Like the and operators, the operator maps a length signal to a length signal:
Definition: The repeat
times operator is defined for any
by
where , and indexing of is modulo (periodic extension). Thus, the operator simply repeats its input signal times.^{7.11} An example of is shown in Fig.7.8. The example is
A frequency-domain example is shown in Fig.7.9. Figure 7.9a shows the original spectrum , Fig.7.9b shows the same spectrum plotted over the unit circle in the plane, and Fig.7.9c shows . The point (dc) is on the right-rear face of the enclosing box. Note that when viewed as centered about , is a somewhat ``triangularly shaped'' spectrum. We see three copies of this shape in .
The repeat operator is used to state the Fourier theorem
where is defined in §7.2.6. That is, when you stretch a signal by the factor (inserting zeros between the original samples), its spectrum is repeated times around the unit circle. The simple proof is given on page .