# Effects of Fractional Time Delay as a Low-Power True Time Delay Digital Beamforming Architecture
## 3 Design  
Using the system and design constraints discussed in Section 2, a minimum length VFD lowpass FIR filter was developed to provide a large bandwidth (4 GHz) with near-constant group delay throughout the passband. The minimum allowable group delay deviation for a given filter occurs where the functional delay of a +0.5 sample delay filter becomes negative, and \(-0.5\) sample delay filter becomes positive. At this point, implemented element delays across the array begin to increment in a sawtooth fashion rather than linearly, adding significant quantization sidelobes. This undesirable group delay characteristic for \(\pm 0.5\) sample delay filters is shown in Figure 2a for a 5-tap filter design, with the sign change point denoted by the arrow. Therefore, it is necessary to ensure this behavior occurs outside of the desired passband. Through simulation analysis, it is found that designing a system with a maximum 40% group delay variation produces minimal additional sidelobe level, and will meet system requirements. Parameters used for simulation are outlined in Table 1.  
In order to maintain flat response throughout the passband (\(\pm\)0.1 dB maximum) and minimize attenuation at the edge of the passband, the VFD architecture utilizes a \(-6\) dB cutoff frequency of 47% of the sampling frequency, \(F_{s}\). The minimum length filter that satisfies these requirements is found to be 29, as shown in Figure 2b. With this \(N=29\) filter design, the maximum group delay variation was determined to be 3.8%, well within the 40% range deemed allowable. Therefore, for the VFD filter to successfully operate in LEO applications it must have a length of \(N=29\) and cutoff frequency of 0.47\(F_{s}\). Additionally, the VFD filter's impulse response utilizes a Chebyshev window with \(-60\) dB sidelobes to minimize mainlobe width while also providing an equiripple passband [27]. Although such large sidelobes are typically undesirable, decreasing the sidelobe level further has the consequence of increasing mainlobe width and reducing the high-frequency resolution of the system. This window choice will also serve to minimize effects of the Gibbs phenomenon, as discussed previously. In order to meet the \(ENOB\) requirement of 5.65 bits, 5 fractional bits were needed to provide a \(-38\) dB sidelobe floor and meet the general LEO system requirements. This 5-bit system would produce 32 discrete fractional time delay steps, providing acceptable time delay resolution while minimizing implementation size. A final block diagram of the \(N=29\), 5-bit system is shown in Figure 3 with key elements including a variable length First In First Out (FIFO) for coarse integer delays and the VFD filter for fractional delays. A detailed block diagram of the highlighted portions \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Sampling Frequency (\(F_{s}\)) & 10 GHz \\ \(-6\) dB Cutoff Frequency (\(F_{c}\)) & \(F_{s}\times 0.47\) \\ Functional Bandwidth & \(F_{s}\times 0.4\) \\ Signal Sample Length & 2048 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Figure 2 simulation parameters. Figure 2: (**a**): Group delay response for an FIR filter of \(N=5\). Solid trace represents 0 fractional delay and dashed traces represent \(\pm\)0.5 sample fractional delay. Arrow denotes sign reversal of the implemented fractional delay. Black dashed line denotes desired functional bandwidth. Red dashed line denotes filter cutoff frequency. (**b**): Left: filter group delay for a +0.5 sample filter for a range of N. Right: range of fractional group delay values from \(-\)0.5 to +0.5 samples for a range of N.  
in Figure 3 implemented in Simulink is shown in Figure 4. Simulations of the system were performed using a 35-element ULA with \(\frac{\lambda}{2}\) element spacing at a 4 GHz maximum frequency. Beam weights were windowed with a Hamming window to suppress sidelobes an equal amount in the spatial domain [27].