Professor Sampler’s Notes: The Shannon-Nyquist Sampling Theorem Part II

Topics Covered:

  • Time-domain to frequency-domain – a different kind of thinking

Mighty is geometry; joined with art, resistless – Euripides

I realized one fatal flaw in my reasoning. I realized, even if took a million samples, I might miss a house or more if they happened to fall between two samples. How do I get all of them? It dawned upon me the only way to know for sure was to know for sure. With as much courage as I could muster, I asked the only guy who knew: Hertz himself. He was mildly curious as to why I was interested, and flattery satisfied his curiosity.
Postman Hertz's typical path in Wolfcrow
What I instantly noticed was that things were a lot more complicated than previously assumed. For instance, Hertz never walked at the same rate throughout. His speed changed over different terrain. It was fast over the road, and slowed down as he climbed steps or walked on dirt.

It was too difficult to keep track of Hertz’s speed when it was constantly changing – accelerating and decelerating. I couldn’t ethically plant a speed device to his shoe, could I? After all, I’m not Shutterbug. So I decided to reverse engineer the whole thing.

The answer came to me while I was listening to the local news. “What steps were taken…” “Is stepping into his new role…” “Step up the pressure…” “I lost 50 pounds with Exxprexxstep…”

Steps! That was one way to get a handle on things. Here’s how:

I asked Hertz to walk over a small distance, and then measured the length of each step. I also asked Hertz to walk as fast as he could over the same distance to measure his fastest speed. Here are the results:

Length of each step = 2 feet = 0.61 meters
Maximum speed Hertz can reach = 2 feet per second or 0.61 m/s

Now I could confidently plot Hertz’s path this way:
Basic sine-like wave
Whoa! How did I do that? Simple. Imagine the first hill as representing a step by the right foot. The foot rises and then falls again. The second hill is inverted, and below the line, because it represents the left foot. I could very well have done each step this way:
Bump Style
But then how do we distinguish left foot from right? So let’s stick with the former representation, since it’s easier on the eyes. One left and one right make a pattern, or wave feature. If we had used the ‘bump’ graph instead, each step would be a wave feature. It is important to grasp the meaning of the distinction, since it is humans who ultimately decide what a wave feature is. This decision also influences how we measure the wave, i.e., its wave function, wavelength, frequency, etc.

One delightful side-effect of this particular problem is, if you look at Hertz’s path ‘map-wise’, you’ll see a constant (continuous) line from start to finish – a perfect analog signal. On the other hand, if you think of his path in terms of steps, it suddenly transforms itself into a discrete (different steps) signal! You didn’t think I happened to come across this example by accident, did you?

Remember how Fourier transforms were useful in reducing complex analog signals into frequency-domain signals that are easier to get a handle on? Fourier mathematics claims every such signal is transformable! That’s why it’s such a big deal. It’s like a boot that fits every foot in the world, and still never goes out of fashion!

We can now say that Hertz takes one step per second at his fastest, or one cycle (two steps, a round trip or wave feature) per two seconds. From our earlier explanation of frequency, we know that this means that Hertz is walking at a frequency of 0.5 Hertz (Hz)! If we calculate our steps without considering left or right feet, we could easily say Hertz is walking at 1 Hz too. Both are valid systems, and choosing which system to follow is usually dependent on the characteristics of that system, so don’t get confused.

In our system we decided to keep left and right facing different directions and our wave pattern (Remember that each wave pattern must be repeated for it to have a frequency) is two hills instead of one. Each wave pattern is two steps, right and left feet together, and the fastest Hertz can travel is 0.5 Hz.

Is this line of thought good enough to get me my 20/20? Let’s find out.

Takeaways:

  • There are more ways than one to approach a problem.

Links for further study:http://en.wikipedia.org/wiki/Sine_wave

Next: Professor Sampler’s Notes: The Shannon-Nyquist Sampling Theorem Part III
Previous: Professor Sampler’s Notes: The Shannon-Nyquist Sampling Theorem Part I