# Pilot-Assisted OFDM for Underwater Acoustic Communication
## 3 System Architecture
### Pilot-Based Channel Estimation
#### 3.1.1 OFDM Transceiver  
A typical CP-OFDM transceiver model working in a UWA channel is shown in Figure 1. Let \(B\) represent the vector of a serial binary bit stream where each element of \(B\) is either a binary 0 or 1. This binary stream is fed into the OFDM transmitter. A serial to parallel converter, represented by S/P in the figure, converts the serial stream into a parallel stream that has \(N\) symbols where \(N\) is the number of subcarriers.  
Each parallel stream of \(N\) samples makes one OFDM symbol where each element of \(N\) consists of \(\log_{2}M\) bits. \(M\) is the modulation order to be used in the next block, i.e., the mapper. For example, if the mapper is BPSK, each symbol at the output of the S/P block will have only one bit. On the other hand, if the mapper is QPSK, each symbol will consist of two bits. The parallel stream of \(N\) symbols then passes through the mapper and is converted into complex numbers based upon the type of mapper used. Let these complex numbers be represented by a vector \(C_{k}\) where \(k\) is the \(k\)th OFDM symbol and each element of the vector \(C_{k}\) is a complex number. Mathematically, this can be shown as:  
\[C_{k}=\left[c_{k,0},\ c_{k,1},\ c_{k,2},\ldots.c_{k,N-1}\right]^{T}. \tag{1}\]  
In the proposed OFDM system, pilots are inserted at predetermined intervals into each OFDM symbol. This is achieved by using certain subcarriers as pilots. Further details of these pilots will be described later in the paper. The parallel stream of complex numbers then passes through the inverse fast Fourier transform (IFFT) that transforms the complex numbers into another set of complex numbers that are orthogonal to each other. Let this new stream of complex numbers be represented by \(X_{k}\), which can be shown mathematically as:  
\[X_{k}=[x_{k,0},x_{k,1},\ \ldots,\ x_{k,N-1}]^{T}. \tag{2}\]  
The P/S block then converts the parallel stream of complex numbers into a serial stream. To combat the effects of a multipath, a cyclic prefix--denoted by CP--is appended to each OFDM symbol before it is converted into an analog signal for the final transmission.  
Figure 1: A typical CP-OFDM transceiver with a channel estimation using pilots.  
If \(T\) represents the duration of the OFDM symbol and \(f_{p}\) the frequency of the \(p\)th subcarrier, then the transmitted signal can mathematically be expressed as:  
\[x_{k}(t)=1/\sqrt{T}\sum_{p=0}^{N-1}c_{k,p}e^{j2\pi f_{p}t}. \tag{3}\]  
In this equation, \(\sqrt{T}\) is a scaling factor. After passing through the UWA channel, the signal is processed by the receiver in reverse order, as shown in Figure 1. Let \(\hat{x}(t)\) be the received signal that is a noisy and distorted version of the transmitted signal. This signal is converted into a digital signal by the A/D block. In the next step, the cyclic prefix is removed. The S/P block coverts the serial stream into parallel streams of \(N\) symbols and the fast Fourier transform (FFT) is applied to each parallel stream. As the signal has been distorted by the UWA channel and noise has been added to it, the complex numbers at the output of the FFT can mathematically be represented as:  
\[\hat{C}_{k}=H_{k}C_{k}+n_{k}. \tag{4}\]  
In the above equation, \(\hat{C}_{k}\) is a vector that represents the noisy and distorted version of the transmitted vector \(C_{k}\). \(H_{k}\) is the channel transfer function affecting the \(k\)th OFDM symbol and \(n_{k}\) is the additive white Gaussian noise affecting the \(k\)th OFDM symbol. The pilots are extracted next and used by the equalizer to neutralize the effects of the UWA channel and estimate the transmitted complex numbers. The de-mapper block coverts the estimated OFDM symbol of complex numbers into digital data and the P/S block converts the parallel stream of digital data into a serial stream.