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On the Signal, Spectrum and Some Fractal Properties of Video-Feedback

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On the Signal, Spectrum and Some Fractal Properties of
Video-Feedback
I.N. Galidakis
University of Crete
Knossos av., Ampelokipoi,
Heraclion 71409, Greece
jgal@math.uoc.gr
1
Contents
1 Abstract 3
2 Introduction 3
3 Simple OELFO 3
3.1 Signal ...................................... 3
3.2 Spectrum .................................... 5
4 Modulation 5
5 Amplification 7
6 Perturbations 8
6.1 CentralObstruction .............................. 9
6.2 Zooming..................................... 9
6.3 Rotational.................................... 12
6.4 Energy...................................... 12
6.4.1 Darkness ................................ 12
6.4.2 Light................................... 12
7 Bibliography 13
8 Appendix 14
2
Figure 1: Simple OELFO
1 Abstract
We derive the signal and calculate the spectrum of an Opto-Electronic Loop-Feedback
Oscillator, commonly known as Video-Feedback oscillator. Additionally we calculate
several parameters of some of the more common fractal optical phenomena which manifest
in such oscillators. Finally, we investigate several perturbation modes and their associated
functions.
2 Introduction
Opto-Electronic Loop-Feedback Oscillators (OELFO) are special oscillators which rely on
optical feedback ([15]). Examples of people who have explored such oscillators are given
in [8]. An overall description of the required setup is given in [2] and simulation examples
with Matlab code are given in [1]. Examples were made popular as early as 1979 in [3,
pp. 490-492]. A gallery of images is shown in [4]. Simulations and several references are
shown in [5]. One complex example was shown on [14]1. Several similar experiments with
such OELFOs were performed in 1990 and 2002 by the author while he was a student and
continue to this day. This thesis examines the signal and spectrum of an OELFO and
explains some of the more common general optical behavior of such systems as chaotic
and fractal, by modelling a simple OELFO.
3 Simple OELFO
A simple OELFO is shown on Fig. 1. It consists of a camera and a monitor which displays
the image captured by the camera and is the simplest type of CCTV ([12]). The circuit
becomes an oscillator when the camera is pointed at the monitor.
This OLEFO has several vibrational modes: A mode which corresponds to the monitor
refresh rate ([16]), approximately at 75 Hz, and a mode which corresponds to the frequency
of the alternating current which powers it ([11]), approximately at 50 Hz. For simplicity
these modes will be ignored as they do not affect the optical behavior of the system.
3.1 Signal
Let us now denote the distance between camera and monitor as AB. The first signal then
travels from Bto Aat v=cand the EM signal travels back from Ato Bapproximately
at v= 2c/3 when using a conventional coaxial-cable RG-59/U ([13]).
Light covers the distance AB in time t1=AB/c, while the EM signal covers the same
distance (via cable) in time t2=AB/(2c/3) = 3AB/(2c). It follows that the signal is
periodic with period 2l=t1+t2=AB/c(1 + 3/2) = 5AB/(2c). Ignoring the vertical
refresh rate of the monitor for simplicity, the oscillator signal is then given by,
1The example shown on this movie is analyzed in section 5.
3
Figure 2: OELFO in equilibrium
s(t) = (c, if 0 tt1,
2c
3, if t1< t t1+t2.(3.1)
Signal (3.1) is associated with an optical signal which manifests on the monitor as
an infinite array of self-engulfing monitor images (Fig. 2, Appendix Fig. 10), because
the resulting (initial) image of the monitor is being fed back into the camera. The
characteristic feature of this manifestation is the ratio λof the tangent of the camera
angle θ1divided by the tangent of the monitor subtended angle θ2at the distance AB
between camera and monitor:
λ=tan θ1
tan θ2
(3.2)
It follows that if the main monitor has radius R1, then the radii Rn,n > 1 of the
images of these monitors in this tunnel on the main monitor, satisfy the relationship:
Rn
Rn+1
=λ,n1 (3.3)
The radii of the images of the monitors then form a geometric progression with com-
mon ratio λ1, hence,
Rn+1 =R1
λn,n1 (3.4)
The progression Rnof monitor images is therefore a fractal. Using Fig. 2,
tan θ2=R1
AB (3.5)
It follows then from equation (3.2),
λ=AB ·tan θ1
R1
(3.6)
Since λis measurable from the monitor image using any two successive pairs Rn,
Rn+1 from the infinite geometric sequence of images, θ1can be solved for in the previous
equation, since AB and R1are known.
Setting fλ(Rn) = Rn=Rn+1, note that when λ > 1, equation (3.3) implies that the
images of the monitor suffer a contraction through the map fλ, hence the entire sequence
converges uniformly on compact subsets to a fixed point R, which depends on the value
of λ.
4
3.2 Spectrum
Signal (3.1) can now be analyzed. s(t) and s(t)+2c/3 will have the same Fourier series
modulo the dc-term2, so we can replace the signal with the signal,
s(t) = (5c
3, if 0 tt1,
0 , if t1< t t1+t2.(3.7)
The Heaviside unit step function ([10]) is defined as,
H(x) = (1 , if x0,
0 , otherwise.(3.8)
Remembering that t1=AB/c and t2= 3AB/(2c), signal (3.7) can be written in terms
of H.
s(t) = 5c
3H(t) (1 H(tt1)) (3.9)
The signal’s period is 2l=t1+t2= 5AB/(2c), so the Fourier coefficients as given in
[7], with s(t) as in (3.9), evaluate with Maple ([6]) as:
a0=1
lZ2l
0
s(t)dt =4c
3
an=1
lZl
l
s(t) cos µnπt
ldt =5csin ¡4
5¢
3
bn=1
lZl
l
s(t) sin µnπt
ldt =5c¡cos ¡4
5¢1¢
3
(3.10)
We note that the coefficients are independent of AB. The corresponding Fourier series
([9]) will then be:
F(t) = a0
2+
X
n=1 ·ancos µnπt
l+bnsin µnπt
l¶¸ (3.11)
Signal (3.7) and its Fourier series F25(t) are shown on Fig. 3.
Setting cn=pa2
n+b2
n, we find the amplitude of the harmonics as,
cn=5cq22 cos ¡4
5¢
3(3.12)
The signal’s spectrum for AB = 1 and c= 3 ·108m/s is then shown on Fig. 4. The
oscillator broadcasts at a frequency f= 1/(t1+t2)1.2·108Hz, or 120 MHz (which is
in the VHF range) and its dominant harmonic has an amplitude of 3 ·108. All harmonics
of order 5n,n∈ {1,2,3, . . .}are zero.
4 Modulation
The OELFO can be modulated, by using configurations with Ncamera-monitor pairs.
The configuration for N= 3 is shown on Fig. 5. The configuration loops back to itself, so
each camera Cifeeds its signal to the corresponding monitor Mi. Monitor MNis watched
by camera C1.
2Note that if F(t) is the Fourier series of the signal f(t), then F(t) = F(t) + Cis the Fourier series
of the signal f(t) + Cwhere Cis a constant.
5
Figure 3: OELFO signal f(t) (red) and Fourier series F25(t) (green)
Figure 4: OELFO signal spectrum for AB = 1 and c= 3 ·108m/s
6
Figure 5: N= 3 modulated OELFO
Without loss of generality, we may assume that all distances between cameras and
monitors are equal to AB. It is clear in this case that the system will have a period
2l=N·(t1+t2) = N·5AB/(2c) or Ntimes the period of a N= 1 OELFO.
If s(t) is the signal of the N= 1 OELFO given in (3.7), it follows that the signal of
the N-modulated OELFO will be given by:
sN(t) = s(Nt) (4.1)
It then follows that if F(t) is the Fourier series of the signal s(t), the Fourier series
for sN(t) will be given as:
FN(t) = a0
2+
X
n=1 ·ancos µNnπt
l+bnsin µNnπt
l¶¸ (4.2)
Hence, the Fourier coefficients anand bnof the N-modulated signal sN(t) will be
identical with those of the signal s(t). So the N-modulated OELFO has the same spectrum
as the N= 1 OELFO.
The optical behavior of the OELFO for this configuration will manifest again as an
infinite geometric progression of self-engulfing monitors, which will now consist of N
geometric sub-progressions interleaved, with the interleaving happening on every N-th
image in the original progression. Therefore, if Rnare the images of any specific monitor
Ri,i∈ {1,2, . . . , N }on a given monitor, these images will satisfy:
Rn+1 =Ri
λNn ,n1 (4.3)
Note that the period of the N-modulated OELFO can be continuously varied by
altering the distances ABibetween monitor Miand camera Ci. For general ABi’s then,
the period of the signal will be:
2l=5
2c
N
X
i=1
ABi(4.4)
5 Amplification
The OLEFO can be amplified using the configuration shown on Fig. 6. There are N
camera-monitor pairs and each camera feeds a specific monitor and each camera “views”
all Nmonitors simultaneously.
7
Figure 6: N-amplified OELFO
Assuming again without loss of generality that the distance between cameras and
monitors is AB, it follows that the period of the Namplified OELFO is 2l=t1+t2=
5AB/(2c).
The spectrum of the N-amplified configuration is then identical to that for the N= 1
OELFO, but with one important difference: Each pair of camera-monitor broadcasts with
intensity3I1, but in this configuration there are Nbroadcasters, hence the broadcasting
intensity of the N-amplified OELFO will be N-times the broadcasting intensity of the
N= 1 OELFO4:
IN=N·I1(5.1)
As in the previous section (4), the period can be continuously varied by altering the
distances ABibetween monitor Miand camera Ci. For general ABiin this case, the
period will be given as:
2l=5
2cLCM (ABi: 1 iN) (5.2)
Note that if the ABiare different5, the system signal will need to be analyzed sepa-
rately, by considering an appropriate superposition of all the component signals within
the period 2l.
The N-amplified configuration can be extended in two dimensions, with N=Nh×Nv
cameras stacked as a rectangle of hhorizontal and vvertical cameras, all pointing at the
Nmonitors, arranged similarly. For simplicity one may assume that there is only one
row of cameras and one row of monitors, stacked horizontally.
The optical behavior of the OELFO in this case will again manifest as an infinite
sequence of self-engulfing monitors of common ratio λ, with an interesting twist: Each
monitor image will display Nmonitors inside it (instead of only one) stacked horizontally,
giving a total of Ninfinite geometric progressions.
Depending on where one looks on the main monitors, those Nsequences will subdivide
and lead to the appropriate self-engulfing monitor sequences.
6 Perturbations
Various types of perturbations may be introduced in the OELFO. Depending on the type
of the perturbation, the system engages in a specific behavior.
3Or power per unit area.
4Assuming the source is spherical, broadcasting intensity falls off as 1/r2, where ris the distance from
the source. This means for example that a N= 10 amplified OELFO at a distance of r=10 3.16m,
broadcasts with the same intensity a N= 1 OELFO broadcasts at the distance of r= 1m.
5If the ABiare not integral, write ABi=ai/bi. Write M= LCM(bi: 1 iN) and ni=Mai/bi.
Then, the period will be given as 2l= 5/(2c) LCM(ni: 1 iN)/M .
8
Figure 7: Central Obstruction
6.1 Central Obstruction
The first type of perturbation is a central obstruction OH, which varies either as A
OBor as 0 OH R1(Fig. 7). In the first case it is easy to see that the critical
distance AO where the OELFO will start oscillating will be such that,
OH
AO =R1
AB = tan(θ2) (6.1)
The OELFO’s signal will then be:
sO(t) = (s(t) , if OH
AO <tan(θ2),
0 , if OH
AO tan(θ2).(6.2)
Knowing the height of the obstruction OH,AO is immediately estimated as a function
of the obstruction height OH.
In the second case, if one keeps AO constant and varies 0 OH R1, the resulting
signal is identical to (6.2).
The optical behavior for these two cases is fairly simple: When OH/AO tan(θ2)
the monitor will simply display an image of the obstructor and nothing else. When
OH/AO < tan(θ2), the monitor will display the usual infinite geometric sequence of
self-engulfing monitors and additionally will display infinitely many images Onof the
obstructor, which will satisfy the following relations:
Rn+1
On
=tan(θ2)
¡OH
AO ¢=µ, 1 µtan(θ2)
¡OH
AB ¢=R1
OH (6.3)
In other words, the obstructor images on screen Onare related to the monitor images
Rnas,
On=Rn+1
µ,n1 (6.4)
Rnis a geometric progression, therefore so is On.
6.2 Zooming
Zooming perturbations occur when the user engages the lens-zoom. In cameras, typically
this zoom is indicated as a magnification range 1XM X . A magnification of M, amounts
to shortening AB as AB0=AB/M for the visual signal. Hence the communication time
t1changes to t0
1=t1/M, but t2is left unchanged. This changes signal (3.7) as follows:
sM(t) = (5c
3, if 0 tt0
1,
0 , if t0
1< t t0
1+t2.(6.5)
9
n M1M2M3
1 1
2 2
310
33.33
410
91.11 14
34.66
514
91.55 6
6 2 22
37.33
76
51.222
92.44 26
38.66
Table 1: Zoom values Mgiving harmonic maxima for n= 1 to n= 7
The period then changes from t1+t2= 5AB/(2c) to t0
1+t2=AB(1/M + 3/2)/c.
The new signal’s Fourier coefficients are found using Maple and are given as,
aM
0=20c
3(2 + 3M)
aM
n=
5csin ³4
2+3M´
3
bM
n=
5c³cos ³4
2+3M´1´
3
(6.6)
The Fourier coefficients of the signal are again independent of AB, but depend on
the zooming factor M, hence zooming affects the OELFO’s spectrum. When M= 1 the
coefficients reduce to the Fourier coefficients aM
n=anand bM
n=bnof signal (3.7).
Setting again cM
n=qaM
n
2+bM
n
2, we find the amplitude of the harmonics as,
cM
n=
5cr22 cos ³4
2+3M´
3(6.7)
Therefore, when zooming occurs, the OELFO will broadcast at different frequencies.
On Fig. 8 we show the variation on the amplitude of the first seven harmonics as a
function of a zooming factor varying continuously from M= 1 to M= 10. Solving the
equation dcM
n/dM = 0 for M, we obtain the maxima shown on (6.8) for n > 1. Some of
these maxima are shown on Table 1.
M=2
3µ2n
2k+ 1 1, 0 k < n (6.8)
This perturbation manifests optically again as an infinite progression of self-engulfing
monitors as in section 3.1, but now λwill have the value,
λ=³tan(θ1)
M´
tan(θ2)=³tan(θ1)
M´
¡R1
AB ¢=tan(θ1)
R1
(AB
M)
=tan(θ1)
¡R1
AB0¢(6.9)
When M= 1, λ > 1. Because λis a monotone decreasing function of Mwith limit 0,
there will be a value M=M0>1, such that λ= 1. This M0satisfies,
tan(θ1)
M0
=R1
AB (6.10)
10
Figure 8: Amplitude variation for the first seven harmonics, as a function of continuous
zoom, 1 M10
Increasing M > M0results in λ < 1. In this case the oscillator will experience a
transition from discreetness to continuity and chaotic optical phenomena will start taking
place, because the infinite sequence of monitors reverses its orientation as a self-engulfing
sequence and a continuous vortex of fractal patterns which are the residuals of the light
signal of the sequence monitors will form an infinite array of patterns. Combined with
rotation, these patterns will be spiral like (Appendix, Fig. 12), with the spiral nature
described in the next section. Occasionally, in this mode, new and unknown patterns
emerge (Appendix, Fig. 15).
Figure 9: Rotating the OELFO camera clockwise by an angle φ
11
6.3 Rotational
Rotating the OELFO camera by an angle φclockwise while it points at the monitor, is
equivalent to rotating the first monitor image by φcounterclockwise. This will then cause
the infinite geometric sequence of self-engulfing monitors to rotate counterclockwise, each
image by φ. If again, Rnis the n-th image on the monitor sequence then,
(Rn+1, Rn) = φ,n1 (6.11)
This means that
(Rn, R1) = (n1)φ,n1 (6.12)
Calling φthe phase angle, if one simultaneously zooms to M > M0(see section 6.2)
while the phase angle is φ, one generates galaxy-like spiral patterns, whose number of
arms will be a function of φAdditional details about the rotation mapping are given in
[2] (Appendix, Fig. 11).
These phenomena can again be explained as in section 3.1. Setting fφ,λ(Rn) = Rn=
Rn+1, when λ > 1, the images of the monitor suffer a contraction through the map fφ,λ,
hence the entire sequence converges uniformly on compact subsets to a fixed point R,
which depends on the value of λand φ(Appendix, Fig. 16).
Rotational perturbations combined with zooming perturbations will induce chaotic
“motion”. Such motion takes place when a secondary vibration takes place, one which de-
pends on information being communicated in all the corridors of the infinite self-engulfing
sequence of monitors.
In the case of both zooming and rotation, the motion vector field is described by the
red vectors on Fig. 9. These vectors will change direction when M > M0.
6.4 Energy
Energy perturbations are created when the OELFO is already in a state of transmitting
at a specific frequency, either in rotational or zooming mode. The energy is external and
can be distinguished into “darkness” and “light”. Darkness is a very brief interruption of
the signal. Light is the introduction of an additional light source into the signal.
6.4.1 Darkness
A quick darkness perturbation takes place when an obstructor moves quickly radially
across the visual field of the camera, at some point blocking the signal completely as in
6.1. When this happens, the signal is interrupted, but after the obstructor moves out
of the view field the OELFO eventually re-stabilizes with a new vibrating signal and
comes again to equilibrium. The central obstruction provides the feed for the bulk of the
manifest optical phenomena, which this time feed on the shape of the obstructor and the
time it takes to move across the visual field (Appendix, Fig. 13).
6.4.2 Light
A quick light perturbation takes place when an additional light source moves across the
visual field of the camera. When this happens, the light will travel down the infinite
corridor of self-engulfing tunnels, and the OELFO will again re-stabilize when the signal
from the additional light source is visible in all corridors of the tunnel. If the OELFO is
in a sensitive state, a new vibration will start, one which creates chaotic patterns out of
the new source light, using it as a dynamics seed (Appendix, Fig. 14).
12
7 Bibliography
References
[1] T. Burt and M.G. Lagoudakis. Non-Linear Dynamics of Video Feedback, 1987.
[2] J. Crutchfield. Space-Time Dynamics in Video Feedback. Physica, pages 191–207,
1984.
[3] D. Hofstadter. odel, Escher, Bach: An eternal golden braid. Vintage Books, 1980.
[4] Peter Henry King. Video Feedback Fractal Genesis. Website, 2004.
http://www.sweetandfizzy.com/fractals/diagrams.html.
[5] Jason Rampe. Softology - Video Feedback. Website, 2004.
http://softology.com.au/videofeedback/videofeedback.htm.
[6] D. Redfern. The Maple Handbook. Springer-Verlag, 1996.
[7] P.I. Rentzeperis. Introduction to Fourier Analysis. Aristotelian University of Thes-
saloniki, 1967.
[8] Unknown. The Ultimate Video Feedback Page. Website, 2004.
http://www.videofeedback.dk/World/.
[9] E.W. Weisstein. Fourier Series. Website, 2004.
http://mathworld.wolfram.com/FourierSeries.html.
[10] E.W. Weisstein. Heaviside Step Function. Website, 2004.
http://mathworld.wolfram.com/HeavisideStepFunction.html.
[11] Wikipedia. Alternating Current. Website, 2004.
http://en.wikipedia.org/wiki/Alternating current.
[12] Wikipedia. Closed Circuit Television. Website, 2004.
http://en.wikipedia.org/wiki/CCTV.
[13] Wikipedia. Coaxial Cable. Website, 2004. http://en.wikipedia.org/wiki/Coaxial
cable.
[14] Wikipedia. The Matrix Reloaded. Website, 2004. http://en.wikipedia.org/wiki/The
Matrix Reloaded.
[15] Wikipedia. Optical Feedback. Website, 2004. http://en.wikipedia.org/wiki/Optical
feedback.
[16] Wikipedia. Refresh Rate. Website, 2004. http://en.wikipedia.org/wiki/Refresh rate.
13
8 Appendix
The following photographs display some of the chaotic optical phenomena described in
the previous sections. They were all taken while the OELFO was oscillating in various
modes. Induced rotary motion or secondary light-intensity motion are not visible, because
these take place in real time.
Figure 10: OELFO in equilibrium
Figure 11: OELFO in rotational equilibrium
Figure 12: OELFO in rotational-zooming equilibrium
14
Figure 13: Darkness perturbation in OELFO in rotational-zooming mode
Figure 14: Light perturbation in OELFO in rotational-zooming mode
Figure 15: Unknown chaotic patterns in OELFO in rotational-deep-zooming mode
Figure 16: Fixed contractive/rotational point in OELFO in rotational-zooming mode
15
... HS(t) is called a Harmonic Phasor Sum and such sums always generate a periodic signal 2 . The period of such a signal is given in [2] 3 and is T = LCM(T j : 1 ≤ j ≤ n). The latter is defined because the T i are commensurable. ...
... 6. 2 The Eigenvalues of C in R 2 ...
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Introduction Feedback is one of the most direct routes to complicated nonlinear dynamics and chaos in practical applications. Sometimes the feedback process is beneficial, e.g. the mechanism by which the human retina adjusts to a range of illumination conditions. Other times of course the process is detrimental, for instance washing out some kind of signal we want to isolate or amplify. Using very simple equipment paired with computer modeling, we have specifically studied the dynamics of video feedback, which is important in contexts ranging from neural systems and image processing to graphics and art. Our observations are consistent with Crutchfield's findings from 1984 (see [1]), and we have furthermore demonstrated similar behavior using computational models. 2 Equipment and Procedure In actuality, a very short list of basic items makes it possible to study a wide range of simple video feedback experiments: TV, video camera (or digital came
Video Feedback Fractal Genesis
  • King Peter Henry
Peter Henry King. Video Feedback Fractal Genesis.
Softology -Video Feedback. Website
  • Jason Rampe
Jason Rampe. Softology -Video Feedback. Website, 2004. http://softology.com.au/videofeedback/videofeedback.htm.
Introduction to Fourier Analysis. Aristotelian University of Thessaloniki
  • P I Rentzeperis
P.I. Rentzeperis. Introduction to Fourier Analysis. Aristotelian University of Thessaloniki, 1967.
  • E W Weisstein
E.W. Weisstein. Fourier Series. Website, 2004. http://mathworld.wolfram.com/FourierSeries.html.
Heaviside Step Function. Website
  • E W Weisstein
E.W. Weisstein. Heaviside Step Function. Website, 2004. http://mathworld.wolfram.com/HeavisideStepFunction.html.