Periodic Functions
A DT signal x[n] is called periodic if there exists an integer N such that x[n+N] = x[n] for all n.
A CT signal x(t) is called periodic if there exists an integer T > 0 such that x(t+T) = x(t).
x(t) = $ e^{n*j*w*t} $ can be a periodic function. The period is $ (2 * pi) / w $. If $ w / (2 * pi) $ is a rational number then the exponential function is periodic.
Periodic Function:
$ e^{(n* j * pi / 6)} $ , has w =$ pi / 6 $ therefore: $ w/(2*pi) = 1/12 $ which is a rational number, thus proving that the function is periodic.
Nonperiodic Function:
$ e^{(n* j / 6)} $ , has w =$ 1 / 6 $ therefore: $ w/(2*pi) = 1/(12*pi) $ which is not a rational number. Accordingly the function is nonperiodic.
