Exercises 1.10 2003/07/24 08:40:01 GMT-5 2004/01/19 06:10:40.583 US/Central Anders Gjendemsjo gjendems@NO-SPAM.tele.ntnu.no Anders Gjendemsjo gjendems@NO-SPAM.tele.ntnu.no Exercise Problems Signals >Exercises to TTT4110: Information and Signal Theory, Signals Problems related to the Signals chapter. Find the digital frequency of x n 2 3 n . Is the signal periodic? If so, find the shortest possible period. Write 2 3 n as 2 f n , where f is the digital frequency. We see that the digital frequency is 3 . For a trigonometric signal to be periodic the digital frequency has to be a rational number, i.e 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. Find the digital frequency of x n 2 4 n . Is the signal periodic? If so, find the shortest possible period. Write 2 4 n as 2 f n , where f is the digital frequency. We see that the digital frequency is 4 2 . For a trigonometric signal to be periodic the digital frequency has to be a rational number, i.e 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 2 1 , hence the signal is periodic. The period, N, is given by N m f m 2 . Since N has to be an integer, we obtain the shortest possible period letting m 2 , which yields N 1 . Find the digital frequency of x n 2 1.5 n . Is the signal periodic? If so, find the shortest possible period. Write 2 1.5 n as 2 f n , 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 m f 2 m 3 . Since N has to be an integer, we obtain the shortest possible period letting m 3 , which yields N 2 . Referring to example 2 find the analog and digital frequency of x1 t and x2 n respectively. 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.