The downsampler is a linear system, but it is not time-invariant. (Figure 1)

Figure 1

y⁢n=x⁢2⁢n y n x 2 n The downsampler can also be represented in matrix form. ⋮y0y1y2y3⋮=( ⋮⋮ …1000000… …0010000… …0000100… …0000001… ⋮⋮ )⁢⋮x0x1x2x3x4x5x6⋮ ⋮ y0 y1 y2 y3 ⋮ ⋮ ⋮ … 1 0000 00… … 0 0100 00… … 0 0001 00… … 0 0000 01… ⋮ ⋮ ⋮ x0 x1 x2 x3 x4 x5 x6 ⋮ where D=( ⋮⋮ …1000000… …0010000… …0000100… …0000001… ⋮⋮ ) D ⋮ ⋮ … 1 0000 00… … 0 0100 00… … 0 0001 00… … 0 0000 01… ⋮ ⋮ .

We can write y=D⁢x y D x where DD is the infinite matrix having the form shown. Note that the matrix DD 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 DD, which has the following form: DT=( ⋮⋮ …1000… …0000… …0100… …0000… …0010… …0000… …0001… ⋮⋮ ) D ⋮ ⋮ … 1 000 … … 0 0 00 … … 0 1 00 … … 0 0 00 … … 0 0 10 … … 0 0 00 … … 0 0 01 … ⋮ ⋮ The matrix DTD represents upsampling: z=DT⁢y⇔z⁢n=[↑2]⁢y⁢n ⇔ z D y z n [↑2] y n Note that DT⁢D≠I D D I so therefore upsampling is not an orthogonal transformation. Indeed, when the matrix product DT⁢D D D is computed, we find that DT⁢D=( ⋮⋮ …100000… …000000… …001000… …000000… …000010… …000000… …000001… ⋮⋮ ) D D ⋮ ⋮ … 1 0000 0… … 0 0000 0… … 0 0100 0… … 0 0000 0… … 0 0001 0… … 0 0000 0… … 0 0000 1… ⋮ ⋮ So z=(DT⁢D)⁢x⇔z⁢n=[↑2]⁢[↓2]x⁢n={x⁢n  if  n is even0  if  n is odd ⇔ z D D x z n [↑2] [↓2] x n x n n is even 0 n is odd Equivalently, Z⁢z=0.5⁢(X⁢z+X⁢−z) Z z 0.5 X z X z

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Last edited by Charlet Reedstrom on Aug 13, 2005 11:11 am -0500.