The Fourier Transform: Periodicity

{Diagrams Forthcoming}

DEFINITION:
Periodicity:  f(t) is periodic with period T if  there is some constant T such that
     f(t) = f(t + nT),    for every integer n, for all t.

EXAMPLE:
    Let y(t) = sin(t).
    Is y(t) periodic?  If so, what is the period?

I find a geometric solution the most straightforward.  Draw a circle with radius  r = 1.  Let O be the center.  For convenience, let O also be the origin of an x-y coordinate space.

Make a space ship.  Attach it to the circle where it intersects the positive x-axis: (1,0).    This is my starting position.  I have not moved yet.    Draw an arrow showing the direction the ship will travel around the circle.  I have chosen to go counter-clockwise.

My ship needs a destination.  Inscribe a right triangle in the circle, with central angle θ; hypotenuse r.    The legs of the triangle are then cos(θ), sin(θ).  It's up to me what I want the range of θ to be, around the circle.  I could divide the circle up into 360 evenly spaced, radial increments. That would be degrees, and super inconvenient right now.

So I'll do this.  From my starting position, trace the circle counter-clockwise to the vertex of the inscribed triangle.  Suppose I can measure that arc length, s, as a fraction of the whole circle.  I'll use that number for θ, too:
     Let θ = s, the arc length.  This is radians.
I want to make this as easy as possible, so I'll travel at a constant speed of 1 radian/sec.   After 1 second, my angle measure and distance along the circle are also 1:     .

The ship has an (x, y) position.  Call this position r.   Right now that position  is   r = (1,0).   As functions of the angle θ  from our starting position, the coordinates are:    x(θ) = cos(θ),    y(θ) = sin(θ).   But I've decided to let θ = s. Hence, the position r is given by:
       
Now I need a way to measure arc length along a circle.

Hahahaha just kidding.  The circle has a diameter, D=2.  To get from (1,0) to (-1,0), in a straight line (along the diameter), the distance is 2.   But for the space ship traveling along the circle, the distance from (1,0) to (-1,0) opens two hemispheres: one whole circle.  If we don't know the distance around the circle, we should assign it a letter.   For a Diameter of 1, I'll call the length around the whole circle π.   ... Right?

That is, the ratio of the Circumference to the Diameter of a circle  is  π.  Our circle has a diameter of 2, and so its circumference is 2π.  From any point on the circle, if you travel this distance in a single direction, the whole circle is traversed, and you are back where you started.   You may go around as many times as you want.  In other words,
       
But I made s = t.  So,
       

That is, 

The functions

are periodic with period
 

In radians, as long as the circle has a radius of 1, angle measure always equals arc length.  But time doesn't.  I can change the relationship to time by changing speeds:

■ Scalar multiplication of t:     y(t) = sin(ct) ,   c an arbitrary constant.
Now we are traveling at a constant speed c around the circle.  Now our position at time t is
       
And     Distance = Rate × Time:
       
Note that the position vector is still r(s) = (cos(s), sin(s)).

Calculate the Period: 
If our current position is s₀ and the time is t₀ how long does it take to circle once and get back to s₀?
      
This calculation is independent of the values of t₀ and s₀.  Both the circumference and our speed are constant.  If complete n circles, our final position is still s₀.   As above, this holds no matter where we are on the circle:    .   However, time will march onward:
       
So what time is it when we finish our first lap, anyway?
       
Time to check myself.  Using ordinary substitution, I should get r(t) = r(t + 2π/c).

Good.

I'd like to talk about speed, and trips around the circle, without writing π everywhere.  Let's agree right now to measure our speed as some multiple of 2π:

■ Speed normalized to the circle:   y(t) = sin(2πct)
2πc is still just a constant.  The problem is the same.
       
Calculate the Period: 
How long does it take to complete one lap?  Distance = Rate × Time:
      
And what time is it when we complete one trip around the circle?
       
Much better.  Our speed is  c circles/second, and it takes 1/c  seconds to complete one circle.  This is not a magical property of numbers, I've done it on purpose to describe the problem.
How can my little spaceship problem be useful?  Change the word circles to cycles, and this is the Hertz scale.  But for the spaceship to follow a signal front around, I  have a lot more work to do.  Onward.

■ Time offset :    y(t) = sin(2πct + δ)
δ has no effect on the relationship between t and s.
The period  is also unaffected by δ.  s + 2πn = s holds for all s.    It holds for s + δ.  Thus,

The functions

 are periodic with period

■ Additional Considerations
Say we are both at (1,0).  I am trapped in my spaceship; you may walk freely about.  If you and I want to meet at (-1,0),  you have one, fastest way to get there: walk straight left along the x-axis. For me there are two, equal paths, through positive y and  negative y.   When we both stand at (-1, 0), for the return trip I again have two choices.  To perform a round-trip, you have one possible course and I have four.

Now suppose you stand at (1,0), and I zoom by.  If you only measure from my current position, I may be traveling in any of the infinite number of other unit circles which pass through the point (1,0).  By intersecting my circle at two points (in a secant, such as a diameter) you can determine which of these courses I traverse.  Without this information, you cannot accurately assess the distance between us and direction at any point in time.  Your distance calculations will be fine if you stand still, but once you begin to move about, both the distance and direction are unknown.  The same information can be derived (! differentiation) from my curve of motion.  Observing my spaceship in the near vicinity of (1,0) should suffice.

■ Next Question:  When is the sum of two periodic functions periodic?