Downsampling as a Matrix Operation2.12003/03/192003/03/26IvanSelesnickselesi@taco.poly.eduErinMaloneyemaloney@rice.eduIvanSelesnickselesi@taco.poly.edudecimationdownsamplingmatrix
The downsampler is a linear system, but it is not
time-invariant. ()
ynx2n
The downsampler can also be represented in matrix form.
⋮y0y1y2y3⋮⋮⋮…1000000……0010000……0000100……0000001…⋮⋮⋮x0x1x2x3x4x5x6⋮ where
D⋮⋮…1000000……0010000……0000100……0000001…⋮⋮.
We can write
yDx
where D is the
infinite matrix having the form shown. Note that the matrix
D that represents
downsampling is not a Toeplitz matrix -- it is not constant
along the diagonals. As we know, the downsampling operation can
not be represented as a convolution.
Let's see what the transpose of the downsampler is. We can do
this by observing the form of the transpose of the matrix
D, which has the
following form:
D⋮⋮…1000……0000……0100……0000……0010……0000……0001…⋮⋮
The matrix D
represents upsampling:
⇔zDyzn[↑2]yn
Note that
DDI so therefore upsampling is not an orthogonal
transformation. Indeed, when the matrix product
DD is computed, we find that
DD⋮⋮…100000……000000……001000……000000……000010……000000……000001…⋮⋮
So
⇔zDDxzn[↑2][↓2]xnxnn is even0n is odd
Equivalently,
Zz0.5XzXz