## 2 Proposed Method
### System Model  
This paper considers a diffusion-based molecular system that comprises two nanomachines, i.e., a transmitter and a receiver, and a fluid propagation channel. To transmit information, the transmitter emits a single type of molecules. The emitted molecules propagate through a diffusion-based fluid medium under the action of Brownian motion. The receiver absorbs and decodes the molecules according to the time-varying molecular concentrations.  
We assumed that \(M\) bits were available to send a symbol with equal time interval \(T_{e}\) during symbol duration \(T_{s}\), i.e., \(T_{s}=MT_{e}\). Let \(d_{i}=\{b_{0},b_{1},\cdots,b_{M-1}\}\)\((i=0,\cdots,\infty)\) be the \(i\)-th symbol, which is a binary information sequence in which \(b_{k}\in\{0,1\}\)\((k=0,\cdots,M-1)\). The transmitter emits an impulse of \(Q\) molecules to send a bit \(b_{k}=1\). However, no molecules are released to send \(b_{k}=0\). The transmission distance between the two nanomachines is denoted as \(r\). According to [PERSON]'s law of diffusion [20], the molecular concentration measured at the receiver at time \(t\) is expressed as follows:  
\[f(t)=\frac{Q}{(4\pi Dt)^{\frac{3}{2}}}\exp(-\frac{r^{2}}{4 Dt}), \tag{1}\]  
where \(D\) is the diffusion coefficient. According to (1), the concentration at a point at fixed distance \(r\) from the transmitter reaches its peak at \(t_{m}=\frac{r^{2}}{6D}\), and the peak value of \(f(t)\) is \(f_{\text{max}}=\left(\frac{3}{2\pi t}\right)^{1.5}\frac{Q}{r^{3}}\). After sending the \(k\)-th bit in \(d_{i}\), the molecular concentration measured by the receiver at time \(t\)\((KT_{e}\leq t<(k+1)T_{e})\) becomes  
\[c(t)=\sum_{j=0}^{k}b_{j}[f(t-jT_{e})+n_{j}(t)], \tag{2}\]  
where \(n_{j}(t)\) represents the counting noise induced by the \(j\)-th emitted molecules, as modelled by the Brownian noise on the basis of random molecular motion. Many studies have assumed that peak time \(t_{m}\) remains constant, which implies that when an initial molecular impulse is emitted, the receiver always detects the signal and decodes it after \(t_{m}\) has elapsed. However, a constant detection time cannot be maintained due to propagation delays or random molecular movements under time-varying channel conditions. A small error at the start of the detection adversely affects the detection of the signals. Moreover, if the system parameters are unknown, the value of \(t_{m}\) is not available to the receiver. Therefore, to correctly decode the information, an estimation of the detection time is necessary.  
Figure 1 shows the concentration of the received molecules at various distances \(r\) and molecules \(Q\). Figure 1a shows that the concentration inversely decays with the transmission distance. The peak time when the concentration reaches its maximum becomes slower with the distance. In addition, the concentration level at the peak time sharply decreases with the distance, which limits the signal decoding performance. Figure 1b shows that the peak time was independent of \(Q\), but the concentration level at the peak time decreased as \(Q\) decreased. In other words, the changes in peak time and maximal concentration level were affected by the system parameters.