Saturday, August 20, 2011

Mathematical foundations - linearity


Linearity is a property that a system has when:
  • The effects of the whole of the inputs is equal to the sum of the effects of the parts: f(x+y) = f(x) + f(y) (the additivity requirement).
  • The effects of a multiple of an input is equal to that same multiple of the effects of the unmultiplied input: f(α.x) = α.f(x) (the homogeneity requirement).
For an example of a linear system, wait until it's dark, then, with the light off, play with your favourite image editing software.  As far as you can, make the entire background of the screen black.  Use this all-black image as your desktop background if you have to.  Use the "stretch" setting to make it fill the whole screen.  Use your image editor in full screen mode if you can.  Draw the curtains, and generally do as much as you can to make the blobs of colour you are about to draw, the only sources of light in the room.
  • Draw a fairly large blob of fully saturated red on the screen - fill about a quarter of the space available.  Look around you - there should be a distinctly red glow in the room.
  • Now add a similarly-sized cyan (100% blue + 100% green) blob.  Your room should be back to a dimly-lit white, and noticeably brighter than with just the red blob.  You have just demonstrated additivity, in adding the effects of a blob of combined blue and green light, to the effects of a blob of red light.  The result is the same as if you had drawn an all-white blob the same size as each of your coloured blobs.  If you can, look closely at your display, with a magnifying glass if you have one, near a white area on the screen: you should see little pixels (either as coloured stripes or round dots in a triangular pattern) composed of red, green and blue elements.  When you look at the room, it doesn't matter whether your coloured blobs were a few large ones or many small ones!
  • Get your red and cyan blobs on the screen again.  Look at the room again and remember the somewhat dim neutral colour the blobs cast into your room.  Now make copies of each of these two blobs: you should have two red blobs and two cyan blobs.  Back to the room: it is now lit, considerably more brightly, but still white!  This is homogeneity in action.  (Almost: you'd have to bring in numbers to do it properly.)  Twice as many coloured blobs lead to twice as much light in the room.  (But your eyes are not good at showing you that there is specifically and exactly twice as much light; and also, the eye is a distinctly non-linear device - that is part of what makes it difficult to adequately capture some scenes on camera.)
UPDATE: I drew the blobs for you.  Animated GIF with 5 seconds between frames.  Notice how the room takes on first a red glow, then white, then a brighter white.  Stretch image and view in full-screen mode for best results.

Why it is important

Linearity allows us to simplify systems, so we can understand them more easily.  Because of additivity, we can examine each cause separately and add all the effects together afterward.  Homogeneity in turn lets us examine a small cause with a small effect, and know the large effect of a large cause.

A circuit composed entirely of resistors, inductors, and capacitors (the three "passives") is a linear system: the current through each is a linear function of the potential difference across its terminals.  If we know the contributions to the potential difference due to various causes, we can consider each cause separately.  This property enables super useful tools like Thévenin's theorem or the related Norton's theorem: both allow you to reduce complex networks of linear components to just two.  That makes networks much easier to analyze, which you'll need to do to make sure your circuits work as intended.

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