3. Performance comparison

change between Eqs. (9a) and (9b) is performed by switch S<sup>1</sup> Therefore, the switching period of S<sup>1</sup> in Figure 2(b) is equal to N · Ts, where T<sup>s</sup> is the sampling period, and its duty cycle is equal to one sample. It is worth mentioning that the effect of the nonlinear operation of D&S algorithm in

There is an alternative way of avoiding the reduction in accuracy generated by the damping factor, without compromising stability. SDFT implementation in Eq. (2) is marginally stable, however, for the particular case of k ¼ 0 (DC component estimation). It takes the following

represented with finite precision, allows to implement the recursive expression without the damping factor r. Therefore, the recurrence in Eq. (10) is unconditionally stable and does not accumulate errors. The modulated sliding discrete Fourier transform (mSDFT) algorithm uses the Fourier modulation property to effectively shift the DFT bin of interest to the position k ¼ 0 and then use Eq. (10) for computing that DFT bin output. This is accomplished by the multiplication of the input signal <sup>x</sup>½n� by the modulation sequence <sup>W</sup><sup>−</sup><sup>k</sup> <sup>n</sup> <sup>N</sup> . This approach allows to exclude the complex twiddle factor from the resonator and avoids accumulated errors and

<sup>k</sup> <sup>½</sup>n�1�−x½n�N�W<sup>−</sup>kðn�N<sup>Þ</sup>

modulation moves the desired kth-bin to k ¼ 0 (0 Hz). The relation between the desired Xk ½n�

requires DFT magnitude estimation, the complex multiplication in Eq. (11b) is unnecessary

depicted in Figure 3(a). In contrast of traditional recursive DFT algorithms, the mSDFT method

Figure 3. (a) Guaranteed-stable mSDFT implementation as IIR filter as given by (11). (b) Guaranteed-stable mSDFT

<sup>N</sup> X<sup>0</sup>

<sup>k</sup> ½n� is a complex constant related to the phase of the complex twiddle factor, since the

<sup>k</sup> j is equal to jXðkÞj. The filter structure of the mSDFT algorithm in Eq. (11) is

Xk <sup>½</sup>n� ¼ <sup>W</sup>k n

potential instabilities [9]. The recursive realization of the mSDFT is:

<sup>k</sup> <sup>½</sup>n� ¼ <sup>X</sup><sup>0</sup>

X0

X0½n� ¼ X0½n�1�−x½n�N� þ x½n� (10)

<sup>N</sup> coefficient, which typically leads to stability issues when it is

<sup>N</sup> <sup>þ</sup> <sup>x</sup>½n�W<sup>−</sup>k n

<sup>k</sup> ½n� is given by Eq. (11b). It is worth noticing that if the application only

<sup>N</sup> (11a)

<sup>k</sup> ½n� (11b)

the dynamic response is negligible as it only changes its structure every N samples.

2.4. Modulated sliding discrete Fourier transform

30 Fourier Transforms - High-tech Application and Current Trends

form:

where X<sup>0</sup>

because <sup>j</sup>X<sup>0</sup>

and the computed X<sup>0</sup>

implementation as IIR filter as given by (12).

The absence of the W<sup>k</sup>

This section discusses the key features of each of the Sb-SDFT that were presented in Section 2. The aim of this analysis is to find underlying similarities and differences between these methods. To this end, a study on statistical efficiency and accuracy is presented in the following subsections. Finally, the section ends with a discussion over the limitations and inaccuracies of the Sb-SDFT inherited by every DFT-based method.

### 3.1. Statistical efficiency

It is common knowledge that the statistical efficiency and noise performance of estimators is determined by comparison with the Cramer-Rao lower bound (CRLB). The CRLB deals with the estimation of the quantities of interest from a given finite set of measurements that are noise corrupted. It assumes that the parameters are unknown but deterministic, and provides a lower bound on the variance of any unbiased estimation. The CRLB is useful because it provides a way to compare the performance of unbiased estimators. Furthermore, if the performance of a given estimator is equal to the CRLB, the estimator is a minimum variance unbiased (MVU) estimator [10].

Computer simulations have been performed to evaluate the performance of the SDFT, the SGT, the mSDFT and D&S algorithm for a single real sinusoid polluted with white Gaussian noise:

$$\mathbf{x}[n] = A\cos\left(\omega n + \phi\right) + \mathbf{w}\mathbf{gn}\ \left[n\right] \tag{13}$$

where A and φ are the amplitude and initial phase, respectively, n is the time domain index, ω denotes the normalized angular frequency (ω ¼ 2πf <sup>o</sup>=f <sup>s</sup>) and wgn[n] is a zero-mean white Gaussian noise of variance σ<sup>2</sup> n. For this case the CRLB for amplitude estimation is approximated by Kay [10]:

$$\text{CRLB}\_A = \frac{2\sigma\_n^2}{N} \tag{14}$$

Parameters were assigned to A ¼ 1, f <sup>o</sup> ¼ 50Hz, f <sup>s</sup> ¼ 6:4 KHz N ¼ 128 and φ is a constant uniformly distributed between <sup>½</sup>0, <sup>2</sup>πÞ. The signal-to-noise ratio (SNR) is equal to <sup>A</sup><sup>2</sup> <sup>=</sup>ð2σ<sup>2</sup> nÞ, whereas different SNR levels were obtained by properly scaling the noise variance σ<sup>2</sup> <sup>n</sup>. All simulation results provided are the averages of 1000 independent runs.

Figure 4(a) and (b) shows the variance in the estimate of A ðσ <sup>A</sup>^ <sup>Þ</sup> versus SNR for two different damping factors. In Figure 4(a), the damping factor was fixed at r ¼ 0:999 for SDFT, SGT and D&S algorithm. In this figure, for SNR levels below −10 dB can be observed that the σ <sup>A</sup>^ values are beneath the CRLB limit. Therefore, beyond this threshold level, the estimations made by the Sb-SDFT techniques cease to be consistent with those of an unbiased estimator. From this threshold level and up to 15 dB, the Sb-SDFT algorithms are efficient MVU estimators, because their σ <sup>A</sup>^ values reach the CRLB. For higher levels of SNR, the <sup>σ</sup> <sup>A</sup>^ for SDFT and SGT remains

Figure 4. (a) Variance of Â versus SNR levels for the analyzed estimators with N ¼ 128 and r ¼ 0:999. (b) Variance of Â versus SNR levels for the analyzed estimators with N ¼ 128 and r ¼ 0:9999. (c) Variance of Â versus r for the four estimators at SNR=80 dB. (d) Variance of Â versus N for the four estimators, with r ¼ 0:9999 and SNR=30 dB.

above the CRLB and asymptotically approximate the −43.5 dB bound. This is mainly due to the fact that the inaccuracy caused by the damping factor in Eqs. (5) and (8) is more relevant than the consequence of SNR level. The D&S algorithm exhibits the same behavior, but beginning at SNR = 60 dB and with σ <sup>A</sup>^ asymptotically approaching the <sup>−</sup>91 dB bound for higher levels. When compared to the performances of the SDFT and the SGT, the D&S algorithm behaves as an MVU estimator for a wider range of SNR, at the cost of a slightly increased computational complexity and a nonlinear functioning. For the range of SNR levels shown in Figure 4(a) beyond the threshold, the variance in Â computed by the mSDFT remains on CRLB curve, so its performance corresponds to an MVU estimator.

This test was repeated for r ¼ 0:9999, and the results are shown in Figure 4(b). It is seen that the performances of the SDFT, SGT and D&S algorithm are better than exhibited in the previous case. This improvement is reflected through an increase in the range of SNR values for which the estimations correspond to an MVU estimator. The results obtained for mSDFT are consistent with those obtained previously, because this estimator does not require a damping factor to ensure stability.

The effect of the damping factor on the σ <sup>A</sup>^ is shown in Figure 4(c). The simulation is performed for SNR = 80 dB because at this level, SDFT, SGT and D&S algorithms do not lie on CRLB curve and have converged to their final values listed in Figure 4(b). For this scenario, the σ <sup>A</sup>^ of the mSDFT is constant and equal to the CRLB, because it does not required a damping factor to achieve stability. Instead, for r ! 1 and SNR beyond threshold level, the σ <sup>A</sup>^ for SDFT, SGT and D&S algorithm approximates the CRLB as it is reflected by Figure 4(c). From the analysis of this figure, it is possible to conclude that for the ideal situation (r ¼ 1) and SNR levels beyond the threshold, all reviewed algorithms reach the CRLB and therefore their statistical efficiency is identical.

Finally, the σ <sup>A</sup>^ versus <sup>N</sup> at SNR = 30 dB are illustrated in Figure 4(d). As expected, <sup>N</sup> increase, that is, the length of the sliding window reduces the variance of Â in the four methods. This is mainly because the estimations are computed in a larger sliding time window, that is, more samples are used for the estimation.

### 3.2. Accuracy analysis

CRLB<sup>A</sup> <sup>¼</sup> <sup>2</sup>σ<sup>2</sup>

Parameters were assigned to A ¼ 1, f <sup>o</sup> ¼ 50Hz, f <sup>s</sup> ¼ 6:4 KHz N ¼ 128 and φ is a constant

damping factors. In Figure 4(a), the damping factor was fixed at r ¼ 0:999 for SDFT, SGT and

are beneath the CRLB limit. Therefore, beyond this threshold level, the estimations made by the Sb-SDFT techniques cease to be consistent with those of an unbiased estimator. From this threshold level and up to 15 dB, the Sb-SDFT algorithms are efficient MVU estimators, because

Figure 4. (a) Variance of Â versus SNR levels for the analyzed estimators with N ¼ 128 and r ¼ 0:999. (b) Variance of Â versus SNR levels for the analyzed estimators with N ¼ 128 and r ¼ 0:9999. (c) Variance of Â versus r for the four

estimators at SNR=80 dB. (d) Variance of Â versus N for the four estimators, with r ¼ 0:9999 and SNR=30 dB.

uniformly distributed between <sup>½</sup>0, <sup>2</sup>πÞ. The signal-to-noise ratio (SNR) is equal to <sup>A</sup><sup>2</sup>

D&S algorithm. In this figure, for SNR levels below −10 dB can be observed that the σ

simulation results provided are the averages of 1000 independent runs.

<sup>A</sup>^ values reach the CRLB. For higher levels of SNR, the <sup>σ</sup>

Figure 4(a) and (b) shows the variance in the estimate of A ðσ

32 Fourier Transforms - High-tech Application and Current Trends

their σ

whereas different SNR levels were obtained by properly scaling the noise variance σ<sup>2</sup>

n

<sup>N</sup> (14)

<sup>A</sup>^ <sup>Þ</sup> versus SNR for two different

<sup>A</sup>^ for SDFT and SGT remains

<sup>=</sup>ð2σ<sup>2</sup> nÞ,

<sup>A</sup>^ values

<sup>n</sup>. All

In this section, the accuracy of the Sb-SDFT methods on the estimation of a single-frequency signal, both in steady-state and dynamics conditions, is analyzed through simulations. The adopted accuracy index is the so-called total vector error (TVE) that combines the effect of magnitude, angle and time synchronization errors on the desired component estimation accuracy. The TVE is defined in the Standard IEEE C37.118.1-2011 [11] as

$$\text{TVE} = 100 \times \sqrt{\frac{\left(\hat{\mathbf{X}}\_{\text{r}} \left[n\right] - \mathbf{X}\_{\text{r}} \left[n\right]\right)^{2} + \left(\hat{\mathbf{X}}\_{\text{i}} \left[n\right] - \mathbf{X}\_{\text{i}} \left[n\right]\right)^{2}}{\mathbf{X}\_{\text{r}} \left[n\right]^{2} + \mathbf{X}\_{\text{i}} \left[n\right]^{2}}} \tag{15}$$

where <sup>X</sup>^ <sup>r</sup> <sup>½</sup>n� and <sup>X</sup>^ <sup>i</sup> <sup>½</sup>n� are the sequences of estimations given by the Sb-SDFT method under test, X<sup>r</sup> ½n� and X<sup>i</sup> ½n� are the sequences of theoretical values of the input signal at the instants of time (n), and the subscripts r and i identify the real and imaginary parts of the desired component, respectively. The TVE is a real number that expresses the Euclidean distance between the true frequency domain complex bin and estimated one.

### 3.2.1. Steady-state condition

At first, the analysis is assessed in steady-state conditions assuming an input signal equal to Eq. (13). Parameters were assigned to A ¼ 1, f <sup>o</sup> ¼ 50Hz, f <sup>s</sup> ¼ 6:4 KHz N ¼ 128 and φ ¼ 0 rad and the damping factor is set to r ¼ 0:9999. The curves plotted in Figure 5(a–d) show the estimated amplitude of the test signal for all Sb-SDFT algorithms in steady state, where the reference value is displayed with a black solid line. Figure 5(e) shows the TVE values as a function of time. SDFTand SGT have the same steady-state TVE values; this error has a mean value with an overlaid ripple that is a direct consequence of the use of a damping factor in Eqs. (5) and (8). For both algorithms, the maximum TVE value is 0.7335%. The D&S algorithm significantly reduces the TVE and maintains the same damping factor than the two previous cases, resulting in improved system performance, with a maximum TVE value of 0.01%. In Figure 5(c), it is shown that when (nmodN) = 0, the estimation is accurate, which is consistent with the period of the fundamental component of the test signal. On the other hand, mSDFT provides precise estimation with a 0% TVE, since it does not require a damping factor to ensure stability.

### 3.2.2. Dynamic condition

The accuracy under dynamic condition of the SDFT, the SGT, the mSDFT and D&S algorithm are evaluated through multiple simulations under the effect of various transient disturbances. The comparison is performed by means of the following test signal:

Figure 5. (a)–(d) Amplitude estimation of the test signal (13) in steady-state condition using the selected Sb-SDFT algorithms with N ¼ 128, r ¼ 0:9999 and f <sup>s</sup> ¼ 6:4 kHz. (e) TVE exhibited by the Sb-SDFT algorithms in steady-state.

$$\mathbf{x}[n] = A\_{\rm o} \left\{ 1 + \delta\_{\rm s} \mathbf{u}[n - n\_{\rm o}] + \delta\_{\rm r} (n - n\_{\rm o}) \mathbf{u}[n - n\_{\rm o}] + \right.$$

$$\delta\_{\rm am} \cos \left[ \omega\_{\rm am} (n - n\_{\rm o}) \right] \mathbf{u}[n - n\_{\rm o}] \left\{ \cos \left( \omega \left. n + \omega\_{\rm g} \right. \right. \right. \right. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \ldots \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right.]$$

where A<sup>o</sup> is the nominal amplitude, δ<sup>s</sup> is the amplitude step depth factor, δ<sup>r</sup> is the amplitude ramp slope factor, δam is the modulation depth factor, ωam is the normalized modulating angular frequency ðωam ¼ 2πf am=f <sup>s</sup>Þ, ω denotes the normalized nominal angular frequency (ω ¼ 2πf <sup>o</sup>=f <sup>s</sup>), ω<sup>g</sup> is the normalized off-nominal angular frequency offset (ω<sup>g</sup> ¼ 2πf <sup>g</sup>=f <sup>s</sup>) and φ is the initial phase. In the following, the performance of the Sb-SDFT is evaluated under the effect of amplitude step, amplitude ramp, amplitude modulation and static frequency offsets. The accuracy is assessed exhaustively, by varying the test signal parameters over a suitable range, in order to determine the maximum TVE values. This approach leads to a fair performance comparison between the considered techniques. Unless otherwise stated, parameters were assigned to A<sup>o</sup> ¼ 1, f <sup>o</sup> ¼ 50 Hz, f <sup>s</sup> ¼ 6:4 KHz, N ¼ 128, φ ¼ 0 rad, r ¼ 0:9999, δ<sup>s</sup> ¼ 0, δ<sup>r</sup> ¼ 0, δam ¼ 0, ωam ¼ 0, ω<sup>g</sup> ¼ 0 and n<sup>o</sup> ¼ 0.

time (n), and the subscripts r and i identify the real and imaginary parts of the desired component, respectively. The TVE is a real number that expresses the Euclidean distance

At first, the analysis is assessed in steady-state conditions assuming an input signal equal to Eq. (13). Parameters were assigned to A ¼ 1, f <sup>o</sup> ¼ 50Hz, f <sup>s</sup> ¼ 6:4 KHz N ¼ 128 and φ ¼ 0 rad and the damping factor is set to r ¼ 0:9999. The curves plotted in Figure 5(a–d) show the estimated amplitude of the test signal for all Sb-SDFT algorithms in steady state, where the reference value is displayed with a black solid line. Figure 5(e) shows the TVE values as a function of time. SDFTand SGT have the same steady-state TVE values; this error has a mean value with an overlaid ripple that is a direct consequence of the use of a damping factor in Eqs. (5) and (8). For both algorithms, the maximum TVE value is 0.7335%. The D&S algorithm significantly reduces the TVE and maintains the same damping factor than the two previous cases, resulting in improved system performance, with a maximum TVE value of 0.01%. In Figure 5(c), it is shown that when (nmodN) = 0, the estimation is accurate, which is consistent with the period of the fundamental component of the test signal. On the other hand, mSDFT provides precise estimation with a 0% TVE, since it does

The accuracy under dynamic condition of the SDFT, the SGT, the mSDFT and D&S algorithm are evaluated through multiple simulations under the effect of various transient disturbances.

Figure 5. (a)–(d) Amplitude estimation of the test signal (13) in steady-state condition using the selected Sb-SDFT algorithms with N ¼ 128, r ¼ 0:9999 and f <sup>s</sup> ¼ 6:4 kHz. (e) TVE exhibited by the Sb-SDFT algorithms in steady-state.

between the true frequency domain complex bin and estimated one.

The comparison is performed by means of the following test signal:

3.2.1. Steady-state condition

3.2.2. Dynamic condition

not require a damping factor to ensure stability.

34 Fourier Transforms - High-tech Application and Current Trends

First, the step response of the Sb-SDFT estimators is evaluated. For this purpose, the parameters of Eq. (16) are set to: δ<sup>s</sup> ¼ 0:1 and n<sup>o</sup> ¼ 640. Figure 6(a) shows the estimated amplitude (Â)

Figure 6. Transients for the estimation of the amplitude of (16) and the evolution of the TVE for the selected Sb-SDFT algorithms, under different test conditions. (a) A step change in amplitude with δ<sup>s</sup> ¼ 0:1, δ<sup>r</sup> ¼ 0, δam ¼ 0 and ω<sup>g</sup> ¼ 0. (b) A ramp-change in amplitude with δ<sup>s</sup> ¼ 0, δ<sup>r</sup> ¼ 0:1, δam ¼ 0 and ω<sup>g</sup> ¼ 0. (c) A sudden amplitude modulation with δ<sup>s</sup> ¼ 0, δ<sup>r</sup> ¼ 0, δam ¼ 0:1, ωam ¼ 2π=f <sup>s</sup> and ω<sup>g</sup> ¼ 0.

and TVE values as a function of time when the amplitude step occurs in x[n]. Ignoring small differences, related to the damping factor effect, the dynamic response during the transient is the same for all the algorithms. This transient has a duration that is equal to the length of the sliding window for all the Sb-SDFT. After the transient, the TVE values provided by the Sb-SDFT estimators are equal to the steady-state values shown in Figure 5(e). Further, simulation results (not reported here for the sake of brevity) confirm that the TVE value in steady state, due to an amplitude step, is the same regardless of the value of δs.

The accuracy of the considered estimators is analyzed in Figure 6(b), assuming that the waveform x[n] is subjected to linear variation of its amplitude. Therefore, the parameters of Eq. (16) were adjusted as follows: δ<sup>r</sup> ¼ 0:1 and n<sup>o</sup> ¼ 640, to create ramp change in the amplitude of the test signal. Once more, the Sb-SDFT exhibit similar dynamics in their amplitude estimation performance. Figure 7(a) shows the worst-case TVE values, after the transient response, returned by the four considered estimators as a function of δ<sup>r</sup> in the range [0,0.1] p. u.. As can be seen, the maximum TVE value achieved by the Sb-SDFT worsens linearly with this parameter. In addition, a gap of 0.78% is observed, between the SDFT, SGT and the other two algorithms, which remains constant for the analyzed range.

The effect of a modulating signal on the estimation accuracy is analyzed in Figure 6(c). Hence, the parameters of Eq. (16) were adjusted as follows: δam ¼ 0:1, ωam ¼ 2π=f <sup>s</sup> and n<sup>o</sup> ¼ 640. The figure shows the estimated amplitude (Â) and TVE values as a function of time when the amplitude modulation of 10% with a frequency of 1 Hz occurs in x[n]. As expected, the dynamic behavior displayed by the Sb-SDFT estimators is similar, with the mSDFT the most accurate of the reviewed algorithms. The curves in Figure 7(b) show the worst case TVE values

Figure 7. (a) Maximum TVE curves versus amplitude ramp slope factor δr. (b) Maximum TVE curves versus amplitude modulation depth factor δam for a modulating frequency f am of 1 Hz. (c) Maximum TVE curves versus amplitude modulating frequency f am with δam ¼ 0:1 p.u. (d). Maximum TVE curves versus static frequency offset f <sup>g</sup>.

returned by the four considered estimators as a function of δam in the range ½0; 0:1�p: u: with f am ¼ 1 Hz. Figure 7(c) shows the worst case TVE values given by the Sb-SDFT as a function of f am in the range ½0, 5� Hz with δam ¼ 0:1p: u: Note that the TVE increment linearly with δam or f am, and that the behavior of the Sb-SDFT estimators is very similar.

Finally, the influence of a simple static off-nominal frequency offset on the Sb-SDFT estimators' performance is analyzed in Figure 7(d). The figure shows the maximum TVE values, in steady state, when the signal (Eq. 16) phase varies as a function of the off-nominal frequency offset f <sup>g</sup> in the range [−1,1] Hz. As expected, the accuracy of all the considered estimators degrades monotonically as the frequency offset increases due to the spectral leakage effect.

The similarities between the Sb-SDFT algorithms found through Figures 6 and 7 are explained by the fact that all implementations of this type of algorithms result from applying Fourier properties and mathematical operations to standard DFT definition (Eq. 1).

### 3.3. Sb-SDFT limitations

and TVE values as a function of time when the amplitude step occurs in x[n]. Ignoring small differences, related to the damping factor effect, the dynamic response during the transient is the same for all the algorithms. This transient has a duration that is equal to the length of the sliding window for all the Sb-SDFT. After the transient, the TVE values provided by the Sb-SDFT estimators are equal to the steady-state values shown in Figure 5(e). Further, simulation results (not reported here for the sake of brevity) confirm that the TVE value in steady state,

The accuracy of the considered estimators is analyzed in Figure 6(b), assuming that the waveform x[n] is subjected to linear variation of its amplitude. Therefore, the parameters of Eq. (16) were adjusted as follows: δ<sup>r</sup> ¼ 0:1 and n<sup>o</sup> ¼ 640, to create ramp change in the amplitude of the test signal. Once more, the Sb-SDFT exhibit similar dynamics in their amplitude estimation performance. Figure 7(a) shows the worst-case TVE values, after the transient response, returned by the four considered estimators as a function of δ<sup>r</sup> in the range [0,0.1] p. u.. As can be seen, the maximum TVE value achieved by the Sb-SDFT worsens linearly with this parameter. In addition, a gap of 0.78% is observed, between the SDFT, SGT and the other

The effect of a modulating signal on the estimation accuracy is analyzed in Figure 6(c). Hence, the parameters of Eq. (16) were adjusted as follows: δam ¼ 0:1, ωam ¼ 2π=f <sup>s</sup> and n<sup>o</sup> ¼ 640. The figure shows the estimated amplitude (Â) and TVE values as a function of time when the amplitude modulation of 10% with a frequency of 1 Hz occurs in x[n]. As expected, the dynamic behavior displayed by the Sb-SDFT estimators is similar, with the mSDFT the most accurate of the reviewed algorithms. The curves in Figure 7(b) show the worst case TVE values

Figure 7. (a) Maximum TVE curves versus amplitude ramp slope factor δr. (b) Maximum TVE curves versus amplitude modulation depth factor δam for a modulating frequency f am of 1 Hz. (c) Maximum TVE curves versus amplitude

modulating frequency f am with δam ¼ 0:1 p.u. (d). Maximum TVE curves versus static frequency offset f <sup>g</sup>.

due to an amplitude step, is the same regardless of the value of δs.

36 Fourier Transforms - High-tech Application and Current Trends

two algorithms, which remains constant for the analyzed range.

The direct application of Sb-SDFT may lead to inaccuracies due to aliasing and spectral leakage, common pitfalls inherited by every DFT-based method. Aliasing is generally corrected by employing anti-aliasing filters or increasing the sampling frequency to a value that satisfies the Nyquist sampling criterion. Instead, when the sampling is not synchronized with the signal under analysis, the DFT is computed over a noninteger number of cycles of the input signal which leads to the spectral leakage phenomenon [1]. Spectral leakage is typically reduced (not eliminated) by selection of the proper nonrectangular time domain windowing functions, to weigh the sequence data at a fixed sampling frequency [12]. This process increases the computational complexity and does not take advantage of the recursive nature of Sb-SDFT methods. Otherwise, spectral leakage can be avoided entirely by ensuring that sequence of samples is equal to an integer number of periods of the input signal [13].

### 4. Coherent sampling approach

In order to avoid the spectral leakage phenomenon, the sequence of samples within a sliding window of a Sb-SDFT must be equal to an integer number of fundamental periods of the input signal. An integer number of periods will be sampled if and only if the coherence criterion holds:

$$\frac{f\_o}{f\_s} = \frac{m}{N} \tag{17}$$

where f <sup>o</sup> is the signal frequency, f <sup>s</sup> is the sampling frequency, N is the sampled sequence length and m is an integer number. This is equivalent to ensuring that an integer number m of sine periods is present in the data sample of length N, and in that case there is no spectral leakage. If Eq. (17) holds, f <sup>s</sup> is referred to as coherent or synchronous sampling frequency.

A variable sampling period approach, named variable sampling period technique (VSPT), was developed by the authors to design synchronization methods that maintain a coherent sampling with the input signal fundamental frequency [14]. This technique has recently been adapted to dynamically adjust the sampling frequency in a harmonic measurement method based on mSDFT [15]. In Ref. [16], the VSPT is generalized so as to be used with any Sb-SDFT algorithm.

In this section, the technique of variable sampling period is briefly described, and a unified small-signal model, which allows to use the VSPT with any Sb-SDFT, is also presented.

### 4.1. Variable sampling period technique

VSPT allows to adapt the sampling frequency to be N times the fundamental frequency of a given input signal. This technique has proven to be efficient both in three-phase and in singlephase applications yielding a robust synchronization mechanism, whose effectiveness has been tested under different conditions and scenarios [14, 17].

Figure 8(a) illustrates the basic VSPT scheme for single-phase implementation, where the input signal is sampled and the input phase ϕu½n� is extracted by the phase detector. Concomitantly with the input sampling, the reference generator provides a signal called reference phase:

$$
\varphi\_{\text{ref}}\left[n\right] = \frac{2\pi n}{N} \tag{18}
$$

The method achieves a null phase error (eϕ½n�) between ϕref ½n� and ϕ<sup>u</sup> ½n�, by varying the sampling period T<sup>S</sup> ½n� as a function of e<sup>ϕ</sup> ½n�. The controller GcðzÞ provides the value of the sampling period and then the sampling generator produces a clock signal (CLK) that starts the conversion and increments the reference phase. The implementation of the phase detector and phase error calculation is key for the proper functioning of this technique. The operating principle is based on the dynamic adjustment of the sampling frequency. An exhaustive explanation of the key elements of this technique can be found in Refs. [14, 17].

### 4.2. Unified small-signal model

VSPT allows to adapt the sampling rate to a multiple of the fundamental frequency of a given input signal, so the coherence criterion holds, thereby preventing the DFT's shortcomings when is used to analyze nonstationary signals. An error signal, related to the phase difference between the fundamental component of the input signal and the reference phase, is needed to adapt the sampling period. Based on this, phase error is feasible to develop a closed-loop control to synchronize the sampling period.

As mentioned in Section 3, when r ! 1 and for a real input signal, the Sb-SDFT algorithms become equivalent. Therefore, for this scenario and for small-signal conditions, these methods supply the same estimation of the kth-bin of an N-points DFT. Based on this concept, Figure 8(b) shows a phase error estimation scheme that employs an Sb-SDFT algorithm, which allows to estimate the phase difference between the fundamental component of the input signal and the reference phase. This scheme obtains the phase error signal from three basic operations, first an

Figure 8. (a) General scheme of the variable sampling period technique, (b) phase error estimation scheme based on Sb-SDFT and (c) system model for Sb-SDFT with coherent sampling adjustment based on VSPT.

Sb-SDFT algorithm with k ¼ 1 is used to estimate the fundamental component (X1½n�) of an Npoints DFT, from a given input sequence of samples (x[n]). Then the phase of the input signal (ϕu½n�) is estimated by computing the argument of the complex result X1½n�, as stated by Eq (4b). Finally, a simple subtraction operation is used to estimate the phase error (eϕ½n�) between the incoming signal and the reference.

Since all the Sb-SDFT methods are derived from Eq. (1), for small-signal condition, they are mathematically equivalent, and the system phase error (eϕ½n�) for small deviation is approximately equal. Therefore, a mathematical model can be extrapolated for implement the VSPT scheme shown in Figure 8(a) with the phase error estimation scheme shown in Figure 8(b). Figure 8(c) presents the small signal model of a coherent sampling scheme for the Sb-SDFT algorithms based on the VSPT, which allows to avoid the spectral leakage phenomenon. The complete mathematical derivation of this model is available in Ref. [16].

### 4.3. Validation

A variable sampling period approach, named variable sampling period technique (VSPT), was developed by the authors to design synchronization methods that maintain a coherent sampling with the input signal fundamental frequency [14]. This technique has recently been adapted to dynamically adjust the sampling frequency in a harmonic measurement method based on mSDFT [15]. In Ref. [16], the VSPT is generalized so as to be used with any Sb-SDFT

In this section, the technique of variable sampling period is briefly described, and a unified small-signal model, which allows to use the VSPT with any Sb-SDFT, is also presented.

VSPT allows to adapt the sampling frequency to be N times the fundamental frequency of a given input signal. This technique has proven to be efficient both in three-phase and in singlephase applications yielding a robust synchronization mechanism, whose effectiveness has

Figure 8(a) illustrates the basic VSPT scheme for single-phase implementation, where the input signal is sampled and the input phase ϕu½n� is extracted by the phase detector. Concomitantly with the input sampling, the reference generator provides a signal called reference phase:

<sup>ϕ</sup>ref <sup>½</sup>n� ¼ <sup>2</sup>π<sup>n</sup>

The method achieves a null phase error (eϕ½n�) between ϕref ½n� and ϕ<sup>u</sup> ½n�, by varying the sampling period T<sup>S</sup> ½n� as a function of e<sup>ϕ</sup> ½n�. The controller GcðzÞ provides the value of the sampling period and then the sampling generator produces a clock signal (CLK) that starts the conversion and increments the reference phase. The implementation of the phase detector and phase error calculation is key for the proper functioning of this technique. The operating principle is based on the dynamic adjustment of the sampling frequency. An exhaustive

VSPT allows to adapt the sampling rate to a multiple of the fundamental frequency of a given input signal, so the coherence criterion holds, thereby preventing the DFT's shortcomings when is used to analyze nonstationary signals. An error signal, related to the phase difference between the fundamental component of the input signal and the reference phase, is needed to adapt the sampling period. Based on this, phase error is feasible to develop a closed-loop

As mentioned in Section 3, when r ! 1 and for a real input signal, the Sb-SDFT algorithms become equivalent. Therefore, for this scenario and for small-signal conditions, these methods supply the same estimation of the kth-bin of an N-points DFT. Based on this concept, Figure 8(b) shows a phase error estimation scheme that employs an Sb-SDFT algorithm, which allows to estimate the phase difference between the fundamental component of the input signal and the reference phase. This scheme obtains the phase error signal from three basic operations, first an

explanation of the key elements of this technique can be found in Refs. [14, 17].

<sup>N</sup> (18)

algorithm.

4.1. Variable sampling period technique

38 Fourier Transforms - High-tech Application and Current Trends

4.2. Unified small-signal model

control to synchronize the sampling period.

been tested under different conditions and scenarios [14, 17].

The specifications and requirements to be met by the controller (Gc(z)) are determined by the application. Several applications require zero phase error and frequency synchronization for normal operation. In these cases, the controller must be proportional integral to achieve zero phase error in steady state; the resulting system being a type II system.

Then the transfer function for the controller in the z domain is

$$\mathbf{G}\_{\mathbf{c}}(z) = \mathbf{K} \frac{\mathbf{(z-a)}}{z-1} \tag{19}$$

As an example of design, ω ¼ 2π · 50 rad=s and N ¼ 128 are adopted. Concerning dynamics, a phase margin of 45° and maximum bandwidth are adopted as design criteria for GcðzÞ. Based on this, and using the design methodology proposed in Ref. [15], the parameters of the controller are <sup>K</sup> <sup>¼</sup> <sup>1</sup>:<sup>7304</sup> � <sup>10</sup><sup>−</sup><sup>5</sup> and <sup>a</sup> <sup>¼</sup> <sup>0</sup>:9974, with a bandwidth of 5.905 Hz.

The estimations obtained by the Sb-SDFT algorithms with coherent sampling supplied by the VSPT, in situations where the input signal frequency deviates from its nominal value, are evaluated in two possible scenarios. The first simulation analyzes the effect of a frequency step

Figure 9. (a) Evolution of the TVE for the selected Sb-SDFT algorithms when a sudden −0.5 Hz step change in the nominal frequency occurs. (b) Maximum TVE curves versus static frequency offset f <sup>g</sup>.

of −0.5 Hz on the performance of the proposed method. Hence, the parameters of Eq. (16) were adjusted as follows: A<sup>o</sup> ¼ 1, f <sup>o</sup> ¼ 50 Hz, φ ¼ 0 rad, δ<sup>s</sup> ¼ 0, δ<sup>r</sup> ¼ 0, δam ¼ 0, ωam ¼ 0, f <sup>g</sup> ¼ −0:5 Hz and n<sup>o</sup> ¼ 640. The Sb-SDFT algorithms are set with f <sup>s</sup> ¼ 6:4 KHz, N ¼ 128, r ¼ 0:9999 and k ¼ 1. The parameters used in the controller GcðzÞ, for the VSPT close loop, are those presented in the previous example of design. Figure 9(a) depicts the effect of the frequency step change on the TVE values given by the estimated X½n� component. During the transient, an oscillatory behavior is noticed, which may be attributed to spectral leakage given by the noncompliance of the coherence criterion (Eq. 17) at the step change. Variations in the estimated values are extinguished once the sampling frequency is properly adjusted by the VSPT method to f <sup>s</sup> ¼ N · ðf <sup>o</sup>−f <sup>g</sup>Þ. Then, under a steady-state condition, the TVE values given by the four Sb-SDFT are equal to those previous to the frequency step.

To complete the evaluation of the accuracy of coherent sampling achieved by the VSPT, the influence of a simple static off-nominal frequency offset on the Sb-SDFT estimators performance is analyzed in Figure 9(b). The figure shows the maximum TVE values, in steady state, when fundamental frequency of Eq. (16) varies as a function of the off-nominal frequency offset f <sup>g</sup> in the range [−1,1] Hz. Due to the VSPT, in steady-state sampling, frequency is coherent with the fundamental frequency of the test signal, ensuring that exactly one period is present in the data sample of length N, and in that case, the Sb-SDFT avoids the spectral leakage phenomenon. Therefore, compared with the results shown in Figure 7(d), the TVE values do not worsen with f <sup>g</sup>, instead remain constant and equal to those shown in Figure 5(e).

### 5. Conclusions

In this work, a comparative study of four Sb-SDFT algorithms is conducted. The comparison includes filter structure, stability, statistical efficiency, accuracy analysis, dynamic behavior and implementation issues on finite word-length precision systems limitations. Based on theoretical studies as well as on simulations, it is deducted that all reviewed Sb-SDFT techniques are equivalent, primarily due to the fact that they are derived from the traditional DFT, therefore in various applications can be applied indistinctly.

It proves that SDFT and SGT have identical performances, in regard to disturbance rejection and precision on spectral estimation. Both of these techniques are used extensively due to their straightforward implementation, although the two have an error in accuracy due to the use of a damping factor. For applications requiring greater precision, this error can be reduced by using the D&S algorithm. On the other hand, it can be eliminated by using mSDFT due to the absence of damping factor, resulting in better performance. The results of the study have shown that mSDFT is the best option when it comes to precision and noise rejection.

The direct application of a Sb-SDFT may lead to inaccuracies due to the spectral leakage phenomenon, common pitfall inherited by every DFT-based method. Spectral leakage arises when the sampling process is not synchronized with the fundamental tone of the signal under analysis and the DFT is computed over a noninteger number of cycles of the input signal. In this sense, a unified small-signal system model is presented, which can be used to design a generic adaptive frequency loop that is based on a variable sampling period technique. The VSPT allows to obtain a sampling frequency coherent with the fundamental frequency of the analyzed signal, avoiding the error introduced by the spectral leakage phenomenon.
