# decimation by dropping two samples

Started by April 26, 2011
```Hi All,
I was analyzing a decimation scheme where two samples are dropped
alternatively . 0,1 are picked 2,3 are dropped , 4,5 are picked and 6 and 7
are dropped and so on. I have never seen such a decimation scheme .

I would like to know whats the advantage of the this scheme. And
also
will the sampling frequency change here ??
```
```On Apr 26, 9:08&#2013266080;am, "sudarshan_onkar"
<sudarshan.onkar@n_o_s_p_a_m.gmail.com> wrote:
> Hi All,
> &#2013266080; &#2013266080; &#2013266080; &#2013266080; I was analyzing a decimation scheme where two samples are dropped
> alternatively . 0,1 are picked 2,3 are dropped , 4,5 are picked and 6 and 7
> are dropped and so on. I have never seen such a decimation scheme .
>
> &#2013266080; &#2013266080; &#2013266080; I would like to know whats the advantage of the this scheme. And
> also
> will the sampling frequency change here ??

Have no idea what advantages this will give. The analysis will
certainly be a mess. Where did you see this? Is it possible that
somebody screwed up? Misspelled the scheme?

Rune
```
```On Apr 26, 3:08&#2013266080;am, "sudarshan_onkar"
<sudarshan.onkar@n_o_s_p_a_m.gmail.com> wrote:
> Hi All,
> &#2013266080; &#2013266080; &#2013266080; &#2013266080; I was analyzing a decimation scheme where two samples are dropped
> alternatively . 0,1 are picked 2,3 are dropped , 4,5 are picked and 6 and 7
> are dropped and so on. I have never seen such a decimation scheme .
>
> &#2013266080; &#2013266080; &#2013266080; I would like to know whats the advantage of the this scheme. And
> also
> will the sampling frequency change here ??

Sampling theory has no requirement for equally spaced points. I've
done it because of hardware limitations, not because it gave me a
better signal estimate.
```
```On 4/26/2011 12:08 AM, sudarshan_onkar wrote:
> Hi All,
>          I was analyzing a decimation scheme where two samples are dropped
> alternatively . 0,1 are picked 2,3 are dropped , 4,5 are picked and 6 and 7
> are dropped and so on. I have never seen such a decimation scheme .
>
>        I would like to know whats the advantage of the this scheme. And
> also
> will the sampling frequency change here ??

Well, I'd approach it something like this:

Decimate by 4.  This you can likely understand.  What do you get?

Now, delay by 1 sample time (i.e. shift over in the input sequence) and
decimate by 4 again.  This you can likely understand and surely if you
understand the first one.

Add the results together to get the double samples.  Superposition applies.

I think it's easier to envision (although I have not done so) if you
think of it in the frequency domain:

the orginal Fourier Transform pair is
f0(t) <> F0(w)
Now decimate f0(t) by 4 by multiplying by f1(t)
f1(t) <> F1(w)
Now decimate f0(t) by 4 again but one sample time later by multiplying
by f2(t):
f2(t) <> F2(w)

Where we can recognize that f2(t) = f1(t-t0)

The decimation sample sequences are:
for f1, f0(t)
for f2, f0(t - t0) where t0 is the original sample interval.

The Fourier Transforms of the decimating sequences are:
F1(w) and F1(w)e^-jwt0

Since the decimation is a multiplication in time, it's a convolution in
frequency with these last two expressions.

I "think" the sum of the two convolutions comes out:

inf
(1/2pi)int F0(u)F1(w-u)[1 + e^-j(w-u)t0]du
-inf
u

So, the resulting spectrum has this form with the "normal" convolution
part and that 1 + exp part thrown in - a phase term.

I dunno if this helps but it's how I'd go about it in general.

And then you can go figure if it's discrete instead of continuous, and
so forth.....

Fred
```
```On 4/26/2011 12:08 AM, sudarshan_onkar wrote:
> Hi All,
>          I was analyzing a decimation scheme where two samples are dropped
> alternatively . 0,1 are picked 2,3 are dropped , 4,5 are picked and 6 and 7
> are dropped and so on. I have never seen such a decimation scheme .
>
>        I would like to know whats the advantage of the this scheme. And
> also
> will the sampling frequency change here ??

Well, I gave an answer to my own question but not to yours!

What's the advantage?  You tell me!  It's not at all clear in the sense
that it can be viewed as the sum of two closely related sequences
decimated by 4.  You should get more information content but only more
bandwidth via a "trick" I guess.

Will the sampling frequency change?  Well sure.  And, I think
effectively by a factor of 4.  But that could be disputed.
I guess if the phase shift is 90 degrees then you've reinvented a sort
of quadrature decimator.  That's likely it.  Then you treat the two

But, *I* ain't gonna prove it!  :-)

Fred
```
```On Apr 26, 6:43&#2013266080;pm, steve <bungalow_st...@yahoo.com> wrote:
> On Apr 26, 3:08&#2013266080;am, "sudarshan_onkar"
>
> <sudarshan.onkar@n_o_s_p_a_m.gmail.com> wrote:
> > Hi All,
> > &#2013266080; &#2013266080; &#2013266080; &#2013266080; I was analyzing a decimation scheme where two samples are dropped
> > alternatively . 0,1 are picked 2,3 are dropped , 4,5 are picked and 6 and 7
> > are dropped and so on. I have never seen such a decimation scheme .
>
> > &#2013266080; &#2013266080; &#2013266080; I would like to know whats the advantage of the this scheme. And
> > also
> > will the sampling frequency change here ??
>
> Sampling theory has no requirement for equally spaced points.

The theory that discusses 'sampling' as generating a discrete
sequence
x[n] from a continuous signal x(t) on the form x[n] <-> x(nT) is
based
on the assumption of equally spaced points. As far as I am aware,
that's
the form of sampling all of DSP is based on.

> I've
> done it because of hardware limitations, not because it gave me a
> better signal estimate.

For practical reasons, one might deviate from the ideal scheme.
Not because it gives any form of 'advantege' - it might not even be
'good' - but because that's all that is available to you.

Rune
```
```>On Apr 26, 6:43=A0pm, steve <bungalow_st...@yahoo.com> wrote:
>> On Apr 26, 3:08=A0am, "sudarshan_onkar"
>>
>> <sudarshan.onkar@n_o_s_p_a_m.gmail.com> wrote:
>> > Hi All,
>> > =A0 =A0 =A0 =A0 I was analyzing a decimation scheme where two samples
a=
>re dropped
>> > alternatively . 0,1 are picked 2,3 are dropped , 4,5 are picked and 6
a=
>nd 7
>> > are dropped and so on. I have never seen such a decimation scheme .
>>
>> > =A0 =A0 =A0 I would like to know whats the advantage of the this
scheme=
>. And
>> > also
>> > will the sampling frequency change here ??
>>
>> Sampling theory has no requirement for equally spaced points.
>
>The theory that discusses 'sampling' as generating a discrete
>sequence
>x[n] from a continuous signal x(t) on the form x[n] <-> x(nT) is
>based
>on the assumption of equally spaced points. As far as I am aware,
>that's
>the form of sampling all of DSP is based on.

Most, but not all. Outside the mainstream (e.g. in some interesting
military applications) structured non-equally spaced samples are used. In
some applications, which are statistical in nature, relatively slow
pseudo-randomly spaced samples are used to good effect.

A blanket statement that samples are not required to be equally spaced is
obviously bogus. Sampling the first half of a song at 96k/second and
expecting to be able to play back the whole song is probably not going to
end well. :-) If you sample in a non-equally spaced manner, where no 2
samples are more than half a cycle of the high frequency component of the
signal apart, you can recover the signal. However, trying to find practical
algorithms to manipulate (e.g. filter) the sample stream might be a
challenge.

>> I've
>> done it because of hardware limitations, not because it gave me a
>> better signal estimate.
>
>For practical reasons, one might deviate from the ideal scheme.
>Not because it gives any form of 'advantege' - it might not even be
>'good' - but because that's all that is available to you.
>
>Rune

Steve

```
```On 26/04/2011 5:08 PM, sudarshan_onkar wrote:
> Hi All,
>          I was analyzing a decimation scheme where two samples are dropped
> alternatively . 0,1 are picked 2,3 are dropped , 4,5 are picked and 6 and 7
> are dropped and so on. I have never seen such a decimation scheme .
>
>        I would like to know whats the advantage of the this scheme. And
> also
> will the sampling frequency change here ??

Hi Sudarshan,

I agree with others here that you get a real mess with this
sampling scheme.

The set of samples you generate is equivalent to a regularly
sampled set which has been multiplied by the function:
0.5*(1 + sqrt(2)*sin((2*pi*n/4) + pi/4))
where n is the sample index.

This is equivalent to
(i) multiplying the original analog signal by a sinusoidal
function of frequency fs/4 and phase pi/4
(ii) adding in some of the original signal.
(ii) sampling the result at a regular rate fs.

Assuming that the original analog signal was low-pass
band-limited to 0.25*fs (and not the usual ~0.5*fs) then the
result is not totally useless. The band from 0.25*fs to
0.5*fs in the unevenly sampled signal will contain a
frequency-translated version of the original signal which
might be useful, but there are much better ways to do
frequency translation if that is the aim.

Any decimation process such as averaging adjacent samples
would appear to make matters worse.

Regards,
John

```
```Hi All,
I followed Freds explanation until the convolution came and
spoiled the party.
Ok we can narrow down the problem to sine wave , may be then
i can extend it . If i have a sampled sine wave at some fs and let the
frequency of sine wave be fm. Now ill decimate this sampled sine wave using
mentioned scheme.
original sine wave index =  0,1,2,3,4,5,6,7,8,9,10,11,12 ....
decimated sinewave index =  0,1,4,5,8,9,12,13 .....

If i see the spectrum (by FFT) what i am expected to see? should i be
seeing single peak corresponding to fm or some other frequency will also
appear ?

```
```On Apr 27, 11:13&#2013266080;am, "sudarshan_onkar"
<sudarshan.onkar@n_o_s_p_a_m.gmail.com> wrote:
> Hi All,
> &#2013266080; &#2013266080; &#2013266080; &#2013266080; &#2013266080;I followed Freds explanation until the convolution came and
> spoiled the party.
> &#2013266080; &#2013266080; &#2013266080; &#2013266080; &#2013266080;Ok we can narrow down the problem to sine wave , may be then
> i can extend it . If i have a sampled sine wave at some fs and let the
> frequency of sine wave be fm. Now ill decimate this sampled sine wave using
> mentioned scheme.
> original sine wave index = &#2013266080;0,1,2,3,4,5,6,7,8,9,10,11,12 ....
> decimated sinewave index = &#2013266080;0,1,4,5,8,9,12,13 .....
>
> If i see the spectrum (by FFT) what i am expected to see? should i be
> seeing single peak corresponding to fm or some other frequency will also
> appear ?

Don't know what to expect to see, other than it will *not*
be anything like a nice spike as from a sinusoidal.

Rune
```