#### Abstract

This study presents three different sinc estimators and a method to estimate complex-valued exponential tone signal frequency using three DFT samples. The proposed method suggests using sinc interpolation together with the well-known Jacobsen estimator. According to simulation results, the root mean square error (RMSE) of the proposed algorithm is lower than those of the Jacobsen estimator and its improved version suggested by Candan. The price paid for improvements in the RMSE is a slight increase in computation time.

#### 1. Introduction

The frequency estimation problem plays an essential role in communications, audio, medical, instrumentation, and other applications. Nonparametric and parametric techniques are the main tools used in spectrum estimation methods. Nonparametric techniques do not assume a particular functional form but allow the form of the estimator to be determined entirely by the data. These methods are based on the discrete Fourier transform of either the signal segment or its autocorrelation sequence. In contrast, parametric methods assume that the available signal segment has been generated by a specific parametric model [1] (p. 195). Many nonparametric and parametric methods, such as DFT-based frequency shifting and filtering, phase-locked loop, maximum likelihood, wavelet, Prony’s, Taylor, interpolation, and of multi-DFT-bins, artificial neural network design, have been published [2–5].

In digital signal processing, the discrete Fourier transform (DFT) is a frequently used tool to detect and estimate the frequencies of a signal. Several estimation methods based on DFT have been proposed. The well-known method for frequency estimation is to use N-point Fourier transform after sampling a signal. After applying an N-point DFT, the frequency of interest rarely resides exactly on a DFT bin center. In this case, frequency estimation resolution depends on the bin spacing of the transform. However, we always need better frequency estimation resolution. There are lots of frequency estimation methods to improve that resolution. These methods have been described in [6–21].

Equation (1) is the best-known Jacobsen peak location estimator and has been used in the DSP business for decades, which uses three DFT samples [22] (p. 733). This spectral peak location estimator which is given by considers the bin with the maximum DFT value, and its adjacent right and left bin DFT values, and , respectively, to calculate a correction factor for the peak location (see Figure 1).

**(a)**

**(b)**

The frequency is estimated as with the units of DFT bins or in terms of the normalized frequency. A bias-corrected version of the Jacobsen estimator, proposed by Candan [10], is given in

Clearly, for a large number of data samples (as ), the two estimators coincide.

These estimators perform well when the actual signal frequency happens to be in the center of the bin. When the actual frequency moves toward either end of the bin, RMS error of the estimators increases. In this paper, we developed estimators based on sinc function interpolations and measured their performances to find an answer to the question, how could these sinc function-based estimators be used to increase the performance of the Jacobsen estimator (1) and its bias-corrected version (2).

#### 2. Preliminaries

A complex signal with an amplitude , a phase , and a normalized frequency can be expressed as where is the total number of samples available. The signal frequency in bin values is defined as (see Figure 1). The DFT of the sequence is and can be expressed as

DFT of the is given by Equation (6). The spectrum would be continuous if *N* samples were increased infinitely. But the DFT works for finite-length signals. Selecting more samples means taking a longer signal and multiplying the signal by a rectangular pulse. Since the multiplication of a signal in the time domain results in the frequency domain being convolved with a sinc function, this reduces frequency resolution [23–27]. Therefore, some resolution improvements are needed for better estimation.

The DFT of the signal forms equally spaced samples in the frequency spectrum. Each spectral value is at a different interval of . These frequency intervals define frequency resolution and are denoted frequency bins. Frequency bins are integer values from 0 up to . Bin has the largest DFT magnitude value (Figure 1). Bins next to the bin, , and the previous bin, , are considered in estimating the tone frequency. The actual frequency could be anywhere in the bin , depending on the bin and magnitude values. And this exact frequency can be found by adding or subtracting the correction value to the bin , since changes in the range of .

The DFT magnitudes of , , and for *V,*, , and frequency sitting anywhere in the tenth bin are shown in Figure 2. The signal frequencies in bin are all represented in bin 10. If the actual frequency happens to be in the middle of bin 10, the magnitude of becomes maximum, and the previous and next bin’s magnitude values become zero. On the other hand, when the actual frequency goes toward either end of the bin, the magnitude of decreases while magnitudes of and increase. A closer look will notice how symmetry exists between and . Since the magnitude margins increase toward the bin ends, any estimator which calculates the correction factor value just by using the magnitude values of and would work well, while there is no doubt that it will not work well where the tone frequencies are close to the bin center.

#### 3. Sinc-Based Estimators

Interpolation is a method of finding new data points based on a set of known data. These data points represent the values of a function for a limited value of the independent variable. It is often required to estimate the value of the function for an intermediate value of the independent variable. The DTFT of a single tone has the characteristics of a sinc function. After performing the DFT of the signal, all the DFT values of transformation will be on the continuous spectrum of Fourier transformation in the noise-free cases. We can find the correction factor by solving the sinc function equations defined by the maximum and the second maximum DFT magnitudes, which is the main idea behind sinc function-based estimators [28–30].

##### 3.1. The First Sinc Estimator

The location of the signal frequency we are trying to estimate is between the largest amplitude and the second biggest one adjacent to it. The position of the second largest magnitude could be before or next to the largest amplitude. Both cases should be investigated to find a solution. Let us start to analyze the situation depicted in Figure 1(a), where the second biggest amplitude comes after the maximum. In this case, DFT values at instances and can be found from Equation (6) after substitutions , and , respectively. The following equations are obtained:

The magnitude values of (7) and (8) are as follows:

Equations (9) and (10) can be written as

If we solve Equations (11) and (12) for , we get the following equation:

Let us take the Taylor series expansion of the function around to find a closed-form equation for .

Assuming small and/or large , we obtain the following estimator after taking the first term of the expansion.

On the other end, let us find out how the estimator behaves in the situation depicted in Figure 1(b), where the second biggest magnitude comes before the largest one. In this case, DFT values at and can be found from Equation (6) after substitutions and , respectively. is defined as a negative variable in this region. The following equations are obtained:

After taking the magnitudes of Equations (17) and (18) and solving for , we obtain the following equations:

##### 3.2. The Second Sinc Estimator

Now, let us find an estimator that calculates the correction factor by looking at the DFT magnitude values adjacent to the maximum DFT value. We have to define the magnitude values of and for to get a solution

If we solve Equations (20) and (21) for , we get the following equation:

The magnitude values of and for are defined as follows:

The solution of Equations (24) and (25) for gives the same result as in Equation (22).

##### 3.3. The Third Sinc Estimator

The previous two estimators compute the correction factor from the DFT magnitude values. Now, let us find an estimator that calculates the correction factor by looking at the maximum and second maximum complex DFT values. We have to go through similar steps as before. We obtain the following equations for the situation depicted in Figure 1(a). where is defined as positive

Equations (26) and (27) can be rewritten as follows:

We obtain the third estimator after solving Equations (28) and (29) as follows:

Now, let us find the estimator for the situation depicted in Figure 1(b). We should repeat the same steps to find DFT complex values at and , which can be found from substitutions and where is negative. The following equations are obtained:

After solving Equations (32) and (33) for , we obtain the following equations: which are a similar estimator as in (30) except for a sign difference.

##### 3.4. Decision Criteria

Table 1 summarizes all the sinc-function-based estimators. The first and third estimators consist of two parts. Before computing the correction factor, there must be a decision criterion for identifying which part of the expression should be used. One of the criteria would be a comparison of magnitude values of and as

If the signal frequency is near the bin center, the error of the estimators would be high due to the declining margin between and , as shown in Figure 2. However, the criterion becomes more stable when the signal frequency goes toward the bin ends.

Figure 3 depicts how the DFT complex values of , , and varies vs. bin frequency for a complex signal having an amplitude of *V*, , and a frequency changing from 9.5 to 10.5. Figure 3 shows that has the value of 32 when , while and are zero when the signal frequency is in the bin center. starts decreasing as the actual signal frequency goes from bin 10 to bin , which means that the second biggest DFT magnitude value comes after the maximum value. Meanwhile, and start increasing. When bin frequency goes from bin 10 to bin 10.5, the contour line of begins at point A and goes to point B. At the same time, and contour lines start at the origin, as the first one follows the path toward D, and the second one goes toward point C in the opposite direction. When bin frequency goes from bin 10 to bin 9.5, the contour line of starts at point A, but goes toward point C. Similarly, and contour lines start at the origin, while the first one follows the path toward B, and the second one goes toward point E. In Figure 3, the complex DFT values of the maximum bin and its adjacent bins are depicted in circles for the actual frequency of bin 10.25. On the other hand, the complex DFT values are indicated in squares for the actual frequency of bin 9.75. The phase of the DFT values vs. actual bin frequency is depicted in Figure 4, which shows that the rate of phase change for all DFT values is equal to each other. As goes toward a positive end, the phase change of and is identical but has opposite signs with . The same situation can be seen as the goes toward a negative end.

Let us analyze Figure 3 to find another decision criterion for the estimators and . When the actual signal frequency is in bin 10.25, the complex DFT values of bin ten and its adjacent bins are shown in Figure 3. They have the following complex values: ,, and . If we add the negative of to and divide the sum by , we get , which has a positive real value. The result of this operation is always positive for all because both the nominator and denominator have similar signs. If we make a similar calculation for the actual frequency of bin 9.75, we end up with negative real values for all negative values of because both the nominator and denominator have opposite signs. For this reason, we propose the following decision criterion:

Figure 5 shows a comparison of the decision criteria between and for at dB. This comparison was made by looking at the values of Expressions (35) and (36). If and are true and positive, a positive sign is assigned for this operation; meanwhile, a negative sign is assigned for false and minus values. Both decision criteria should give a positive sign for but a negative sign for . Figure 5 shows that false decisions mainly concentrate around , which is a cause of RMSE deterioration. The decision criterion outperforms in numbers of correct decisions even in small sample sizes and low SNR levels.

Figure 6 shows a comparison of estimators and with the two different criteria in terms of RMSE vs. SNR. Two different criteria were applied to each estimator simultaneously. The RMS errors of the estimators which use the criterion are shown with dashed lines. Solid lines represent the RMS errors of the estimators which use the criterion . This comparison indicates that estimators and have better performance with decision criterion 2.

#### 4. Simulation Results

This section compares sinc estimators, as stated in Table 1, with the Jacobsen estimator (1) and its modified version suggested by Candan (2) in terms of complex and real signals.

##### 4.1. Simulation Results for Complex Signals

During numerical simulations, a complex signal with an amplitude of 1 V, a randomly changing phase between and , and a frequency have been used. The signal has also been contaminated with a white Gaussian noise

Here is a circularly symmetric complex white Gaussian noise with mean zero and variance . The signal-to-noise ratio is defined as .

Figures 7 and 8 examine how sample size and SNR levels affect estimators’ behavior for the signal frequency at bin ten with an offset of . Both figures show that the RMSE decreases as the sample size increases. The first and third sinc estimators outperform the other estimators at all levels of SNR values.

Figure 9 compares the sinc estimators with the estimators of Jacobsen and Candan. The number of samples was reduced to 8, but the signal frequency was kept the same as in Figures 7 and 8. Figure 9 shows that the sinc estimators perform well even at a small sample size of . Also, RMSE improvements are so distinct at low SNR values.

Table 2 shows how much execution time is needed for each estimator at bin frequency 10 for a sample size of . Execution times have been measured in a loop and then averaged to find the time for a single run. The price to be paid for improvements in RMSE is roughly a microsecond.

Figure 10 expresses the RMS errors as a function of the correction factor when the sample size and dB. It shows that as the frequency, we are trying to estimate is at a location toward the bin edges; the RMS errors of the sinc estimators get smaller when is greater than +0.20 or less than -0.20. We examined RMSE variation with for sample sizes ranging from to at different SNR values to get a whole picture of the behavior of the estimators. It has been observed that when the sample size is large, the RMS errors of the sinc estimators become lower than those of the Jacobsen and Candan estimators when is greater than +0.15 and less than -0.15. When the sample size is reduced to as small as , as in Figure 10, the RMS errors of the sinc estimators go below those of the others when is greater than +0.20 or less than -0.20.

Figure 11 shows how the bias of the estimators changes with respect to SNR for and . The sinc estimators and have lower biases than those of the Jacobsen estimator, while their biases are almost the same as those of the Candan estimator.

Figure 12 shows how the RMS errors of the estimators’ changes vs. number of periods for , while the SNR level was kept at 10 dB. The sinc estimators and have lower RMS errors than the Jacobsen and its bias-corrected version estimator for all spectrum ranges. The number of periods does not significantly affect the RMS errors of the estimators.

##### 4.2. Simulation Results for Real Signals

In this section, numerical simulations have been carried out in order to determine how the proposed estimators behave in real signals. A real signal with an amplitude of 1 V, a randomly changing phase between and , and a frequency of have been used.

The signal has also been contaminated with a white Gaussian noise, which has zero mean and variance . The signal-to-noise ratio is defined as .

Figures 13 and 14 examine how the estimators behave in real signals for different sample sizes and SNR levels for the signal frequency at bin ten with an offset of . Both figures show that and estimators give estimation errors of 0.0196 and 0.011 bin widths for , , and , respectively. These RMS errors indicate that and estimators developed for complex signals perform as well as for real signals.

#### 5. Conclusions

The estimator proposed by Jacobsen has drawn wide attention because of its fast and accurate computation. There have been contributions regarding its further improvements. This study is another attempt to increase its accuracy.

All the simulations performed showed that the improvements suggested by the sinc estimators, and , are remarkable even for small sample sizes and in a wide range of SNR values. When the sinc estimators and are compared with Jacobsen and Candan estimators for dB, and an offset of , estimation errors are roughly 0.0150, 0.0327, and 0.0285 bin widths for the sinc, Jacobsen, and Candan estimators, respectively. So, we propose a frequency estimation method for the Jacobsen estimator (1), as stated in Table 3. The price paid for the reduced RMS errors is a slight increase in the computational workload.

#### Data Availability

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available. Corresponding author: Hasan Bayazit Bursa Uludag University, Bursa 16059, Turkey, [email protected], tel.: 90-551-139 8848.

#### Conflicts of Interest

The authors declare no conflict of interest.

#### Authors’ Contributions

The conceptualization was prepared by E.D. and H.B.; the methodology was prepared by X.X.; The software analysis was done by H.B.; the validation was prepared by E.D. and H.B.; the formal analysis was prepared by H.B.; the investigation was prepared by E.D.; the resources were gathered by H.B.; the data curation was prepared by H.B.; the writing of the original draft preparation was prepared by H.B.; the writing of the review and editing was prepared by H.B. and E.D.; the visualization was done by H.B.; the supervision was done by E.D.; and the project administration was done by E.D. All authors have read and agreed to the published version of the manuscript.