Unlike traditional narrow-band wave signals (e.g., continuous waves, CW) and wide-band wave signals (e.g., linear frequency modulation, LFM), AFC consists of a series of modes with the same amplitude, phase, and evenly distributed frequencies, as shown in Figure 1 . In the time domain, the AFC signals have narrow pulse width, high stability of frequency, large instantaneous power and good coherence, which can be mathematically expressed by Equations 1–3:

where r e c t is the rectangular pulse function, t is the time, T is the pulse period, n is an integer, m is the number of modes, A is the power amplification factor, E n is the amplitude of individual mode, f n   is the frequency of individual mode, θ n is the initial phase, f c e o is the initial frequency, f_rep is the interval frequency.

The strictly evenly distributed frequency intervals and the extremely precise amplitude and phase lead to the stability of the measurement results with f c e o and f r e p well referenced to an Rb clock. Thus, through the coherent superposition of each highly stable frequency component, the pulse signal with a narrow pulse width and high peak power in the time domain can be formed. Each pulse contains all harmonic components and the corresponding frequency and phase information can be calculated by Fourier transform. The echo waveforms of AFC can be expressed as Equation 4:

where a is the gain coefficient of the echo intensity, Δ is the Doppler factor, R is the distance between the signal source and the moving target, c is the velocity of sound, and w ( t ) is white noise (Equation 5).

where v is the radial velocity of a moving target.

Assume that at time t 0 , the pulse front of the signal hits a fast-moving target with speed ν and reflects it back. After the time interval of t d , i.e. at moment t 1 , the pulse back edge hits the moving target. During this time, the target moves the distance v t d . According to the quantitative relationship in Figures 2A, B , the following equations can be obtained:

where c is the velocity of sound, τ and τ ′ are the duration of the emitted and received pulse. From Equation 6 and Equation 7, it can be easily deduced as

Suppose the number of periods of the transmitted signal is N T , obviously with no change in two moments. The f r e p and f r e p ' can be determined using the following equations:

According to Equations 8–10, the equation can be deduced as Equation 11

The direction of movement of the target is set as positive. As can be seen from Figure 2C , the Doppler effect leads to compression of the AFC signal in the time domain when the target moves toward the transmitter and the receiver. The severity of the Doppler effect can be reflected via the peak intervals of the signal envelope.

Fourier transform is performed on the received signal,

According to the linear property and time-shift property of Fourier transform, the Equation 12 can be written as

Due to the time-domain scaling property of the Fourier transform, Equation 13 can be further rewritten as

Set e − j ω 2 R c · a 1 + Δ as β, and make ω 1 + Δ as ω ′ , and the Equation 14 can be written as

ω ′ is changed to ω , and Equation 15 is simplified to

Applying the discrete Fourier transform to Equation 16, we can get

where T_s is the sampling interval.

It can be seen from Equation 17, the frequency domain characteristics of the echo signal are the linear superposition of individual phonon signals. Since the phonon modes are Doppler-sensitive single-tone signals, the Doppler factor can be determined by each independent component. This is equivalent to obtaining multiple Doppler factor measurements from just one set of transmitting and receiving pulse signals. The frequency of the received signal can also be expressed as Equation 18:

According to the relationship between the repetition frequency and period of the AFC signal, it is also consistent with Equation 11.

The AFC signal consists of multiple signals with separate single-tone components, each affected by the Doppler effect, resulting in the Doppler frequency shift phenomenon. The frequency shift of each phonon mode can be directly calculated by f d n = f r n − f n . f r n can be quickly determined by the FFT of the received signals followed by peak detection, and f n is the given information of the broadcast signal.

As can be seen from Figure 3A , the frequency shift between the first frequency component of the transmitted waveform and the received waveform is 200 Hz, and the last is 350 Hz. It varies linearly with the frequency of the emitted phonon mode ( Figure 3B ). The phonon modes of the transmitted signals differ from each other. Still, the variable size of the frequency shift f d n for each frequency component can be determined, corresponding to the same Doppler factor. This means the Doppler factor can be calculated multiple times with just one measurement, leading to a more accurate estimate.

Different frequency components with the same frequency resolution lead to different deviations for different phonons ( Figure 4A ). The variance of Δ n can be deduced as: D ( Δ n ) = 1 / ( 12 f n 2 ) , assuming that the frequency resolution is 1 Hz and the f d n is evenly distributed in the range from -0.5 Hz to 0.5 Hz. To obtain a more precise estimate, we calculate the weight of each measurement by taking the inverse of the variance of Δ n . The sum of these inverses is then utilized to standardize the weight of each measurement. Further derivation of the Doppler factor is as Equation 19

To further clarify the principle of the AFC-based Doppler estimation method, an AFC signal with an initial frequency of 100 kHz, an interval frequency of 500 Hz, a cutoff frequency of 200 kHz, and an initial phase of 0 rad for all phonon modes is constructed. The Doppler factor is set to 0.02 with a frequency resolution of 1 Hz, and the duration is set at 50 ms. The received signal in the time domain is transformed by FFT. The uncertainty of each individual phonon mode can be induced by the frequency resolution, and all Doppler shifts correspond to the same Doppler factor. This distortion can be eliminated by multiple unequal precision measurements ( Figure 4B ), a mode similar to the principle of the CUU estimation method.

As shown in Figure 5 , as δ ranges from 0.05 to 0.45, which is the ratio of offset frequency to frequency resolution, the bias of single-tone signal after FFT does not exceed 3.2×10^-5. While the AFC-based method keeps a bias of less than 1.1×10^-5. Traditional Doppler factor estimation using single-tone waveforms is limited by the number of samples of the Fourier transform N, which inevitably leads to the spectral leakage. However, the use of AFC waveforms can enable the acquisition of multiple different Doppler factor values across different phonon modes, which is equivalent to carrying out numerous measurements of the identical Doppler factor simultaneously. In this way, the utilization of the spectrum has been significantly improved. It can be further reduced by increasing the number of phonon modes, and there is almost no additional computational effort.

To demonstrate the effectiveness of the proposed method, two sets of Monte Carlo experiments were conducted using the methods listed in Table 1 , which were repeated 1000 times. In the experiments, the duration of the single-tone signals and the AFC signal were both set at 50 ms. The AFC was set with a starting frequency of 10 kHz, a cutoff frequency of 17.5 kHz, and an interval frequency of 0.5 kHz, the frequency of the single-tone signal was set to 14 kHz, the sampling rate and the sampling number were set to 96 kHz. The conditions for two series of tests were set as follows:

1) Keep δ of the single-tone signals unchanged at 0.45, convert it to an equivalent Doppler factor, and gradually increase the input signal-to-noise ratio (SNR) from -2 dB to 10 dB. Observe the bias and the root mean square error (RMSE);

2) The number of sampling points was reduced to half of the previous, and other parameters remained the same as in experiment 1. Observe the RMSE with SNR.

The Cramer-Rao bound (CRB) can be used to indicate the best-estimated performance, although it is not typically applicable for biased estimators. It can be expressed by Equation 20

where f_s is the sampling frequency. The variance of the Doppler factor can be obtained through the variance transfer formula.

As shown in Figure 6 , the deviation of the true Doppler factor by the CUU method based on AFC does not exceed 1.1×10^-5 when δ is 0.45. In contrast, alternative frequency estimation methods have a minimal bias of 1.5×10^-5 and a maximal bias of 5.3×10^-5. It is noteworthy that the CUU method has a bias with a maximum value that is smaller than the minimum value observed in other frequency estimation methods.

Figure 7 demonstrates that at a frequency resolution of 1 Hz, the RMSE of the proposed method is no greater than 1.5×10^-5, significantly lower than 2.9×10^-5 observed with other frequency estimation methods. At a frequency resolution of 2 Hz, the RMSE for the proposed method is lower than 1.8×10^-5, better than 6.0×10-5 for other methods, as shown in Figure 8 . In both scenarios, the RMSE of the AFC-based method is closer to the CR bound, showing the high accuracy and robustness of the CUU method based on AFC.

In order to provide further validation for the effectiveness of the proposed method, Watermark, a widely accessible benchmark for physical-layer techniques in underwater acoustic communications, was employed. This benchmark is based on empirical measurements of the time-varying impulse response collected at sea (van Walree et al., 2017).

The Norway-Oslofjord (NOF1) channel with the large available bandwidth was selected to validate the practical efficacy of underwater communication. In the single-input single-output (SISO) scenario, the time-varying impulse response (TVIR) of the transmitted signal affected by the Doppler effect is obtained, and the signal packet is retrieved through serial acquisition processes. The signal has been established with a SNR of 20 dB, a pulse length of 50 ms, a frequency resolution of 1Hz, and other parameters consistent with the first Monte Carlo experiment in Section 4.1. Figure 9A demonstrates that due to the frequency-selective fading of underwater acoustic channels, the magnitude of each frequency component varies, while the AFC signal spectrum remains uniformly spaced. By calculating the Doppler shift based on frequency rather than phase, the impact of multipath effects is limited. Contrary to conventional methods that primarily focus on spectral peaks and their adjacent lines while disregarding the remaining spectrum, our proposed approach considers signals from different frequencies under the influence of the same Doppler effect, leading to distinct spectral peaks at various positions on the spectrum. Unlike the singular spectral peak observed in CW signal circumstance ( Figure 9B ), each spectral peak of the AFC signal effectively reflects the potency of the Doppler effect. The cumulative effect of multiple measurements leads to the estimation of the Doppler factor aligning closely with the true value, thereby enhancing the accuracy of estimation and resulting in improved estimation precision ( Figure 9C ). The difference of the CUU method utilizing the AFC signal is below 3.4×10^-6, notably lower than that of the 3.2×10^-5 induced by spectral leakage.

Table 2 shows the calculation scales of the methods. It can be seen that the computational cost of the proposed algorithm is lower than that of other frequency estimation methods for single-tone signals. The proposed method does not require additional complex multiplication and complex addition, and it achieves a better Doppler estimation performance. Essentially, this is because the CUU method makes full use of spectrum information.