How to find inverse Fourier transform of summ of delta functions?
Using Duality Property, we have $X(j\omega) = \delta(\omega-\omega_{0})$. By using this and rewriting our function using $1-2k = -\omega_{0}$, we get:
$$\begin{align}
x(t) &amp;= \frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(\omega-\omega_{0})e^{j\omega t}d\omega \\
 &amp;= \frac{1}{2\pi}e^{j \omega_0 t}
\end{align}$$
$$
\sum_{k=-\infty}^{\infty}\delta(\omega-\omega_{k})\leftrightarrow \frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{j\omega_{k}t}
$$
Hence:
$$
\sum_{k=-\infty}^{\infty}\delta(\omega+1-2k)\leftrightarrow \frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{j(2k-1)t}
$$