Ideal Low Pass Filter

$x[n] = \frac{\sin(2\pi f_c n)}{\pi n}, \quad f_c = 0.15 \\ X(f) = \begin{cases} 1, & |f| \leq f_c < \frac{1}{2}, \\ 0, & f_c < |f| \leq \frac{1}{2} \end{cases}$

time domain

frequency domain - real and imaginary parts

Note: $x[n]$ is real and $X(f)$ has Hermitian symmetry ( $\Re\{X(f)\}$ is even and $\Im\{X(f)\}$ is odd).

magnitude frequency response

Note: $x[n]$ is real and $|X(f)|$ is even.

log magnitude frequency response

wrapped phase response

unwrapped phase response