Shift Operator

The shift operator is defined by

$$\hbox{\sc Shift}_{\Delta,n}(x) \isdef x(n-\Delta)$$

and $\hbox{\sc Shift}_{\Delta}(x)$ denotes the entire shifted signal. Note that since indexing is modulo $N$, the shift is circular (or ``cyclic''). However, we normally use it to represent time delay by $\Delta$ samples. We often use the shift operator in conjunction with zero padding (appending zeros to the signal $x$) in order to avoid the ``wrap-around'' associated with a circular shift.

Figure: Successive one-sample shifts of a sampled periodic sawtooth waveform having first period $[0,1,2,3,4]$.

Figure 7.2 illustrates successive one-sample delays of a periodic signal having first period given by $[0,1,2,3,4]$.

Example: $\hbox{\sc Shift}_1([1,0,0,0]) = [0,1,0,0]\;$ (an impulse delayed one sample).

Example: $\hbox{\sc Shift}_1([1,2,3,4]) = [4,1,2,3]\;$ (a circular shift example).

Example: $\hbox{\sc Shift}_{-2}([1,0,0,0]) = [0,0,1,0]\;$ (another circular shift example).