Incoherent integration of hyperbolic frequency modulated pulses by the Radon transform

Zhaoning Gu 1

Key Laboratory of Underwater Acoustic Signal Processing of Ministry of Education Southeast University, Nanjing ,210096,China

Shiliang Fang 2

Key Laboratory of Underwater Acoustic Signal Processing of Ministry of Education Southeast University, Nanjing ,210096,China

Hongli Cao 3

Key Laboratory of Underwater Acoustic Signal Processing of Ministry of Education Southeast University, Nanjing ,210096,China

ABSTRACT We herein investigate the incoherent integration of hyperbolic frequency modulated (HFM) pulses for improvement of detection performance under heavy ocean noise. In active sonar systems, combining wideband transmit signals and matched filtering o ff ers considerable processing gain. However, when relative motion exists between the target and sonar system, the processing gain decreases due to the mismatch caused by the Doppler distortion. As a Doppler invariant signal, the HFM waveform gets the property at the cost of a bias in the matched filter output. We propose an incoherent pulse integration method based on the Radon transform to take advantage of the property of HFM signals and handle the bias in outputs of matched filters. In the proposed method, return pulses are first matched filtered, and the Radon transform is performed on the outputs afterward. After that, range bias compensation is performed based on the transform results. The proposed method o ff ers reliable detection performance under heavy ambient ocean noise thanks to the processing gain obtained by both pulse integration and matched filtering. Also, with the help of the Radon transform, detection, bias compensation, and target parameter (range and radial velocity) estimation could be carried out simultaneously.

1 230189079@seu.edu.cn

2 slfang@seu.edu.cn

3 honglicao@seu.edu.cn This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant No. 2242021k30019, in part by State Key Laboratory of Hydrology-Water Resources And Hydraulic Engineering under Grant No. 2021490811, and in part by the National Natural Science Funds of China under Grant Nos. 91938203.

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

1. INTRODUCTION

A combination of wideband transmit waveform with matched filters gives sonar systems considerable processing gain against ambient sea noise. However, the Doppler distortion will introduce mismatch and degrade the performance of matched filters. Thus, as a Doppler invariant waveform, the hyperbolic frequency modulated (HFM) waveform can maintain the processing gain in the outputs of matched filters under Doppler distortion. Based on this advantage, the HFM is of great importance in underwater applications. However, the HFM grants this property at the cost of a bias in the output of the matched filter. The bias in outputs of matched filters should be properly handled before taking advantage of the invariant property. Methods that aim at addressing the bias problem have been an ongoing topic for years. In [1], Yao proposed to estimate the parameters of HFM via short time Fourier transform. Wang [2] proposed to estimate the compensate the bias through composite waveform. On that basis, Wei [3] proposed to use a speed spectrum scanning method, and Zhao [4] proposed to estimate the Doppler factor form the auto-correlation of outputs of matched filters. However, these methods are all single pulse-based and requires higher signal-to-noise ratio (SNR). To address the applications with both low SNR and heavy Doppler distortion, we propose to use the multi-pulse returns of the HFM signal. In the proposed approach, multiple return HFM pulses form a unbiased velocity and a biased range estimation with the help of the Radon transform. Because the velocity is unbiased, we can compensate the biased range estimation through the coupling relationship between velocity and range estimation. Furthermore, because we make use of multiple return pulses, the proposed approach is more reliable against heavy ambient sea noise.

2. PROBLEM FORMULATION

The HFM waveforms are valued for their Doppler-invariant property. However, as shown in Figure. 1, the invariant property is accompanied by a bias in the output of matched filter. In this section, we briefly introduce the joint range-Doppler estimation based on the composite HFM waveform and its peak-pair selection problem under low SNR conditions.

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v = 0 m/s v = 30 m/s

v = 0 m/s v = 30 m/s

Amplitude

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(B) (A) Figure 1: Matched filter outputs of linear frequency-modulated (LFM) and HFM waveforms with frequency ranging from 10 to 20 kHz at di ff erent velocities (Doppler factor α ≤ 1, and the velocity of sound propagation is 1500 m / s). (A) 25 ms LFM waveform. (B) 25 ms HFM waveform. The decrease in peak amplitude can be solved by using extended matched filtering [5].

An HFM signal s ( t ) with frequency ranging from f 1 to f 2 can be written as

s ( t ) = cos " 2 π

k ln(1 + k f 1 t ) # , t ∈ [0 , T ] , (1)

where k = ( f 1 − f 2 ) / ( f 1 f 2 T ) and T is the signal length. The instantaneous frequency f i ( t ) of s ( t ) is

f i ( t ) = ∂

" 1

k ln(1 + k f 1 t ) # f 1 1 + k f 1 t . (2)

∂ t

Similarly, the instantaneous frequency f α ( t ) of the Doppler-distorted waveform s ( α t ) is

f α ( t ) = f 1 / [1 + k f 1 ( t + 1 − α

α k f 1 )] . (3)

A comparison of (2) with (3) shows that the Doppler distortion leads to a shift in the instantaneous frequency, which, in turn, causes a bias in the output of matched-filter. Thus, if we want to take advantage of the HFM waveform, we should handle the bias first.

3. PROPOSED METHOD

If the target is simplified as a nonfluctuating point, the received signal can be written as As ( α t − τ ) [6], where τ = 2 R 0 / ( c + v ), R 0 is the range of the target, c is the speed of sound, v is the target’s radial velocity, and v is positive when the target is moving towards the system. Thus, we can get a peak in the output of the matched filter at

t = τ/α + ( α − 1) /α k f 1 . (4)

Thus, once we obtain the Doppler factor α , we can compensate the bias utilizing (4). According to (4), suppose we have M consecutive pulses, the locations of the peaks in the outputs of matched filters are

t m = ( τ 0 + m ∗ ∆ τ ) /α + ( α − 1) /α k f 1 , m = 0 , · · · , M − 1 , (5)

where ∆ τ = T s v

c is the increment of delay in each pulse, T s is the time interval between each pulse, and v is the velocity of the target. Thus, as shown in Figure. 1, these peaks locate on a line with slope determined by the velocity of the target. Based on this observation, we can utilize the Randon

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20 40 60 80 100 120 140 160 Range (m)

Figure 2: Matched filter outputs of 20 consecutive HFM pulses. The HFM waveform ranges from 18 kHz to 20 kHz, the pulse duration is 20 ms and the sample frequency is 50 kHz.

transform to estimate the velocity of the target. Once the we obtain the velocity, we can compensate the bias and estimate the true location of the target using (4). Suppose the matched filter output shown

in Figure. 2 are represented as S ( x , y ), where x stands for the range bin, and y stands for the pulse index. The Radon transform can be performed as

Z ∞

R ( r , θ ) = Z ∞

−∞ S ( x , y ) δ ( x cos θ + y sin θ − r ) dxdy . (6)

−∞

The result of the Radon transform is shown in Figure. 3, and the line results in a bright point in the result of the Radon transform. From the location of the brightest point R ( r 0 , θ 0 ), we can estimate the

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r (pixels from center)

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88 88.2 88.4 88.6 88.8 89 89.2 89.4 89.6 (degree)

Figure 3: The result of the Radon transform of the matched filter output in Figure. 2. The line in Figure. 2 results in bright point in the output of the Radon transform.

velocity as v = tan ( θ 0 ) × c f s T s , (7)

where f s is the sample frequency, T s is time interval between pulses, and c is the speed of sound in water. Once we obtain the speed of target, we can compensate the bias and estimate the trajectory of the target accordingly.

4. NUMERICAL RESULTS

We test the e ff ectiveness of the proposed approach in this section. We set the velocity of the target to 8 m / s, the frequency of HFM ranges from 18 kHz to 20 kHz, the pulse duration is set to 25 ms, the time interval between pulses is set to 200 ms, and the sample frequency is 50 kHz. In this section we take a total 20 pulses, and test our approach under di ff erent SNRs. To get a smooth result, we perform 1000 times Monte Carlo runs, and the results are shown in Figure. 4.

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Figure 4: The mean absolute error (MAE) of the proposed approach under di ff erent SNRs. (A) The MAE of radial velocity estimation. (B) The MAE of range estimation.

From the results in Figure. 4, the proposed method o ff ers acceptable performance under low SNR conditions. This is because the proposed approach realizes the incoherent integration of multiple

return pulses. In Figure. 4, the error becomes flat at low SNR because we set the possible velocity ranges form 0 to 30 m / s.

5. CONCLUSION

Unlike other wideband waveforms, the HFM waveform has the property of Doppler invariant. Thus, the HFM waveform can o ff er una ff ected performance against moving target under heavy ambient sea noise. Moreover, the proposed approach realizes the non-coherent integration of multiple pulses. Thus, the requirement for SNR is further lowered. The numerical simulation proves that the proposed method can o ff er reliable estimation of radial range and velocity of the target under heavy ambient sea noise.

REFERENCES

[1] Shuai Yao, Shiliang Fang, Xiaoyan Wang, and Li Wang. Parameter estimation for HFM signals using combined STFT and iteratively reweighted least squares linear fitting. Signal Process. , 99:92–102, 2014. [2] Fangyong Wang, Shuanping Du, Wei Sun, Qiang Huang, and Jiao Su. A method of velocity estimation using composite hyperbolic frequency-modulated signals in active sonar. J. Acoust. Soc. Am. , 141(5):3117–3122, May 2017. [3] R. Wei, X. Ma, S. Zhao, and S. Yan. Doppler estimation based on dual-HFM signal and speed spectrum scanning. IEEE Signal Process. Lett. , 27:1740–1744, August 2020. [4] S. Zhao, S. Yan, and L. Xu. Doppler estimation based on HFM signal for underwater acoustic time-varying multipath channel. In IEEE Int. Conf. Signal Process., Commun. Comput., ICSPCC , pages 1–6, September 2019. [5] J. J. Murray. On the Doppler Bias of Hyperbolic Frequency Modulation Matched Filter Time of Arrival Estimates. IEEE J. Ocean. Eng. , 44(2):446–450, April 2019. [6] Jin Qu, Kon Max Wong, and Zhiquan Luo. The estimation of time delay and Doppler stretch of wideband signals. IEEE Trans. Signal Process. , 43(4):904–916, April 1995.