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Course by: Douglas L. Jones. E-mail the author

# Interpolation, Decimation, and Rate Changing by Integer Fractions

Module by: Douglas L. Jones. E-mail the author

## Interpolation: by an integer factor L

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

X 1 ω={ X 0 Lω  if  |ω|<πL0  if  πL|ω|π X 1 ω X 0 L ω ω L 0 L ω
(1)
We wish to accomplish this digitally. Consider Equation 2 and Figure 1.
ym={ X 0 mL  if  m=0±L±2L0  otherwise   y m X 0 m L m 0 ± L ± 2 L 0
(2)
The DTFT of ym y m is
Yω=m=ωyme(jωm)=n= x 0 ne(jωLn)=n=xne(jωLn)= X 0 ωL 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
(3)
Since X 0 ω X 0 ω is periodic with a period of 2π 2 , X 0 Lω=Yω X 0 L ω Y ω is periodic with a period of 2πL 2 L (see Figure 2). By inserting zero samples between the samples of x 0 n x 0 n , we obtain a signal with a scaled frequency response that simply replicates X 0 ω X 0 ω LL times over a 2π 2 interval!

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

x 1 m=ym* h L m x 1 m y m h L m
(4)
Given H L m={1  if  |ω|<πL0  if  πL|ω|π 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 (Figure 3).

## Decimation: sampling rate reduction (by an integer factor M)

Let ym= x 0 Lm y m x 0 L m (Figure 4)

That is, keep only every LLth sample (Figure 5) In frequency (DTFT):
Yω=m=yme(jωm)=m= x 0 Mme(jωm)=n= x 0 nk=δnMke(jωnM)|n=Mm=n= x 0 nk=δnMke(j ω n)| ω =ωM=DTFT x 0 n*DTFTδnMk 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
(5)
Now DTFTδnMk=2πk=0M1Xkδ ω 2πkM DTFT δ n M k 2 k 0 M 1 X k δ ω 2 k M for |ω|<π ω as shown in homework #1 , where Xk X k is the DFT of one period of the periodic sequence. In this case, Xk=1 X k 1 for k01M1 k 0 1 M 1 and DTFTδnMk=2πk=0M1δ ω 2πkM DTFT δ n M k 2 k 0 M 1 δ ω 2 k M .
DTFT x 0 n*DTFTδnMk= X 0 ω *2πk=0M1δ ω 2πkM=12πππ X 0 μ (2πk=0M1δ ω μ 2πkM)d μ =k=0M1 X 0 ω 2πkM 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
(6)
so Yω=k=0M1 X 0 ωM2πkM Y ω k 0 M 1 X 0 ω M 2 k M i.e., we get digital aliasing.(Figure 6) Usually, we prefer not to have aliasing, so the downsampler is preceded by a lowpass filter to remove all frequency components above |ω|<πM ω M (Figure 7).

## Rate-Changing by a Rational Fraction L/M

This is easily accomplished by interpolating by a factor of LL, then decimating by a factor of MM (Figure 8).

The two lowpass filters can be combined into one LP filter with the lower cutoff, Hω={1  if  |ω|<πmaxLM0  if  πmaxLM|ω|π H ω 1 ω L M 0 L M ω Obviously, the computational complexity and simplicity of implementation will depend on LM L M : 2/3 23 will be easier to implement than 1061/1060 10611060!

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