We can picture periodic sequences as having discrete
points on a circle as the domain
Shifting by mm,
fn+m
f
n
m
, corresponds to rotating the cylinder
mm notches ACW (counter
clockwise). For
m=2
m
2
, we get a shift equal to that in the following
illustration:
To cyclic shift we follow these steps:
1) Write
fn
f
n
on a cylinder, ACW
2) To cyclic shift by mm, spin
cylinder m spots ACW
fn→f
((
n
+
m
))
N
→
f
n
f
((
n
+
m
))
N
If
fn=01234567
f
n
0
1
2
3
4
5
6
7
, then
f
((
n

3
))
N
=34567012
f
((
n

3
))
N
3
4
5
6
7
0
1
2
It's called circular shifting, since we're moving
around the circle. The usual shifting is called "linear shifting"
(shifting along a line).
f
((
n
+
N
))
N
=fn
f
((
n
+
N
))
N
f
n
Spinning NN spots is the same
as spinning all the way around, or not spinning at all.
f
((
n
+
N
))
N
=f
((
n

(
N

m
)
))
N
f
((
n
+
N
))
N
f
((
n

(
N

m
)
))
N
Shifting ACW mm is equivalent to
shifting CW
N−m
N
m
f
((

n
))
N
f
((

n
))
N
The above expression, simply writes the values of
fn
f
n
clockwise.