# Resampling by multistage decimation (Richard Lyons book)

Started by June 1, 2011
```Hello everybody,

I am trying to understand the multistage decimation problem stated by
Richard Lyons in his book "understanding digital signal processing" 1st
edition, page 306. It is as follows:

- We have an input data arriving at 400 kHz and we want to decimate by 100,
but we are going to do this in two stage: 50 and 2.

- The original signal bandwidth is something greater than 100 kHz, and our
range of interest is 0 - 1.8 kHz. This means we need to lowpass to avoid
aliasing.

The solution says:  " if the first low-pass filter LPF1 (we are doing a
decimation of 50 first) has a cutoff frequency of 1.8 kHz and its stopband
is defined to begin at 6 kHz, the ouput of the D = 50 decimator will have a
spectrum [figure is shown..] Notice that there is aliasing between 2 and 4
kHz - Not to worry, LPF2 will take care of this (the LPF2 is the one
applied in the second decimation = 2) "

My explanation:
I am supposing the LPF1 is done with a Fs = 400 kHz, which makes sense,
since the stopband is defined to begin at 6 kHz. After it is done, we have
a signal with frequencies up to 6 kHz (because the transition band also
accounts, i.e from 1.8 kHz to 6 kHz). However, the ouput of the filter is
decimated with D = 50, which means that the new frequency is 400/50 = 8
kHz. This also means that the maximum frequency without aliasing is 4 kHz,
but HOWEVER we had (from the lowpass filtering) some signal spectra between
4 and 6 kHz, so aliasing occurs.

Two questions:
1- Am I supposing right if I think that the low pass filters work at the
frequency of the sampled input (400 kHz the first one and 8 kHz the second
one, because its input would be the signal decimated with a factor of 50).

2- For those who have the book (im sure a lot), dont you think the figure
7-5c is bad? I mean, the "LPF1 fequency response" and it's should have a
common part centered around 4 kHz (the aliasing I mentioned before).

Kind regards,

Mario
```
```sorry a mistake:

With respect to question 2: "..it&acute;s IMAGE (centered 8 kHz, which is the
first replica since the sampling frequency is 8 kHz after the decimation).

>Kind regards,
>
>Mario
>
```
```> Two questions:
> 1- Am I supposing right if I think that the low pass filters work at the
> frequency of the sampled input (400 kHz the first one and 8 kHz the second
> one, because its input would be the signal decimated with a factor of 50).
>

Yes, you are.

> 2- For those who have the book (im sure a lot), dont you think the figure
> 7-5c is bad? I mean, the "LPF1 fequency response" and it's should have a
> common part centered around 4 kHz (the aliasing I mentioned before).
>
I don't have the first edition of the book, so i can't answer this
question.

--

Alexander
```
```Thanks Alexander,

--Mario

>
>> Two questions:
>> 1- Am I supposing right if I think that the low pass filters work at
the
>> frequency of the sampled input (400 kHz the first one and 8 kHz the
second
>> one, because its input would be the signal decimated with a factor of
50).
>>
>
>Yes, you are.
>
>> 2- For those who have the book (im sure a lot), dont you think the
figure
>> 7-5c is bad? I mean, the "LPF1 fequency response" and it's should have
a
>> common part centered around 4 kHz (the aliasing I mentioned before).
>>
>I don't have the first edition of the book, so i can't answer this
>question.
>
>--
>
>Alexander
>
```
```On 6/1/2011 3:48 AM, MRR wrote:
> 2- For those who have the book (im sure a lot), dont you think the figure
> 7-5c is bad? I mean, the "LPF1 fequency response" and it's should have a
> common part centered around 4 kHz (the aliasing I mentioned before).

Yes, it looks like there's a small problem with the figure.

I think it should look like this:

|                               |                                 |
x-------------+                 x                 -+--------------x
|             |---              |              --- |              |
|             |   ---           |           ---    |              |
|              |     ---        |        ---      |               |
|              |        ---     |     ---         |               |
|              |           ---  |  ---            |               |
|              |              --+--               |               |
|              |             ---| ---             |               |
|              |          ---   |    ---          |               |
|               |      ---      |       ---      |                |
|               |   ---         |          ---   |                |
|               |---            |             ---|                |
+------------------------------------------------+-----------------
0
8kHz            2kHz             4kHz              6kHz

Now the diagonal transition lines go down from 2kHz to 6kHz and from
6kHz to 2kHz.  This makes the aliasing that's mentioned evident.

You should tell Rick if he doesn't see this.
There's a similar treatment in the 2nd Ed. with factors of 25 and 4
instead of 50 and 2.  And, there's a similar figure but I didn't bother
to analyze it.

Fred

```
```Well, now everything makes sense!! Thanks a lot for taking the time for
analyzing the figure Fred.

Regards,

Mario

>On 6/1/2011 3:48 AM, MRR wrote:
>> 2- For those who have the book (im sure a lot), dont you think the
figure
>> 7-5c is bad? I mean, the "LPF1 fequency response" and it's should have
a
>> common part centered around 4 kHz (the aliasing I mentioned before).
>
>Yes, it looks like there's a small problem with the figure.
>
>I think it should look like this:
>
>
>    |                               |                                 |
>    x-------------+                 x                 -+--------------x
>    |             |---              |              --- |              |
>    |             |   ---           |           ---    |              |
>    |              |     ---        |        ---      |               |
>    |              |        ---     |     ---         |               |
>    |              |           ---  |  ---            |               |
>    |              |              --+--               |               |
>    |              |             ---| ---             |               |
>    |              |          ---   |    ---          |               |
>    |               |      ---      |       ---      |                |
>    |               |   ---         |          ---   |                |
>    |               |---            |             ---|                |
>    +------------------------------------------------+-----------------
>    0
>8kHz            2kHz             4kHz              6kHz
>
>Now the diagonal transition lines go down from 2kHz to 6kHz and from
>6kHz to 2kHz.  This makes the aliasing that's mentioned evident.
>
>You should tell Rick if he doesn't see this.
>There's a similar treatment in the 2nd Ed. with factors of 25 and 4
>instead of 50 and 2.  And, there's a similar figure but I didn't bother
>to analyze it.
>
>Fred
>
>
```
```On Wed, 01 Jun 2011 05:48:52 -0500, "MRR"
<mario.ruzruiz@n_o_s_p_a_m.gmail.com> wrote:

>Hello everybody,
>
>I am trying to understand the multistage decimation problem stated by
>Richard Lyons in his book "understanding digital signal processing" 1st
>edition, page 306. It is as follows:
>
[Snipped by Lyons]

Hello Mario,
Fred is correct, sadly, there's a mistake in
Figure 7-5(c) in early printings of the
1st Edition of the book.

On page 307, the "6" in the denominator of the first ratio
in Eq. (7-7) should have been a "4".  Thus the result of
Eq. (7-7) should be 665 stages.  In Figure 7-5(c), those
two diagonal "transition region" lines should not overlap
each other, they should meet on the horizontal axis
at the Freq = 4 kHz point.  (Those two errors were
corrected in later printing of the 1st Edition.)

Mario, I'll send you the errata for the 1st Eidtion if
you E-mail me and tell me what 'Printing Number' you have.
You can determine the "Printing Number" of the American
version of the 1st Edition of the book by looking at
the page just before the "Dedication" page.

On that page (before the Dedication) you'll see all
sorts of publisher-related information.  Down toward
the bottom of the page you may see lines printed
something like:

ISBN 0-201-63467-8
6 7 8 9 10 MA 02 01 00 99

That initial 6 on the second line above means
the "Sixth Printing".

A later printing shows something like:

0-201-63467-8
910-MA-02

That initial 9 on the second line above means
the "Ninth Printing".

An even later printing shows something like this:

0-201-63467-8
11 12 13 14 15 16 17 18 MV 05 04 03

That initial 11 on the second line above means
the "Eleventh Printing".

Regards,
[-Rick-]
```
```Dear Rick,

Thank you. I do not own the book, it is from my university. Anyways, the
main point for me is that now everything makes sense.

Btw, i dont know if there is an spanish version of your book, i think it
would worth...

Thanks a lot,

Mario
>On Wed, 01 Jun 2011 05:48:52 -0500, "MRR"
><mario.ruzruiz@n_o_s_p_a_m.gmail.com> wrote:
>
>>Hello everybody,
>>
>>I am trying to understand the multistage decimation problem stated by
>>Richard Lyons in his book "understanding digital signal processing" 1st
>>edition, page 306. It is as follows:
>>
>   [Snipped by Lyons]
>
>Hello Mario,
>   Fred is correct, sadly, there's a mistake in
>Figure 7-5(c) in early printings of the
>1st Edition of the book.
>
>On page 307, the "6" in the denominator of the first ratio
>in Eq. (7-7) should have been a "4".  Thus the result of
>Eq. (7-7) should be 665 stages.  In Figure 7-5(c), those
>two diagonal "transition region" lines should not overlap
>each other, they should meet on the horizontal axis
>at the Freq = 4 kHz point.  (Those two errors were
>corrected in later printing of the 1st Edition.)
>
>Mario, I'll send you the errata for the 1st Eidtion if
>you E-mail me and tell me what 'Printing Number' you have.
>You can determine the "Printing Number" of the American
>version of the 1st Edition of the book by looking at
>the page just before the "Dedication" page.
>
>On that page (before the Dedication) you'll see all
>sorts of publisher-related information.  Down toward
>the bottom of the page you may see lines printed
>something like:
>
>   ISBN 0-201-63467-8
>   6 7 8 9 10 MA 02 01 00 99
>
>That initial 6 on the second line above means
>the "Sixth Printing".
>
>A later printing shows something like:
>
>   0-201-63467-8
>   910-MA-02
>
>That initial 9 on the second line above means
>the "Ninth Printing".
>
>An even later printing shows something like this:
>
>   0-201-63467-8
>   11 12 13 14 15 16 17 18 MV 05 04 03
>
>That initial 11 on the second line above means
>the "Eleventh Printing".
>
>Regards,
>[-Rick-]
>
```