Hi all, I have a time continuous sinewave which is passed through a sample and hold function. So the output is like a stair-case. The frequency spectrum shows the input at fin and the aliases at fs-fin,fs+fin,2fs-fin,2fs+fin and so on.. (where fin is the frequency of the sinewave and fs is the sampling frequency).The aliases are ofcourse attenuated by the sinc response. My question is what is the phase relation between fin and the aliases? Is there any mathematical description someone can guide me to. regards, Sam
Phase of Alias
Started by ●February 19, 2009
Reply by ●February 19, 20092009-02-19
SammySmith wrote:> Hi all, > I have a time continuous sinewave which is passed through a sample and > hold function. So the output is like a stair-case. The frequency spectrum > shows the input at fin and the aliases at fs-fin,fs+fin,2fs-fin,2fs+fin and > so on.. (where fin is the frequency of the sinewave and fs is the sampling > frequency).The aliases are ofcourse attenuated by the sinc response. > My question is what is the phase relation between fin and the aliases? > Is there any mathematical description someone can guide me to.There are at least two misconceptions here. For one, what you have are a train of images, not aliases. Aliases are all at or below fs/2 and arise when the staircase is reduced to amplitude numbers; that is, sampled. For another, "phase" is well defined only for signals of precisely the same frequency. In all other cases, you must state unambiguously the meaning you give the term. There is no general standard. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 19, 20092009-02-19
>SammySmith wrote: >> Hi all, >> I have a time continuous sinewave which is passed through a sample and >> hold function. So the output is like a stair-case. The frequencyspectrum>> shows the input at fin and the aliases at fs-fin,fs+fin,2fs-fin,2fs+finand>> so on.. (where fin is the frequency of the sinewave and fs is thesampling>> frequency).The aliases are ofcourse attenuated by the sinc response. >> My question is what is the phase relation between fin and the aliases?>> Is there any mathematical description someone can guide me to. > >There are at least two misconceptions here. For one, what you have are a>train of images, not aliases. Aliases are all at or below fs/2 and arise>when the staircase is reduced to amplitude numbers; that is, sampled. > >For another, "phase" is well defined only for signals of precisely the >same frequency. In all other cases, you must state unambiguously the >meaning you give the term. There is no general standard. > >Jerry >-- >Engineering is the art of making what you want from things you can get. >����������������������������������������������������������������������� >Hi Jerry, You are right, I was actually referring to images rather than aliases, Regarding the phase,u know if you have a nonlinear system described by lets say y(t)=a.x(t)+b.x(t)^2+c.x(t)^3...and a sine wave is input to this system, the phase of the 2nd harmonic at the output(=y) would be twice the phase of the input tone and similarly the 3rd harmoinc has 3 times the phase of the input tone. but if instead of this nonlinear system, I input the sine wave to a sample and hold system , than i dont get harmonics but only images as previoulsy mentioned.so the question is whether this image has a similar kind of phase relationship with the input sinewave as the harmonics of the nonlinear system have with the input. My guess is that probably if a relation does exists its not as simple as twice or thrice the input phase. Thanks, Sam
Reply by ●February 19, 20092009-02-19
On Feb 20, 6:45�am, "SammySmith" <eigenvect...@yahoo.com> wrote:> >SammySmith wrote: > >> Hi all, > >> I have a time continuous sinewave which is passed through a sample and > >> hold function. So the output is like a stair-case. The frequency > spectrum > >> shows the input at fin and the aliases at fs-fin,fs+fin,2fs-fin,2fs+fin > and > >> so on.. (where fin is the frequency of the sinewave and fs is the > sampling > >> frequency).The aliases are ofcourse attenuated by the sinc response. > >> My question is what is the phase relation between fin and the aliases? > >> Is there any mathematical description someone can guide me to. > > >There are at least two misconceptions here. For one, what you have are a > >train of images, not aliases. Aliases are all at or below fs/2 and arise > >when the staircase is reduced to amplitude numbers; that is, sampled. > > >For another, "phase" is well defined only for signals of precisely the > >same frequency. In all other cases, you must state unambiguously the > >meaning you give the term. There is no general standard. > > >Jerry > >-- > >Engineering is the art of making what you want from things you can get. > > > > Hi Jerry, > > You are right, I was actually referring to images rather than aliases, > Regarding the phase,u know if you have a nonlinear system described by lets > say y(t)=a.x(t)+b.x(t)^2+c.x(t)^3...and a sine wave is input to this > system, the phase of the 2nd harmonic at the output(=y) would be twice the > phase of the input tone and similarly the 3rd harmoinc has 3 times the > phase of the input tone. > but if instead of this nonlinear system, I input the sine wave to a sample > and hold system , than i dont get harmonics but only images as previoulsy > mentioned.so the question is whether this image has a similar kind of phase > relationship with the input sinewave as the harmonics of the nonlinear > system have with the input. My guess is that probably if a relation does > exists its not as simple as twice or thrice the input phase. > > Thanks, > SamWith a zero-order hold the phase relationship is perfectly linear. It jumps by -pi radians if I remember at the sampling frequency and continues. This is because of the sine term which changes sign every pi radians. Hardy
Reply by ●February 19, 20092009-02-19
SammySmith wrote:>> SammySmith wrote: >>> Hi all, >>> I have a time continuous sinewave which is passed through a sample and >>> hold function. So the output is like a stair-case. The frequency > spectrum >>> shows the input at fin and the aliases at fs-fin,fs+fin,2fs-fin,2fs+fin > and >>> so on.. (where fin is the frequency of the sinewave and fs is the > sampling >>> frequency).The aliases are ofcourse attenuated by the sinc response. >>> My question is what is the phase relation between fin and the aliases? > >>> Is there any mathematical description someone can guide me to. >> There are at least two misconceptions here. For one, what you have are a > >> train of images, not aliases. Aliases are all at or below fs/2 and arise > >> when the staircase is reduced to amplitude numbers; that is, sampled. >> >> For another, "phase" is well defined only for signals of precisely the >> same frequency. In all other cases, you must state unambiguously the >> meaning you give the term. There is no general standard.> Hi Jerry, > > You are right, I was actually referring to images rather than aliases, > Regarding the phase,u know if you have a nonlinear system described by lets > say y(t)=a.x(t)+b.x(t)^2+c.x(t)^3...and a sine wave is input to this > system, the phase of the 2nd harmonic at the output(=y) would be twice the > phase of the input tone and similarly the 3rd harmoinc has 3 times the > phase of the input tone.Don't you mean "frequency" where you write "phase"? What does it mean the phase of one frequency to be twice as much as the phase of another? Suppose the phase of the original is zero. What is the phase of the second harmonic?> but if instead of this nonlinear system, I input the sine wave to a sample > and hold system , than i dont get harmonics but only images as previoulsy > mentioned.so the question is whether this image has a similar kind of phase > relationship with the input sinewave as the harmonics of the nonlinear > system have with the input. My guess is that probably if a relation does > exists its not as simple as twice or thrice the input phase.You correctly wrote that the images are fs-fin, fs+fin, 2fs-fin, 2fs+fin etc. That relation is exact. Just do the arithmetic. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 19, 20092009-02-19
Jerry Avins wrote:> SammySmith wrote: >> Hi all, >> I have a time continuous sinewave which is passed through a sample and >> hold function. So the output is like a stair-case. The frequency spectrum >> shows the input at fin and the aliases at >> fs-fin,fs+fin,2fs-fin,2fs+fin and >> so on.. (where fin is the frequency of the sinewave and fs is the >> sampling >> frequency).The aliases are ofcourse attenuated by the sinc response. >> My question is what is the phase relation between fin and the aliases? >> Is there any mathematical description someone can guide me to. > > There are at least two misconceptions here. For one, what you have are a > train of images, not aliases. Aliases are all at or below fs/2 and arise > when the staircase is reduced to amplitude numbers; that is, sampled. > > For another, "phase" is well defined only for signals of precisely the > same frequency. In all other cases, you must state unambiguously the > meaning you give the term. There is no general standard. > > JerryI disagree on one point in the statement: "Aliases are all at or below fs/2 and arise when the staircase is reduced to amplitude numbers; that is, sampled." Sampling does not take place when the signal is "...reduced to amplitude numbers..." This is the process of digitising. If there is a 'staircase'present then the signal has already been sampled, and the samples exist in analog form. Because each sample is represented by a finite-width pulse (of duration 1/fs) then the spectrum of the sampled signal is modified by a sinc function,as the OT mentions. Regards, John
Reply by ●February 20, 20092009-02-20
John Monro wrote:> Jerry Avins wrote: >> SammySmith wrote: >>> Hi all, >>> I have a time continuous sinewave which is passed through a sample and >>> hold function. So the output is like a stair-case. The frequency >>> spectrum >>> shows the input at fin and the aliases at >>> fs-fin,fs+fin,2fs-fin,2fs+fin and >>> so on.. (where fin is the frequency of the sinewave and fs is the >>> sampling >>> frequency).The aliases are ofcourse attenuated by the sinc response. >>> My question is what is the phase relation between fin and the >>> aliases? Is there any mathematical description someone can guide me to. >> >> There are at least two misconceptions here. For one, what you have are >> a train of images, not aliases. Aliases are all at or below fs/2 and >> arise when the staircase is reduced to amplitude numbers; that is, >> sampled. >> >> For another, "phase" is well defined only for signals of precisely the >> same frequency. In all other cases, you must state unambiguously the >> meaning you give the term. There is no general standard. >> >> Jerry > > > I disagree on one point in the statement: "Aliases are all at or below > fs/2 and arise when the staircase is reduced to amplitude numbers; that > is, sampled." > > Sampling does not take place when the signal is "...reduced to amplitude > numbers..." This is the process of digitising.OK. I'll go with your semantics. Making the staircase is quantizing. Assigning a number to the height of a step is digitizing. You named the process, while I described it.> If there is a 'staircase' present then the signal has already been > sampled, and the samples exist in analog form. Because each sample is > represented by a finite-width pulse (of duration 1/fs) then the spectrum > of the sampled signal is modified by a sinc function,as the OT mentions.Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 20, 20092009-02-20
On Feb 19, 11:06�pm, Jerry Avins <j...@ieee.org> wrote: ...> Making the staircase is quantizing.it might be ZOHing. in fact, as i review the OP: "I have a time continuous sinewave which is passed through a sample and hold function", in this case, it was a ZOH that was making the staircase. so, i s'pose in that case, we ask the OP's question with the ZOH undone, for first consideration. then the phase of some alias component is the same as the phase of the component (likely the negative frequency component) that got folded over to the alias. then to reapply the ZOH, it's a 1/2 sample delay, which is a linear-phase (w.r.t. frequency) component added to that of the folded component. r b-j
Reply by ●February 20, 20092009-02-20
robert bristow-johnson wrote:> On Feb 19, 11:06 pm, Jerry Avins <j...@ieee.org> wrote: > ... >> Making the staircase is quantizing. > > it might be ZOHing. in fact, as i review the OP: "I have a time > continuous sinewave which is passed through a sample and hold > function", in this case, it was a ZOH that was making the staircase. > > so, i s'pose in that case, we ask the OP's question with the ZOH > undone, for first consideration. then the phase of some alias > component is the same as the phase of the component (likely the > negative frequency component) that got folded over to the alias. then > to reapply the ZOH, it's a 1/2 sample delay, which is a linear-phase > (w.r.t. frequency) component added to that of the folded component.Hold on! There are no aliases in the staircase or as long as the images exist. An analysis of the quantized will show aliases, but the staircase waveform fed into an analog waveform analyzer won't. Anyway, ZOHing is one way to achieve the quantizing and is clearly what the OP described. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 20, 20092009-02-20
On Feb 19, 11:55�pm, Jerry Avins <j...@ieee.org> wrote:> robert bristow-johnson wrote: > > On Feb 19, 11:06 pm, Jerry Avins <j...@ieee.org> wrote: > > ... > >> Making the staircase is quantizing. > > > it might be ZOHing. �in fact, as i review the OP: "I have a time > > continuous sinewave which is passed through a sample and hold > > function", in this case, it was a ZOH that was making the staircase. > > > so, i s'pose in that case, we ask the OP's question with the ZOH > > undone, for first consideration. �then the phase of some alias > > component is the same as the phase of the component (likely the > > negative frequency component) that got folded over to the alias. �then > > to reapply the ZOH, it's a 1/2 sample delay, which is a linear-phase > > (w.r.t. frequency) component added to that of the folded component. > > Hold on! There are no aliases in the staircase or as long as the images > exist.well, i was assuming that the OP meant "aliases" (as i understand the term). images are another issue. images didn't get moved around in frequency. but the ZOH does attenuate the images some amount. if the OP meant images. then what i said didn't apply. r b-j> An analysis of the quantized will show aliases, but the staircase > waveform fed into an analog waveform analyzer won't. Anyway, ZOHing is > one way to achieve the quantizing and is clearly what the OP described.
