# Estimate of Passive Time Reversal Communication Performance in Shallow Water
## 1 Introduction  
Underwater acoustic communication presents a challenging problem arising from the environmental properties of underwater acoustic waveguides. Acoustic waves propagating through shallow-water channels experience multiple interactions with the sea surface and ocean bottom. This causes a significant, time-varying delay spread called ISI (inter-symbol interference) [1]. ISI results in serious distortion of communication signals and makes demodulation difficult [2]. Thus, the ISI causes significant degradation in communication performance [3; 4; 5; 6]. In addition, the temporal variations in communication channels resulting from sea surface movement and source/receiver movements result in a short coherent time and large Doppler spread, which degrade communication performance [7; 8; 9]. Up to the 1990s, a channel equalization process was mainly used in coherent acoustic communication to compensate for the ISI [2; 10; 11; 12]. Furthermore, a single-channel receiver was extended to the multichannel receiver system to improve the communication performance. However, there is a significant shortcoming that means multichannel equalization requires the high computational complexity [5; 12; 13]. In the early 2000s, the time reversal technique was introduced as an alternative to the equalization process [14; 15; 16]. Although the time reversal technique also employs an array of spatially separated receivers, it is relatively simple, effectively reduces the ISI in a multipathenvironment, and restores the original communication signal, thereby increasing the signal-to-noise ratio (SNR). Recently, passive time reversal processes have been widely used in underwater communication due to the simplicity of the system, in which an array is required only at the receiver [17; 18; 19]. Passive time reversal is related to the reciprocity of the active time reversal. Thus, the two methods are theoretically equivalent except that the direction of the communication link is reversed [19]. Recently, passive time reversal has been applied on a block-by-block basis and was combined with a phase lock loop (PLL) to overcome the difficulties caused by the time-varying characteristics of the channel [20; 21]. The performance of the time reversal can be characterized by the behavior of the \(q\)-function, which is the sum of the autocorrelation of each channel impulse response (CIR) for each receiver [18; 19]. The passive time reversal \(y(t)\) is expressed by the equation below. \[y(t) = s(t)*\left[\sum_{i}^{N}h_{i}(t)*h_{i}(-t)\right], \tag{1}\]  
Here, \(s(t)\) is a signal transmitted from a source. \(h_{i}(t)\) and \(h_{i}(-t)\) are the CIR at the \(i\)-th receiver and its time reversed form, respectively. \(N\) is the number of receivers. The expression inside the bracket is defined as the \(q\)-function. In general, the \(q\)-function depends on the complexity of the communication channel, the number of receivers in the array, and their spacing [19]. In an ideal case in which the estimated CIR has infinite bandwidth and no multipath, the \(q\)-function becomes a delta function, and the time reversal process recovers the exact original communication signal [14; 15; 16; 17; 18; 19]. However, in a real ocean waveguide, the \(q\)-function is no longer the delta function, but has a main lobe and side lobes, producing a residual ISI of time compressed signal. As the energy of the side lobes becomes larger or the width of main lobe becomes wider, the bit error rate (BER) of the communication signal after the time reversal process tends to increase. Therefore, if the residual ISI is not negligible even after the time reversal process, additional equalization processes, such as an adaptive decision feedback equalizer, are needed to eliminate the residual ISI. Such properties of the \(q\)-function affecting communication performance were investigated by [PERSON] [18]. However, the parameter describing quantitatively the relation between the \(q\)-function and communication performance was not proposed in his work.