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Interpolation: by an integer factor l

Interpolation means increasing the sampling rate, or filling in in-between samples. Equivalent to sampling abandlimited analog signal L times faster. For the ideal interpolator,

X 1 ω X 0 L ω ω L 0 L ω
We wish to accomplish this digitally. Consider [link] and [link] .
y m X 0 m L m 0 ± L ± 2 L 0
The DTFT of y m is
Y ω m ω y m ω m n x 0 n ω L n n x n ω L n X 0 ω L
Since X 0 ω is periodic with a period of 2 , X 0 L ω Y ω is periodic with a period of 2 L (see [link] ).
By inserting zero samples between the samples of x 0 n , we obtain a signal with a scaled frequency response that simply replicates X 0 ω L times over a 2 interval!

Obviously, the desired x 1 m can be obtained simply by lowpass filtering y m to remove the replicas.

x 1 m y m h L m
Given H L m 1 ω L 0 L ω In practice, a finite-length lowpass filter is designed using any of the methods studied so far ( [link] ).

Interpolator block diagram

Decimation: sampling rate reduction (by an integer factor m)

Let y m x 0 L m ( [link] )

That is, keep only every L th sample ( [link] )
In frequency (DTFT):
Y ω m y m ω m m x 0 M m ω m n M m n x 0 n k δ n M k ω n M ω ω M n x 0 n k δ n M k ω n DTFT x 0 n DTFT δ n M k
Now DTFT δ n M k 2 k 0 M 1 X k δ ω 2 k M for ω as shown in homework #1, where X k is the DFT of one period of the periodic sequence. In this case, X k 1 for k 0 1 M 1 and DTFT δ n M k 2 k 0 M 1 δ ω 2 k M .
DTFT x 0 n DTFT δ n M k X 0 ω 2 k 0 M 1 δ ω 2 k M 1 2 μ X 0 μ 2 k 0 M 1 δ ω μ 2 k M k 0 M 1 X 0 ω 2 k M
so Y ω k 0 M 1 X 0 ω M 2 k M i.e. , we get digital aliasing .( [link] )
Usually, we prefer not to have aliasing, so the downsampler is preceded by a lowpass filter to remove all frequencycomponents above ω M ( [link] ).

Rate-changing by a rational fraction l/m

This is easily accomplished by interpolating by a factor of L , then decimating by a factor of M ( [link] ).

The two lowpass filters can be combined into one LP filterwith the lower cutoff, H ω 1 ω L M 0 L M ω Obviously, the computational complexity and simplicity of implementation will depend on L M : 2 3 will be easier to implement than 1061 1060 !

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Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
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