The Convolution Sum I

{UNDER CONSTRUCTION
 Note: I do not lik this representation of the Dirac Delta.  If I use the left-handed rectangle, I incur an offset of a single sample under certain operations.  In the limit, this extra offset vanishes, but for practical computation it is an error and requires fiddly accounting that is better resolved another way.  I think the Delta is best represented as symmetric about the origin, and sample positions centered in the rectangles.  Below I use Oppenheim's approach in Signals and Systems, but I will change it shortly }

TASK:  Reconstruct an arbitrary continuous function x(t) from discrete samples.

• Assume x(t) is differentiable and integrable.
• Sample at evenly spaced intervals in time:  Between two adjacent samples k and k+1, the change in time is then
  
• For convenience, assume a sample falls exactly at time t = 0.  Then the kth sample falls at time t = kΔ.  
• Let each sample be represented by a rectangle of width Δ and height x(kΔ).  Again, for convenience, let the rectangles be left-handed.
• Let a time t_k fall between two adjacent samples, t = (k −1)Δ  and   t = kΔ.

If  x(t) is continuous, there is a point in time between every two samples where the slope of x(t) is the same as the line connecting those two samples.  Between two samples, (k−1)Δ and kΔ, call this time ξ_k.  Formally,

And we also have, for any t,

THE DASTARDLY DEEDS:

Assumptions which permit formal approximation.

• Let t_k be any time in this same interval. Then, arguing from (1) and (2),
  
• I want to write my approximation as 'x(t)' and operate on it formally, as a function.  Consider the sum
  
  All terms of x(t) are added together.  But for any given time t, I only need one of these terms.  I have the following curious puzzle:
            
  ...where the & indicate composition, and not addition.  Similarly,
            
  The discrete terms are made continuous by holding each value until the next sample (multiplying each sample by Δt = Δ) .
• I require:
       1.  A memory system.  I have written down evenly spaced values of x(t), and I would like to carry them around in a container.  Given a particular value of t, say t = t₀ I want my function to go to the sheet of paper, look up the value I have written there, and retrieve only the value x(t₀) from the list.
    2. The approximation of x(t) to be formally differentiable and integrable.

Assumptions are independent of argument.  I have made an assumption which permits me arbitrary, and wrongful, inference.   What is it?**

The rest is by construction.  I will not introduce symbols and arguments to "understand" their behavior or marvel over the results.  I will define them purposefully: assigning them the properties necessary to carry out the desired tasks.

I am interested in sound.  The assumption:

    that x(t) is a composition of functions over a homogeneous medium, each of whose behavior is reducible to a a disturbance propagating at a constant rate in all directions, from a fixed point of origin

accurately describes my data.

MAKE AN INDEX FUNCTION
Let


Using the FTC, by direct manipulation of the sums, we can verify that
          
In words, u(t) is the unit step function. δ(t) is the unit impulse function, also called the Dirac Delta.
O I am a clever monster, said the Dirac Delta.  Just you wait.

Fig. 1: Unit impulse (left) and step (right) functions

Say I wish to know the value of x(t) at some time t = mΔ.  Consider the expression:

         

Evaluated at t = mΔ,

         

By the definition of δ_Δ(t), x(t) has the value x(mΔ) for the whole interval  mΔ ≤ t < (m+1)Δ.  For all other values of t,  x(t) = 0.   I can now retrieve individual terms from the sum (3):


If I am not manipulating audio these labels are likely irrelevant.  Comprenne qui voudra. Fair warning.
Now let Δ → 0 and n → ∞:
          
For any given t, there is only one nonzero term in the sum (see below).  In the limit, the difference between the area under the approximation and the original function vanishes, if the limit exists.

   ■ kΔ is an index: at a fixed time t_i, the (kΔ)th term contributes x(kΔ)δ_Δ(t_i − kΔ)Δ to the total sum.

Let kΔ = τ, Δ = dτ.  Then

In the limit, as in equation (6), there is exactly one instant when the sum is nonzero, for any fixed t_i.  As a dummy variable, τ is not just a formality: τ iterates through the terms of summation.  For ∫δ(t − τ)dτ all nonzero values occur when t = τ, but for convolution in general, the distinction is meaningful.

Example:
Let x(t) = u(t).
          
And we can formally integrate stepwise functions.

   ■ Proposition: x(t) can be taken out of the summation; the integral sign:


Formal Proof:
Note that, vary t as we like, δ(t − τ) is nonzero only when t = τ.  Hence,
          
The variable of integration is τ, x(t) is now constant for the integration:
          

The Long Way Around Shouldn't I check all of this?  Yes.  Consider the summation of δ_Δ(t) alone:

Note that
     -The sum has a single nonzero term, at k = 0.
     -The sum is a constant.  As Δ → 0, the limit of Δ(1/Δ) exists and is 1.
     -If n = 0, there is a single term, δ_Δ(0)Δ = 1.
     -For finite n > 0,
               
     -Letting n→ ∞ has no effect on the sum; all new terms are 0.
That is, (8) is unchanged for the simultaneous limits n→∞, Δ→0.
But,

Likewise, for the summation of δ_Δ(t−τ),

where k_iΔ < t_i < (k_i+1)Δ.
Again, the sum in insensitive to the simultaneous limits n→∞, Δ→0, and

for all t.

Which is to say, with a lot of squiggles,
          

I told you I was a clever monster.  Oh yes,  said the Dirac Delta.


■ Substitution of variable:
{To DO}

____________________

** Assumption: x(t) is a function.
This is called begging the question.  The question was, are the data before me a function?  I pretend it has been answered.  Once I have decided this, all formal obstacles can be brushed aside.   In general, calling the data an unknown function is a confession.

I am satisfied that there exist phenomena which satisfy the wave equation.  The above arguments are ideally suited to the description of such phenomena, and I will use them.  I find the arguments, and the math, beautiful, often bewildering.