Re: About DFT

Fred Marshall wrote:
"John Monro" <johnmonro@xxxxxxxxxxxxxxx> wrote in message news:4340e1a2$0$21275$afc38c87@xxxxxxxxxxxxxxxxxxxxxxx

Fred Marshall wrote:

I like the semantics too - so here goes....

****one topic:

It bothers me when y'all use the term "shift" when you mean something like "creates an image", "sum and difference frequencies", etc. A frequency shift is just that and its time dual is multiplication by e^jw0t where w0 is the amount of shift. Note that a bonal fide "frequency shift" makes everything move in the same direction - *not* "shift down above zero" and "shift up below zero"! It also makes the time sequence thus frequency-shfited a complex time sequence in general. Thus my comment.....

*****new subject: 

(snip)

Fred


Fred,
When ah say "shift" ah mean "shift," same as in the DSP books.

For the record, my argument goes along these lines:

1. When you sample a real sinusoidal signal of frequency F, you are sampling a signal that can be usefully visualised in complex form with frequency components either side of zero, at -F and +F.


If you start with a real signal and sample it with a real sampling waveform then were is the "complex form" you refer to? Not in time, right?
The sinusoid has components at -F and +F before sampling, right?

Fred,
To answer your questions,
a. The 'complex form' refers to the way the signal is represented in the frequency domain.

b. The sinusoid does indeed have -F and +F components in the frequency domain representation before and after sampling.

2. The sampling waveform has components at 0, Fs, 2Fs, 3Fs ... (and a corresponding set of negative-frequency components which don't add anything to the discussion.)


Yup


3. The sampling process is a multiplication process and each frequency component in the unsampled signal is multiplied by each frequency component in the sampling waveform.


Ummmmm.. I would say the sampling process is a multiplication in time and a convolution in frequency. That is a *huge* difference.

- the sampling waveform (for simplicity) can be viewed as a sequence of unit samples.
- the FT of the sampling waveform is a sequence of samples at ...-2*Fs, -Fs, 0, Fs, 2*Fs...
The latter is convolved and not multiplied by the FT of the original continuous signal.

From your comments to (4) you do acknowledge that the result of the multiplication in the time domain is that the spectral components in the unsampled signal are shifted up by 0, Fs, 2Fs etc. A quick fiddle with diagrams on some scraps of paper confirms that the result is consistent with convolution in the frequency domain, so there is no problem here.

4. Multiplication in the time domain produces shifting in the frequency domain and as you say, "makes everything move in the same direction." Considering multiplication by the Fs component, the result is that:
a. The (-F) component moves up to (-F+Fs)
which is the (Fs-F) component.
b. The (+F) component moves up to (+F+FS)
which is the (Fs+F) component.


I take back what I said about the process not consisting of frequency "shifts". I learned something here. So the discussion was useful for me at least.

A simple frequency shift requires multiplication by a complex exponential that does not resolve to a real sine or cosine. And, the multiplication results in a complex time series.
Yes, but it 'aint all that simple.' The reason you use the complex signal is to suppress the negative frequencies to prevent them from playing havoc with the new signal. For example we want to shift the audio band up by 10Hz to minimise acoustic feedback in a sound system. The 1000 Hz components for example go to 1010 Hz, but if there are any negative frequency components present they go to 990 Hz, which results in a 20 Hz AM modulation effect.

If you're doing modulation with a real sinusoid then that's a sum of two complex exponentials and, by superposition, you get two shifted versions of the original spectrum. One is shifted up and one is shifted down. That's where "sum *and* difference frequencies" come from I do believe.
No, multiplication by Fs shifts all components up by Fs. The terms 'sum' and 'difference' are only a description of the result, not a prescription for achieving it.

So, if you have a signal -F and +F and you multiply by a cosine at Fs which is the sum of two complex exponentials at -Fs and +Fs, then you get a term which is a shift by Fs and a term which is a shift by -Fs. The number of spectral components must double if they don't overlap. You get:

-F-Fs, F-Fs which is the shifted spectrum due to multiplication by e^-j2*pi*Fs*t.
-F+Fs, F+Fs which is the shifted spectrum due to multiplication by e^j2*pi*Fs*t.

When you line these up in frequency you get terms at:

-Fs-F, -Fs+F,  Fs-F, Fs+F

And the possibly expected terms at -Fs and +Fs don't show up at all because this is multiplication by a complex exponential and not multiplication by a constant plus a sinusoid - as in normal AM modulation.
But of course the DC term is present in the sampling waveform, so the -F and +F components still pop up!

This is a good place to address your conclusion. You address the situation where F>Fs/2.
In that case we get the same result as above of course but we see that:
-Fs+F is between -Fs/2 and 0.
Fs-F is between 0 and Fs/2.
So, there are two terms in the result in the range -Fs/2 to Fs/2 - one due to the +F term at negative frequency and one due to the -F term at positive frequency.
Since this is still a real waveform or sequence then we must continue to carry the negative frequency term which is created by the positive frequency term +F.

Sure, there is no problem with that. 

Notes:
Both of the components shift in the same direction and by the same amount.


***And in the opposite direction and by the same amount.
None of your argument supports the above conclusion, and there is no mechanism for generating this effect.


Multiplication by the '0' sampling waveform component leaves a copy of the original -F and +F components. 

Yup


Multiplication by sampling component 2Fs, 3Fs ... generates components higher than the region of interest and are not considered further.

***Only if F<Fs/2. e.g. if F=1.9*Fs then you get a term at 0.1Fs.
True, but that was an explanation of why I was ignoring those cases. I think a reasonable assumption would be that we would not be attempting to sample a signal that high.

5. Apart from the baseband signal, the Fs-F component is the only component that present that can extend into the region between 0 and Fs/2 (depending on the magnitude of F).


Yabbut ... you ignore the necessary negative frequency component -Fs+F
To paraphrase what you've said above: "the -Fs+F component is the only compoent that's present that can extend into the region beween -Fs/2 and 0/"

You are quite right.
Although I thought it was safe to ignore the negative part of the sampled spectrum, it turns out that it is not. The output reconstruction filter is a simple real filter of course and includes all the component between -Fs/s and +Fs/2 (approx.) and so the negative components can't be ignored. The positive frequency intrudes into that region, as you point out. The embarrassing thing is that I think you or Jerry referred to that issue way back in the thread, so I must apoligise for being a little slow on the uptake.>
6. The Fs+F component can never go below Fs. 

Seems a trivial observation...


Conclusion: The Fs-F component is the sole cause of aliasing and is entirely due to the -F component in the unsampled signal.
QED. 


Nope ... as above.

I'm wondering if we aren't seeing things in different "frames". I'm avoiding complex time series, analytic signals, etc. If your frame of reference includes these things then that may be from where your viewpoint derives. Then the discussion needs to be a bit different if only for clarity.

Fred

I think what happened was that when I was thinking about the complex sampling case by background assumption was that only the components from 0 to Fs/s would contribute to the final output. You can do that in complex case, but it does not apply to real sampling and reconstruction.

Thanks for the enlightenment!

Regards,
John


.

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