1.6 Exercises

Write $\cos (2\pi \sqrt{3}n)$ as $\cos (2\pi fn)$ , where $f$ is the digital frequency. We see that the digital frequency is $\sqrt{3}$ . For a trigonometric signal to be periodic the digital frequency has to be a rational number, i.e $f=\frac{m}{N}$ , 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 $\cos (2\pi \sqrt{4}n)$ as $\cos (2\pi fn)$ , where $f$ is the digital frequency. We see that the digital frequency is $\sqrt{4}=2$ . For a trigonometric signal to be periodic thedigital frequency has to be a rational number, i.e $f=\frac{m}{N}$ , where both m,N are integers. N is the signal period. In this case the digital frequency is a rational number, $f=\frac{2}{1}$ , hence the signal is periodic. The period, N, isgiven by $N=\frac{m}{f}=\frac{m}{2}$ . Since N has to be an integer, we obtain the shortest possible period letting $m=2$ , which yields $N=1$ .

Write $\sin (2\pi \times 1.5n)$ as $\sin (2\pi fn)$ , where $f$ 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 $N=\frac{m}{f}=\frac{2m}{3}$ . Since N has to be an integer, we obtain the shortest possible period letting $m=3$ , which yields $N=2$ .

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