easily be masked by strong components, and their accurate calculations are often constrained by the performance of the window.
Blackman-Harris window 512
Second order HSCW 256
Fourth order HSCW 512
A0 (V)
−4.0 × 10−4
1.7×10−7
3.21×10−7
8.43×10−8
1.54×10−8
1.4×10−13
A1 (V)
−1.0 × 10−3
1.2×10−8
2.40×10−7
7.52×10−8
4.96×10−8
2.13×10−9
−7
−7
−8
−8
1.7×10−14
−4
f1 (Hz)
2.0 × 10
ϕ1 (rad)
7.0 × 10−3 −3
A3 (V)
−2.0 × 10
f3 (Hz)
−6.0 × 10−4
ϕ3 (rad)
472
3.0 × 10
−3
1.6×10
5.02×10
2.64×10
2.06×10
−1.0×104
6.37×10−6
2.69×10−7
7.94×10−7
2.20×10−11
−6
−7
−9
−9
2.87×10−9
2.06×10−8
1.7×10−14
−5.5×10
−9.0×10−6 7.0×10
−4
5.51×10
3.91×10
5.02×10−7
2.64×10−8
−7
−6
1.53×10
7.85×10
7.78×10
−1.87×10
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−6
1.84×10−13
Assume that the complex signals containing 2nd― 37th harmonics are of the form: 37 ⎡ ⎛ 2πhf 0 n ⎞⎤ x ( n ) = ∑ ⎢ Ah sin ⎜ + ϕh ⎟ ⎥ , h =1 ⎢ ⎝ fs ⎠ ⎥⎦ ⎣
(26)
where the fundamental frequency f0 = 49.8 Hz, the sampling rate fs = 6400 Hz, and the amplitudes and phases of the fundamental and the harmonics are as given in Table 3. The fourth order HSCW with length of N4=4M=1024 is used to weight the signals given by eq. (26). The HSCW-based phase difference correction algorithm is then applied to estimate the harmonics parameters, giving the amplitudes and phases of the fundamental and the harmonics. The relative errors of fundamental and harmonics amplitudes and phases are listed in Table 3. The relative fundamental frequency error is 1.45×10−9%. As shown in Table 3, the fundamental amplitude is about 2000 times of those of the 35th and 37th harmonics, and there is another strong component, the 36th harmonic, which is present near the weak components. Because of the good side lobe performance of the HSCW, the spectral leakage is strongly suppressed, interferences between harmonics are reduced, and the signal harmonics parameters, even those of weak components, can be accurately estimated. 4.3 Simulation under frequency fluctuation and white noise
If the signal frequency fluctuates, it is strictly difficult to achieve synchronous sampling even adopting phaselocked loop (PLL) system. When asynchronous sampling is used, the change in frequency can lead to the
change of the spectral leakage among the signal harmonics, and thus can affect the accuracy of the signal harmonics parameter estimation. With the fundamental frequency fluctuation between 49.5―50.5 Hz, HSCWbased phase difference correction algorithm was carried out to estimate the parameters of the signals formulated by eq. (26). The results are shown in Figure 3. Considering the signal as shown in eq. (26), the fundamental frequency f0 varies from 49.5 Hz to 50.5 Hz with an interval of 0.1 Hz. The fourth-order HSCW with length of N=4M=512 was used. The relative errors of phases and amplitudes versus frequency fluctuations are given in Figure 3. As shown in Figure 3, when signal frequency shifts, the impact of the shift on harmonics parameter estimation can be effectively eliminated by weighting the signal with the HSCW. The relative errors of the amplitudes and phases of the fundamental and harmonics exhibit moderate fluctuations. If white noise is present, the spectral leakage among signal harmonics can also change and affect the estimation. Simulation was done on signals formulated by eq. (25) corrupted by white noise. The Blackman window, the Blackman-Harris, and the second order and the fourth order HSCWs with Hanning window length of M=128 were selected for the simulation. Figure 4 illustrates the absolute error distributions of the fundamental phases and amplitudes with the presence of white noise. It can be seen in Figure 4 that if the noise is relatively strong, i.e., SNR
