# Performance Aspects of Retrodirective RFID Tags +
## 2 Materials and Methods  
In the current section, the spatial signature equipped, pinhole channel model for RF tags is studied. Backscatter radio systems operate in a channel that is a cascade connection of two conventional ones: the reader's transmitter tag and the tag reader's receiver channels. This is the so-called pin- (key-)hole two-way channel. Compared to the usual wireless channel, it exhibits deeper fades, thus requiring extra measures to be classified as reliable. Previous works have shown that these extra measures can be based on the diversity of the rich scattering environment. This, of course, holds when the tag antennas are uncorrelated, which is not the case when the tag is equipped with an array (phased) antenna. Now, the antenna elements are in close proximity to one another, and the pinhole diversity fails. In the current work, we show that even in this case, when the tag is armed with a (say, two-element) array antenna, a performance increase can still be achieved by properly designing the array's passive feeding network. Let us present the channel model. The forward part of the keyhole channel \(\mathbf{H}_{f}\) models the \(\text{N}_{\text{TX}}\)\(\text{Tx}\) antenna-initiated propagation to the \(\text{N}_{\text{TAG}}\) antennas of the tag. The backscatter part of the keyhole channel \(\mathbf{H}_{b}\) models tag-initiated propagation to the \(\text{N}_{\text{RX}}\)\(\text{Rx}\) antennas of the reader's receiver. The received signal is given as follows [12]:  
\[\mathbf{z}=\frac{1}{2}\mathbf{H}_{b}\cdot\mathbf{S}\cdot\mathbf{H}_{f}\cdot\mathbf{x}+\mathbf{n}. \tag{1}\]  
Here, we use time-invariant channel. In (1), the elements of the channel matrices are assumed to be complex Gaussian variables. Their real and imaginary parts are \(\sim\!\!N\!\left(0,\frac{\sigma_{\gamma_{\mathbf{z}}}^{2}}{2}\right)\). \(\mathbf{z}\) is \(N_{RXXt}\) column vector of the received baseband signal. \(\mathbf{H}_{f}\) is the \(N_{TAGX}N_{TTX}\) matrix of the forward channel, \(\mathbf{H}_{b}\) is the \(N_{RXX}N_{TAG}\) matrix of the backward channel, \(\mathbf{S}\) is the \(N_{TAGX}N_{TAG}\) tag's signaling matrix, \(\mathbf{x}\) is the transmitted \(N_{TX}xI\) column vector, and the noise vector \(\mathbf{n}\) is in the form of an \(\text{N}_{\text{RXX}1}\) column. \(\mathbf{S}\) is defined as the \(\text{N}_{\text{TAG}}\) port scattering matrix [13]. Here, this matrix is identified as the scattering matrix of the array antenna's passive feeding network. In general, dependent both on the circuitry and the antennas, this matrix has various distinct forms. For example, when no active circuitry is present (meaning we focus on passive tags) and each antenna elementary radiator of the tag array is terminated with its own load termination (no signals are intercoupled between the antenna elements), then this matrix is proportional to the \(\mathbf{I}_{N_{TAG}}\)\(N_{TAG}xN_{TAG}\) identity matrix as follows:  
\[\mathbf{S}=\Gamma\cdot\mathbf{I}_{N_{TAG}}, \tag{2}\]  
Figure 2: Comparison of omnidirectional and retrodirectional answers of the interrogated tag arrays: (**a**) the tag array response to omnidirectional radiation; (**b**) the tag array response to retrodirectional radiation. where \(\Gamma\) is a diagonal matrix with each diagonal element being the respective array antenna's reflection coefficient. Up to now, only this matrix has been used in applications. Nevertheless, if the tag array antennas respond to the incoming wave in a way in which the signals are intercoupled between the antennas, then the use of off-diagonal elements in the representation of the \(\mathbf{S}\) matrix is mandatory. This slight increase in analysis complexity compensate the designer with additional possible solutions. Indeed, an application in which all, not only the diagonal, elements of \(\mathbf{S}\) are employed is identified in the current work. Here, the scattering matrix is the one that forms a retrodirective array [14].