Hi, When an analog signal is sampled, we get replications in the digtal frequency domain. But when the same analog signal is analysed in analog frequency domain, we do not have those replications. I know this can be proved mathematically. But I want to know the physical explanation for the replications after sampling the analog signal. Thanks., Sandeep
Physical significance: Replications of analog signal in digtial domain
Started by ●September 23, 2004
Reply by ●September 23, 20042004-09-23
sandeep_mc81@yahoo.com (Sandeep Chikkerur) wrote in message news:<d5d88eb5.0409221945.66654d46@posting.google.com>...> Hi, > > When an analog signal is sampled, we get replications in the digtal > frequency domain. But when the same analog signal is analysed in > analog frequency domain, we do not have those replications. > > I know this can be proved mathematically. But I want to know the > physical explanation for the replications after sampling the analog > signal.There is no *physical* explanation. The physical world is analog. The "discrete" or "sampled" world is mathematical, not physical, so only the rules of mathemathics (as opposed to the rules of physics) apply to sampled signals. Sampled signals happen to belong to that part of the mathemathical world that has no physical counterpart, which in turn means that there are no *physical* interpretation of the peculiarities of sampled (as opposed to analog) signals. Ouch! After having re-read what I just wrote, I am glad the weekend is not all that far away! Rune
Reply by ●September 23, 20042004-09-23
Sandeep, No doubt you're familiar with the duality between time and frequency. For example, a sinc function in time is a rectangle in frequency and a rectangle in time is a sinc in frequency. The list goes on and on... You've just stumbled onto another big example of duality that many people never realize. That is, sampling in time causes periodicity in frequency and similarly sampling in frequency causes periodicity in time. You may want to read that one slowly a few times. ;) A couple examples of this are the Fourier Series as well as the Discrete Fourier Transform. With the Fourier series you are starting with a periodic function in time and that corresponds to a summation of discrete frequencies. Starting to see the connection? To go even further, remember that when multiplying the DFTs of two time domain signals you end up with the periodic-convolution of the two signals. That's because the DFT is taking SAMPLES of the full frequency response (i.e. the DTFT) which in turn results in periodicity of your time domain sequence. Great question! I hope this helps clarify! Brad "Sandeep Chikkerur" <sandeep_mc81@yahoo.com> wrote in message news:d5d88eb5.0409221945.66654d46@posting.google.com...> Hi, > > When an analog signal is sampled, we get replications in the digtal > frequency domain. But when the same analog signal is analysed in > analog frequency domain, we do not have those replications. > > I know this can be proved mathematically. But I want to know the > physical explanation for the replications after sampling the analog > signal. > > Thanks., > Sandeep
Reply by ●September 23, 20042004-09-23
Sandeep Chikkerur wrote:> Hi, > > When an analog signal is sampled, we get replications in the digtal > frequency domain. But when the same analog signal is analysed in > analog frequency domain, we do not have those replications. > > I know this can be proved mathematically. But I want to know the > physical explanation for the replications after sampling the analog > signal. > > Thanks., > SandeepAll those replications appear in the analog spectrum of a signal chopped at the same frequency used in the sampling. They are not mathematical fictions; you can tune a radio to one of them. It's called "modulation". Jerry -- ... they proceeded on the sound principle that the magnitude of a lie always contains a certain factor of credibility, ... and that therefor ... they more easily fall victim to a big lie than to a little one ... A. H. �����������������������������������������������������������������������
Reply by ●September 23, 20042004-09-23
Rune Allnor wrote:> sandeep_mc81@yahoo.com (Sandeep Chikkerur) wrote in message news:<d5d88eb5.0409221945.66654d46@posting.google.com>... > >>Hi, >> >>When an analog signal is sampled, we get replications in the digtal >>frequency domain. But when the same analog signal is analysed in >>analog frequency domain, we do not have those replications. >> >>I know this can be proved mathematically. But I want to know the >>physical explanation for the replications after sampling the analog >>signal. > > > There is no *physical* explanation. The physical world is analog. > The "discrete" or "sampled" world is mathematical, not physical, so > only the rules of mathemathics (as opposed to the rules of physics) > apply to sampled signals. Sampled signals happen to belong to that > part of the mathemathical world that has no physical counterpart, > which in turn means that there are no *physical* interpretation of > the peculiarities of sampled (as opposed to analog) signals.I respectfully disagree. Sampling of real signals is accomplished by periodically opening and closing a switch. The replications exist in the sampled signal, not just in the numbers that represent the samples. Drive the control input of a CMOS transmission gate with a pulse train while a signal is connected to its signal input. Connect the signal output to a spectrum analyzer. You will see the replications. They are created by the switching process, not by equations. Jerry -- ... they proceeded on the sound principle that the magnitude of a lie always contains a certain factor of credibility, ... and that therefor ... they more easily fall victim to a big lie than to a little one ... A. H. �����������������������������������������������������������������������
Reply by ●September 23, 20042004-09-23
On Thu, 23 Sep 2004 12:55:36 GMT, "Brad Griffis" <bradgriffis@hotmail.com> wrote:>Sandeep, > >No doubt you're familiar with the duality between time and frequency. For >example, a sinc function in time is a rectangle in frequency and a rectangle >in time is a sinc in frequency. The list goes on and on... > >You've just stumbled onto another big example of duality that many people >never realize. That is, sampling in time causes periodicity in frequency >and similarly sampling in frequency causes periodicity in time. You may >want to read that one slowly a few times. ;)Nice explanation, I've not seen that mentioned in a while. Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions. http://www.ericjacobsen.org
Reply by ●September 23, 20042004-09-23
Eric Jacobsen wrote:> On Thu, 23 Sep 2004 12:55:36 GMT, "Brad Griffis" > <bradgriffis@hotmail.com> wrote: > > >>Sandeep, >> >>No doubt you're familiar with the duality between time and frequency. For >>example, a sinc function in time is a rectangle in frequency and a rectangle >>in time is a sinc in frequency. The list goes on and on... >> >>You've just stumbled onto another big example of duality that many people >>never realize. That is, sampling in time causes periodicity in frequency >>and similarly sampling in frequency causes periodicity in time. You may >>want to read that one slowly a few times. ;) > > > Nice explanation, I've not seen that mentioned in a while. >It is why rotating machines, periodic in rotational position, only have discrete frequencies. And wheels go "backwards" on film because the time is discrete even though the frequencies can be above the sampling rate. When the Fourier theory is presented you have continuous unbounded time and continuous unbounded frequencies. So you have the Fourier Integral. Then the theory is applied to rotating machines. The result is Fourier Series. Continuous periodic time and discrete unbounded frequencies. Then the theory is applied to sampled time. The result if often called Fourier Sequences. Discrete unbounded time and continuous periodic frequencies. Except only the first period of frequencies gets discussed to any extent. The theory skips the uninteresting case of sampled periodic time because it is just matrix multiplication and one uses computers to do that sort of trivial work. But then the complexity folks had a field day and the result is the Fast Fourier Transform. (Historically the FFT the one of the things that showed that complexity was a real subject but who cares about real history. ;-)) Discrete periodic time and discrete periodic frequency. No interesting math (i.e. limits) because it is all finite. But there is this curious folding in both time and frequency so that the Fourier Integral "intuition" can give results that are a bit corrupted. Of course computers can not "do" continuous things so this is the only version that actually gets implemented.> > Eric Jacobsen > Minister of Algorithms, Intel Corp. > My opinions may not be Intel's opinions. > http://www.ericjacobsen.org
Reply by ●September 23, 20042004-09-23
Rune Allnor wrote:> There is no *physical* explanation. The physical world is analog. > The "discrete" or "sampled" world is mathematical, not physical, so > only the rules of mathemathics (as opposed to the rules of physics) > apply to sampled signals. Sampled signals happen to belong to that > part of the mathemathical world that has no physical counterpart, > which in turn means that there are no *physical* interpretation of > the peculiarities of sampled (as opposed to analog) signals.In quantum mechanics, which describes the physical world, many things are discrete (sampled), not continuous (analog). Question: (I saw this on a real homework assignment.) How many different velocities can a baseball pitcher pitch a ball inside a baseball stadium. You can supply numbers for the mass of the ball and diameter of the stadium. There is a discrete set of velocities, depending on the size of the stadium and the mass of the ball, at least in the horizontal directions. -- glen
Reply by ●September 24, 20042004-09-24
Jerry Avins <jya@ieee.org> wrote in message news:<4152df71$0$4044$61fed72c@news.rcn.com>...> Rune Allnor wrote: > > > sandeep_mc81@yahoo.com (Sandeep Chikkerur) wrote in message news:<d5d88eb5.0409221945.66654d46@posting.google.com>... > > > >>Hi, > >> > >>When an analog signal is sampled, we get replications in the digtal > >>frequency domain. But when the same analog signal is analysed in > >>analog frequency domain, we do not have those replications. > >> > >>I know this can be proved mathematically. But I want to know the > >>physical explanation for the replications after sampling the analog > >>signal. > > > > > > There is no *physical* explanation. The physical world is analog. > > The "discrete" or "sampled" world is mathematical, not physical, so > > only the rules of mathemathics (as opposed to the rules of physics) > > apply to sampled signals. Sampled signals happen to belong to that > > part of the mathemathical world that has no physical counterpart, > > which in turn means that there are no *physical* interpretation of > > the peculiarities of sampled (as opposed to analog) signals. > > I respectfully disagree. Sampling of real signals is accomplished by > periodically opening and closing a switch. The replications exist in the > sampled signal, not just in the numbers that represent the samples. > Drive the control input of a CMOS transmission gate with a pulse train > while a signal is connected to its signal input. Connect the signal > output to a spectrum analyzer. You will see the replications. They are > created by the switching process, not by equations.You really made me think there, Jerry. The explanation I came up with, is that what you describe is some sort of 'degenerated' Amplitude Modulation of the signal, where the sum of the multiple AM carriers happen to make up a pulse train[*]. As far as I can see (proviso to how clear one can see at 5 AM, when I woke up and found myself thinking about this), the 'pulse AM' explanation checks out with your observations and also with the sampling theorem, since these short pulses in the limit become Dirac delta functions. At least before reaching the Dirac limit, the explanation checks out with the pulse train having lots of figher harmonics, thus generating 'aliased' images. So what we have achieved here, is to equate 'sampling' to 'weird AM modulation'. I'm not sure how helpful that would be to somebody who struggles to understand the properties of sampled signals, though... Nah, I prefer to stick with my "weird mathemathical world" view on sampled signals. Rune [*] A train of continuous-time rectangular pulses has a Fourier representation that contains lots of odd harmonics. This is breafly mentioned on pages 121-123 in Haykin's "Communication systems", 1983, in the section "Switching [AM] modulator". Haykin actually makes a point of that the higher harmonics of the switch period has to be filtered out to produce a clean AM signal.
Reply by ●September 24, 20042004-09-24
On Thu, 23 Sep 2004 17:08:34 -0700, glen herrmannsfeldt <gah@ugcs.caltech.edu> wrote:> >(snipped)> >In quantum mechanics, which describes the physical world, >many things are discrete (sampled), not continuous (analog). > >Question: (I saw this on a real homework assignment.) > >How many different velocities can a baseball pitcher pitch >a ball inside a baseball stadium. You can supply numbers >for the mass of the ball and diameter of the stadium. > >There is a discrete set of velocities, depending on the size >of the stadium and the mass of the ball, at least in the >horizontal directions. > >-- glenHi Glen, fascinating, fascinating, fascinating! Glen, can ya' give me a brief explanation of this amazing (to me at least) idea? I'm (very) vaguely familiar with the ideas that energy is quantized, and mass is quantized. But is distance also quantized? Is there such a thing as an "absolute minimum distance"? Can it be(!)? Thanks, [-Rick-]
