Research on choices of spectral bins in energy-based frequency estimators under intensive noise

Abstract

This paper presented the problem of wrong choice of spectral bins in the energy-based frequency estimation method under intensive noise, and studied its impact on the frequency estimation. Based on the theory of energy-based method, the causes for wrong location of spectral bins in the traditional estimation method were analyzed. In order to reduce wrong choice rate of spectral bins, two optimizational location strategies of spectral bins were introduced, and the effects of them were confirmed by computer simulation. A numerical test under intensive noise was carried out, in which estimation errors returned by optimizational spectral location strategies were compared. It was demonstrated that the estimation accuracy under intensive noise can be remarkably improved by using optimizational spectral location strategies. In particular, Macleod’s optimizational strategy is strongly suggested because of its prominent advantage in reducing occurrences of wrong location as well as its best performance in frequency estimation.

Keywords

Frequency estimation discrete Fourier transform mistaken location energy-based method

Introduction

Spectral energy-based parameter estimation method implemented in spectrum domain is a commonly used method in the signal processing, such as detection of engine rotational speed and power quality measurement.^1,2 Initially proposed in D. Petri’s study of the measurement of signal parameters,³ this method was introduced to analyze the A/D converter performance by using windows with minimum side-lobe energy.⁴ In 2003, the energy-based frequency estimation method (EBM) was employed in the measurement of laser Doppler velocity to achieve higher precision.⁵ In the past few years, this method has been widely applied to improve the precision in parameter identification. In 2001, Ding proved that the energy center of the power spectrum of the symmetric window function was located in the coordinate origin or near the origin, and consequently developed this approach into the energy-based method.⁶ The conclusion not only provided a theoretical basis for EBM, but also indicated that EBM was suitable for all symmetric windows. The effect of number of points on the EBM accuracy was investigated by Lin and Ding in 2009.⁷ Theoretical formula for frequency estimation error was further derived.^8,9 Recently, the influencing factors on the accuracy of EBM, including systematic errors, interference from neighboring spectral components, and additive wide band noise, have been theoretically quantified by Belega.^9,10

Apart from EBM, interpolation DFT (IpDFT) algorithm and phase difference method (PDM) are also two commonly used parameter estimation algorithms in spectrum domain. IpDFT algorithm is strongly dependent on the category of the weighting function, and interpolation formulas for different windows usually differ from each other.^11–14 Moreover, there are no analytic expressions for most windows.¹⁵ Obviously, IpDFT algorithm would be inconvenient for some windows with a lower side-lobe level or a faster decay rate of side-lobe.^15,16 PDM can be applied for many kinds of windows, but it requires the phase difference of two time-domain signal segments to achieve estimation.^17–20 Therefore, it will inevitably increase calculation amount and decrease the estimation speed, limiting its application in engineering fields.¹⁸ In contrast, EBM can provide an accurate and efficient estimator with only a few number of samples. EBM has a uniform formula for different windows.⁶ More importantly, EBM can reduce the frequency estimation errors caused by the spectral leakage to certain extent.⁶ It should be pointed out that EBM is also effective for the averaged power spectrum. In practice, averaged power spectrum rather than one power spectrum is often taken into analysis because of random additive noise. As a result, EBM is a good candidate to achieve fast and accurate frequency estimation in spectral analysis.

However, spectral bins are often incorrectly located in the traditional EBM under intensive noise. For instance, it is extremely easy to mistake the fourth spectral bin in four-point EBM because of its relatively smaller magnitude.⁹ The wrong location of spectral bin will, in turn, result in considerable estimation errors and lead to difficulties in achieving high accurate parameters estimation. As a result, it is of critical significance to overcome wrong choice of spectral bins for enhancing estimation accuracy.

The aim of this paper is to study how to decrease the occurrence of wrong location of spectral lines and consequently to improve the capability against additive noise in EBM. The remaining part of this paper is organized as follows. In Section ‘Theoretical background of energy based method’, theoretical background of EBM is briefly presented. In Section ‘Mistaken location of bins in energy based method’, the problem of mistaken location of spectral bins in EBM is discussed. In Section ‘Optimizational strategies for determination of spectral bins’, two optimizational strategies for determination of spectral bins are introduced and occurrences of wrong location with these two strategies are shown. In Section ‘Determination of spectral bins before weighting’, in order to further study the problem of mistaken location, the aforementioned location strategies with rectangle window are investigated. In Section ‘Simulations’, the estimation errors of four-point EBM with different location strategies are compared by means of computer simulation. Finally, some conclusions are drawn in Section ‘Conclusion’.

Theoretical background of energy based method

Assume a multi-frequency discrete-time signal is written as below

x (n) = \sum_{i = 1}^{P} A_{i} \cos (2 π \frac{f_{i}}{f_{s}} n + θ_{i}), n = 0, 1, 2, 3 \dots N - 1

(1)

where

f_{i}, A_{i}, θ_{i}

respectively stand for frequency, amplitude and phase of the ith cosine wave.

f_{s}, N

represent sampling rate and the number of samples, respectively. The Nyquist sampling theorem can be satisfied when

f_{s} > 2 f_{i}

. To distinguish two adjacent frequency components, it should meet

| f_{i} - f_{i + 1} | > 2 Δ f

(2)

where

Δ f = f_{s} / N

denotes frequency resolution. Before applying fast Fourier transform (FFT) to equation (1), the samples are usually weighted by a window function to suppress the leakage effect. Then, one can obtain

X (k) = \sum_{i = 1}^{P} \frac{A_{i}}{2} e^{j θ_{i}} W_{N} (k - λ_{i}) + \frac{A_{i}}{2} e^{- j θ_{i}} W_{N} (k + λ_{i})

(3)

where

W_{N} (\cdot)

denotes the discrete time Fourier transform (DTFT) of the adopted window. For an acquired record of N samples, the ratio between the frequency of a certain component f_i and the sampling rate f_s is expressed as

\frac{f_{i}}{f_{s}} = \frac{λ_{i}}{N} = \frac{l_{i} + δ_{i}}{N}

(4)

where

λ_{i}

refers to the normalized frequency expressed in frequency bin. l_i and

δ_{i} (- 0.5 < δ_{i} < 0.5)

are respectively the integer and the fractional part of

λ_{i}

. Usually, l_i can be determined by a local maximum in spectrum if the product of sampling number N and signal-to-noise ratio (

SNR

) is higher than the threshold.^21,22 In this paper, it only considers the case in which

N \cdot SNR

is higher than the threshold. When

δ_{i} = 0

, it means coherent sampling, otherwise it indicates non-coherent sampling. In practice application, it is quite difficult to obtain a coherent sampling. As a result, the normalized frequency often lays between two adjacent FFT bins, causing considerable errors in parameters estimation. The estimation errors can be compensated by the characteristics of spectral energy distribution so that a more accurate estimate can be obtained. The energy of a spectral bin r can be written as

Y (r) = | X (r) |^{2}

(5)

The second term on the right side of equation (3) represents the image part of the spectrum, which would be very small compared with the first term and can be neglected when $λ_{i} > 5$ and $λ_{i} < N / 2 - 5$ . Moreover, if a proper window is adopted, the leakage effect from neighboring frequency components can be also ignored and thus each frequency component can be treated as an individual. Therefore, an individual peak in equation (3) can be further simplified to

X (r) ≅ \frac{A_{i}}{2} e^{j θ_{i}} W_{N} (r - λ_{i})

(6)

Combining (6) with (5), the spectral energy $Y (r)$ can be approximated as

Y (r) = | X (r) |^{2} = \frac{A_{i}^{2}}{4} | W (r - λ_{i}) |^{2}

(7)

For a certain spectral peak, the spectral energy can be expressed as

Y (l_{i}) = \frac{A_{i}^{2}}{4} | W (l_{i} - λ_{i}) |^{2} = \frac{A_{i}^{2}}{4} | W (- δ_{i}) |^{2}

(8)

If we set

G (δ) = | W (- δ) |^{2}

(9)

it can be proved that the spectral energy distribution of any symmetric window function meets the following equation^6,22

{lim}_{P \to \infty} \sum_{p = - P, p \in Z}^{P} G (p + δ) (p + δ) ≅ 0

(10)

The above equation reveals a very important conclusion that when $P \to \infty$ the energy barycenter of spectral bins approaches the coordinate origin. Furthermore, since the energy of a window function is almost concentrated in the main-lobe, the energy of the side-lobes can be nearly neglected in practice. In other words, it is possible to make the barycenter of spectral energy locate around the coordinate origin with only a few spectral bins in the main-lobe. As a result, equation (10) can be simplified as

\sum_{p = - M_{1}}^{M_{2}} G (p + δ) (p + δ) ≅ 0

(11)

After some algebraic manipulations, equation (11) can be reformulated as

δ \sum_{p = - M_{1}}^{M_{2}} G (p + δ) ≅ - \sum_{p = - M_{1}}^{M_{2}} G (p + δ) p

(12)

where either

M_{2} = M_{1}

M_{2} = M_{1} \pm 1

On the other hand, the sum of energy of several spectral lines around a spectral peak $| X (l_{i}) |$ can be written as

\sum_{p = - M_{1}}^{M_{2}} Y (l_{i} + p) (l_{i} + p) = \frac{A_{i}^{2}}{4} \sum_{p = - M_{1}}^{M_{2}} G (- δ_{i} + p) (l_{i} + p) = \frac{A_{i}^{2}}{4} [\sum_{p = - M_{1}}^{M_{2}} G (- δ_{i} + p) p + \sum_{p = - M_{1}}^{M_{2}} G (- δ_{i} + p) l_{i}]

(13)

Combining (12) and (13), we can get

\sum_{p = - M_{1}}^{M_{2}} Y (l_{i} + p) (l_{i} + p) ≅ \frac{A_{i}^{2}}{4} (l_{i} + δ_{i}) \sum_{p = - M_{1}}^{M_{2}} G (- δ_{i} + p) = λ_{i} \sum_{p = - M_{1}}^{M_{2}} Y (l_{i} + p)

(14)

Finally, the spectral energy based estimator can be written as

λ_{i} ≅ \frac{\sum_{p = - M_{1}}^{M_{2}} Y (l_{i} + p) (l_{i} + p)}{\sum_{p = - M_{1}}^{M_{2}} Y (l_{i} + p)}

(15)

The estimator indicates that the normalized frequency can be accurately determined by spectral energy of several DFT bins around the spectral peak. The correction value $δ_{i}$ can be written as

δ_{i} ≅ \frac{\sum_{p = - M_{1}}^{M_{2}} pY (l_{i} + p)}{\sum_{p = - M_{1}}^{M_{2}} Y (l_{i} + p)}

(16)

From equation (15), we can see the accuracy of EBM is closely related to the properties of window. Higher precision could be obtained by turning to those windows which have lower side-lobes level, faster rate of side-lobes falloff, and more energy centered around the main-lobe. Moreover, we could take the advantage of adjustable windows, such as Dolph–Chebyshev window and Kaiser–Bessel window. Dolph–Chebyshev window could minimize the norm of the side-lobes for a given main-lobe width.¹⁵ Kaiser–Bessel window has the best energy concentration in the mainlobe. It should be stressed that a smaller side-lobes level may decrease the spectral leakage, however, it would, at the same time, cause a wider main-lobe, leading to weakened ability of resolving disparate signals and worse equivalent noise bandwidth (ENBW). The number of spectral bins used in EBM, the theoretical deviation of frequency, and the noise level can also pose considerable influence on the accuracy. In the following sections, we take signals weighted by Hanning window as an example to illustrate the estimation problem in EBM.

Mistaken location of bins in energy based method

It is clear in equation (15) that the spectral bins used in EBM directly determine the final value of estimate. When an odd number M of spectral bins are adopted, the accounted spectral lines include the spectral peak and $(M - 1) / 2$ bins on each side. For example, three-point EBM takes the spectral bin with the highest energy and the additional one on each side. In this case, equation (16) can be written as

{\hat{δ}}_{i} = \frac{Y (l_{i} + 1) - Y (l_{i} - 1)}{Y (l_{i} - 1) + Y (l_{i}) + Y (l_{i} + 1)}

(17)

When $δ_{i} \approx 0$ , it can achieve a very good performance because of $Y (l_{i} - 1) \approx Y (l_{i} + 1)$ , and thus ${\hat{δ}}_{i} \approx 0$ can be resulted. However, when $| δ_{i} | \approx 0.5$ , it would be easy to have a wrong choice of the spectral peak because of $Y (l_{i} - 1) \approx Y (l_{i})$ or $Y (l_{i} + 1) \approx Y (l_{i})$ . If the spectral peak is not correctly located, the estimate will be calculated by

{\hat{λ}}_{i} = l_{i} + 1 + \frac{Y (l_{i} + 2) - Y (l_{i})}{Y (l_{i}) + Y (l_{i} + 1) + Y (l_{i} + 2)} = l_{i} + \frac{2 Y (l_{i} + 2) + Y (l_{i} + 1)}{Y (l_{i}) + Y (l_{i} + 1) + Y (l_{i} + 2)}

(18a)

for

δ_{i} \approx 0.5

, or

{\hat{λ}}_{i} = l_{i} - 1 + \frac{Y (l_{i}) - Y (l_{i} - 2)}{Y (l_{i}) + Y (l_{i} - 1) + Y (l_{i} - 2)} = l_{i} + \frac{- 2 Y (l_{i} - 2) - Y (l_{i} - 1)}{Y (l_{i}) + Y (l_{i} - 1) + Y (l_{i} - 2)}

(18b)

for

δ_{i} \approx - 0.5

. The problem, of course, will introduce an uncertain and considerable error.

The similar problem can be found, and probably be even worse, when even number of spectral bins is adopted. The accounted even number of spectral lines includes the spectral peak and the several bins near the spectral peak. It should be pointed out that either the left side or the right side would have one more spectral bin than the other side. For example, the four-point EBM realizes frequency correction by employing the spectral line with the highest energy and the additional three spectral lines around it.⁹ If we set $M_{1} = 1$ and $M_{2} = 2$ in equation (15), the normalized frequency can be estimated by

λ_{i 1} = l_{i} + \frac{Y (l_{i} + 1) + 2 Y (l_{i} + 2) - Y (l_{i} - 1)}{Y (l_{i} - 1) + Y (l_{i}) + Y (l_{i} + 1) + Y (l_{i} + 2)}

(19a)

If we let $M_{1} = 2$ and $M_{2} = 1$ , the normalized frequency can be estimated by

λ_{i 2} = l_{i} + \frac{Y (l_{i} + 1) - 2 Y (l_{i} - 2) - Y (l_{i} - 1)}{Y (l_{i} - 2) + Y (l_{i} - 1) + Y (l_{i}) + Y (l_{i} + 1)}

(19b)

Since $λ_{i}$ is unknown, we can only determine equation (18) or equation (19) by considering the energy of spectral lines. The traditional method of choosing spectral bins by the four-point EBM is dependent on the energy relationship between $Y (l_{i} - 2)$ (the second spectral lines near spectral peak on the left side) and $Y (l_{i} + 2)$ (the second spectral lines near spectral peak on the right).⁹ In other words, once the local spectral peak l_i is determined, the fourth spectral line could be determined by comparing the energy relation between $Y (l_{i} - 2)$ and $Y (l_{i} + 2)$ . Hence, we have the estimator

{\hat{λ}}_{i} = {λ_{i 1}, ifY (l_{i} - 2) < Y (l_{i} + 2), λ_{i 2}, ifY (l_{i} - 2) \geq Y (l_{i} + 2) (condition 1)

However, $Y (l_{i} - 2)$ and $Y (l_{i} + 2)$ would be very small and almost have the same value when $δ \approx 0$ . As a result, it is easy to have a wrong choice between $Y (l_{i} - 2)$ and $Y (l_{i} + 2)$ when additive random noise is involved. Consequently, it would lead to a wrong choice between $λ_{i 1}$ and $λ_{i 2}$ , thus resulting in considerable errors in the final estimate. This problem extremely weakens the resistance against the noise. Figure 1 shows the occurrence of mistaken location as a function of $δ_{i}$ when a cosine wave is sampled. In simulation, the sampling frequency was 256, the number of samples was 256, frequency varied within the range [63.5, 64.5] while phase was random. The cosine wave was corrupted by Gaussian noise with SNR of −5,0, and 5, respectively. The SNR expressed in dB unit is defined as $SNR = 10 \log_{10} A^{2} / (2 σ^{2})$ , where $σ^{2}$ is the variance of the additive noise. For a specified SNR, a total of ten thousand instances were performed.

Figure 1.

Occurrence of mistaken location in traditional EBM when Hanning window adopted (a) mistaken location of the spectral peak (b) mistaken location of the fourth spectral bin.

The occurrences of mistaken location for three-point and four-point EBM are described in Figure 1. As shown in Figure 1(a), the three-point EBM has a highest level at $| δ_{i} | = 0.5$ , where the wrong choices account for nearly 50%. The occurrence of wrong location declines with the decrease of $| δ_{i} |$ , and achieves the minimum level at $| δ_{i} | = 0$ . At a certain deviation, the value reduces with the increase of SNR, which implies that the noise level directly affects the location of spectral lines. Regarding four-point EBM in Figure 1(b), similar problem appears. However, being different from the results in Figure 1(a), the four-point EBM reaches the highest level at $| δ_{i} | = 0$ and the lowest level at $| δ_{i} | = 0.5$ . It should be noted that the occurrence of wrong location is larger than 0.3 in the whole range of $| δ_{i} |$ , which is much higher than that in Figure 1(a) when SNR is equal to -5. Even in the weaker noise environment where SNR is equal to 5, the occurrence still remains significant. The results indicate that reducing the occurrence of mistaken location, especially in four-point EBM, is an indispensable measure to enhance the accuracy of estimation under strong noise condition.

Optimizational strategies for determination of spectral bins

Since the problem of mistaken location in four-point EBM is much more profound than that in three-point EBM, in this part, we only focused on the problem in four-point EBM, and proposed two optimizational strategies for determination of spectral bins.

Determination of spectral bins with larger energy

The traditional four-point EBM has poor resistance against noise because of wrong choices of spectral bins, resulting from direct comparison between $| Y (l_{i} - 2) |$ and $| Y (l_{i} + 2) |$ in terms of their energy. Since $| Y (l_{i} - 2) |$ and $| Y (l_{i} + 2) |$ are both close to the margin of the mainlobe, the values are much smaller compared with $| Y (l_{i} + 1) |$ , $| Y (l_{i}) |$ and $| Y (l_{i} + 1) |$ , which will cause a serious degradation of SNR. In order to enhance the resistance against noise, we considered applying spectral lines with larger energy, i.e., the second and third largest DFT energy $| Y (l_{i} - 1) |$ and $| Y (l_{i} + 1) |$ to determine the required spectral bins. So, we get the modified estimator as below

{\hat{λ}}_{i} = {λ_{i 1}, if | Y (l_{i} + 1) | > | Y (l_{i} - 1) | λ i_{2}, if | Y (l_{i} + 1) | < | Y (l_{i} - 1) | (condition 2)

Since $| Y (l_{i} - 1) |$ and $| Y (l_{i} + 1) |$ are neighbors of spectral peak, they certainly would be much larger than $| Y (l_{i} - 2) |$ and $| Y (l_{i} + 2) |$ . Therefore, $| Y (l_{i} - 1) |$ and $| Y (l_{i} + 1) |$ are more robust against the additive random noise than $| Y (l_{i} - 2) |$ and $| Y (l_{i} + 2) |$ under the same noise level. Figure 2(a) shows the occurrence of mistaken location under condition 2. Compared with Figure 1(b), it is obvious that the number of wrong choices in Figure 2(a) is much smaller. Particularly, the occurrence of wrong location is zero for $0.35 < | δ_{i} | < 0.5$ , when SNR = −5, while the relative ranges corresponding to SNR = 0 and SNR = 5 are $0.2 < | δ_{i} | < 0.5$ and $0.1 < | δ_{i} | < 0.5$ , respectively. The maximum ratio of mistaken location is still up to 0.5. It is demonstrated again that the noise distortion and the frequency deviation are two major factors that affect the location of spectral bins in EBM. The simulation results indicate that the optimizational approach can indeed alleviate the problem of mistaken location, especially when $| δ_{i} | \approx 0.5$ . However, the occurrence of wrong location still maintains at a relatively high level.

Figure 2.

Occurrence of mistaken location for two optimizational strategies when Hanning window adopted (a) optimizational strategy using spectral bins near the spectral peak (b) Macleod's optimizational strategy.

Determination by Macleod’s optimizational strategy

Similar to the EBM, IpDFT also requires several bins with larger modulus to achieve frequency estimation. Therefore, it should be considered that some excellent strategies, which were taken in IpDFT, can be applied in the EBM to promote the estimation accuracy. In 1999, Macleod studied IpDFT algorithms and suggested an optimizational strategy containing the information about phase and modulus by aid of the peak sample and its two neighbors.²³ The occurrence of mistaken location was efficiently reduced by Macleod’s optimizational strategy when rectangle window was used and consequently the resistance of IpDFT algorithm against noise was strengthened. By introducing Macleod’s optimizational strategy, the four-point energy based estimator can be written as

λ_{i} = {λ_{i 1}, ifR (- 1) > R (1) λ_{i 2}, ifR (- 1) \leq R (1) (condition 3)

where

R (- 1)

and

R (1)

denote the spectral components with a phase reference relative to the spectral peak. They are defined as

R (\pm 1) = Re [X (l \pm 1) X^{*} (l)]

(20)

where

X^{*} (l)

is conjugate complex number of

X (l)

. Figure 2(b) describes the occurrence of mistaken location with Macleod’s optimizational strategy. The trends of three curves are almost the same as those with optimizational strategy using spectral bins near the spectral peak. It implies that the noise interfere still exerts dominant influence to the determination of spectral lines even though Macleod’s optimizational strategy is adopted, and the determination performance is not so significantly improved compared with that when the optimizational strategy is adopted in Section ‘Determination of spectral bins with larger energy’.

Determination of spectral bins before weighting

In the above section, it is acknowledged that the inference from random noise exerts an great influence on the correct determination of spectral lines. In spectrum analysis, due to truncation of weighting function, wide-band noise would be accumulated within each frequency bin, therefore, the magnitude of a spectral bin at a specified frequency will be biased by wide-band noise. The biased magnitudes not only interfere the correct selection of spectral lines, but also cause some fluctuations of frequency estimates around the theoretical frequency. The bias of magnitude is dependent on the noise level on one hand, on the other hand, it is also dominated by ENBW of the adopted weighting function. A lower SNR or a wider ENBW of the tapered window may lead to larger amplitude offsets, which may increase the occurrence of wrong location of spectral bin, and thus result in larger estimation errors. Consequently, we considered to locate spectral bins before tapering, because the rectangle window has a narrower main-lobe, which not only means a better ability of peak discrimination in multifrequency signals, but also implies a stronger noise immunity.

Figure 3 shows the occurrences of wrong location of spectral bins under traditional strategy, the optimization strategy with spectral bins near the spectral peak, and Macleod’s optimization strategy respectively before Hanning window is adopted. When the traditional strategy is taken, the performance in Figure 3(a) is similar to that with Hanning window in Figure 1(b). When the optimizational strategy with larger spectral bins is taken, the performance in Figure 3(b) is improved compared with that in Figure 3(a), but much worse than that with Hanning window in Figure 2(a). This confirms the fact again that the occurrence of wrong location is closely concerned with the magnitudes of bins, and a high probability of wrong choices would occur if the bins involved in EBM algorithm are in small magnitude. When Macleod’s optimizational strategy is taken, the occurrence of wrong choices in Figure 3(c) is much smaller than that with Hanning window in Figure 2(b). By comparison, it can be seen that Macleod’s optimizational strategy also shows better performance when rectangle window is used compared with that when traditional strategy and the optimizational strategy with spectral bins near the spectral peak are adopted. This may be due to that the traditional strategy and the optimizational strategy with spectral bins near the spectral peak only use magnitude information, while the Macleod’s optimizational strategy uses both magnitude and phase information. The results reveal that the application of Macleod’s optimizational strategy before windowing has a clear advantage over the other strategies. As a result, Macleod’s optimizational strategy is strongly recommended in practice to ensure a correct choice of spectral bins.

Figure 3.

Occurrence of mistaken location for the traditional and optimizational strategies before weighting. (a) traditional strategy (b) optimizational strategy using spectral bins near the spectral peak (c) Macleod's optimizational strategy.

Simulations

In this section, in order to investigate the effects of different location strategies of spectral bin on the energy based estimator, and to compare the estimation accuracies of these strategies under strong noise, some numerical tests were performed. For simplicity but without loss of generality, cosine waves contaminated by Gaussian white additive noise with zero mean were generated. In the simulations, the signal’s frequency was distributed randomly in the range [127.5, 128.5]. The phase of signal was scattered in the range $[- π, π]$ . The sampling rate f_s was set to 512. The number of samples was $N = 512$ . In order to obtain a high occurrence of wrong location, standard deviation of the noise was characterized by SNR = − 5. A total of 2,000,000 simulation cosine waves were produced. Three location strategies of spectral bins were respectively performed before and after weighting. Once the orders of spectral bins were determined, the energy of bins with signals weighted by Hanning window would be adopted to achieve estimation. In order to clearly compare the performances, correction errors for each strategy were counted and rearranged in a descending order. Figure 4 shows the correction errors returned by four point EBM with different spectral bin location strategies, including traditional strategy (C1 and Cr1 respectively for after and before weighting), optimizational strategy with spectral bins near the spectral peak (C2 and Cr1 respectively for after and before weighting), and Macleod’s optimizational strategy (C3 and Cr3 respectively for after and before weighting). The results obtained by four point EBM without mistaken location (TC) were also presented for comparison.

Figure 4.

Correction errors for the four-point EBM with different spectral bin location strategies.

As shown in Figure 4, the numerical simulation on the noise-contaminated cosine wave is in good agreement with the theoretical analysis. The traditional strategy is implemented after windowing and shows the highest the error curve, which indicates its high sensitivity to additive random noise. The traditional strategy has the worst accuracy with maximum error over 0.6 bin. Even it is implemented before windowing, the error curve still remains at a high level. In the case where the optimizational strategy is taken, in which determination of spectral bins is depended on the second and third largest bins, the results would become much better no matter before or after windowing. It is worth noticing that if determination of spectral bins is conducted after windowing, the result of the optimizational strategy is approximately the same to that of the Macleod’s optimizational strategy, which is consistent with the occurrence of wrong choices depicted in Figure 2. The best choice is to take Macleod’s optimizational strategy before windowing, because it achieves the lowest error curve. The simulation reveals that the noise immunity can be greatly enhanced by reducing wrong choices of spectral bin in EBM.

Conclusion

In this paper, we investigated the phenomenon of wrong location of DFT bin due to the additive random noise and its influence on the spectral energy based frequency estimator. The reasons for mistaken location of spectral bins for the three-point and the four-point energy-based method were studied. Both the theoretical analysis and numerical simulation indicate that the problem of mistaken location is one of the major influencing factors to estimation accuracy. Two optimizational strategies were presented to decrease the occurrence of mistaken location of spectral bins. Studies showed that the occurrence of wrong choices could be remarkably reduced by adopting the optimizational strategies. Numerical simulations indicate that Macleod’s optimizational strategy with rectangle window enjoys the best performance. The comparative studies also reveal that taking Macleod’s optimizational strategy before windowing in EBM can enhance the resistance to the additive noise and achieve better estimation results. Therefore, Macleod’s optimizational strategy is highly recommended in EBM to improve the estimation accuracy, especially for the engineering with involvement of intensive noise.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (Grant No. 51404051), Chongqing Research Program of Basic Research and Frontier Technology (Grant No. cstc2017jcyjAX0033) and also partly supported by Scientific and Technological Research of Chongqing Municipal Education Commission (Grant No. KJ1600428).

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