# Undersampling

Started by June 7, 2007
```I am on a team developing a system that will record and process large
(1 GHz) bandwidth pulses.  Doing some research I have came across some
one that suggested sampling in the 2nd nyquist zone, for two main
reasons.  Sampling the signal at a higher frequency makes our analog
front end much better and simpler deign.  It also reduces harmonics
within our frequencies of interest allowing more integration gain.
What I mean is that if you sample at 2GHz your typical frequency range
would be from DC-1GHz, but the 100MHz harmonic is in your frequency
range of interest.  If you sample from 1GHz-2GHz all harmonics are out
of band.  I know there is aliasing, but you know about that and it can
be fixed by shifting the FFT assuming you analog filtered correctly.
I know you will lose some quality of the A/D because the frequency
response of the A/D drops in the second nyquist range, but the font
end will allow higher frequencies to pass through.  I would think
there are some problems and that I am overlooking something because it
can't be this simple.  Any comments?  Thanks.

```
```birdforsale@gmail.com wrote:
> I am on a team developing a system that will record and process large
> (1 GHz) bandwidth pulses.  Doing some research I have came across some
> one that suggested sampling in the 2nd nyquist zone, for two main
> reasons.  Sampling the signal at a higher frequency makes our analog
> front end much better and simpler deign.  It also reduces harmonics
> within our frequencies of interest allowing more integration gain.
> What I mean is that if you sample at 2GHz your typical frequency range
> would be from DC-1GHz, but the 100MHz harmonic is in your frequency
> range of interest.  If you sample from 1GHz-2GHz all harmonics are out
> of band.  I know there is aliasing, but you know about that and it can
> be fixed by shifting the FFT assuming you analog filtered correctly.
> I know you will lose some quality of the A/D because the frequency
> response of the A/D drops in the second nyquist range, but the font
> end will allow higher frequencies to pass through.  I would think
> there are some problems and that I am overlooking something because it
> can't be this simple.  Any comments?  Thanks.

I would not presume to tell you that you don't know what you're about,
but it's clear at least that I don't. For one thing, we don't use the
same terms, and I can only get what yours mean. "Second Nyquist zone" is
a new one for me.

How does one sample from 1 to 2 GHz? Samples are taken at a single
steady rate. How one construes the result is up to the designer, and it
/is/ possible to undersample provided the sample rate exceeds twice the
bandwidth of interest and certain other conditions are met.*

The 100MHz harmonic of what?

An alias is an out-of-band component that is indistinguishable from an
in-band component. There's no way to fix it after the fact, but it can
be prevented by proper filtering /before sampling/.

Why did FFT join the party? Aside from apparently fruitless "shifting",
there's no mention of its purpose.

I hope you can restate your question more clearly.

Jerry
__________________________________________
* There's a clear exposition in section 2.3, (Bandpass Sampling) of R.
Lyons, "Understanding Digital Signal Processing".
--
Engineering is the art of making what you want from things you can get.
&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;
```
```> I would not presume to tell you that you don't know what you're about,
> but it's clear at least that I don't. For one thing, we don't use the
> same terms, and I can only get what yours mean. "Second Nyquist zone" is
> a new one for me.
>
> How does one sample from 1 to 2 GHz? Samples are taken at a single
> steady rate. How one construes the result is up to the designer, and it
> /is/ possible to undersample provided the sample rate exceeds twice the
> bandwidth of interest and certain other conditions are met.*

Let me try to state this more clearly.  The A/D would always run at
2GHz, the frequencies of interest would be mixed to the 1-2GHz range
which fails nyquist, because you are alisised/undersampled.  Typically
we would mix the frequencies of interest to DC-1GHz to meet nyquist.
I see your * and will try to find that book you are referring to for
guidance.

>
> The 100MHz harmonic of what?
If the input bandwidth frequencies are mixed to vary from DC-1GHz and
the input signal has a 100MHz component (which it will) the harmonic
is within the bandwidth you care about.  If your input frequencies are
mixed from 1GHz-2Ghz then a 1.1Ghz harmonic should be filtered out by
our analog filter before the A/D.  Both would be sampled with the same
2GHz A/D.

>
> An alias is an out-of-band component that is indistinguishable from an
> in-band component. There's no way to fix it after the fact, but it can
> be prevented by proper filtering /before sampling/.
I think this is what I was talking about above, but I am not sure.

>
> Why did FFT join the party? Aside from apparently fruitless "shifting",
> there's no mention of its purpose.
My understanding is that if sampled at 2GHz a 1.1GHz signal would
appear to be a frequency of 900MHz after sampled and an FFT was
taken.  A 1.9GHz signal would appear as a 100MHz signal if you took an
FFT.  If you sampled a signal with two tones wouldn't you have to flip
the FFT for the correct image to result?  I am assuming you analog
filtered below 1GHz and above 2GHz so the only frequencies allowed in
the A/D should be from 1GHz-2GHz.

>
> I hope you can restate your question more clearly.
By no means do I claim to be a DSP expert so I was hoping I could get
some guidance here.  I hope things are explained more clearly, and
would welcome comments on problems with this approach that I am
missing and downsides to doing this other that a slight (according to
the A/D spec sheets I still want to do more research on this) hit on A/
D ENOB.

>
> Jerry
> __________________________________________
> * There's a clear exposition in section 2.3, (Bandpass Sampling) of R.
> Lyons, "Understanding Digital Signal Processing".
> --
> Engineering is the art of making what you want from things you can get.
> =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
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=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF- Hide qu=
oted text -
>
> - Show quoted text -

```
```>I am on a team developing a system that will record and process large
>(1 GHz) bandwidth pulses.  Doing some research I have came across some
>one that suggested sampling in the 2nd nyquist zone, for two main
>reasons.  Sampling the signal at a higher frequency makes our analog
>front end much better and simpler deign.  It also reduces harmonics
>within our frequencies of interest allowing more integration gain.
>What I mean is that if you sample at 2GHz your typical frequency range
>would be from DC-1GHz, but the 100MHz harmonic is in your frequency
>range of interest.  If you sample from 1GHz-2GHz all harmonics are out
>of band.  I know there is aliasing, but you know about that and it can
>be fixed by shifting the FFT assuming you analog filtered correctly.
>I know you will lose some quality of the A/D because the frequency
>response of the A/D drops in the second nyquist range, but the font
>end will allow higher frequencies to pass through.  I would think
>there are some problems and that I am overlooking something because it
>can't be this simple.  Any comments?  Thanks.
>
>

I am working on a team on a similiar approach.  We are sampling an IF
centered at 750 MHz with 1 Gsps A/D's.  Therefore, we are in the 2nd
'Nyquist Zone'.  I think people use that terminology just to confuse
others/sound smart.  Basically, 1st Nyquist zone is 0-Fs/2, 2nd zone is
Fs/2-Fs, 3rd zone is Fs-(3*Fs/2), etc.  The key thing to remember is that
even zones will be spectrally inverted.  I think that is what you were
referring to when you brought up the FFT.  We correct this when we do our
digital quadrature mixing by mixing with Fs/4 (as oppossed to -Fs/4).
Undersampling is a common technique so I don't think you have anything to
```
```birdforsale@gmail.com wrote:
>> I would not presume to tell you that you don't know what you're about,
>> but it's clear at least that I don't. For one thing, we don't use the
>> same terms, and I can only get what yours mean. "Second Nyquist zone" is
>> a new one for me.
>>
>> How does one sample from 1 to 2 GHz? Samples are taken at a single
>> steady rate. How one construes the result is up to the designer, and it
>> /is/ possible to undersample provided the sample rate exceeds twice the
>> bandwidth of interest and certain other conditions are met.*
>
> Let me try to state this more clearly.  The A/D would always run at
> 2GHz, the frequencies of interest would be mixed to the 1-2GHz range
> which fails nyquist, because you are alisised/undersampled.  Typically
> we would mix the frequencies of interest to DC-1GHz to meet nyquist.
> I see your * and will try to find that book you are referring to for
> guidance.

http://tinyurl.com/263z2j Disclaimer: Lyons is a regular contributor
here with whom I have sat over beer.

Let's get clear about the Nyquist criterion; there's no circumventing
it. It states that the sample rate must *exceed* twice the bandwidth of
interest -- equal isn't good enough. It does *not* say that the band of
people to think they get around a fundamental rule. "Bandpass sampling"
is more descriptive and less misleading.

Now, with a 2 GHz sample rate, you can sample just about any
nearly-GHz-wide band you like, provided everything outside that band is
filtered out and you are, in some cases, prepares to unscramble the
result. Given a signal limited to 1 to 2 GHz sampled at 2 GHz, and run
through a 1 GHz low-pass filter, the spectrum will be inverted; 2 GHz
will show up at DC and 1 GHz will show up as 1 GHz.

>> The 100MHz harmonic of what?
> If the input bandwidth frequencies are mixed to vary from DC-1GHz and
> the input signal has a 100MHz component (which it will) the harmonic
> is within the bandwidth you care about.  If your input frequencies are
> mixed from 1GHz-2Ghz then a 1.1Ghz harmonic should be filtered out by
> our analog filter before the A/D.  Both would be sampled with the same
> 2GHz A/D.

Is your sampler nonlinear? If not, how is your harmonic generated? If it
is in the desired bandwidth and you shift the whole band up a GHz, the
harmonic will be shifted with the rest and sampled.

>> An alias is an out-of-band component that is indistinguishable from an
>> in-band component. There's no way to fix it after the fact, but it can
>> be prevented by proper filtering /before sampling/.
> I think this is what I was talking about above, but I am not sure.
>
>> Why did FFT join the party? Aside from apparently fruitless "shifting",
>> there's no mention of its purpose.
> My understanding is that if sampled at 2GHz a 1.1GHz signal would
> appear to be a frequency of 900MHz after sampled and an FFT was
> taken.  A 1.9GHz signal would appear as a 100MHz signal if you took an
> FFT.  If you sampled a signal with two tones wouldn't you have to flip
> the FFT for the correct image to result?  I am assuming you analog
> filtered below 1GHz and above 2GHz so the only frequencies allowed in
> the A/D should be from 1GHz-2GHz.

So what makes the harmonic disappear if it's already in band?

Flip, yes; shift no. I see what you mean now. There might be a
time-domain way to invert the spectrum. Doesn't negating alternate
samples do that?

>> I hope you can restate your question more clearly.
> By no means do I claim to be a DSP expert so I was hoping I could get
> some guidance here.  I hope things are explained more clearly, and
> would welcome comments on problems with this approach that I am
> missing and downsides to doing this other that a slight (according to
> the A/D spec sheets I still want to do more research on this) hit on A/D
> ENOB.
>
>> Jerry
>> __________________________________________
>> * There's a clear exposition in section 2.3, (Bandpass Sampling) of R.
>> Lyons, "Understanding Digital Signal Processing".
>> --
>> Engineering is the art of making what you want from things you can get.
>> &#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;- Hide quoted text -
>>
>> - Show quoted text -
>
>

--
Engineering is the art of making what you want from things you can get.
&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;
```
```> My understanding is that if sampled at 2GHz a 1.1GHz signal would
> appear to be a frequency of 900MHz after sampled and an FFT was
> taken.  A 1.9GHz signal would appear as a 100MHz signal if you took an
> FFT.  If you sampled a signal with two tones wouldn't you have to flip
> the FFT for the correct image to result?  I am assuming you analog
> filtered below 1GHz and above 2GHz so the only frequencies allowed in
> the A/D should be from 1GHz-2GHz.
>

You're right, the 1.1 GHz component (the translated harmonic that you
don't want) will appear to be at a frequency of 900 MHz. But how does
after you sample, then now you need a lowpass/notch filter and you
need to undo the spectral inversion. If you don't frequency-translate
the signal before you sample, the harmonic will appear to be at 100
MHz, and you can highpass/notch filter the signal to remove the
harmonic without having to un-invert the spectrum.

It's unclear what you're actually trying to accomplish by sampling in
that way.

Jason

```
```> >> The 100MHz harmonic of what?
> > If the input bandwidth frequencies are mixed to vary from DC-1GHz and
> > the input signal has a 100MHz component (which it will) the harmonic
> > is within the bandwidth you care about.  If your input frequencies are
> > mixed from 1GHz-2Ghz then a 1.1Ghz harmonic should be filtered out by
> > our analog filter before the A/D.  Both would be sampled with the same
> > 2GHz A/D.
>
> Is your sampler nonlinear? If not, how is your harmonic generated? If it
> is in the desired bandwidth and you shift the whole band up a GHz, the
> harmonic will be shifted with the rest and sampled.

The sampler is mostly linear.  There is some variance with frequency
across the A/D, but wouldn't most A/Ds have some variance across 1GHz
of bandwidth?  Where the harmonics are coming from I am not exactly
sure.  I know the transmitter generates large harmonics but they
should be filtered before the first mixer.  I think some come from the
mixers and other various places.  The idea I was going after was that
if the IF is DC-1GHz a 400MHz signal has a harmonic of 800MHz which is
in the frequencies of interest; there is nothing I can do about that.
If the IF is 1GHz-2Hz than that same 400MHz signal would come in at
1.4GHz (by changing the analog mixing) and the harmonic would be at
2.8GHz.  The 2.8GHz value can be filtered before the A/D to remove the
unwanted signal.  This seems valid to me just wanted to make sure I
wasn't missing something, based on all the comments on this one
section I think I may be overlooking something.

```
```LearningDSP wrote:
>>>> The 100MHz harmonic of what?
>>> If the input bandwidth frequencies are mixed to vary from DC-1GHz and
>>> the input signal has a 100MHz component (which it will) the harmonic
>>> is within the bandwidth you care about.  If your input frequencies are
>>> mixed from 1GHz-2Ghz then a 1.1Ghz harmonic should be filtered out by
>>> our analog filter before the A/D.  Both would be sampled with the same
>>> 2GHz A/D.
>> Is your sampler nonlinear? If not, how is your harmonic generated? If it
>> is in the desired bandwidth and you shift the whole band up a GHz, the
>> harmonic will be shifted with the rest and sampled.
>
> The sampler is mostly linear.  There is some variance with frequency
> across the A/D, but wouldn't most A/Ds have some variance across 1GHz
> of bandwidth?

Linearity is not related to frequency response. A time-invariant linear
system (I assume that the properties of your sampler are time-invariant)
has the property that the sum of its responses to two separate stimuli
is the same as its response to the sum of the stimuli. Such a system
generated no harmonics.

>             Where the harmonics are coming from I am not exactly
> sure.  I know the transmitter generates large harmonics but they
> should be filtered before the first mixer.  I think some come from the
> mixers and other various places.  The idea I was going after was that
> if the IF is DC-1GHz a 400MHz signal has a harmonic of 800MHz which is
> in the frequencies of interest; there is nothing I can do about that.
> If the IF is 1GHz-2Hz than that same 400MHz signal would come in at
> 1.4GHz (by changing the analog mixing) and the harmonic would be at
> 2.8GHz.  The 2.8GHz value can be filtered before the A/D to remove the
> unwanted signal.  This seems valid to me just wanted to make sure I
> wasn't missing something, based on all the comments on this one
> section I think I may be overlooking something.

The frequency of the spur will depend on its cause. You may be right,
but maybe not. Whatever, the spur originates in the analog front end.

Jerry
--
Engineering is the art of making what you want from things you can get.
&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;
```