Concept:
The DTFT of an aperiodic signal x(n) is defined as:
\(X\left( {{e^{j\omega }}} \right) = \mathop \sum \limits_{n = - \infty }^{ + \infty } x\left( n \right){e^{ - j{\rm{\omega }}n}}\)
Application:
\(h\left( n \right) = \left\{ {\frac{1}{3},\frac{1}{3},\frac{1}{3}} \right\}\)
\(H\left( {{e^{j\omega }}} \right) = \frac{1}{3} + \frac{1}{3}{e^{ - j\omega }} + \frac{1}{3}{e^{ - j2\omega }}\)
\(= \frac{1}{3}{e^{ - j\omega }}\left[ {{e^{j\omega }} + {e^{ - j\omega }}} \right] + \frac{1}{3}{e^{ - j\omega }}\)
\(= \frac{2}{3}{e^{ - j\omega }}\cos \omega + \frac{1}{3}{e^{ - j\omega }}\)
\(= \frac{1}{3}{e^{ - j\omega }}\left[ {1 + 2\cos \omega } \right]\)
At ω = ω0
\(H\left( {{e^{j{\omega _0}}}} \right) = 0\)
We can, therefore, write:
1 + 2 cos ω0 = 0
cos ω0 = -1/2
\(\cos {\omega _0} = \cos \frac{{2\pi }}{3}\)
\(\omega_o = \frac{{2\pi }}{3} = 2.09\;radians\)