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Downsampling as a Matrix Operation

Module by: Ivan Selesnick. E-mail the author

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

Figure 1
Figure 1 (sec3fig1.jpg)

yn=x2n 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=Dx 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=DTyzn=[2]yn z D y z n [2] y n Note that DTDI D D I so therefore upsampling is not an orthogonal transformation. Indeed, when the matrix product DTD D D is computed, we find that DTD=( 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=(DTD)xzn=[2][2]xn={xn  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, Zz=0.5(Xz+Xz) Z z 0.5 X z X z

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