Download Circular Shift Operator: Delaying and Circularly Shifting Signals and more Study notes Computer Security in PDF only on Docsity!
Circular Shift Operator
Richard Baraniuk
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License †
Abstract
Introduction to the circular shift operator.
The concept of a time shift or delay is crucial to DSP.
Example 1
y [n] = x [n − 2], which means "delay x by 2 time units" (Figure 1).
Figure 1: N = 8.
There are now 2 issues:
- What to put in the rst two positions in the graph of y (Figure 1)?
- What to do with values that slide o the end of y (Figure 1)?
A solution: stu the values that slide o the end into the beginning (Figure 2).
Figure 2
∗ Version 1.2: Jun 15, 2004 11:59 am GMT- † http://creativecommons.org/licenses/by/1.
1 Notes
This is equivalent to putting x on a wheel/circle with N ticks and spinning it 2 ticks. x = (1, 1 , 1 , 1 , 0 , 0 , 0 , 0)
T
(Figure 3).
Figure 3: N = 8.
To delay x by 2 units, spin the wheel 2 ticks counter-clockwise and read o y. y = (0, 0 , 1 , 1 , 1 , 1 , 0 , 0)
T
(Figure 4).
Figure 4
So, we call y a circular shift of x.
This is also equivalent to viewing x as one period of an in nite-length periodic vector xp.
x =
xp =
We can then shift xp 2 units and read o y
y =
i.e.: we can view a C N signal as one period of a periodic signal.
Finally we can write y in terms of x using modulo arithmetic.
y [n] = x [(n − 2) modN ] (1)
for 0 ≤ n ≤ N − 1.
3.1 Notes
- Cm is a "permutation" matrix.
- Columns of Cm are shifted δ basis.
Indeed...
Cm =
δ[m] mod N
δ[m+1] mod N
... δ[m+N −1] mod N
Example 4
C 3 for N = 5. Apply to x = (1, 2 , 3 , 4 , 5)
T (Figure 6).
Figure 6
4 Aside on Circular Shifts
Cmx circularly shifts the column vectorx down m units.
Example 5
C 1
x
T Cm
T circularly shifts the row vectorx
T right m units.
Example 6
C 1
T
