Reply by Egler, Mark ●August 12, 20022002-08-12
Bala,
I think your question about sub-Nyquist sampling was in reply to my posts
that Jeff kindly referenced. Your question used kilohertz (kHz) as units for
frequency while my post used only hertz (Hz), so if you thought I was saying
that you can successfully sample a 15-30 kHz signal with only 30 samples per
second (sps) then you're wrong; it would require 30 kilosamples per second
(ksps). The sample rate must be twice the band-WIDTH, i.e. Fs >
2*(Fmax-Fmin), AND the resulting aliased spectrum must map every input
frequency to a unique aliased frequency.

So in the first case, 15-30 Hz, sampling at 30 sps gives a reverse-mapped
spectrum from 15 to 0 Hz, i.e. 30 Hz input will map to 0 Hz, and 15 Hz input
maps to 15 Hz. Since the sampled signal has a periodic spectrum, the signal
can be reconstructed by converting back to continuous-time (i.e with a D-A
converter) and analog filtering with a band-pass filter with brick-wall
cut-off frequencies at 15 and 30 Hz. You probably need to draw the periodic
spectrum of the sampled signal to visualize this.

Now, in the second case, 30-45 Hz, sampling at 30 sps causes the 30 Hz input
to map to 0 Hz (DC), and the 45 Hz input becomes 15 Hz. This alias-mapping
is non-reversed, but as my second post noted, this does not really matter
because the same reconstruction approach still works: just change the
brick-walls of the analog band-pass filter to 30 Hz and 45 Hz.

Does that make sense?

Regards,
Mark

> -----Original Message-----
> From: Jeff Brower [mailto:]
> Sent: Tue, August 06, 2002 11:29 AM
> To: Daniar Hussain
> Cc: Bala Sekhar;
> Subject: Re: [matlab] sampling rate ???? > Daniar, Bala-
>
> Hey guys, Mark Egler's post reply to Prijit Debnath (copy
> below) from a few days ago
> is fantastic on this subject. I suggest you read it before
> you try to re-cover the
> ground he has already explained in detail.
>
> Jeff Brower
> DSP sw/hw engineer
> Signalogic
>
> > In order to understand the Nyquist rate, it is easiest to
> do so in the
> > FREQUENCY domain rather than in the time domain, which you
> are trying to
> > do.
> >
> > In the Freq. domain, the signal must be BAND-limited -- ie:
> zero beyond
> > some max. frequency. You must sample at TWICE this max.
> frequency in
> > order to capture all information. This is because in DT,
> the frequency
> > domain is periodic, and in order for frequencies not to
> overlap, you need
> > to sample at twice the max. frequency.
> >
> > I hope this makes sense.
> >
> > Daniar
> >
> > On Mon, 5 Aug 2002, bala sekhar wrote:
> >
> > > Hi all,
> > >
> > > can you please explain me some thing about sub-nyquist
> > > sampling.
> > >
> > > I cannot understand the concept of sub-nyquit
> > > sampling.
> > >
> > > suppose,the bandwidth is 15 Khz,sampling rate is 30
> > > sps
> > >
> > > if you need to sample in the range 15-30khz,it could
> > > be done without
> > > downsampling the signal to 0-15 khz.I am aware that
> > > the the frequency
> > > gets mapped properly and could be retrived back.
> > >
> > > In the case of 30-45Khz ,i read there is NO MAPPING of
> > > signals...
> > >
> > > questions:
> > > Could some one explain in detail how is this done?
> > > what is the exatly is happening in
> > > 1.15-30khz
> > > 2.30-45khz
> > >
> > > thank you
> > >
> > > regards,
> > > Bala.
>
Reply by Jeff Brower ●August 6, 20022002-08-06
Daniar, Bala-

Hey guys, Mark Egler's post reply to Prijit Debnath (copy below) from a few days
ago
is fantastic on this subject. I suggest you read it before you try to re-cover
the
ground he has already explained in detail.

Jeff Brower
DSP sw/hw engineer
Signalogic

> In order to understand the Nyquist rate, it is easiest to do so in the
> FREQUENCY domain rather than in the time domain, which you are trying to
> do.
>
> In the Freq. domain, the signal must be BAND-limited -- ie: zero beyond
> some max. frequency. You must sample at TWICE this max. frequency in
> order to capture all information. This is because in DT, the frequency
> domain is periodic, and in order for frequencies not to overlap, you need
> to sample at twice the max. frequency.
>
> I hope this makes sense.
>
> Daniar
>
> On Mon, 5 Aug 2002, bala sekhar wrote:
>
> > Hi all,
> >
> > can you please explain me some thing about sub-nyquist
> > sampling.
> >
> > I cannot understand the concept of sub-nyquit
> > sampling.
> >
> > suppose,the bandwidth is 15 Khz,sampling rate is 30
> > sps
> >
> > if you need to sample in the range 15-30khz,it could
> > be done without
> > downsampling the signal to 0-15 khz.I am aware that
> > the the frequency
> > gets mapped properly and could be retrived back.
> >
> > In the case of 30-45Khz ,i read there is NO MAPPING of
> > signals...
> >
> > questions:
> > Could some one explain in detail how is this done?
> > what is the exatly is happening in
> > 1.15-30khz
> > 2.30-45khz
> >
> > thank you
> >
> > regards,
> > Bala.


-------- Original Message --------
Subject: RE: [matlab] sampling rate ????
Date: Fri, 2 Aug 2002 11:29:40 -0400
From: "Egler, Mark" <>
To: "'" <>

On re-reading my post, I realized that the spectrum-reversed case is no more
difficult than the un-reversed case. The reconstruction in both situations
is to filter the sample stream with a band-pass filter that passes only the
original signal spectrum.

Mark Egler
> -----Original Message-----
> From: Egler, Mark
> Sent: Fri, August 02, 2002 11:03 AM
> To: '
> Subject: RE: [matlab] sampling rate ???? > Since no one else is forthcoming, I'll take a stab at this.
> In theory, the
> minimum sampling frequency required to capture all the
> information required
> to reconstruct the signal uniquely is twice the signal's bandwidth.
>
> For your example, 10-25 Hz is a bandwidth of 15 Hz, so you
> ought to be able
> to sample at 30 sps, not 50 sps as required if the signal was
> "low-pass" or
> "baseband", as you said. But there's a catch: the band-pass
> signal must
> first be down-converted (translated in frequency) from 10-25
> Hz down to 0-15
> Hz. In this case, that would be more trouble than the savings
> in sample rate
> would justify. The down-conversion is required so that no two
> frequencies in
> the signal spectrum alias to the same frequency between 0 and
> Fs/2. If each
> frequency has a unique alias frequency, there's no ambiguity in
> recontruction.
>
> But there are cases where the down-conversion isn't
> necessary. Suppose the
> signal was 15-30 Hz. Now you could sample at 30 sps and the
> resulting signal
> would be "aliased" to 0-15 Hz, but the aliasing doesn't cause
> information to
> be lost because the frequencies map unambiguously: 30 Hz
> becomes 0, 20 Hz
> becomes 10 Hz, 15 Hz becomes 15 Hz. The spectrum is reversed.
> To reconstruct
> this signal, you of course need to know that the samples
> represent a 15-30
> Hz signal, not a 0-15 Hz signal.
>
> Even easier is the case 30-45 Hz: just sample at 30 sps, and
> your samples
> will give the spectrum un-reversed, so reconstruction is more
> straight-forward. This approach is often referred to as "sub-Nyquist
> sampling".
>
> I think it's also important to point out that the theoretical
> minimum of
> twice the bandwidth is not a practical sample rate because to
> reconstruct a
> signal this way you would need an infinite number of samples, and an
> infinite delay for a causal system. In practice, you need to
> use a sample
> rate somewhat higher in order to recontruct with reasonable
> filters and
> delays. The other problem is that real signals are almost never truly
> band-limited, they typically have some energy beyond the
> stated bandwidth
> (especially if the "bandwidth" is given by -3dB points!).
>
> There are a lot of subtleties in this subject, witness the extended
> correspondence to the editor a few years ago in the IEEE
> Signal Processing
> Magazine (July 1995, Jan 1996, Sept 1996) when someone
> published an article
> claiming that he had "broken the Nyquist barrier" by using the above
> technique.
>
> Hope thats enough detail. If not, look up the references.
>
> Mark Egler
>
> > -----Original Message-----
> > From: prijit debnath [mailto:]
> > Sent: Wed, July 31, 2002 3:08 PM
> > To:
> > Subject: [matlab] sampling rate ????
> >
> >
> > hello,
> > i am a new student of dsp. we know for a
> > bandlimited signal,in order to avoid aliasing the
> > samplig rate should be >= 2* maxfrequency.....
> > but what should be the sampling rate for a
> > signal whose spectrum is from (10-25)hz (say)...
> > i.e. for a non-baseband signal....???????
> > Kindly reply in detail.
> > regards,
> > prijit



Reply by Daniar Hussain ●August 6, 20022002-08-06

In order to understand the Nyquist rate, it is easiest to do so in the
FREQUENCY domain rather than in the time domain, which you are trying to
do.

In the Freq. domain, the signal must be BAND-limited -- ie: zero beyond
some max. frequency. You must sample at TWICE this max. frequency in
order to capture all information. This is because in DT, the frequency
domain is periodic, and in order for frequencies not to overlap, you need
to sample at twice the max. frequency.

I hope this makes sense.

Daniar On Mon, 5 Aug 2002, bala sekhar wrote:

> Hi all,
>
> can you please explain me some thing about sub-nyquist
> sampling.
>
> I cannot understand the concept of sub-nyquit
> sampling.
>
> suppose,the bandwidth is 15 Khz,sampling rate is 30
> sps
>
> if you need to sample in the range 15-30khz,it could
> be done without
> downsampling the signal to 0-15 khz.I am aware that
> the the frequency
> gets mapped properly and could be retrived back.
>
> In the case of 30-45Khz ,i read there is NO MAPPING of
> signals...
>
> questions:
> Could some one explain in detail how is this done?
> what is the exatly is happening in
> 1.15-30khz
> 2.30-45khz
>
> thank you
>
> regards,
> Bala. >
>
> _____________________________________
> Note: If you do a simple "reply" with your email client, only the author of
this message will receive your answer. You need to do a "reply all" if you want
your answer to be distributed to the entire group.
>
> _____________________________________
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>
> To Join:
>
> To Post:
>
> To Leave:
>
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>
> More DSP-Related Groups: http://www.dsprelated.com/groups.php3
>
> ">http://docs.yahoo.com/info/terms/ >

Reply by bala sekhar ●August 5, 20022002-08-05
Hi all,

can you please explain me some thing about sub-nyquist
sampling.

I cannot understand the concept of sub-nyquit
sampling.

suppose,the bandwidth is 15 Khz,sampling rate is 30
sps

if you need to sample in the range 15-30khz,it could
be done without
downsampling the signal to 0-15 khz.I am aware that
the the frequency
gets mapped properly and could be retrived back.

In the case of 30-45Khz ,i read there is NO MAPPING of
signals...

questions:
Could some one explain in detail how is this done?
what is the exatly is happening in
1.15-30khz
2.30-45khz

thank you

regards,
Bala.



Reply by Egler, Mark ●August 2, 20022002-08-02
On re-reading my post, I realized that the spectrum-reversed case is no more
difficult than the un-reversed case. The reconstruction in both situations
is to filter the sample stream with a band-pass filter that passes only the
original signal spectrum.

Mark Egler
> -----Original Message-----
> From: Egler, Mark
> Sent: Fri, August 02, 2002 11:03 AM
> To: '
> Subject: RE: [matlab] sampling rate ???? > Since no one else is forthcoming, I'll take a stab at this.
> In theory, the
> minimum sampling frequency required to capture all the
> information required
> to reconstruct the signal uniquely is twice the signal's bandwidth.
>
> For your example, 10-25 Hz is a bandwidth of 15 Hz, so you
> ought to be able
> to sample at 30 sps, not 50 sps as required if the signal was
> "low-pass" or
> "baseband", as you said. But there's a catch: the band-pass
> signal must
> first be down-converted (translated in frequency) from 10-25
> Hz down to 0-15
> Hz. In this case, that would be more trouble than the savings
> in sample rate
> would justify. The down-conversion is required so that no two
> frequencies in
> the signal spectrum alias to the same frequency between 0 and
> Fs/2. If each
> frequency has a unique alias frequency, there's no ambiguity in
> recontruction.
>
> But there are cases where the down-conversion isn't
> necessary. Suppose the
> signal was 15-30 Hz. Now you could sample at 30 sps and the
> resulting signal
> would be "aliased" to 0-15 Hz, but the aliasing doesn't cause
> information to
> be lost because the frequencies map unambiguously: 30 Hz
> becomes 0, 20 Hz
> becomes 10 Hz, 15 Hz becomes 15 Hz. The spectrum is reversed.
> To reconstruct
> this signal, you of course need to know that the samples
> represent a 15-30
> Hz signal, not a 0-15 Hz signal.
>
> Even easier is the case 30-45 Hz: just sample at 30 sps, and
> your samples
> will give the spectrum un-reversed, so reconstruction is more
> straight-forward. This approach is often referred to as "sub-Nyquist
> sampling".
>
> I think it's also important to point out that the theoretical
> minimum of
> twice the bandwidth is not a practical sample rate because to
> reconstruct a
> signal this way you would need an infinite number of samples, and an
> infinite delay for a causal system. In practice, you need to
> use a sample
> rate somewhat higher in order to recontruct with reasonable
> filters and
> delays. The other problem is that real signals are almost never truly
> band-limited, they typically have some energy beyond the
> stated bandwidth
> (especially if the "bandwidth" is given by -3dB points!).
>
> There are a lot of subtleties in this subject, witness the extended
> correspondence to the editor a few years ago in the IEEE
> Signal Processing
> Magazine (July 1995, Jan 1996, Sept 1996) when someone
> published an article
> claiming that he had "broken the Nyquist barrier" by using the above
> technique.
>
> Hope thats enough detail. If not, look up the references.
>
> Mark Egler
>
> > -----Original Message-----
> > From: prijit debnath [mailto:]
> > Sent: Wed, July 31, 2002 3:08 PM
> > To:
> > Subject: [matlab] sampling rate ????
> >
> >
> > hello,
> > i am a new student of dsp. we know for a
> > bandlimited signal,in order to avoid aliasing the
> > samplig rate should be >= 2* maxfrequency.....
> > but what should be the sampling rate for a
> > signal whose spectrum is from (10-25)hz (say)...
> > i.e. for a non-baseband signal....???????
> > Kindly reply in detail.
> > regards,
> > prijit
> >
> >
> >
> > ------------------------ Yahoo! Groups Sponsor
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> > -------~->
> >
> > _____________________________________
> > Note: If you do a simple "reply" with your email client, only
> > the author of this message will receive your answer. You
> > need to do a "reply all" if you want your answer to be
> > distributed to the entire group.
> >
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> >
> > To Join:
> >
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> > To Leave:
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> the author of this message will receive your answer. You
> need to do a "reply all" if you want your answer to be
> distributed to the entire group.
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Reply by Egler, Mark ●August 2, 20022002-08-02
Since no one else is forthcoming, I'll take a stab at this. In theory, the
minimum sampling frequency required to capture all the information required
to reconstruct the signal uniquely is twice the signal's bandwidth.

For your example, 10-25 Hz is a bandwidth of 15 Hz, so you ought to be able
to sample at 30 sps, not 50 sps as required if the signal was "low-pass" or
"baseband", as you said. But there's a catch: the band-pass signal must
first be down-converted (translated in frequency) from 10-25 Hz down to 0-15
Hz. In this case, that would be more trouble than the savings in sample rate
would justify. The down-conversion is required so that no two frequencies in
the signal spectrum alias to the same frequency between 0 and Fs/2. If each
frequency has a unique alias frequency, there's no ambiguity in
recontruction.

But there are cases where the down-conversion isn't necessary. Suppose the
signal was 15-30 Hz. Now you could sample at 30 sps and the resulting signal
would be "aliased" to 0-15 Hz, but the aliasing doesn't cause information to
be lost because the frequencies map unambiguously: 30 Hz becomes 0, 20 Hz
becomes 10 Hz, 15 Hz becomes 15 Hz. The spectrum is reversed. To reconstruct
this signal, you of course need to know that the samples represent a 15-30
Hz signal, not a 0-15 Hz signal.

Even easier is the case 30-45 Hz: just sample at 30 sps, and your samples
will give the spectrum un-reversed, so reconstruction is more
straight-forward. This approach is often referred to as "sub-Nyquist
sampling".

I think it's also important to point out that the theoretical minimum of
twice the bandwidth is not a practical sample rate because to reconstruct a
signal this way you would need an infinite number of samples, and an
infinite delay for a causal system. In practice, you need to use a sample
rate somewhat higher in order to recontruct with reasonable filters and
delays. The other problem is that real signals are almost never truly
band-limited, they typically have some energy beyond the stated bandwidth
(especially if the "bandwidth" is given by -3dB points!).

There are a lot of subtleties in this subject, witness the extended
correspondence to the editor a few years ago in the IEEE Signal Processing
Magazine (July 1995, Jan 1996, Sept 1996) when someone published an article
claiming that he had "broken the Nyquist barrier" by using the above
technique.

Hope thats enough detail. If not, look up the references.

Mark Egler

> -----Original Message-----
> From: prijit debnath [mailto:]
> Sent: Wed, July 31, 2002 3:08 PM
> To:
> Subject: [matlab] sampling rate ???? > hello,
> i am a new student of dsp. we know for a
> bandlimited signal,in order to avoid aliasing the
> samplig rate should be >= 2* maxfrequency.....
> but what should be the sampling rate for a
> signal whose spectrum is from (10-25)hz (say)...
> i.e. for a non-baseband signal....???????
> Kindly reply in detail.
> regards,
> prijit >
> ------------------------ Yahoo! Groups Sponsor
> ---------------------~-->
> Will You Find True Love?
> Will You Meet the One?
> Free Love Reading by phone!
> http://us.click.yahoo.com/7dY7FD/R_ZEAA/Ey.GAA/wHYolB/TM
> --------------------------
> -------~->
>
> _____________________________________
> Note: If you do a simple "reply" with your email client, only
> the author of this message will receive your answer. You
> need to do a "reply all" if you want your answer to be
> distributed to the entire group.
>
> _____________________________________
> About this discussion group:
>
> To Join:
>
> To Post:
>
> To Leave:
>
> Archives: http://www.yahoogroups.com/group/matlab
>
> More DSP-Related Groups: http://www.dsprelated.com/groups.php3
>
> ">http://docs.yahoo.com/info/terms/


Reply by prijit debnath ●July 31, 20022002-07-31
hello,
i am a new student of dsp. we know for a
bandlimited signal,in order to avoid aliasing the
samplig rate should be >= 2* maxfrequency.....
but what should be the sampling rate for a
signal whose spectrum is from (10-25)hz (say)...
i.e. for a non-baseband signal....???????
Kindly reply in detail.
regards,
prijit