Single Real Pole

$x[n] = a^n u[n] = \begin{cases} a^n, & n \geq 0, \\ 0, & n<0\end{cases}, \quad a=0.9 \\ X(f) = \frac{1}{1-a e^{-j2\pi f}}$

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

Note: x[n] is real and $\angle X(f)$ is odd.

unwrapped phase response