An Improved High-precision Frequency Estimation Algorithm

13 Frequency estimation, one of the key technologies in wireless communication and radar 14 science, has been extensively and deeply studied by scholars. Based on a detailed frequency 15 estimation characteristics analysis of two existing algorithms (the fixed Quinn algorithm and 16 the A&M algorithm), this paper proposes a new and improved high-precision frequency 17 estimation algorithm. The proposed algorithm is a two-step estimation. In the first step, we 18 use the fixed Quinn algorithm to calculate coarse frequency errors, and in the second step, we 19 use the results from the first step as the initial iteration in the A&M algorithm. After the first 20 iteration, we obtain the precise frequency errors, and finally, we obtain the frequency estimate 21 value. Through simulation experiments on MATLAB, the experimental results verify that the 22 improved algorithm greatly enhances the estimation accuracy compared to the fixed Quinn algorithm and the A&M algorithm; moreover, the improved algorithm has excellent anti-noise performance and stronger algorithm robustness.


Introduction 29
Frequency estimation of sinusoidal signals submerged in Gaussian white noise has important 30 applications in wireless communication [1], radar science [2], power systems [3-5] and other 31 fields. Regarding the frequency estimation of noisy signals, researchers have proposed 32 various methods. In 1974, when algorithm studies began, Rife and Boorstyn [2] proposed a 33 frequency estimation method that has been widely used and is called the Rife algorithm. 34 However, this method had a high computational complexity. In 1989, Kay  Owing to the shortcomings of limited estimation accuracy from the fixed Quinn 65 algorithm, the unstable anti-noise performance, and the A&M algorithm requiring a large 66 amount of calculation, this paper proposes an improved high-precision frequency estimation 67 4 algorithm based on holistic studies of the frequency estimation characteristics of the two 68 existing algorithms. The algorithm is a two-step estimation, which is implemented by coarse 69 estimation and fine estimation [19]. Through a MATLAB simulation analysis, the improved 70 algorithm is found to greatly enhance the estimation accuracy compared to the fixed Quinn 71 algorithm and the A&M algorithm. As the length of the signal increases, the root mean square 72 error (RMSE) of the algorithm is close to the Cramer-Rao lower bound (CRLB). Under 73 different SNRs, the algorithm also has good noise immunity and stable performance. 74 75

The Fixed Quinn Algorithm 77
The sinusoidal signals interfered with by Gaussian white noise [9, 20] are presented as: 78 79 where is the sequence number, corresponding to the DFT coefficient with the largest 80 amplitude after DFT is performed on the signal. In addition, is the frequency deviation 81 value relative to the maximum spectral line; is the initial phase of the signal; A is the 82 amplitude of the signal; and N is the length of the signal. Furthermore, is a zero-mean 83 complex Gaussian white noise sequence with independent real and imaginary parts and a 84 variance of 2σ 2 . This formula defines as the DFT of the signal, and is the real part of 85 the signal. 86 According to Liao and Chen [9], the frequency deviation value relative to the 87 maximum spectral line can be obtained by the following equation: Step 1: Perform the FFT on the sinusoidal signals interfered with by Gaussian white 95 noise, record the transformation results, and search for the index corresponding to 96 the largest spectral line in the results. 97 Step 2: Suppose that the number of iterations of the A&M algorithm is Q and that the 98 initial iteration value is 0. 99 Step 3: According to formula 5 from Aboutanios and Mulgrew [10], calculate . 100 Step 4: Calculate the result of . 101 Step 5: According to the formula (i = 1..., Q), calculate the frequency 102 deviation value relative to the maximum spectral line. 103 Step 6: Determine the number of iterations P at this time and the number of 104 hypothetical iterations Q. If P≤Q, return to step 3; if P> Q, then calculate as the 105 final frequency estimation error, and then proceed to step 7. 106 Step 7: Insert into the formula , and find the signal frequency. 107 (Here, is the sampling frequency; N is the length of the signal.) 108 109

The Principle of the Improved High-precision Frequency Estimation Algorithm 110
The steps to implement the improved algorithm are as follows: 111 Step 1: Perform the FFT on the sinusoidal signals embedded in Gaussian white noise, 112 record the transformation results, and search for the index corresponding to the 113 largest spectral line in the results. 114 Step 2: Insert the Fourier coefficients at index and into the fixed Quinn 115 algorithm, and record the frequency estimation error result as . 116 Step 3: Insert into the A&M algorithm to find . 117 Step 4: Calculate . 118 Step 5: Calculate the frequency deviation value relative to the maximum spectral line 119 . 120 Step 6: Insert into the formula , and calculate the signal frequency. 121 (Here, is the sampling frequency; N is the length of the signal.) 122 The flowchart of the algorithm is shown in Figure 1. 123 124

The Calculation Analysis of the Improved Algorithm 126
Assuming that the number of iterations of the A&M algorithm is Q and the length of the 127 signal is N, we use the A&M algorithm to perform frequency estimation, which requires 128 approximately 0.5N*log 2 N+2NQ complex multiplications and N*log 2 N+2NQ complex 129 additions (assuming base-2 algorithm (radix-2) [21] is used for FFT implementation) [22]. 130 When we use the fixed Quinn algorithm to perform frequency estimation, the calculation 131 amount of the frequency deviation value relative to the maximum spectral line is not 132 considered by using the derived formula, and then we use approximately 0.5 N*log2N 133

Simulation Analysis 143
According to the improved algorithm described above, we conducted Monte Carlo simulation 144 in MATLAB. The parameters are set as follows: The sampling frequency of the signal fs = 1 145 (the normalized sampling frequency), length of the signal N=64, the amplitude of the signal 146 A=1, the phase = 0, the mean noise value is 0, and the variance is determined by the SNR. 147 We performed 1,000 Monte Carlo simulation experiments on each frequency point. 148 149

The Performance Analysis of Different Algorithms 150
The range of the SNR is 0 dB to 3 dB (step: 1 dB), and the RMSE of the fixed Quinn SNR is in the range of 0 dB to 3 dB (in steps of 1 dB) and , the RMSE of the 159 algorithm demonstrates that even for different SNRs, the algorithm can still maintain high 160 estimation performance, and the results are even closer to the CRLB than those of the other 161 two methods. The difference between the RMSE of the algorithm's frequency estimation and 162 the theoretical RMSE of the frequency estimation is very small, and the proposed algorithm 163 estimation accuracy is high; therefore, the algorithm can be applied in practical situations. 164 The results further verify that the algorithm proposed in this paper has good anti-noise 165 performance and stability under different SNRs. 166 167

The Performance Analysis of Different Signal Lengths 168
In the case of different signal lengths (N= 32 and 128), SNR = 3 dB, and other parameters 169 remaining unchanged, the RMSEs of the fixed Quinn algorithm, the A&M algorithm, and the 170 improved algorithm proposed in this paper are shown in Figures 6 and 7. In the figures, the 171 formula has an interval value of 0.02, and the vertical axis shows the RMSE (unit: 172 dB). 173 In Figures 6 and 7, under different signal lengths, compared with the estimation 174 accuracy of the fixed Quinn algorithm and the A&M algorithm, the algorithm proposed in 175 this paper can maintain high estimation performance over the entire frequency band. Even in 176 the situation of a low SNR, the estimation performance is still stable. As the signal length 177 simulation results show that the calculation burden of the algorithm proposed in this paper is 202 slightly more than that of the fixed Quinn algorithm but less than that of the A&M algorithm. 203 Nevertheless, the proposed algorithm in this paper demonstrates improved overall accuracy, 204 and the mean square error of frequency estimation is very close to the CRLB. In addition, the 205 algorithm proposed in this paper has good anti-noise performance. With increasing signal 206 length, the estimation accuracy of the algorithm in this paper is highly enhanced, and the 207 algorithm can be widely used in many signal processing contexts. 208