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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 |
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sampling rate ????
Started by ●July 31, 2002
Reply by ●August 2, 20022002-08-02
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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 ●August 2, 20022002-08-02
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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 > > ---------------------~--> > > 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/ > > > > > > ------------------------ Yahoo! Groups Sponsor > ---------------------~--> > Access your PC just like Web Mail > http://us.click.yahoo.com/r5uw2C/zncEAA/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/ |
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Reply by ●August 5, 20022002-08-05
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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. |
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Reply by ●August 6, 20022002-08-06
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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. > > _____________________________________ > 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 ●August 6, 20022002-08-06
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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 ●August 12, 20022002-08-12
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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. > |
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