Phase of Alias

>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,
Sam

Jerry Avins wrote:
> 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.

Jerry,
My semantic point was only incidental.  Your description 
didin fact make it quite clear that you were describing 
digitisation.  The main point that I was making was a 
different one.  I was pointing out that it is the original 
analog sampling that produces the images, and not the 
digitisation.

Regards,
John

> 
>> 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

John Monro wrote:

   ...

> Jerry,
> My semantic point was only incidental.  Your description didin fact make 
> it quite clear that you were describing digitisation.  The main point 
> that I was making was a different one.  I was pointing out that it is 
> the original analog sampling that produces the images, and not the 
> digitisation.

I took that for granted. In one post, I mentioned that they would show 
in an analog spectrum analyzer. Of course, if the staircase is itself 
sampled at a sufficient rate, an FFT will show them also.

The form of the images is interesting. R.B-J. pointed out that the ZOH 
attenuates the higher ones. Another way to see this is that the 
difference between the staircase and the original waveform is very 
nearly a train of triangles that vary in amplitude sinusoidally, a 
quarter cycle plus half the width of step delayed from quantized 
sinusoid. This is what one would see if a triangular carrier were 
modulated by a sinusoid. The carrier images fall off as the square of 
the harmonic number, and so the 20th harmonic is 26 dB down without 
filtering.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
> 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.

Jerry,

The semantic point was only incidental, and was not the main 
point of my posting.  Your description had in fact make it 
quite clear that you were describing the process of 
digitising, but as I was going to be using the terms 
'sampling' and 'digitising' in my following argument then 
this semantic point had to be made first.  I went on to say:

"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."

In other words, sampling (and aliasing, if any) occurs when 
the analog samples are taken.  Digitising can not produce 
aliases.

Regards,
John


> 
>> 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

Jerry Avins wrote:
> 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


Jerry,

The semantic point was only incidental, and was not the main 
point of my posting.  Your description had in fact made it 
quite clear that you were describing the process of 
digitising.  As I was going to be using the terms 'sampling' 
and 'digitising' in my following argument then I had to 
first make it clear that I would be using the terms in the 
usual way.

The point I was making was that sampling (and aliasing, if 
any) occurs when the analog samples are taken.  I was 
disputing your claim that digitising can produce aliases.

Regards,
John