b) are presented in Fig. \ref{fig:p111}. The results show both methods for truncating the antenna without sharp ends successfully mitigate the dip in gain caused by the resonance resulting in an $\sim \text{3}$ dB improvement. However, the low frequency realized gain of the novel tip truncated sinuous antenna is lower. This is due to the reduction in antenna size by moving the truncation inward on the antenna which results in less usable bandwidth at low frequency. In order to maintain the same outer radius, and thus similar low frequency performance, as a traditionally truncated sinuous antenna, $\tau$ and $R_1$ must be appropriately scaled. 

\begin{figure}[]
	\centerline{\includegraphics[width=\columnwidth]{p111}}
	\caption{Comparison of co-polarized realized gain at low frequency for sinuous antennas ($\alpha=45^\circ$ and $\tau=0.75$) with different truncation methods.}
	\label{fig:p111}
\end{figure}

\section{Improved Sinuous Antenna}
Building on the results presented in Sections \ref{sec:res} and \ref{sec:trunc}, a sinuous antenna was designed in an effort to produce similar performance as the reference antenna illustrated in Fig. \ref{fig:1}b while mitigating both log-periodic and truncation related resonances. The value of $\alpha$ is chosen as 45$^\circ$ in order to prevent log-periodic resonances while the outer region is truncated by the method proposed in Section \ref{sec:novel}. The antenna in Fig. \ref{fig:trunc}a, $\alpha=\text{45}^\circ$ and sharp ends clipped, could have also been used to provide similar resonance mitigation; however, a fully self-complimentary design was desired. The new antenna is shown in Fig. \ref{fig:10}a and the design parameters are listed in Table \ref{tbl:improv}. Both $\tau$ and $R_1$ have been adjusted in order for the truncation radius to be equal to the outer radius of the reference design ($R_T = R_1^{ref} = \text{5 cm}$). In addition, the radius of the bowtie element feed is  kept to a constant 0.5 cm. The number of cells $P$ is increased by one in order to have the same number of full cells, once the circular truncation is applied to the outermost cell tip, as the reference antenna. From the results presented in Fig. \ref{fig:p107}, one might have based the improved design on the 12 cells antenna with 50$^\circ$ angular width; however, for comparison purposes, particularly in the time-domain, the improved design was kept as close to the reference antenna as possible.

\begin{table}
	\caption{Improved Sinuous Antenna Design Parameters}
	\setlength{\tabcolsep}{3pt}
	\centering
	\begin{tabular}{p{90pt}p{30pt}p{50pt}}
		\toprule
		Parameter & Symbol & Value(s) \\
		\midrule
		Number of Arms & $N$ & 4 \\
		Number of Sinuous Cells & $P$ & 9 \\
		Angular Width & $\alpha$ & 45$^\circ$\\
		Outer Radius & $R_1$ & 5.72 cm \\
		Outer Truncation Radius & $R_T$ & $5$ cm \\
		Inner Radius (feed) & $R_{in}$ & 0.5 cm \\
		Expansion Ratio & $\tau$ & 0.7628 \\
		Arm Sweep Angle & $\delta$ & 22.5$^\circ$ \\
		\bottomrule
	\end{tabular}
	\label{tbl:improv}
\end{table}

For further comparison, the reference antenna was trimmed according the methods proposed in \cite{kang2013experimental} (shown in Fig. \ref{fig:10}b) in order to mitigate the resonances. This required the removal of the sharp ends via clipping as well as a 20$^\circ$ trim of all the sinuous cell tips. This antenna is compared against the reference as well as the improved design.

\begin{figure}[!tbp]
	\centerline{\includegraphics[width=\columnwidth]{fig10-crop}}
	\caption{Sinuous antenna designs modified in order to mitigate the resonances (a) improved design, (b) ends trimming technique.}
	\label{fig:10}
\end{figure}

\subsection{Frequency Domain Analysis}
The simulated boresight co-polarized realized gains of the three antennas are shown in Fig. \ref{fig:p114}. The results show that the improved design successfully mitigates both the log-periodic and truncation resonances while providing gain of similar (sometimes higher) magnitude. In comparison, the trimmed design is also able to mitigate the resonances but the realized gain is not as smooth. Similarly, the group delay is significantly smoother with the removal of the resonances as shown in Fig. \ref{fig:p117}. The simulated match ($S_{11}$) of the antennas to an ideal 267 $\Omega$ port is shown in Fig. \ref{fig:p115}. The different designs all have $S_{11} \le -\text{10 dB}$ starting at approximately 1.5 GHz ($\pm\,\text{50 MHz}$) with the improved design having an overall better match than the trimmed version which is no longer self-complimentary due to trimming. 

\begin{figure}[]
	\centerline{\includegraphics[width=\columnwidth]{p114}}
	\caption{Comparison of co-polarized realized gain for traditional sinuous with resonances and modified versions that mitigate the resonances.}
	\label{fig:p114}
\end{figure}

\begin{figure}[]
	\centerline{\includegraphics[width=\columnwidth]{p117}}
	\caption{Comparison of group delay for traditional sinuous with resonances and modified versions that mitigate the resonances.}
	\label{fig:p117}
\end{figure}

\begin{figure}[]
	\centerline{\includegraphics[width=\columnwidth]{p115}}
	\caption{Comparison of match ($S_{11}$) for traditional sinuous with resonances and modified versions that mitigate the resonances.}
	\label{fig:p115}
\end{figure}

\subsection{Time Domain Analysis}
Resonances present in the gain of the sinuous antenna ultimately result in ringing when the antenna is used in pulsed type applications which can be particularly troublesome for close in sensing applications such as GPR. Examination of the radiated fields in the time domain is necessary to determine the extent of such ringing. Fig. \ref{fig:p108} shows the radiated pulses (time-shifted to start at $t=\text{0}$) from the three sinuous antennas when driven by an UWB pulse. The excitation pulse used was a Differentiated Gaussian which is defined by
\begin{equation}\label{eq:pulse}
v_{pulse}(t) = -v_{peak}\frac{(t - \mu)}{\sigma} \exp \left [ 0.5 - \frac{(t - \mu)^2}{2 \sigma^2} \right ], 
\end{equation}
where $\mu$ represents an arbitrary time shift and $\sigma$, defined as $2.3548/(2\pi f_{BW})$, controls the width of the pulse. In the presented analysis  $v_{peak}$ was set to 1 V and $f_{BW}$ to 7.5 GHz resulting in peak spectral energy at 3.2 GHz.

As shown in Fig. \ref{fig:p108}, the resonances produce high frequency ringing following the pulse. Mitigation of the resonances significantly reduces this ringing. The improved design has a slightly different pulse shape compared to the other designs which is due to the change in $\tau$. Both the improved and trimmed designs provide similar performance improvements i.e., less ringing. However, the improved design is able to achieve this while maintaining a self-complimentary structure.

\begin{figure}[]
	\centerline{\includegraphics[width=\columnwidth]{p116}}
	\caption{Comparison of radiated pulses (time shifted to zero) for traditional sinuous with resonances and modified versions that mitigate the resonances.}
	\label{fig:p108}
\end{figure}

\section{Conclusion}
In this paper, an analysis of the relationship between the sinuous antenna design parameters and the log-periodic resonances in the radiation was presented. It was determined that these log-periodic resonances may be eliminated by proper selection of the arm angular width $\alpha$. By choosing a value of $\alpha$ that reduces the interleaving of the arms, the log-periodic resonances were subsequently reduced. The optimal value of $\alpha$ was slightly impacted by the choice of expansion ratio $\tau$ but converged at approximately 45$^\circ$ for all the designs investigated. These results are corroborated by design guidance provided in \cite{duhamel1993frequency}. Mitigation of the log-periodic resonances by proper choice of design parameters provides advantages over other techniques proposed in the literature since the antenna remains self-complimentary and does not