An Exact Analysis of Fine Resolution Frequency Estimation Method from Three DFT Samples: Windowed Data Case

The estimation of the frequency in a Discrete Fourier Transform (DFT) or Fast Fourier Transform (FFT) is a fundamental and well-studied problem in signal processing and communications. A variety of approaches to this problem, distinguished primarily by estimation accuracy, computational complexity, and processing latency, have been developed. For a signal containing an integer number of periods there are no differences in the application of the DFT algorithm and passing the representation of the waveform in the frequency spectrum and vice versa. Transform and reverse application generates errors only due to the calculation algorithm. The direct method to get a correct estimation of a peak frequency is to increase spectrum resolution up to the value of interest. But, unfortunately, this approach requires an additional processing time, which might be considered unacceptable, especially for portable systems without a powerful computing unit. A second solution to solve this problem is to implement a method to estimate the frequency value between spectral lines to correct dispersion effect due to the signal processing by Discrete Fourier Transform (DFT). The literature describes a number of methods to determine the estimate frequency of a spectral peak and its amplitude, but most of them involve a volume of high computation and enabling the calculation only are both the real and the imaginary parts of spectrum. This paper proposes a new frequency estimator, similar with another well-known algorithm, but more accurate having also a low computational complexity.

Phase correction of discrete Fourier transform coefficients to reduce frequency estimation bias of single tone complex sinusoid

Sinusoidal signal frequency estimation is one of the fundamental problems in signal processing, and it is widely used in wireless communication, signal processing, navigation, radar and so on. In this paper, an interpolation frequency estimation algorithm based on Discrete Fourier Transform (DFT) and cosine windows is proposed. Firstly, the sampling sequence of the signal is multiplied by a cosine window. Then, N-point DFT is used to search the position of the maximum spectral line and get the coarse estimation of frequency. Finally, the accurate frequency estimation is obtained by DFT interpolation of the maximum spectral line and the two Discrete-Time Fourier Transform (DTFT) samples on the left and right of the maximum spectral line. According to the simulation results, the performance of the proposed algorithm is better than that of MV-IpDTFT(3) algorithm, MV-IpDTFT(2) algorithm and Candan algorithm. The effect of harmonic interference on the frequency estimation results can be effectively suppressed.KeywordsFrequency estimationInterpolationDFTCosine windowsSignal processing

Frequency estimation from discrete Fourier transform (DFT) coefficients of a rectangular windowed signal under the influence of additive white noise is a well studied problem in signal processing. In its simplest form, the process involves finding the spectral peaks. When higher frequency resolution is required, a frequency offset can be found from the interpolation of DFT coefficients. However, most of the past researches focus on monotonic cisoid signals. In practical situations where multiple harmonics are present, the sidelobes from other harmonics interfere with the estimation of the harmonic being considered. In this case, windows with smaller sidelobes such as Hanning window are preferred over rectangular window. Given the increased mathematical complexity of Hanning window, analytical solution has not yet been available for DFT interpolation. In this paper, we derive an exact analytical solution of the estimated frequency from DFT interpolation of Hanning windowed signal. In experiments, we show that the new analytical solution is accurate for monotonic cisoid signal and can considerably reduce the effect of interharmonic interference as compared to previous rectangular windowed methods.

The method for detection of complex sinusoids in additive white Gaussian noise and estimation of their frequencies is proposed. It contains two stages: 1) sinusoid detection (model order estimation) and coarse frequency estimation, and 2) fine frequency estimation. The proposed method operates in the frequency domain, i.e., it uses the discrete Fourier transform (DFT) as the main tool. Sinusoid detection is performed so that a fixed probability of false alarm is provided (Neymann–Pearson criterion). For both coarse and fine frequency estimations, the three-point periodogram maximization approach is used. Simulations are carried out for variable signal-to-noise ratio, variable frequency displacement between the sinusoids and variable offset from the frequency grid. The proposed method meets the Cramer-Rao lower bound in frequency estimation and practically does not depend on the frequency displacement except for very small displacement values. In terms of model order estimation accuracy, it outperforms the state-of-the-art approaches. The most expensive operation in the method is the calculation of the DFT. Therefore, in terms of calculation complexity, the proposed method is on par with the most efficient algorithms for multiple frequency estimations.

A frequency estimation algorithm of sinusoidal signal was presented. It consists of a coarse frequency estimation followed by fine frequency estimation. Discrete Fourier transform (DFT) determines the coarse frequency estimation and least squares estimator (LSE) makes it possible to get the optimal frequency difference estimation. Subsequently, we derive the relationship between the asymptotic error variance (AEV) and the Cramer-Rao bound (CRB). Simulation results show that the performance of the proposed algorithm approaches the CRB of the sinusoid when the signal-to-noise ratio (SNR) is higher than the SNR threshold.

The estimation of real sinusoid frequency is a significant problem in many scientific fields. The positive- and negative-frequency components of a real sinusoid interact with each other in the frequency spectrum. This leads to estimation bias. In this paper, we proposed an algorithm which is based on maximum sidelobe decay (MSD) windows. Firstly, the coarse frequency estimate is obtained by using Discrete Fourier Transform (DFT) and MSD windows. Then the negative-frequency component is removed by frequency shift. At last, the fine frequency estimation is performed by a high-precision frequency estimation algorithm. Simulation results show that the proposed algorithm has higher accuracy and better frequency estimation performance than AM algorithm, Candan algorithm, and Djukanovic algorithm.KeywordsFrequency estimationReal sinusoidWindowsDFT

Fine resolution frequency estimation from three DFT samples: Case of windowed data

Frequency estimator of sinusoid based on interpolation of three DFT spectral lines

Influence of the noise on DFT-based sine-wave frequency and amplitude estimators

A classic problem in signal processing is that of analysing empirical data in order to extract information contained within that data. The primary goal of this article is to employ the discrete Fourier transform (DFT) techniques for approximating, to a prescribed accuracy, the response of a shift-invariant recursive linear operator to a finite-length excitation. In this development, the required properties of the Fourier transform (FT) are first reviewed with particular attention directed toward the stable implementation of shift-invariant recursive linear operators. This is found to entail the decomposition of such operators into their causal and anticausal component operators. Subsequently, relevant issues related to the approximation of the FT by the DFT are examined. This includes the important properties of the non-uniqueness of mapping between a sequence and a given set of DFT coefficients. In the unit-impulse response approximation, DFT is shown to provide a useful means for approximating the unit-impulse response of a linear recursive operator. This includes making a partial fraction expansion of the operator's frequency-response. The error incurred in using the DFT for effecting the unit-impulse response approximation is then treated. This error analysis involves the introduction of one-sided exponential sequences and their truncated mappings that arise in a natural fashion when employing the DFT. These concepts form the central theme of the article.

Frequency estimator of monotone complex sinusoid is a fundamental problem in signal processing. One of the approaches is to directly estimate the frequency from interpolation of the discrete Fourier transform coefficients. These direct approaches try to find a first-order approximation to the frequency and the remaining higher-order terms lead to estimation bias. In this paper, instead of considering the higher-order term as the source of error, we propose to use the higher-order terms to our advantage by solving the higher-order polynomial equation. We derive the conditions to obtain two third-order polynomial equations and the optimal conditions to apply the two equations. Experiments show that the new estimators can reduce the estimation bias by seven orders at large frequency offset.

Fine frequency estimation of a single complex sinusoid is considered. We propose to refine the frequency estimation provided by the state-of-the-art methods based on three discrete Fourier transform (DFT) samples around the DFT maximum. To that end, parabolic interpolation of the periodogram peak is used. With the calculation of only three additional periodogram samples, the Cramér-Rao lower bound is met. Moreover, the proposed method's performance does not depend on the frequency displacement. Due to its simplicity, efficiency and low computational burden, the proposed method is suitable for real-time applications.

A sinusoidal frequency estimator based on interpolated Discrete Fourier Transform (DFT) algorithm by using Maximum Sidelobe Decay (MSD) windows is proposed in this paper. Firstly, the received sinusoid is weighted by an appropriate MSD window. Then DFT is carried out on the weighted sinusoid and the coarse estimation is acquired by finding the position of the maximum DFT sample. Different from all the existing algorithms, the presented estimator adopts the maximum DFT sample and two Discrete Time Fourier Transform (DTFT) spectral lines which are on the same side of the maximum DFT sample in the fine estimation step. MSE formulas of the presented estimator in additive white noise are derived. Simulation results indicate that the presented estimator outperforms the competing estimators.

The study on the frequency estimation of a real sinusoid has been addressed in this article. Discrete‐Fourier‐transform (DFT)‐based frequency estimation always uses the maximum bin of DFT as a coarse estimation followed by a fine estimation algorithm, which computes a correction term to compensate for the initial frequency offset caused by the coarse estimation. To enhance the frequency estimation accuracy, a fine estimation algorithm based on DFT samples and fuzzy logic (FL) is proposed in this study. Firstly, a modified phase‐based algorithm is presented to estimate the sign of the correction term. Then, a main‐lobe coefficient and a side‐lobe coefficient are constructed using the amplitude of three spectral lines to calculate the correction term. A FL controller is utilised to generate a weighted factor to adjust the weight of the main‐lobe coefficient and the side‐lobe coefficient in the formula of the correction term. Compared with other existing fine algorithms, the proposed algorithm improves the correct probability of sign estimation by 10%–20%, and improves the estimation accuracy by 5%–8% at lower carrier‐to‐noise ratios (CNRs). In addition, the performance of the proposed algorithm is less affected by the initial frequency offset than other algorithms.