# Fractional Decimation in DSP

Started by March 6, 2010
```Hi, Friends!
Recently, I have some questions about Fractional Decimation.
I think about them for a lot of time, but can't find a good solution.

It comes from my reading Douglas W.Barker's paper "Efficient Resampling
Implementations "  presented on "Tips&Tricks" IEEE Signal Processing
Magazine,2008.
I found some interrelated materials in Mr. Fred Harris's book << Multirate
signal processing for communication systems >> .
This book gives me excellent analysis in how to determine the Interpolation
factor. But the polyphase interpolation does not only contain the
interpolation, but also the decimation. I have following questions about
fractional decimation
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when decimation, how the decimated stopband attenuation will be affect by
the decimation. I think for the reason that the original outband noise is
folded into the decimated stopband and passband, they will both be degraded.
Can we *Accurately* analyze their degradations with mathematic model?
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And there are some more questions

Question One:
On page 145 of Chapter 7 in his book, he said:
"If we up sample by N and down sample by Q the worst case cumulative gain
due to
the folding is (Q/(Q&#2013266088;C1), which results in worst case folded spectral levels
of Q/(Q&#2013266088;C1) * 1/(2N). "
(1/2N is a calculated noise, he is analyzing the decimation affect to the
interpolated signal)
**********How to get the (Q/(Q&#2013266088;C1)?  And is there any paper about it?

Question Two:
On page 53, Chapter 3
(He designs a low-pass filter with the equiripple method, the passband is
about normalized 0.025 Fs, the transition is very narrow, and the stopband
attenuation is 60dB and equiripple)
For a specific example, the filter presented in Figure 3.20 designed for
60dB side-lobe levels is
used in a 32-to-1 down sampling application. If the side-lobes are
equiripple at 60-dB the integrated
side-lobe level is -36.1dB which, when distributed over the remaining
bandwidth of 1/32 (-15.1 dB) of input sample rate, results in an effective
alias side-lobe suppression of -51 .2dB, equivalent.
********** Where does the   "integrated side-lobe level is -36.1dB " come
from? How can we get it?

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Shenzhi Ph.D candidate
EE Engineering
Huazhong Uni.Sci.Tech.
Wuhan,China
email: markkknd@hotmail.com
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