# sampling ...

Started by July 6, 2013
```Sirs,

I have some questions related to sampling and subsequent re-construction of
signals. I am hsitoping to get answers ...

1) Assume a signal consists of two signals of frequencies f1 and f2
such that frequncy f1 < f2. Sampling is done to ensure fs >= 2xf1 and fs <
2xf2.

Since the sampling rate does not meet the nyquist rate for f2, it would
lead to f2 aliasing into f1. When the discrete signal is re-constructed
back, how would the content of f2 contribute to the recovered signal?

I am assuming that since fs does not meet required sampling rate for f2,
the recovered signal will not look like original signal ...

2) Continuing on question 1 above, assuming that aliasing effects are to be
avoided, I am assuming that filtering with low pass filter will solve the
problem. Is the correct time to filter is after sampling is carried out?

3) Assume that signal has signals of f1, f2, f3 frquencies and f1 < f2 <
f3. If signal of interest is f2 , what would be the practical way to
extract this signal. I have been told that in practical applications,
bandpass filters (where f2 lies) are generally avoided.

4) When a signal is described as containing different frequency components,
I have generally seen literature describing them as ADDITION of individual
signals. Is my understanding correct that addition operation is the basis
for getting the composite signal?

thanks, manish

_____________________________
Posted through www.DSPRelated.com
```
```On Sat, 06 Jul 2013 13:16:16 -0500, "manishp" <58525@dsprelated>
wrote:

>Sirs,
>
>I have some questions related to sampling and subsequent re-construction of
>signals. I am hsitoping to get answers ...
>
>1) Assume a signal consists of two signals of frequencies f1 and f2
>such that frequncy f1 < f2. Sampling is done to ensure fs >= 2xf1 and fs <
>2xf2.
>
>Since the sampling rate does not meet the nyquist rate for f2, it would
>lead to f2 aliasing into f1. When the discrete signal is re-constructed
>back, how would the content of f2 contribute to the recovered signal?

There will be an in-band signal <fs/2 that is the alias of f2.

>I am assuming that since fs does not meet required sampling rate for f2,
>the recovered signal will not look like original signal ...

It will have the alias of f2 added.

>2) Continuing on question 1 above, assuming that aliasing effects are to be
>avoided, I am assuming that filtering with low pass filter will solve the
>problem. Is the correct time to filter is after sampling is carried out?

Sampling is where the alias happens, so it cannot be removed after
sampling without also removing all other non-aliased energy around the
alias of f2.

This is why an anti-alias filter appears before the ADC, usually just
before.

>3) Assume that signal has signals of f1, f2, f3 frquencies and f1 < f2 <
>f3. If signal of interest is f2 , what would be the practical way to
>extract this signal. I have been told that in practical applications,
>bandpass filters (where f2 lies) are generally avoided.

Why avoid a bandpass filter if it does what you need?   Bandpass
filters can be tricky to implement sometimes, but sometimes they're
exactly what you need.    Lots of tradeoffs apply.

You can also perform a complex downconversion so that f2 is at
baseband (i.e., near DC), so that f1 and f3 can be removed with
low-pass filters.

>4) When a signal is described as containing different frequency components,
>I have generally seen literature describing them as ADDITION of individual
>signals. Is my understanding correct that addition operation is the basis
>for getting the composite signal?

Generally.

>thanks, manish
>
>
>
>
>_____________________________
>Posted through www.DSPRelated.com

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```
```manishp <58525@dsprelated> wrote:

> I have some questions related to sampling and subsequent re-construction of
> signals. I am hsitoping to get answers ...

> 1) Assume a signal consists of two signals of frequencies f1 and f2
> such that frequncy f1 < f2. Sampling is done to ensure fs >= 2xf1 and fs <
> 2xf2.

> Since the sampling rate does not meet the nyquist rate for f2, it would
> lead to f2 aliasing into f1. When the discrete signal is re-constructed
> back, how would the content of f2 contribute to the recovered signal?

Sampling and aliasing is usually described in terms of frequency
bands. If you have discrete frequencies, the discussion is different.

> I am assuming that since fs does not meet required sampling rate for f2,
> the recovered signal will not look like original signal ...

Depends on how you recover it.

> 2) Continuing on question 1 above, assuming that aliasing effects are to be
> avoided, I am assuming that filtering with low pass filter will solve the
> problem. Is the correct time to filter is after sampling is carried out?

If the problem is as described, there is no aliasing, unless f1 and the
prospective alias are especially close. When doing reconstruction,
you need an appropriate filter. If the filter has pass bands around
f1 and f2, and not around the aliases of f1 or f2, then the result
will be reconstructed just fine.

> 3) Assume that signal has signals of f1, f2, f3 frquencies and f1 < f2 <
> f3. If signal of interest is f2 , what would be the practical way to
> extract this signal. I have been told that in practical applications,
> bandpass filters (where f2 lies) are generally avoided.

Again, with discrete frequencies it is likely that the original can
be reconstructed. Note that Nyquist depends on the signals being
of infinite (time) extent. If they aren't, you should consider them
as narrow, but not infinitely narrow, in bandwidth. Even so, it is
likely that the aliases don't overlap, and so the original can
be reconstructed.

> 4) When a signal is described as containing different frequency components,
> I have generally seen literature describing them as ADDITION of individual
> signals. Is my understanding correct that addition operation is the basis
> for getting the composite signal?

Generally it is the way to describe them, but not the only way
to generate them. For example, some operations that generate
frequency components are more like multiplication.

-- glen
```
```Sir,

>This is why an anti-alias filter appears before the ADC, usually just
>before.

Based on this comment, Am I correct to say that,

1) filtering has to be done in analog domain to remove HF/f2 content or

2) if filtering in analog domain is not feasible then sample at a higher
rate to meet the rate for f2, do filtering after that and then do
downsampling to store and do rest of the processing with lower sample rate
...

Case 2) above is basically my assumption ... I dont know how much of this
is correct ...

>Why avoid a bandpass filter if it does what you need?   Bandpass
>filters can be tricky to implement sometimes, but sometimes they're
>exactly what you need.    Lots of tradeoffs apply.
>
>You can also perform a complex downconversion so that f2 is at
>baseband (i.e., near DC), so that f1 and f3 can be removed with
>low-pass filters.

This was one of my assumption too. f2 and f3 frequencies would come down
but I was not sure what complex multiplication would do to f1 which is
already at a low frequency. I will look into further into texts for this.

Thanks once again, manish

_____________________________
Posted through www.DSPRelated.com
```
```Sir,

>If the problem is as described, there is no aliasing, unless f1 and the
>prospective alias are especially close. When doing reconstruction,
>you need an appropriate filter.

Does this mean that aliasing is does not always take place due to
insufficient sampling.

>If the filter has pass bands around
>f1 and f2, and not around the aliases of f1 or f2, then the result
>will be reconstructed just fine.

One of my original question was what would an aliasing do to the signal.
The hint I am getting is that due to insufficient sampling, we could
actualy end up with not just f1 and f2 but many other frequencies which are
aliases of f2. Am I correct?

>Generally it is the way to describe them, but not the only way
>to generate them. For example, some operations that generate
>frequency components are more like multiplication.

Thanks. If this is the case, would normal filtering method work for such
signals?

thanks once again, manish

_____________________________
Posted through www.DSPRelated.com
```
```manishp <58525@dsprelated> wrote:

(snip, I wrote)
>>If the problem is as described, there is no aliasing, unless f1 and the
>>prospective alias are especially close. When doing reconstruction,
>>you need an appropriate filter.

> Does this mean that aliasing is does not always take place due to
> insufficient sampling.

In the DSP context, a signal with a specified frequency is a
sinusoid. You need amplitude, phase, and frequency to specify
one, so three samples, ignoring aliasing for now, are sufficient.

If you know f1<2*fs, for some fs, you can specify those three points.

For two frequencies, f1 and f2, both less than 2*fs, six samples
will be enough, unless you are particularly unlucky.

But most real problems have signals within a band, that is,
continuous in frequency space for some region of frequency space.

In that case, the math is different.

While your favorite FM station may say that the frequency is 91.1MHz,
it is actually a band between 91.0MHz and 91.2MHz, possibly slightly
less to allow for a guard band. (Actually, an FM signal has
infinite extent in frequency space, but falls off fast enough that
one can approximate it over such a range.)

Nyquist is normally more useful for such banded signals, less
useful for single frequencies.

-- glen
```
```On Sun, 07 Jul 2013 05:36:52 -0500, "manishp" <58525@dsprelated>
wrote:

>Sir,
>
>>This is why an anti-alias filter appears before the ADC, usually just
>>before.
>
>Based on this comment, Am I correct to say that,
>
>1) filtering has to be done in analog domain to remove HF/f2 content or
>
>2) if filtering in analog domain is not feasible then sample at a higher
>rate to meet the rate for f2, do filtering after that and then do
>downsampling to store and do rest of the processing with lower sample rate
>..
>
>Case 2) above is basically my assumption ... I dont know how much of this
>is correct ...

Both of those are correct.  Note that ANY energy present at the input
to the converter above half the sample rate will alias, so an analog
anti-alias  filter before the converter is almost always used.

>>Why avoid a bandpass filter if it does what you need?   Bandpass
>>filters can be tricky to implement sometimes, but sometimes they're
>>exactly what you need.    Lots of tradeoffs apply.
>>
>>You can also perform a complex downconversion so that f2 is at
>>baseband (i.e., near DC), so that f1 and f3 can be removed with
>>low-pass filters.
>
>This was one of my assumption too. f2 and f3 frequencies would come down
>but I was not sure what complex multiplication would do to f1 which is
>already at a low frequency. I will look into further into texts for this.

The complex mix is linear, so it affects all frequencies the same.
If it is a complex downconversion all frequencies will be moved an
equal amount toward minus infinity.    It is possible that f1 could
become a negative frequency, and if it moves outside of the pass band
of a low-pass filter it will be attenuated by the LPF.

>Thanks once again, manish
>
>_____________________________
>Posted through www.DSPRelated.com

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com
```