SIGNALS AND SYSTEMS-VI PERIODIC (CIRCULAR) SHIFT OF SEQUENCES

EDITOR: B. SOMANATHAN NAIR

1. INTRODUCTION

We can introduce shift in periodic sequences. This is illustrated in Example 1.

Example 1: Figure 1 shows the sequence x(n). Using this sequence, perform the operations: (a)  x[((n ­-  1))₄]     (b) x[((n ­+ 1))₄]             (c)  x[((- n­))₄]

Solution: The symbol x((n)) is used to indicate that x(n) is a periodic sequence and the symbol x[((n­))_N],  read as x(n modulo N), indicates that x(n) is periodic and limited to N samples.

(a). The easiest way to solve problems of this kind is to use our tabulation method. Table 1 is prepared based on the data given. We tabulate the sequence in the second row. In the third row, we show x((n - 1)), which is obtained by shifting x((n)) to  the right by one cell. In the fourth row, we show x[((n - 1))₄], which is obtained by limiting x((n - 1)) to four places from n = 0 to n = 3.

 

(b) In Table 2, we show x((n)) in row 2 and x((n+1)), which is x((n)) shifted to the left by one cell, in row 3. In row 4, we show x[((n+1))₄].

(c) In Table 3, we show x((n)) in row 2 and x((‒n)), which is x((n)) inverted  in row 3. In row 4, we show x[((‒n))₄].