The Fourier Transfom - Periodicity II

THE SUM OF TWO PERIODIC FUNCTIONS

Given two elementary periodic functions,
          

When is the sum    periodic?  If it is periodic, what is the period?

1.  WHEN IS THE SUM PERIODIC?

f_i(t) is periodic with period T_i= 1/c_i
f_j(t) is periodic with period T_j = 1/c_j

If the sum is periodic then there exist two integers k₁, k₂ such that
          
That is, some multiple of T_i is also a multiple of T_j.   Take two tires, place them next to each other.  Mark a vertical lines down the side of each tire. Now spin them at two different speeds, 2πc_i and 2πc_j.  If those two lines are ever vertical at the same time in the future, then each wheel has at that moment completed a revolution.  For the tire f_i this is the k₁^th revolution.  For the the tire f_j this is the k₂^th revolution.

If no such integers exist, the sum is not periodic.  Putting the k's on one side:
        

So we have a simple rule:
If T_i/T_j is irrational, then there are no such k₁, k₂ and y(t) is not periodic.

2-1. DETERMINE THE PERIOD

If we can find on pair k₁, k₂ then we can find an infinite number of them:  multiply both sides by some integer n:
       
nk₁, nk₂ are integers which satisfy our query. To determine the fundamental period of y(t), we seek the least integers that satisfy k₁, k₂.  I'll do this by reducing fractions as I work. In short,

■ Procedure:   Given 
          
   1. Reduce m/r and n/s to lowest terms:    M/R,  N/S.
   2. Set c₁= the greatest common factor of the numerators M and N.
           c₂ = the greatest common factor of the denominators R and S.
 Then y(t) is periodic with period
          

2-2. THE DETAILS:

Introduce positive integers m, n, r, s;
Rational numbers P = m/r,      Q = n/s;
Distinct irrational numbers φ, θ.   (with all rational factors removed)

Reduce m/r and n/s (remove common factors, if any), and let
     
Expressed as fractions in lowest terms,
     
Introduce elementary periodic functions f_i(t), and corresponding periods T_i:
       

       

       

Determine  the period of y(t), for the following periods of the f(t). 
(a)  Integer periods. 

Let   .
The fundamental period is given by the least integers k₁, k₂ such that:
       
We have
       
Remove common factors of m and n:
       
Using T = k₁T₁ . . . y(t) is periodic with period
          

(b) Rational periods.  

Let  .  
If y(t) is periodic, there are integers k₁ and k₂ such that   T = k₁T₃ = k₂T₄.
       
To put k₁/k₂ in lowest terms, reduce S/R and M/N. (S/N and M/R are already reduced).
       
Using T = k₁T₃,  . . . y(t) is periodic with period
       

(c) One signal irrational.  

Let  .
We know that the ratio of periods cannot be rational.  y(t) is not periodic.
Specifically, we must have
       
But φ didn't stop being irrational. There are no such k₁, k₂.

(d) Both signals irrational

Let  .
If φ and θ were multiples of the same common irrational number, then the sum is still periodic.  (In the examples above, all coefficients of t have the common factors 2 and π.  This does not affect whether or not their sums are periodic.)

 θ and φ are distinct rational numbers;  f₅(t) + f₆(t) is not periodic.

■Next Question: At what time t_m does the maximum value of y(t) occur?

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?