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Operators

Module by: Richard Baraniuk. E-mail the author

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Summary: Introduction to operators.

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Consider the processing setup in Figure 1.

Figure 1: x∈ℂN x N and y∈ℂN y N .
Figure 1 (op.png)
We have studied some special classes of operators O O.

Transforms

O O is an N×N N N ONB matrix or its transpose

y=BHx y B x (1)
where xx is in the "time domain" and y y is in the "BB domain." BB is invertible, B-1=BH B B , and no information is lost going from xx to yy and from yy to xx.

Normal Matrix Systems

Figure 2
Figure 2 (systems.png)
A A in Figure 2 is not an ONB matrix in general, but a normal matrix: AHA=AAH A A A A .

This is more general than an orthogonal matrix of ONB.

Note:

Some information could be lost going from x x to yy; i.e.: A-1 A might not exist.
Usually we think of both xx and yy as living in the "time domain."

This set studies these kinds of systems...

Definition 1: Operator
An operator is a mapping that takes a vector x∈ℂN x N and produces a vector y∈ℂN y N (Figure 3).
Figure 3
Figure 3 (opDef.png)

Example: DC Offset

y=x+711⋮1 y x 7 1 1 ⋮ 1 (2)
where the vector of ones has N N ones (Figure 4).

Figure 4
Figure 4 (DCoff.png)

Example: Squarer

yn=x2n y n x n 2 (3)
where 0≤n≤N−1 0 n N 1 (Figure 5).

Figure 5
Figure 5 (squarer.png)

Example: Simple 2-point Averaging Smoothing Filter

yn=xn+xn−1modN2 y n x n x n 1 N 2 (4)
(Figure 6).
Figure 6: N=9 N 9 .
Figure 6 (smoother.png)
Why the mod N N? 1

Example: More Extensive Smoothing Filter

yn=xn+1modN+2xn+xn−1modN4 y n x n 1 N 2 x n x n 1 N 4 (5)
(Figure 6).

Example: Median Smoother

yn=medianxn−1modNxnxn+1modN y n x n 1 N x n x n 1 N (6)
where the median is the "middle" number in an ordered list. e.g.: The median of 1-73 1 -7 3 is 1. (Figure 6).

Example: Edge Detector

yn=xn−xn−1modN y n x n x n 1 N (7)
(Figure 6).

Example: Matrix Multiply

y=Ax y A x (8)
(Figure 2).
y=∑k=0N−1akxk y k 0 N 1 a k x k (9)
where ak a k are the columns of matrix AA, A=a0a1…a N - 1 A a 0 a 1 … a N - 1 or
yn=∑k=0N−1Ankxk y n k 0 N 1 A n k x k (10)
where Ank A n k means row nn, column kk entry of A=11111-11010200201020100000 A 1 1 1 1 1 -1 1 0 1 0 2 0 0 2 0 1 0 2 0 1 0 0 0 0 0 . (Figure 7).
Figure 7: N=5 N 5 .
Figure 7 (matmult.png)

Footnotes

  1. Because we have to take a ℂN N signal to ℂN N signal. i.e.: same length.

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