Analytical solutions for frequency estimators by interpolation of DFT coefficients

Highlights

• Introduce a new set of unbiased analytical estimators for the frequency of cisoid.

• Experiments show that the new estimators are more accurate than the previous estimators.

• With noise, some estimators work better at low frequency offset and vice versa.

Abstract

For the estimation of the frequency of a complex sinusoid, previous methods interpolate DFT coefficients to obtain the frequency offset. Most of these interpolation methods can be considered as linear approximation of a nonlinear function. Up to date, only a few previously proposed estimators have been shown to have analytical expressions. In this paper, we show that almost all the interpolated estimators proposed to date, both direct and iterative, have simple analytical solutions for the frequency offset. Thus, the estimation bias of these estimators can be better understood and a whole new set of unbiased estimators using the analytical solutions is also introduced.

Introduction

Frequency estimation of a complex sinusoidal signal is a fundamental problem in digital signal processing and has applications in areas such as biomedical signal processing, radar analysis, signal detection and telecommunications [1]. The signal can be described as

$$s\left[n\right]=A_0{e}^{j(2\pi (f_0/f_s)n+{\theta }_0)}+w\left[n\right],n=0,1,\dots ,N-1$$

where A₀, f₀, and θ₀ are the amplitude, frequency, and phase of the signal, respectively, the term $w\left[n\right]$ is an additive noise, f_s is the sampling frequency, and N is the number of samples.

The ML estimator for the frequency estimation under additive white Gaussian observation noise is the peak search of the periodogram corresponding to the input signal $s\left[n\right]$ [2]:

$${\widehat{f}}_{0,\mathit{ML}}=\arg \underset{f}{\max }\left\{\left|\sum _{n=0}^{N-1}s\left[n\right]e^{-j2\pi \mathit{nf}}\right|\right\}.$$

The Cramér Rao lower bound (CRLB) of the mean squared error (MSE) is

$${\sigma }_f^2=\frac{6f_s^2}{4{\pi }^{2}N(N^2-1)\rho }$$

where ρ is the SNR of $s\left[n\right]$.

The search for the frequency is usually divided into two steps: a coarse search and a fine search. The coarse search finds the index of the peak magnitude, k_p, of the discrete Fourier transform (DFT):

$$S\left[k\right]=\sum _{n=0}^{N-1}s\left[n\right]e^{-j(2\pi /N)\mathit{nk}}.$$

$$k_p=\arg \underset{k}{\max }\left|S\right[k\left]\right|.$$

The fine search finds a frequency offset $\widehat{\delta }$ in the vicinity of k_p with the constraint $\left|\widehat{\delta }\right|<1/2$. The estimated frequency is

$${\widehat{f}}_0=(k_p+\widehat{\delta })\frac{f_s}{N}.$$

In the fine search, $\widehat{\delta }$ is calculated by interpolating the DFT coefficients in the neighborhood of $S\left[k_p\right]$ [3], [4], [5], [6], [7], [8], [9], [10], [11].

Since the exact relationship between the frequency offset and the DFT coefficient interpolation is nonlinear, most of these direct interpolation approaches are inevitably approximation methods by linearizing the nonlinear function and cause estimation bias. In this case, the MSE has two components [12]:

$$\mathrm{MSE}={\sigma }_e^2+{\mathrm{bias}}^2.$$

The first component, σ_e², is contributed by the additive noise $w\left[n\right]$ in (1) and its lower bound is characterized by CRLB. The second component, bias, is caused by the linearization of the interpolation. From the composition of MSE, we can see that the performance of an estimator can be divided into two regions according to the noise level. At low SNR (${\sigma }_e⪢\mathrm{bias}$), σ_e is the dominant term and the performance of an estimator is determined by SNR. On the other hand, at high SNR (${\sigma }_e⪯¡\mathrm{bias}$), the bias term is the determining factor and the performance of an estimator is limited by the bias. A general discussion on these two limiting factors can be found in [12]. To make the estimation more accurate, either σ_e or bias has to be reduced. In this regard, many iterative approaches [2], [6], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] have been proposed to refine the solution. As the number of iteration increases, the bias can either be neglected or eliminated and Aboutanios and Mulgrew [17] have shown that σ_e² of their iterative estimator can approach CRLB asymptotically in two iterations. The bias can also be reduced by zero padding [23], [24] or solving its nonlinear function [23], [24], [25]. However, the most efficient way to completely eliminate the bias is to remove the approximation from the estimation.

Therefore, to obtain a truly unbiased estimator, nonlinear analytical solution is necessary. In this regard, three analytical solutions have been proposed to date. The first was proposed by Bertocco et al. [26], [27], [28], [29] using two neighboring DFT coefficients. The second was derived by Aboutanios [27] from their iterative estimator. The third was proposed by Candan [12] during submission of this paper. It shows that Jacobsen estimator has an analytical solution. In this paper, we will show that analytical solutions are not limited to these two cases and almost all the interpolated estimators proposed to date have simple analytical solutions for the frequency offset $\widehat{\delta }$. These analytical expressions have both practical and theoretical values. Practically, these expressions allow us to have estimators that are accurate without iteration and are only sensitive to noise. Thus, a whole new set of unbiased estimators is introduced in the paper. Theoretically, the solutions provide a basis to understand the estimation bias caused by the approximation. In the experiments, we show that these analytical solutions do indeed provide accurate estimation without bias when noise is not present.

The remainder of the paper is organized as follows. In the second section, we present the theoretical analysis that leads to the analytical solutions of the estimators and discusses possible variations of the analytical estimators. In the third section, we show the experimental results. Finally, in the fourth section, we conclude.