dbd wrote:
...
> I wouldn't bet that any scheme couldn't be patented, but that says
> more about the patent system than the scheme.
Curses! Foiled again!
Jerry
--
Engineering is the art of making what you want from things you can get.
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Nyquist constrain and IQ represented signal
Started by ●November 14, 2007
Reply by ●November 14, 20072007-11-14
Reply by ●November 14, 20072007-11-14
"Ron N." <rhnlogic@yahoo.com> writes:> On Nov 14, 2:19 pm, Randy Yates <ya...@ieee.org> wrote: >> What sample rate is required for a signal with a 100-Hz bandwidth? > > How long do you plan on sampling relative to the sampling > error or noise level?The question was intended to be within the context of an ideal mathematical system using infinite-precision quantization, i.e., time-quantization only. -- % Randy Yates % "Rollin' and riding and slippin' and %% Fuquay-Varina, NC % sliding, it's magic." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Living' Thing', *A New World Record*, ELO http://www.digitalsignallabs.com
Reply by ●November 15, 20072007-11-15
"Steve Underwood" <steveu@dis.org> wrote in message news:fhg74v$o3f$1@home.itg.ti.com...> Fred Marshall wrote: >> "RobR" <masked@gmx.de> wrote in message >> news:-cydnXVvT4TpnqbanZ2dnUVZ_vOlnZ2d@giganews.com... >>> Hello, >>> >>> there is a question that bugs me for quite a long time: >>> >>> You can read about Nyquist constrain online, that to reconstruct all >>> frequencies within a signal, it has to be sampled with at least twice >>> the >>> bandwidth _or_ maximum frequency. >>> Maybe this _or_ is already the problem... >>> >>> Let's see, >>> I have an IQ branched digital signal. >>> So my maximum positive signal is not equal to the bandwidth of the >>> signal, >>> since also the negative frequencies are present, so the bandwidth of the >>> signal is twice the maximum positive frequency. >>> >>> Now what is the Nyquist frequency for such a signal representation? >>> Is it twice the max positive frequency or twice the bandwidth (4 times >>> max >>> pos. frequency)? >>> >>> From what i have seen, it is twice the max positive frequency. >>> But how can this be? Is it the IQ demodulation, that gives me twice the >>> information in contrast to a signal in "as it is" representation (one >>> real >>> stream)? >>> In fact, i need a clarification between nyquist adopted to a real signal >>> and nyquist adopted to a complex IQ representation signal. >>> >>> Best regards, >>> Robert >> >> First, try to keep it simple and also keep the terminology straight. >> >> I don't know what an IQ "branched" signal is.... anything would be >> conjecture. >> >> Start out with purely real signals. Let's call the bandwidth B - in the > > "Purely" real? Don't you mean degenerate complex signals, where the > permissible set of values is highly constrained? :-) > > It puzzles me when people treat complex like its a special case. All > numbers are complex. Reals are a subset of complex, where the j part is > always zero. Integers are a further subset, where the fractional part is > always zero. Cardinals are a further subset, which by act of faith > believes negative values don't exist - what else would you expect from a > Cardinal? :-). > > SteveComplex is a special case too, isn't it? i.e. what about i,j,k and other higher order spaces? Take your pick. So, not so puzzling after all. Fred
Reply by ●November 15, 20072007-11-15
RobR wrote: (snip)> If I take signal, first and second derivative, then I would need at least > third of nyquist? This steps could be carried on to infinity derivatives, > where I would need zero samples (asymptotically), right?I would have said one, but zero is pretty close to one. I once thought that the Weather Channel should, in addition to the current temperature also show the derivative. Maybe also the second derivative would be nice. And the third... -- glen
Reply by ●November 15, 20072007-11-15
Steve Underwood <steveu@dis.org> wrote in news:fhg74v$o3f$1@home.itg.ti.com:> It puzzles me when people treat complex like its a special case. All > numbers are complex. Reals are a subset of complex, where the j part > is always zero. Integers are a further subset, where the fractional > part is always zero. Cardinals are a further subset, which by act of > faith believes negative values don't exist - what else would you > expect from a Cardinal? :-). > > Steve >There is a book that does a great job of explaining why imaginary numbers are no more imaginary as real numbers. It's called "Asimov on Numbers". Its a great book for people who have learned way to much math by rote, and never really understood why. I gave my copy to a niece who was taking her third high school math course. Of course, what do we know; we all believe in imaginary numbers and negative frequencies, etc..... Al Clark Danville Signal Processing, Inc.
Reply by ●November 15, 20072007-11-15
On 14 Nov, 15:17, "RobR" <mas...@gmx.de> wrote:> Hello, > > there is a question that bugs me for quite a long time: > > You can read about Nyquist constrain online, that to reconstruct all > frequencies within a signal, it has to be sampled with at least twice the > bandwidth _or_ maximum frequency. > Maybe this _or_ is already the problem...No. The Nyquist criterion relates to bandwidth only. However, most people tend to automatically think in terms of baseband signals, in which case "bandwidth" and "maximum frequency" are the same.> Let's see, > I have an IQ branched digital signal. > So my maximum positive signal is not equal to the bandwidth of the > signal, > since also the negative frequencies are present, so the bandwidth of the > signal is twice the maximum positive frequency.Wrong. The usual form of Nyquist's criterion is based on the assumption that the signal is real-valued. That's the reason why the sampling frequency needs to be *twice* the bandwidth.> Now what is the Nyquist frequency for such a signal representation? > Is it twice the max positive frequency or twice the bandwidth (4 times max > pos. frequency)?Nyquist's criterion for a single sideband IQ signal (which formally is complex-valued) is Fs >= BW. Rune
Reply by ●November 15, 20072007-11-15
Fred Marshall wrote:> "Steve Underwood" <steveu@dis.org> wrote in message > news:fhg74v$o3f$1@home.itg.ti.com... >> Fred Marshall wrote: >>> "RobR" <masked@gmx.de> wrote in message >>> news:-cydnXVvT4TpnqbanZ2dnUVZ_vOlnZ2d@giganews.com... >>>> Hello, >>>> >>>> there is a question that bugs me for quite a long time: >>>> >>>> You can read about Nyquist constrain online, that to reconstruct all >>>> frequencies within a signal, it has to be sampled with at least twice >>>> the >>>> bandwidth _or_ maximum frequency. >>>> Maybe this _or_ is already the problem... >>>> >>>> Let's see, >>>> I have an IQ branched digital signal. >>>> So my maximum positive signal is not equal to the bandwidth of the >>>> signal, >>>> since also the negative frequencies are present, so the bandwidth of the >>>> signal is twice the maximum positive frequency. >>>> >>>> Now what is the Nyquist frequency for such a signal representation? >>>> Is it twice the max positive frequency or twice the bandwidth (4 times >>>> max >>>> pos. frequency)? >>>> >>>> From what i have seen, it is twice the max positive frequency. >>>> But how can this be? Is it the IQ demodulation, that gives me twice the >>>> information in contrast to a signal in "as it is" representation (one >>>> real >>>> stream)? >>>> In fact, i need a clarification between nyquist adopted to a real signal >>>> and nyquist adopted to a complex IQ representation signal. >>>> >>>> Best regards, >>>> Robert >>> First, try to keep it simple and also keep the terminology straight. >>> >>> I don't know what an IQ "branched" signal is.... anything would be >>> conjecture. >>> >>> Start out with purely real signals. Let's call the bandwidth B - in the >> "Purely" real? Don't you mean degenerate complex signals, where the >> permissible set of values is highly constrained? :-) >> >> It puzzles me when people treat complex like its a special case. All >> numbers are complex. Reals are a subset of complex, where the j part is >> always zero. Integers are a further subset, where the fractional part is >> always zero. Cardinals are a further subset, which by act of faith >> believes negative values don't exist - what else would you expect from a >> Cardinal? :-). >> >> Steve > > Complex is a special case too, isn't it? i.e. what about i,j,k and other > higher order spaces? Take your pick. So, not so puzzling after all. >Are you thinking of things like Quaternions when you say i,j,k? They are a pretty bastardised form of number at best. I think they are far from being a more generalised form of number than a complex one. Complex numbers are pure and wholesome. Steve
Reply by ●November 15, 20072007-11-15
On Nov 14, 2:17 pm, "RobR" <mas...@gmx.de> wrote:> Hello, > > there is a question that bugs me for quite a long time: > > You can read about Nyquist constrain online, that to reconstruct all > frequencies within a signal, it has to be sampled with at least twice the > bandwidth _or_ maximum frequency. > Maybe this _or_ is already the problem... > > Let's see, > I have an IQ branched digital signal. > So my maximum positive signal is not equal to the bandwidth of the > signal, > since also the negative frequencies are present, so the bandwidth of the > signal is twice the maximum positive frequency. > > Now what is the Nyquist frequency for such a signal representation? > Is it twice the max positive frequency or twice the bandwidth (4 times max > pos. frequency)? > > From what i have seen, it is twice the max positive frequency. > But how can this be? Is it the IQ demodulation, that gives me twice the > information in contrast to a signal in "as it is" representation (one real > stream)? > In fact, i need a clarification between nyquist adopted to a real signal > and nyquist adopted to a complex IQ representation signal. > > Best regards, > RobertThe replies to this post are already more clever than I am, but let me explain what I think I know. The sampling theorem, as expressed by Brigham (in "The Fast Fourier Transform") says that if the Fourier Transform of a function h(t) is zero for all frequencies greter than fc, then h(t) can be uniquely determined from its sampled values, where the sample interval Y =1/2fc or less. Brigham then goes on to use the term band-limited (rather than bandwidth), and explains 'band llimited' as "a shortened way of saying the Fourier transform is zero for |f| > fc (that is, abs(f) > fc). So this clearly defines the bandwidth as 2 * fc. Now to the complex versus real. The Fourier transform is defined in terms of sinusoids, specifically complex sinusoids that can be expressed in the form exp(jwt+p) where j is sqrt(-1). I have for some time taken this to imply that all signals processed by an FT are assumed to have complex values. If a signal has only real values (strictly, if its imaginary values are all zero), then its Fourier transform is even. That is, it is symmetric abou tthe centre frequency. If the centre frequency is 0 then it is symmetric about zero. So we can get into the habit of thinking only of the positive frequencies, in which case we might think of the bandwidth as fc rather than 2*fc. I think the symmetry may be what is referred to elsewhere in this thread as 'unused frequencies'. They are 'unused' in the sense that, if we only measure real values then we have to assume a special kind of signal whose imaginary values are zero. If we think of a complex sinusoid as a vector (in Argand space) rotating at the angular frequency, then a signal with only real values either has very peculiar amplitude modulation or is composed of contra-rotating complex sinusoids such that the imaginary parts always exactly cancel out. Hence, we have assumed something about the spectrum and in doing so have restricted the possible interpretations. (Of course if the signal really does have only real values then the assumption is correct - the point is, that the assumption is an imposition). If we measure a complex signal (for instance measuring I and Q) then we are measuring more information and not making an assumption. This then lets us distinguish positive from negative frequencies, and so the negative half of the spectrum is no longer a simple reflection of the postiive. I do not think this is new information because of the complex samples. Rtaher, I think the imposition of zero imaginary values restricts the set of functions that can be matched such that their spectra are symmetric. A symmetric frequency spectrum does not have 'less' information: it has the valid information that the signal has a symmetric spectrum. This only becomes confusing if in fact we 'know' the signal does not have a symmetric spectrum - in which case we should have done the I,Q sampling or equivalent. Of course if the signal really is real only, then other transfoms may be appropriate - such as a cosine transform. I hope this helps, Chris ====================== Chris Bore BORES Signal Processing www.bores.com If the signal has complex values, then its transform is not symmetric and
Reply by ●November 15, 20072007-11-15
Steve Underwood wrote:> Fred Marshall wrote: >> "Steve Underwood" <steveu@dis.org> wrote in message >> news:fhg74v$o3f$1@home.itg.ti.com... >>> Fred Marshall wrote: >>>> "RobR" <masked@gmx.de> wrote in message >>>> news:-cydnXVvT4TpnqbanZ2dnUVZ_vOlnZ2d@giganews.com... >>>>> Hello, >>>>> >>>>> there is a question that bugs me for quite a long time: >>>>> >>>>> You can read about Nyquist constrain online, that to reconstruct all >>>>> frequencies within a signal, it has to be sampled with at least >>>>> twice the >>>>> bandwidth _or_ maximum frequency. >>>>> Maybe this _or_ is already the problem... >>>>> >>>>> Let's see, >>>>> I have an IQ branched digital signal. >>>>> So my maximum positive signal is not equal to the bandwidth of the >>>>> signal, >>>>> since also the negative frequencies are present, so the bandwidth >>>>> of the >>>>> signal is twice the maximum positive frequency. >>>>> >>>>> Now what is the Nyquist frequency for such a signal representation? >>>>> Is it twice the max positive frequency or twice the bandwidth (4 >>>>> times max >>>>> pos. frequency)? >>>>> >>>>> From what i have seen, it is twice the max positive frequency. >>>>> But how can this be? Is it the IQ demodulation, that gives me twice >>>>> the >>>>> information in contrast to a signal in "as it is" representation >>>>> (one real >>>>> stream)? >>>>> In fact, i need a clarification between nyquist adopted to a real >>>>> signal >>>>> and nyquist adopted to a complex IQ representation signal. >>>>> >>>>> Best regards, >>>>> Robert >>>> First, try to keep it simple and also keep the terminology straight. >>>> >>>> I don't know what an IQ "branched" signal is.... anything would be >>>> conjecture. >>>> >>>> Start out with purely real signals. Let's call the bandwidth B - in >>>> the >>> "Purely" real? Don't you mean degenerate complex signals, where the >>> permissible set of values is highly constrained? :-) >>> >>> It puzzles me when people treat complex like its a special case. All >>> numbers are complex. Reals are a subset of complex, where the j part >>> is always zero. Integers are a further subset, where the fractional >>> part is always zero. Cardinals are a further subset, which by act of >>> faith believes negative values don't exist - what else would you >>> expect from a Cardinal? :-). >>> >>> Steve >> >> Complex is a special case too, isn't it? i.e. what about i,j,k and >> other higher order spaces? Take your pick. So, not so puzzling after >> all. >> > Are you thinking of things like Quaternions when you say i,j,k? They are > a pretty bastardised form of number at best. I think they are far from > being a more generalised form of number than a complex one. > > Complex numbers are pure and wholesome.Wholesome, but not whole. Fred is right; what about the third dimension? If you don't like quaternions, what about vector analysis? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 15, 20072007-11-15
Rune Allnor wrote: ...> Nyquist's criterion for a single sideband IQ signal > (which formally is complex-valued) is Fs >= BW.Almost. Fs > BW. When the inequality isn't large enough, there are at least two practical difficulties. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
