Calculating aliasing components at a particular sampling frequency

Started by November 1, 2011
I'm not sure how to calculate aliasing components as I've only dealt with
systems that are sampled above fs >= 2B. So if I want to make sure that I
have aliasing but the components due to aliasing are at most 20dB below
the maximum amplitude of the signal I don't know how to do that.

On Nov 1, 12:21=A0am, "Benjamin S." <niaci...@yahoo.com> wrote:
> I'm not sure how to calculate aliasing components as I've only dealt with
> systems that are sampled above fs >=3D 2B. So if I want to make sure that=
I
> have aliasing but the components due to aliasing are at most 20dB below
> the maximum amplitude of the signal I don't know how to do that.

You can work backwards from any given post-sampling frequency to
determine what
analog frequencies map into that point - then see what signal energies
are available
at those analog frequencies.  If you can't do that (or some filter
operation equivalent)
you're stuck - what aliasing means is that you can't distinguish those
frequency
components once sampling has occurred.

On Nov 1, 3:21=A0am, "Benjamin S." <niaci...@yahoo.com> wrote:
> I'm not sure how to calculate aliasing components as I've only dealt with
> systems that are sampled above fs >=3D 2B. So if I want to make sure that=
I
> have aliasing but the components due to aliasing are at most 20dB below
> the maximum amplitude of the signal I don't know how to do that.

I'm not sure what you are asking.  Aliasing does not change the
amplitude of the signal, just the frequency.  If your amplitude
changes it is because there is a filter in the path that attenuates
the signal before aliasing perhaps.  Or if you have a post alias
filter it can attenuate the signal because the signal frequency is in
the skirts of the filter.

Maybe you can restate the question or at least tell us more about that
you want to do.

Rick

On 11/1/11 12:38 PM, rickman wrote:
> On Nov 1, 3:21 am, "Benjamin S."<niaci...@yahoo.com>  wrote:
>> I'm not sure how to calculate aliasing components as I've only dealt with
>> systems that are sampled above fs>= 2B. So if I want to make sure that I
>> have aliasing but the components due to aliasing are at most 20dB below
>> the maximum amplitude of the signal I don't know how to do that.
>
> I'm not sure what you are asking.

i dunno for sure what he wants, but it may be the aliases resulting from
practical reconstruction where the anti-imaging filter does not beat the
images down to zero.  then, when resampled, these images (beaten down to
some level) get moved around in frequency.

>  Aliasing does not change the
> amplitude of the signal, just the frequency.

another example would be sampling a sinusoid of frequency at exactly
Nyquist.  the different aliases differ by amplitude and phase and not
frequency.

>  If your amplitude
> changes it is because there is a filter in the path that attenuates
> the signal before aliasing perhaps.

yup, the reconstruction filter.

>  Or if you have a post alias
> filter it can attenuate the signal because the signal frequency is in
> the skirts of the filter.

and it's the location in frequency where we likely discriminate between
pre or post-alias.  if it's moved from the "original", it must be an
image that fell in the baseband when resampled.

> Maybe you can restate the question or at least tell us more about that
> you want to do.

that would be good.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."


On Tue, 01 Nov 2011 07:21:53 +0000, Benjamin S. wrote:

> I'm not sure how to calculate aliasing components as I've only dealt
> with systems that are sampled above fs >= 2B. So if I want to make sure
> that I have aliasing but the components due to aliasing are at most 20dB
> below the maximum amplitude of the signal I don't know how to do that.

Wow.  I knew exactly what I was going to say, then I read everyone else's
responses and I got all confused.

At any rate, what I _think_ you want to know is how to calculate where
your aliased spectrum lands in your desired spectrum, in a way that lets
you figure out if the aliased spectrum is going to cause a problem -- yes?

http://www.wescottdesign.com/articles/Sampling/sampling.pdf

Then, take a close look at Figure 3 and the surrounding text.  Figure 3
is doing exactly what I think you want: it's showing how the aliased
spectra fall in the "real" spectra, which in turns lets you figure out
their effect on your desired signal.

To do the same thing mathematically, assume that you have a signal
spectrum S(f), and that you're sampling that signal at f_s.  Then the
spectrum of your sampled signal will be:

S_s(f) = sum_{k=-\infty}^\infty S(f + k f_s)

Or in other words, your spectrum at a given frequency will be the sum of
the prototype's spectrum at that frequency and all integer harmonics of
your sampling frequency.  The component at 0*f_s is usually your intended
signal, while the components at k*f_s, k != 0 are usually your unwanted
aliases.  So if you know your prototype signal's spectrum you can easily
find the aliased spectrum.

Check back in if this did not help.

--
www.wescottdesign.com