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    By: Daniel McKennaAs a part of collection: "Fundamentals of Signal Processing"

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Upsampling

Module by: Phil Schniter. E-mail the author

Summary: (Blank Abstract)

Upsampling

The operation of upsampling by factor L∈N L describes the insertion of L−1 L 1 zeros between every sample of the input signal. This is denoted by " ↑⁢L ↑ L " in block diagrams, as in Figure 1.

Figure 1
Figure 1 (m10403fig1.png)

Formally, upsampling can be expressed in the time domain as y⁢n={x⁢nL  if  nL∈Z0  otherwise   y n x n L n L 0 In the zz-domain, Y⁢z=∑ny⁢n⁢z−n=∑n,nL∈Zx⁢nL⁢z−n=∑kx⁢k⁢z(−k)⁢L=X⁢zL Y z n y n z n n nL x n L z n k x k z k L X z L and substituting z=ej⁢ω z ω for the DTFT,

Y⁢ej⁢ω=X⁢ej⁢ω⁢L Y ω X ω L
(1)
As shown in Figure 2, upsampling compresses the DTFT by a factor of LL along with the ωω axis.

Figure 2
Figure 2 (m10403fig2.png)

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