## 2 Research Methodology
### Adaptive Pulse Width Frequency Shifter  
In traditional backscatter systems, self-interference is a critical issue that needs to be addressed. Figure 2b illustrates an example of such an interference. When the reader's receiving end captures the modulated signal reflected by the tag, it also receives the signal transmitted from the reader, which occupies the same frequency band as the backscattered signal. Furthermore, the signal strength of the transmitted signal is significantly greater than that of the backscattered signal, resulting in strong interference. Figure 3 demonstrates the amplitudes of the received backscattered signal and the self-interference signal at the Rx. From the figure, it is evident that the self-interference signal is 30 dB stronger than the backscattered signal at Rx. Figure 2: Illustration of different backscatter systems. (**a**) Harmonic interference resilient backscatter system. (**b**) Traditional backscatter system. When the tag switches the RF switch on and off at a frequency \(f\), it effectively modulates the incident signal with a standard square wave signal \(S_{\text{tag}}(ft)\) through a multiplication operation, resulting in frequency spectrum shifting. The standard square wave signal can be represented by a Fourier series, as shown in Equation (1) [9]. Assuming that the incident signal received at the tag, represented as \(S_{\text{incident}}\), can be simplified to a sinusoidal form, \(r(t)\) in Equation (2) can describe the backscatter process in which the tag generates a standard square wave for frequency shifting. From Equation (2), we can observe that due to the presence of harmonic components in the standard square wave, the frequency-shifting process can unintentionally shift the frequency points of the incident signal to undesired locations. Notably, the harmonic components at \(f_{c}\pm 3f\) possess relatively significant amplitudes, which can lead to potential harmonic interference with nearby tags operating at other frequency points. \[S_{\text{tag}}\left(ft\right)=\frac{4}{\pi}\sum_{n=1,3,5...odd}^{\infty}\frac{ 1}{n}\sin(2\pi nft), \tag{1}\]  
\[ \begin{split} r(t)&=S_{incident}\times S_{\text{ tag}}\left(ft\right)\\ &=\sin(2\pi f_{c}t)\times S_{\text{tag}}\left(ft\right)\\ &=\frac{4}{\pi}\sum_{n=1,3,5...odd}^{\infty}\frac{1}{n}\sin(2\pi f _{c}t)\times\sin(2\pi nft)\\ &=\frac{2}{\pi}\sum_{n=1,3,5...odd}^{\infty}\frac{1}{n}\{\cos(2 \pi(f_{c}-nf)t)\\ &-\cos(2\pi(f_{c}+nf)t)\}.\end{split}  \tag{2}\]  
We propose using an adaptive pulse-width square wave \(S(t)\) for frequency shifting, which is composed of two different pulse widths, x and y, as illustrated in Figure 4. The Fourier series of the adaptive pulse-width square wave \(S(t)\) is presented in Equation (3). The coefficients \(a_{k}\) are presented in Equation (4). From Equation (4), it can be inferred that by altering the values of the pulse widths x and y, the amplitude of a specific harmonic component of the square wave can be adjusted. \[S(t)=\sum_{k=-\infty}^{+\infty}a_{k}e^{jk\omega_{0}^{\prime}t}, \tag{3}\]  
Figure 3: Received signal strength at Rx. (**a**) Power spectrum of received signals at Rx. (**b**) Power-level analysis of the received signals. \[a_{k} =\frac{1}{T^{\prime}}\int_{0}^{T^{\prime}}S(t)e^{-j\kappa\omega_{0}^{ \prime}t}dt \tag{4}\] \[=\frac{1}{T^{\prime}}\left(\int_{0}^{x}e^{-j\kappa\omega_{0}^{ \prime}t}dt+\int_{\mathrm{T}}^{y+T}e^{-j\kappa\omega_{0}^{\prime}t}dt\right)\] \[=\frac{e^{-j\kappa\omega_{0}^{\prime}x}+e^{-j\kappa\omega_{0}^{ \prime}(y+T)}-e^{-j\kappa\omega_{0}^{\prime}T}-1}{-j\kappa\omega_{0}^{\prime} T^{\prime}}.\]  
An adaptive pulse-width square wave can be obtained by performing an XOR operation on square waves with different duty cycles. Specifically, we can first simulate a triangular wave through the incremental and decremental operation of a counter, and we can then compare the dynamically updated value of the counter with an adjustable threshold to generate a square wave with a tunable duty cycle, as shown in Figure 5a.