## Multistage decimation by two with halfband filters

Started by 5 years ago4 replieslatest reply 5 years ago280 views
Rick Lyons book Understanding Digital Signal Processing third edition chapter 10.12.1 Half-band Filtering Fundamentals states:

"It’s normal to use the same half-band filter in multistage decimation by two as was done in Figure 10-25(b)." and

"However, in multistage interpolation by factors of two it would be computationally inefficient to use the same half-band filter in each stage."

I fully agree with the second statement, but have questions about the first.

Let me give an example of decimation by 8 from 44100 Hz with interesting band 0-1800Hz.
The first filter going from sampling frequency 44100 to 22050 Hz has a transition window of (22050-2*1800)/22050=0.8367 which needs 25 taps for a stop band attenuation of 140 dB.
The second filter going from sampling frequency 22050 to 11025 Hz has a transition window of (11025-2*1800)/11025=0.6735 which needs 35 taps for a stop band attenuation of 150 dB.
The third filter going from sampling frequency 11025 to 5512.5 Hz has a transition window of (5512.5-2*1800)/5512.5=0.3469 which needs 55 taps for a stop band attenuation of 140 dB.

Suppose we would take 3 times the same filter, we would need the third filter and 3*55=165 taps in stead of with 3 different filters 25+35+55=115 taps. So for the same reason of computational efficiency, I prefer 3 different filters.

But probably I am missing something?
[ - ]

You are exactly right, I don't think you've missed anything.  Curious what application needs 140dB of attenuation...

In interpolation I have use the same filter for the last 2 or 3 spots.  But it was already down to a 5 tap filter (2 multipliers) with CSD coefficients and minimized to a really tiny implementation.  I couldn't do any better even with the really large transition bands so I just left it.

[ - ]

Hello bertramaerts. You have NOT missed anything! You are correct and your uncertainty is my fault. And for this I beg your pardon. For an explanation please see my blog at:

By the way, in case if you haven't seen it, please see the following web page.

[ - ]