1. The Cepstral Bend on Fourier
The Fourier Transform is built for measuring ‘everyday’ oscillations;
In particular, periodicity in sine-waves,
in general periodicity in waveform,
indirectly, the underlying size of a periodic waveform in oscillation.
The Fourier Transform is sometimes pushed a little too far,
especially when the signals are not sinusoids, but,
sometimes by pushing the Fourier a little farther,
say,
into the Cepstrum domain,
It moves from being an indirect measure into
a very direct and insightful measure,
indeed,
the measure we are looking for.
The Cepstrum, is in effect, a cascade of two Fourier Transforms

2. A basic unit circle, with a continuous blue outline.
Our study has very little to so with continuity.
Our analysis is with sequences of number,
which are realistic, discrete, bounded,
and most importantly, periodic

3. It is key to know when to begin and stop a sequence.
Here we have 100 discrete points, actually, separated in space.
How close you want to get hose points simply is a matter of cost,
And of practicality. In any event, no matter how close we can get the points,
They will always be discrete and separated.
1st point on x=axis
We have pictured 100
discrete points around
the unit circle,
Counter-clock-wise
Last point just
before x-axis

4. Peeling off ‘continuous looking’ (x,y) coordinates will form the
‘continuous looking’ sine and cosine curves.
But we are not going to do that.
(x, y)’s

5. Just emphasizing the discrete and separateness of a realistic sampling of something
that we want to measure. Sometimes, what we want to measure appears continuous, like a
Mercury temperature gauge, and sometimes, what we want to measure is something
That is already discrete, like a reordering of the numbers 1 through 5, like {1, 4, 2, 5, 4}. .
(x,
y)’s
Here we are simply splitting the pairs of x, y coordinates into
the first 30 x’s in RED, and the first 30 y’s in Blue, counter-clock-wise.
The first coordinate on the graph to split, if we start on the x-axis, (green dot),
is (1,0) which is split by sending 1 to the red discrete curve, and 0 to the blue discrete curve.
For just about the remainder of this PPT, we will use the blue discrete curve.

6. Just getting use to seeing discrete points
Again, we will be focusing just on the blue sine curve
7th pt pair
1st pt pair

7. The period does not end at the starting point,
Unfortunately, like I use to draw willy-nilly in Math class
The first few starting
Sine curve points,
magnified at the
Beginning point
AT ZERO
The last few starting
Sine curve points,
Magnified at the
ending pt, just
BEFORE ZERO

8. Three sets of sine waves with their start stops accentuated.
The points are real, the continuous looking colored lines draw
through them, are not. They are inserted to actually emphasize the dots.

9. Sine curves, as we will use them, have to be exactly periodic, and,
Only 1 of 4 of the sinusoid looking curves is exact:
5 pts
too
long
4 pts
too
short
1 pt
too
Long
(sorry)
Just
Right
(by being
just right
below
the zero)

10. Possible and so ‘Probable’ quantum-sized molecular orbit
Begin
orbit
End
orbit

11. 100 points distributed over one cycle on TOP
100 points distributed over six cycles on BOTTOM
Note the ‘apparent’ spreading in the y axis direction, even though spacing
on the axis is the same.

12. We are going to look at a basis set of 10 Sine waves.
The purpose is to get a feel for the sine wave portion of the Fourier Transform
When teaching high school Trig, I included a last dot to ‘touch’ the X-axis. I don’t do that anymore.

13. What does a 1 Hz wave look like compared to other frequencies

14. What does a 7 Hz wave look like compared to other frequencies

15. Interesting is that with correlation, it is a cumulative sum, show left to right in the third plot, using the cumulative set of points from the second plot, and the Cumulative Sum, at the end, sums to 0, which says different frequency do not correlate, they are in orthogonal, in this measure

17. Correlating the same frequency adds up to a positive number

19. Now consider forming a 1 , 2, and 8 Hz signal

20. Overlap
Add them
Correlate the addition against the bank of frequencies 1 thru 10 and
only the 2nd ,3rd ,and 8th frequencies show evidence of presence. 3 is larger because
the wave was bigger in the previous slide.

21. Taking 6 Hz. If the waves were exactly periodic, then the correlations should be zero when correlating on anything but 6 Hz. However, the correlation is NOT zero, when we correlate any time, no matter how small before or after the exact period, we don’t get zero. That’s why finding the true period of a wave form is important. Anything else is only an approximation. That’s it.

22. Now, correlating a 6 Hertz signal outside of its true period still is relatively high
compared to the other frequencies, but but cause the waveform extending beyond
The true period, this plot says that, we really don’t have a 6 Hz wave, there are other
Close by waves that are needed to reconstruct this extended (and non-periodic) wave.

24. Part 2
The Cepstrum

25. Example 1: The basics of Cepstrum, but with a fabricated signal, NOT a physical signal.
Starting with what is known as the sampling function, the Shah function, the Comb function, here we
60 timing points with one of 6 combing bristles every 10 points.
2. Then, in the Fourier domain, The Fourier transform of (1), we get an inverse relation,
60 frequency points with one of 10 combing bristles every 6 points.
And then, in the Cepstrum domain, The Fourier transform of (2), the outcome appears to reflect back
to (1), but these are known as Cepstral components, and they indicate the spacing between the
samples on (1). In a sense, (3) is actually misleading although it is correct. Example 2 is better, in a way. 1. 2. 3.

26. Example 2: The basics of Cepstrum, but with a (real) physical signal.
Yet, the obvious repetition could be taken as a repeated fabrication of a single base waveform, but it isn’t.
The Fourier transform and the Cepstrum, as shown in the next slide, will not be overtly like
the Combing function, but it is, in smaller pieces. Note the Dots.

27. The waveform with 2350 points sampled.
The Fourier Transform of (1)
The Cepstrum domain of (1) or the Fourier Transform of (2).
The combing in 2 and 3 is not spread across the plots, but it is in pieces, as shown in next slide
As we zoom in on the Fourier tranform and the Cepstrum

28. The original signal with the 10 RED arrows indicating the starting point of a basic waveform spread across 2350 pts.
The Zoomed in Fourier Transform indicates there are indeed 10 cycles of a base waveform and there is combing
effect seen at 20, and ~30 and ~ 40.
The Zoomed in Cepstrum indicates various bumps but the comb bristles we are interested in will relate to
the ratio 2350/10 == 235, and its first multiple of 470 as indicated. Return to previous slide to see fuller comb.
235
470

29. Tying Examples 1 and 2 together, we have (1) our waveform which exhibits a series of periodic base
waveforms of size 235 samples found by using the Cepstrum. Some of the waveforms are overlaid in (2).
THUS The waveform in (1) can be thought as a single base waveform in (2) being centrally placed on top
of the combing bristles in (3) spaced 235 points apart. 1. 3. 2.