Signal Processing - November 2017 - 162

4

4

2

2

0

0

-2

-2

is small while the phase shift i (t) and
i (t + Dt) is large, which makes the
curve round.

Mystery curve in 3-D

-4
-4

-2

0

2

-4
-4

4

-2

0
(a)

2

4

-2

0
(b)

2

4

4

FIGURE 8. Another mystery curve made by polar
form, as shown in mysterycurve_made2.m [3].

2
8,000
0

6,000
4,000

-2

2,000
-4
-4

0
-40 -20

0

20

40

60

FIGURE 9. The spectrum of the complex signal
c(t) shown in Figure 8.

FIGURE 10. (a) Preserving the lowest two bins
in Figure 9. (b) Preserving the lowest five bins.

to Figure 8, but there are only five nonzero
terms in the frequency response.
From the view of the polar form, we can
also notice that the x (t) = r (t) cos (i (t))
and y (t) = r (t) sin (i (t)) are actually
AM-FM signals, which are widely used
in speech analysis [8]-[10] and other
areas [11], [12]. In particular, the r (t)
is called the AM part and di/dt is the
instantaneous frequency that corresponds to the FM part. In Figure 11(b),

an AM-FM signal is given. The envelope
[AM part, r (t) ] is given in Figure 11(a).
The mystery curve corresponding to the
AM-FM signal is given in Figure 11(c).
This mystery curve, compared with the
other curves mentioned in this article, is
less sharp. The reason is that the AM signal has a slow changing envelope (r (t))
and a rather fast phase (i (t)) variation.
In other words, in a small amount of time
Dt, the difference of r (t) and r (t + Dt)

3.5

Like the case in 2-D, the mystery curve
in 3-D can be defined by time shift
and rotation operation. Recall that, by
the Euler rotation theorem, any combination of three dimension rotations
is equivalent to a single 2-D rotation
[13] about an axis. So, without loss
of generality, we can assume the axis
to be the z-axis. Therefore, the x (t) and
y (t) must satisfy the same constraints
as the 2-D case, and z (t) is a periodic
signal with period 2r/N. For example,
Figure 12(a) is a 2-D mystery curve,
and if we put it into 3-D by setting
z (t) = 1 + cos (5t), the result is shown
in Figure 12(b). The code is given in
mysterycurve_3D.m [3].
Because the rotation axis is z, it
is easy to generate a 3-D fivefold symmetric curve whose shape looks like a
vase or pineapple, as we can observe
in Figure 12(c), which contains only the
color red, and Figure 12(d), which is distinguished by five colors.
Another example is given in Figure 12(e) and (f). In Figure 12(e),
we illustrate a threefold 3-D mystery curve whose rotation axis is not
z. To show the symmetry, we color
the curve in red, green, blue (RGB)
and provide another view angle in
Figure 12(f ). The code is given in
mysterycurve_3D2.m [3].

4

4

2

2

0

0

-2

-2

3
2.5
2
1.5
1
0.5

0

1,000 2,000 3,000 4,000 5,000 6,000
(a)

-4

0

1,000 2,000 3,000 4,000 5,000 6,000
(b)

-4
-4

-2

0
(c)

2

4

FIGURE 11. (a) The AM part r (t) = 2 + 1.5 cos (3t) . (b) The AM-FM signal r (t) cos (i (t)) where r (t), indicated as the blue dash line, is the same as (a) and
i (t) = 11t + 3 cos (3t) . (c) The mystery curve corresponding to the same r (t) and i (t) . The code is given in myster_AMFM.m [3].

162

IEEE SIGNAL PROCESSING MAGAZINE

|

November 2017

|
Table of Contents for the Digital Edition of Signal Processing - November 2017
