# Signal Detection for Ambient Backscatter Communication with OFDM Carriers
## 2 System Model and Problem Formulation
### Overall System Architecture  
The overall system architecture utilizing the AmBC over OFDM carriers is illustrated in Figure 1. In this system, we consider two communication components coexist: the legacy OFDM system and the AmBC system. In the legacy OFDM system, the RF source transmits OFDM signals to its legacy users, while in the AmBC system a backscatter tag transmits its modulated signals to the reader over ambient OFDM carriers from the RF source. Note that this tag is equipped with a switch that can split the received signal into two parts: information decoder (ID) and energy harvester (EH). Assume that they are all connected to a single antenna and use the same RF signals. The RF-powered passive tag communicates with the reader by switching its antenna impedance of its backscattered signals. The energy harvester collects the energy from the ambient OFDM signals and uses it to provide a small amount power required for the communication and performing tasks at the tag. Finally, the backscattered signal is received and decoded by the reader.  
Mathematically, the RF source transmit a passband signal \(\bar{s}(n)=\text{Re}\left\{\sqrt{P_{s}}s(n)\exp\{j\omega 2\pi f_{c}\frac{n}{f_{s}} \}\right\}\), where \(s(n)\) is the equivalent complex baseband signal with unit power, \(P_{s}\) is the average transmit power, \(f_{c}\) is the carrier frequency, and \(f_{s}\) is the OFDM bandwidth. The tag receives the RF source signal and transmit its modulation signal \(\bar{c}(n)\) to the reader. When we add the CP, the ambient OFDM signals are converted to a serial form and transmitted through a wireless channel. Suppose that the channel impulse response of a multipath channel is modeled as a finite impulse response filter with a certain number of taps. We denote \(N_{st},N_{tr},\) and \(N_{sr}\) as the number of taps corresponding to \(h_{st}(n)\), \(h_{tr}(n)\), and \(h_{sr}(n)\), respectively. Here, we define the maximum delay of the multipath channels as  
Figure 1: A communication system utilizing the AmBC over OFDM carriers. The AmBC system consists of three main components: RF source (e.g., TV tower), ambient backscatter transmitter (e.g., AmBC tag), and ambient backscatter receiver (e.g., reader), while the legacy OFDM system consists of several legacy receivers (e.g., mobile phones).  
\(L\triangleq\max\{N_{sr},N_{st}+N_{tr}-1\}\). Let \(N\) be the number of subcarriers of OFDM signals \(s(n)\) and \(N_{cp}\) be the CP length. In order to gain perfect timing alignment and frequency synchronization at the receiver side, we assume that the maximum delay spread of the channel is less than the length of the CP, i.e., \(L\ll N_{cp}\). Theoretically, increasing the number of OFDM subcarriers \(N\) leads to larger delay spread. With a fixed available bandwidth, it may cause the frequency mismatch problem between two neighbor subcarriers because the subcarrier spacing is very small. On the other hand, the number of OFDM carriers should be proportional to the CP length in practice. Thus, a tradeoff between the CP length and the subcarrier spacing must be obtained for a reasonable design.  
Let \(x(n)\) be the tag data signal, thus, the received signal at the reader is given as  
\[y(n)=\underbrace{\big{[}\eta c(n)x(n)\big{]}h_{tr}(n)}_{y_{n}(s):\text{ received backscattered}}+\underbrace{\sqrt{P_{s}}s(n)h_{sr}(n)}_{\text{from the $\text{RF source}$}}+w(n), \tag{1}\]  
where \(w(n)\sim\mathcal{CN}(0,\sigma^{2})\) and \(\eta\) are the noise and the signal attenuation parameter inside the tag, respectively.