# Average the averages

Started by October 9, 2006
```I am making an application where several inputs has to be filtered. The
filtering is a simple moving average filtering with a variable filter
length. The problem is what happens when I connect these moving average
filters in cascade? I know that I will get some aliasing, but how big is
the aliasing problem? The scenario is as follows:

The fastest sampling rate of the system is 100 Hz. I connect a number of
filters in cascade, i.e. is use the output of the first filter as input to
the next. Between the filters I downsample the signal with factor M (M = fs
x k, where k is an integer larger than 1). I know that one are suppose to
use anti aliasing filters when downsampling to avoid aliasing, but if I
use these  AA filters, the beatiful simplicity of the system is lost. So I
am willing to accept some aliasing, but I cannot figure out how much
aliasing I will get. To make my problem clearer, here is an example: Two
filters connected in cascade with downsampling in between. Both filters
are 2 point moving average filters. The input to the first filter is
sampled at 100 Hz. The averaged output of the first filter is downsampled
by a factor 2, to 50 Hz. This signal is used as input to the next filter.

Input to filter 1: x1, x2, x3, x4, x5, x6, x7, ...

Output of the first filter: (x1+x2)/2, (x2+x3)/2, (x3+x4)/2, (x4+x5)/2,
...

after downsampling by 2 (reomve every second sample): (x1+x2)/2,
(x3+x4)/2, (x5+x6)/2, ...

which is input to the second filter.

Output of the second filter:(x1+x2)/2 + (x3+x4)/2  (x5+x6)/2 + (x7+x8)/2
---------------------, ---------------------,..
2		       2

which eaquals: x1/4 + x2/4 + x3/4 + x4/4, x5/4 + x6/4 + x7/4 + x8/4,
.....

It seems to me as the two 2-point MA filters in cascade with downsampling
inbetween results in a 4-point MA filter. So to sum up: How big a problem
is the aliasing?

```
```The frequency response of your "filter" should be a si function, as you
use a rect time window. Correct?

Best regards,

Andre

sorenbirk wrote:
> I am making an application where several inputs has to be filtered. The
> filtering is a simple moving average filtering with a variable filter
> length. The problem is what happens when I connect these moving average
> filters in cascade? I know that I will get some aliasing, but how big is
> the aliasing problem? The scenario is as follows:
>
> The fastest sampling rate of the system is 100 Hz. I connect a number of
> filters in cascade, i.e. is use the output of the first filter as input to
> the next. Between the filters I downsample the signal with factor M (M = fs
> x k, where k is an integer larger than 1). I know that one are suppose to
> use anti aliasing filters when downsampling to avoid aliasing, but if I
> use these  AA filters, the beatiful simplicity of the system is lost. So I
> am willing to accept some aliasing, but I cannot figure out how much
> aliasing I will get. To make my problem clearer, here is an example: Two
> filters connected in cascade with downsampling in between. Both filters
> are 2 point moving average filters. The input to the first filter is
> sampled at 100 Hz. The averaged output of the first filter is downsampled
> by a factor 2, to 50 Hz. This signal is used as input to the next filter.
>
> Input to filter 1: x1, x2, x3, x4, x5, x6, x7, ...
>
> Output of the first filter: (x1+x2)/2, (x2+x3)/2, (x3+x4)/2, (x4+x5)/2,
> ..
>
> after downsampling by 2 (reomve every second sample): (x1+x2)/2,
> (x3+x4)/2, (x5+x6)/2, ...
>
> which is input to the second filter.
>
> Output of the second filter:(x1+x2)/2 + (x3+x4)/2  (x5+x6)/2 + (x7+x8)/2
> 			    ---------------------, ---------------------,..
> 				       2		       2
>
> which eaquals: x1/4 + x2/4 + x3/4 + x4/4, x5/4 + x6/4 + x7/4 + x8/4,
> ....
>
> It seems to me as the two 2-point MA filters in cascade with downsampling
> inbetween results in a 4-point MA filter. So to sum up: How big a problem
> is the aliasing?
>
>
>

--

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```
```>I am making an application where several inputs has to be filtered. The
>filtering is a simple moving average filtering with a variable filter
>length. The problem is what happens when I connect these moving average
>filters in cascade? I know that I will get some aliasing, but how big is
>the aliasing problem? The scenario is as follows:
>
>The fastest sampling rate of the system is 100 Hz. I connect a number of
>filters in cascade, i.e. is use the output of the first filter as input
to
>the next. Between the filters I downsample the signal with factor M (M =
fs
>x k, where k is an integer larger than 1). I know that one are suppose
to
>use anti aliasing filters when downsampling to avoid aliasing, but if I
>use these  AA filters, the beatiful simplicity of the system is lost. So
I
>am willing to accept some aliasing, but I cannot figure out how much
>aliasing I will get. To make my problem clearer, here is an example: Two
>filters connected in cascade with downsampling in between. Both filters
>are 2 point moving average filters. The input to the first filter is
>sampled at 100 Hz. The averaged output of the first filter is
downsampled
>by a factor 2, to 50 Hz. This signal is used as input to the next
filter.
>
>Input to filter 1: x1, x2, x3, x4, x5, x6, x7, ...
>
>Output of the first filter: (x1+x2)/2, (x2+x3)/2, (x3+x4)/2, (x4+x5)/2,
>...
>
>after downsampling by 2 (reomve every second sample): (x1+x2)/2,
>(x3+x4)/2, (x5+x6)/2, ...
>
>which is input to the second filter.
>
>Output of the second filter:(x1+x2)/2 + (x3+x4)/2  (x5+x6)/2 + (x7+x8)/2
>			    ---------------------, ---------------------,..
>				       2		       2
>
>which eaquals: x1/4 + x2/4 + x3/4 + x4/4, x5/4 + x6/4 + x7/4 + x8/4,
>.....
>
>It seems to me as the two 2-point MA filters in cascade with
downsampling
>inbetween results in a 4-point MA filter. So to sum up: How big a
problem
>is the aliasing?
>
>
>
>

Sorry about the layout. The output of the second filter is

(x1+x2)/2 + (x3+x4)/2  (x5+x6)/2 + (x7+x8)/2
---------------------, ---------------------,...
2		        2
```
```Hi Andre

Yes, that is correct.

>The frequency response of your "filter" should be a si function, as you
>use a rect time window. Correct?
>
>Best regards,
>
>Andre
>
>sorenbirk wrote:
>> I am making an application where several inputs has to be filtered.
The
>> filtering is a simple moving average filtering with a variable filter
>> length. The problem is what happens when I connect these moving
average
>> filters in cascade? I know that I will get some aliasing, but how big
is
>> the aliasing problem? The scenario is as follows:
>>
>> The fastest sampling rate of the system is 100 Hz. I connect a number
of
>> filters in cascade, i.e. is use the output of the first filter as input
to
>> the next. Between the filters I downsample the signal with factor M (M
= fs
>> x k, where k is an integer larger than 1). I know that one are suppose
to
>> use anti aliasing filters when downsampling to avoid aliasing, but if
I
>> use these  AA filters, the beatiful simplicity of the system is lost.
So I
>> am willing to accept some aliasing, but I cannot figure out how much
>> aliasing I will get. To make my problem clearer, here is an example:
Two
>> filters connected in cascade with downsampling in between. Both
filters
>> are 2 point moving average filters. The input to the first filter is
>> sampled at 100 Hz. The averaged output of the first filter is
downsampled
>> by a factor 2, to 50 Hz. This signal is used as input to the next
filter.
>>
>> Input to filter 1: x1, x2, x3, x4, x5, x6, x7, ...
>>
>> Output of the first filter: (x1+x2)/2, (x2+x3)/2, (x3+x4)/2,
(x4+x5)/2,
>> ..
>>
>> after downsampling by 2 (reomve every second sample): (x1+x2)/2,
>> (x3+x4)/2, (x5+x6)/2, ...
>>
>> which is input to the second filter.
>>
>> Output of the second filter:(x1+x2)/2 + (x3+x4)/2  (x5+x6)/2 +
(x7+x8)/2
>> 			    ---------------------, ---------------------,..
>> 				       2		       2
>>
>> which eaquals: x1/4 + x2/4 + x3/4 + x4/4, x5/4 + x6/4 + x7/4 + x8/4,
>> ....
>>
>> It seems to me as the two 2-point MA filters in cascade with
downsampling
>> inbetween results in a 4-point MA filter. So to sum up: How big a
problem
>> is the aliasing?
>>
>>
>>
>
>
>--
>
>For emailing me, please exchange abuse by
>a . l o d w i g
>
>

```