Filtering out negative frequencies Is it actually possible to filter out the negative frequencies for the baseband signal is we take into consideration that the filter's impulse response is symmetric around the zero frequency?
I'm not sure if you ask about symmetry of the impulse response or symmetry of the frequency response. All filters with a real impulse response have conjugate symmetric frequency responses and symmetrical magnitude frequency responses. A symmetrical magnitude frequency response would affect the magnitudes of both frequencies $\omega$ and $-\omega$ identically so it could not possibly remove or attenuate a negative frequency while keeping the positive frequency intact. So the impulse response has to be complex.

The discrete impulse response $b_k$ of an ideal do-nothing filter is:

$$b_k = \frac{\int_{-\pi}^{\pi}e^{i\omega k}d\omega}{2\pi} = \left\{\begin{array}{ll}1 &amp;\text{ if }k = 0\\\frac{\sin(\pi k)}{\pi k}&amp;\text{ if } k \ne 0\end{array}\right.\\ = [\dots, 0, 0, 0, 1, 0, 0, 0, \dots]$$

The middle value $1$ is at $b_0$. Changing where the integral starts gives the ideal negative frequency removal filter:

$$b_k = \frac{\int_{0}^{\pi}e^{i\omega k}d\omega}{2\pi} = \left\{\begin{array}{ll}1/2&amp;\text{ if } k = 0\\\frac{\sin(\pi k)}{2 \pi k} + i\left(\frac{1 - \cos(\pi k)}{2\pi k}\right)&amp;\text{ if }k \ne 0\end{array}\right.= [\dots, 0, -\frac{i}{5\pi}, 0, -\frac{i}{3\pi}, 0, -\frac{i}{\pi}, \frac{1}{2}, \frac{i}{\pi}, 0, \frac{i}{3\pi}, 0, \frac{i}{5\pi}, 0, \dots]$$

The middle value $1/2$ is at $b_0$. The real part of the impulse response is zero at other values of $k$. The imaginary part has odd symmetry with respect to $\operatorname{Im}(b_0) = 0$. It is plain to see from the integral that the frequency response is zero for frequency $\omega &lt; 0$ and constant ($1$) for $\omega &gt; 0$.