One of the cross-media communication cores involves demodulating the underwater sound source frequency from the WSAW. Hence, this section proposes and elaborates on an improved RELAX algorithm to estimate the WSAW spectrum.
3.1. Algorithm Theory
The RELAX algorithm estimates the target signal parameters through a nonlinear least square process, affording robustness. Assuming that the received signal has zero mean Gaussian white noise, the underwater sound source sends
signals of different frequencies. After phase extraction, the phases obtained by demodulation include
signals of different frequencies and random noise.
where
is the sampling point,
is the
complex amplitude, and
is the
frequency. The noise
is complex Gaussian noise with zero mean and variance of
.
From the nonlinear least squares criterion:
where
and
.
We estimate the amplitude and frequency of the
WSAW, and the remaining signals, including the
WSAW signal, can be expressed as:
By minimizing Equations (8) and (9) simultaneously, we obtain the estimated parameter values of the
WSAW signal:
The convergence condition of Equation (11) is that the difference between the value and the value of Equation (9) is less than a given threshold .
When the side lobes of the strong signals are higher than the main lobes of the weak signals, the RELAX algorithm results will be wrong, because the result of RELAX subtracts the estimated maximum value of each cycle without suppressing the side lobe signal. For example, the underwater sound source transmits four different frequencies, i.e., 100 Hz, 130 Hz, 180 Hz, and 300 Hz signals, and the RELAX algorithm fails to detect the 300 Hz signals in
Figure 2. Since the WSAW is a non-stationary signal, the periodogram and the RELAX algorithm are unaware of the time information, and therefore the underwater sound source information cannot be effectively parsed. To solve these problems, we propose the following two improvements.
First improvement: when the measured signal contains some significant signals, the strong signal side lobe overwhelms the weak main lobe signal, affecting the estimation accuracy of the WSAW characteristic parameters.
Windowing is an effective method to suppress side lobes, which, however, brings two problems simultaneously: main lobe broadening and peak drop. The former affects the estimation accuracy, and the latter affects the gain, leading to a significant error during the next iteration. Adding a window can significantly suppress the side lobes and increase the weak signal parameter estimation accuracy. Let
be the coefficient of the corresponding window function. Then the normalized amplitude gain is defined as:
where
is the signal compensation gain,
is the number of sampling points, and
is the sampling moment.
Due to the peak drop problem, the amplitude signal estimation is inaccurate when calculating the error function in Equation (9), significantly affecting the subsequent parameter estimation. Thus, the core idea to overcome this concern is to use the gain of the window function to compensate for the attenuation of the signal peak and to modify Equations (10) and (11) to obtain:
where
is the diagonal matrix of window function coefficients
.
Second improvement: The above analysis applies to stationary signals, but the received signals are cyclostationary, comprising several stationary signals. Additionally, the signals must be processed hierarchically in advance and divided into different sections according to their strengths to achieve accurate frequency estimation.
The received signals are cyclostationary, and the same frequency signal part is stationary. Therefore, determining the stationary signal interval of each same frequency part is necessary. Given that the data is segmented, we calculate the generalized inner product statistics of each data unit sample and compare them against the background statistical characteristics of the entire data to find the strong and weak signal interval points. This is important as the latter points better affect the interval data division of different regions. The specific algorithm flow is presented below:
First, the number of iterations is set., which decreases as the signal SNR increase. To achieve the best possible performance, this work considers 100 iterations.
Next, the generalized inner product is used to calculate the Mahalanobis distance and classify the interval of the different amplitude signals. The adjacent point difference relies on , and a new sequence is obtained that is multi-point stratified with a length of , which in turn is used to obtain the sequence .
Then, the difference of the signal sample sets corresponding to the two frequencies is obtained through the
data and after calculating the generalized inner product. After that, the
variation coefficient is calculated, and the average value and the standard deviation of the coefficient are combined to estimate the interval.
Finally, after obtaining the interval of the different amplitude signals, the improved one is used to estimate the parameters of the signals in each interval, to obtain the underwater sound source parameters
:
where
and
are the WSAW amplitudes, and
and
are the WSAW frequencies.