Write cos2π3n 2 3 n as cos2πfn 2 f n , where ff is the digital frequency. We see that the digital frequency is 3 3 . For a trigonometric signal to be periodic the digital frequency has to be a rational number, i.e f=mN f m N , where both m,N are integers. N is the signal period. Here the digital frequency is not a rational number, hence the signal is not periodic.

Write cos2π4n 2 4 n as cos2πfn 2 f n , where ff is the digital frequency. We see that the digital frequency is 4=2 4 2 . For a trigonometric signal to be periodic the digital frequency has to be a rational number, i.e f=mN f m N , where both m,N are integers. N is the signal period. In this case the digital frequency is a rational number, f=21 f 2 1 , hence the signal is periodic. The period, N, is given by N=mf=m2 N m f m 2 . Since N has to be an integer, we obtain the shortest possible period letting m=2 m 2 , which yields N=1 N 1 .

Write sin2π1.5n 2 1.5 n as sin2πfn 2 f n , where ff is the digital frequency. We see that the digital frequency is 1.5. The digital frequency is a rational number(3/2), hence the signal is periodic. The period, N, is given by N=mf=2m3 N m f 2 m 3 . Since N has to be an integer, we obtain the shortest possible period letting m=3 m 3 , which yields N=2 N 2 .

Using the same reasoning as above we easily see that the analog sine has frequency 1, while the discrete time sine has digital frequency 1/20.