Talk:Aliasing

Contents

1 An example in astronomy 2 "Filming a spoked wheel" 3 Notation and fonts (four kinds of L?) 4 Aliasing and radio crosstalk 5 Section numbering 6 Aliasing in computer science 7 Unidentified request 8 Wikipedia bug (math in headers?) 9 Picture of aliasing 10 Length of technical section 11 Sound example 12 The Nyquist criterion is in fact simplistic...see below

An example in astronomy

I added this section as a further example that the conventional wisdom of sinc filters isn't always true. Sinc filtering the measured image g, regardless which sinc filter is used, does not lead to any accurate measurement of the radius of the star.

"Filming a spoked wheel"

In my experience, this also happens when watching a spoked wheel. Can someone confirm? (or is this maybe evidence that I'm living in the Matrix... hm, my head hurts now ;-) -- Tarquin 15:26 Dec 20, 2002 (UTC)

This could be the case if your eye samples the scene - perhaps periferal vision does this. I sometimes get a related effect if I see a TV screen or other flickering source out of the corner of my eye - the flicker frequency appears to be much less than 50Hz. -- Easter 15:32 Dec 20, 2002 (UTC)

For really freaky effects, try watching a TV screen or CRT monitor while using an electric toothbrush. The image wobbles up and down, it was quite alarming the first time I saw it. -- Tarquin

This is what my colleagues in broadcast engineering call the ginger biscuit effect. Any vibration to the head will do. -- Easter 15:47 Dec 20, 2002 (UTC)

I think we need an article on this! -- Tarquin

This effect is caused by a failure of persistence of vision, although it could also be regarded as a kind of aliasing (with the vibration of your head providing the sampling frequency). I don't believe that the eye normally does any sampling in the time domain, at least not in a periodic way. Devices like CRT monitors rely on your persistence of vision, which is the slow response of your retina to changing or flickering images. This only works if your eye muscles can produce a stationary image on the retina. When you move your head faster than your eye muscles can track, as happens when you eat crunchy food, the TV image is spread out over your retina and appears fragmented, because parts of your retina see one field of the image and other parts see the next field, or the few milliseconds of darkness between fields. I'm guessing that this effect is more noticeable at the edge of the field of view, because the mechanism that stabilises the eyeball mainly uses data from the centre of the retina. -- Heron 11:30, 31 Mar 2004 (UTC)

Notation and fonts (four kinds of L?)

OK, so far, so mathematical, but the article now uses no fewer than four different kinds of L: can the sampling mappings please be called something different, to avoid confusion?

There's <math>L<math> and <math>L^1<math> and <math>L^2<math> and <math>\mathcal L<math>!

That's ok, but I'm not sure that <math>S_{point}<math> attempt is optimal. Perhaps <math>S_0<math> and <math>\mathcal S<math> or <math>S_1<math> would be better. You have to be careful to change all the references to <math>L<math> and <math>\mathcal L<math> if you do that, they are used consistently throughout the article. I'm going to wait and watch, but if you want me to do it, and have a notation suggestion, just let me know here. Loisel 00:40 Jan 28, 2003 (UTC)

I've just reverted my changes: they were worse, not better. I agree, I need to think of something better...

Yes, I agree, <math>S_0<math> and <math>S_1<math> would be better. Could you do it, please, I made a mess of my attempts to fix things last time.

Thank you, that's much better!

You're welcome. For reference purposes, let me add a detail. The standard notation for L^p spaces uses the cursive uppercase L, and it is fairly standard to also use block letter L and <math>\mathcal L<math> for linear maps over L^p (although <math>\Lambda<math> is very popular as well, even if it hints that it is a linear map into the field of scalars.) On the other hand, for a nonspecialist, perhaps using letters and symbol less similar to one another is helpful. Loisel 01:00 Jan 28, 2003 (UTC)

Aliasing and radio crosstalk

I have a question regarding this paragraph:

The term "aliasing" derives from the usage in radio engineering, where a radio signal could be picked up at two different positions on the radio dial in a superheterodyne radio: one where the local oscillator was above the radio frequency, and one where it was below. This is analagous to the frequency-space "wrapround" that is one way of understanding aliasing. However, there is a deeper way of understanding aliasing, based on continuity arguments, which is outlined below as an introduction.

This is very interesting to me. I am not completely certain that this crosstalk between radio stations is covered by the "outlined below as an introduction" portion. Unfortunately, I am not a physicist (or an engineer) so I don't know what's actually going on.

I think it would be great if someone who understands the long-winded L^2 stuff I wrote could tell me how that relates to the radio waves. I mean, the wrapping around of frequencies I describe depends completely on using a simple sampling scheme (like S_0) and so it's mostly for digital signal processing. In the analog world, I'm not exactly sure what's going on.

If anyone can give us more details about the underlying physics of the radio wave crosstalk phenomenon described above, perhaps there's a section in the aliasing article that needs to treat the analog aliasing process separately, which might be different from the dsp aliasing stuff Loisel 01:38 Jan 28, 2003 (UTC)

• Is this origin of the term accurate? Aliasing is usually associated with sampling, rather than AM modulation: namely when a sinusoidal signal is sampled at the wrong rate it becomes identical with (i.e. becomes an "alias" for) a sinusoid of another frequency.

Section numbering

The subsection numbers under "technical discussion" serve a purpose: the introduction to "technical discussion" refers to these sections by number. If you want to remove the subsection numbers, you'll have to change the introduction as well. For reference purposes, Encyclopedia Britannica uses numbering for some of its articles (my 1973 copy of World Wars has a complicated numbering system.) Loisel 18:49 Jan 28, 2003 (UTC)

I'm referring to this: In engineering, the method introduced in the third section is called sampling, while a method such as that introduced in the fifth section is called filtering. This discussion may be viewed as a theoretical introduction to the ideas of anti-aliasing. Loisel 18:51 Jan 28, 2003 (UTC)

• I deleted the following outline, since the automatic table-of-contents makes it largely superfluous:Jorge Stolfi 13:59, 29 Mar 2004 (UTC)

" First we will introduce a formal notion of "continuous signal". Since there are more than one possible choices (depending on the subject at hand), we will give some general outline, but fix our attention on a specific example for the purpose of this article. Second, we will give a notion of similarity of signals. Again, this precise notion depends on the underlying physical problem, but we will provide a common example for the sake of discussion. Third, we will give the most common sampling method as an example, and fourth we will show its failings. Fifth, we will give an improved sampling method that is more in-tune with the similarity notion introduced in the second section.
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In engineering, the method introduced in the third section is called sampling, while a method such as that introduced in the fifth section is called filtering.
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This discussion may be viewed as a theoretical introduction to the ideas of anti-aliasing."

Aliasing in computer science

An article titled "Bishop", after hundreds of words on concept of "bishop" used in religion, had a one-line comment that a piece in chess is called a "bishop", with an appropriate link. I moved that to the beginning of the article where it would actually be seen be anyone interested. I've done the same thing here with the meaning of "aliasing" in computer science. Michael Hardy 19:18 Jan 28, 2003 (UTC)

Unidentified request

Hfastedge, don't pollute carefully written articles with your requests. That's what the talk page is for. Loisel 07:29 30 Jun 2003 (UTC)

Wikipedia bug (math in headers?)

WHAT THE HELL HAPPENED? Loisel 17:55, 29 Jul 2003 (UTC)

If you mean the table of contents, see Wikipedia:Software updates. You can turn it off via your preferences if you don't like it. If you mean something else, you're going to have to say what. --Camembert

What's this 4LIQ9nXtiYFPCSfitVwDw7EYwQlL4GeeQ7qSO business? Evercat 17:59, 29 Jul 2003 (UTC)

These are unique hashes generated by Tomasz' math functions to identify the contents of a math element. I don't know why they appear when you just type "<math>" though. Ask User:Taw.—Eloquence 18:08, 29 Jul 2003 (UTC)

Please don't use <math> in headers. Use a proper substitution.—Eloquence 18:01, 29 Jul 2003 (UTC)

Can someone fix the mathematic notations in the article?

Picture of aliasing

What we need is ONE simple picture showing a sinusoid being sampled at too high rate and matching a lower-frequency sinusoid. Then we can probably delete some 10,000 words... Jorge Stolfi 20:46, 23 Mar 2004 (UTC)

Added such a picture.Jorge Stolfi 22:16, 23 Mar 2004 (UTC)

Length of technical section

The "Technical description" section was way too long and too heavy on math, it confused more than clarified the concept. Thus I have done some rather radical trimming and replanting.

Specifically, I moved most details of the "reconstruction" sections to a new page signal reconstruction, keeping only the definition of the "standard" reconstruction R. I also deleted the following paragraph since it was not germane to "aliasing"; it should go to some other page (Fourier analysis?):

" We note here that there is an efficient algorithm, known as the Fast Fourier transform to convert vectors between the canonical basis of <math>\Bbb C^n<math> and the Fourier basis <math>(d_k)<math>. This algorithm is significantly faster than the matrix multiplication required in the general case of change of basis. On the other hand, wavelets are often defined so that the change of basis matrix is sparse, and so again the change of basis algorithm is efficient. "

The following paragraph was deleted too; perhaps it should go to signal processing:

"The signal could arise from a variety of physical processes. For instance, one could measure the seismic movement of the ground with a seismograph. The output of a seismograph is a strip of paper known as a seismogram. This strip of paper can be interpreted as the graph of a function. This function will be in L² as defined above, and thus we obtain a mathematical signal from a physical process."

The following paragraphs did not seem to make sense: given that "signal" was defined as a *function*, it would seem that <math>S_0<math> is always well-defined in that case. Perhaps this text was assuming that a signal could be a *distribution* (such as, e.g., Dirac's)?

"The domain of <math>S_0<math> includes at least all continuous functions of <math>[0,1]<math>. On the other hand, for technical reasons, it is not clear how to extend <math>S_0<math> to all of <math>L^2<math>. In particular (and perhaps more telling) is that <math>S_0<math> is not continuous as a function on <math>L^2<math>.

Indeed, define <math>f_k<math> by

<math>f_k(x)=\left\{\begin{matrix} 1, & \mbox{if }x\mbox{ is in } \left[ 1-{1 \over k},1 \right], \\ 0, & \mbox{otherwise} \end{matrix}\right.<math>

Then, the norm <math>||f_k||<math> in <math>L^2<math> is <math>1/\sqrt k<math> and so <math>f_k<math> converges to zero. However, for any <math>k>n<math>, the vector <math>S_0f_k<math> is <math>(0,...,0,1)<math> and so <math>S_0f_k<math> does not converge to zero. Hence <math>S_0<math> is not continuous."

Therefore, the sampling function <math>S_0<math> very poorly represents our notion of closeness in our signal space <math>L^2<math>.

The following section has the same problem, and it also assumes non-trivial concepts of wavelets etc., so it should probably go elsewhere, too:

"For instance, it is possible to choose a reconstruction formula based on the Haar basis (see wavelets) in which case <math>S_1<math> does not fold any high frequencies into the lower frequencies. However, this reconstruction formula (or the Haar basis) are inappropriate to most problems.

If one is giving a reconstruction formula in terms of Hilbert bases, as is our case, then one can give a "perfect" filter, which does not fold any frequencies at all, in terms of convolutions.

This sampling method, unlike <math>S_0<math>, is defined over all of <math>L^2<math>. Also, by the Cauchy-Schwarz inequality (for instance,) <math>S_1<math> is also continuous in the root mean square norm. Hence, signals which alias to the same sampled vector will be related as far as the root mean square norm is concerned.

Finally the discussion of the operator <math>S_1<math> does not seem to be very useful. The operator does not eliminate aliasing, it only reduces it. On the other hand, when addressing this topic one MUST discuss the sinc filter (which does eliminate aliasing) and the Gaussian filter (which does a pretty good job, and is free from ringing). In any case, this material should be in anti-aliasing, not here.

No filtering algorithm "eliminates" aliasing. Whatever algorithm you use, you will never be able to recover an unexpectedly complex signal after sampling and filtering. In the classical setting, if you have 1000 samples but the signal is sin(10000x) you will not be able to recognize it. Instead, you will guess that it's some lower frequency signal -- this, regardless of your filtering algorithm. Hence sin(10000x) aliases to something else, regardless of your filtering algorithm. With that in mind, sinc filtering is optimal in the sense I described in the article, which is not always the correct meaning. If you look at my example in the caveats, a normalized linear polynomial over T, you will see that if you apply any of the filters you mention (S_1, Gaussian or sinc) you get a very poor representation of the original function, regardless how many samples you have. In that case, the "optimal" filtering algorithm is to recover z completely using two samples cleverly. Loisel 20:47, 8 Apr 2004 (UTC)

"<math>L^2<math> is contained in <math>L^1([0,1])<math> (see Lp spaces.) Hence, we can define a new "interval averaging" sampling method by

<math>S_1 f := n \left( \int_0^{1/n}f(t)dt, \int_{1/n}^{2/n}f(t)dt, ..., \int_{1-1/n}^1 f(t)dt \right)<math>

In the case of <math>S_1<math>, one can analyze, via convolutions, to which extend the high frequencies are "folded into" the low frequencies. They still are, but to a somewhat lesser extent."

Jorge Stolfi 01:55, 25 Mar 2004 (UTC)

• Jorge, obviously you made many changes without knowing what you were talking about. The paragraphs you mention above as not making any sense are in fact highly relevant. I will therefore undo some of your damage. The sampling function S_0 is in fact ill defined on L^2, which is the space of signals. This need to be mentioned. The remaining text actually said specifically what anti-aliasing was and up to what point it worked. Loisel 06:13, 29 Mar 2004 (UTC)
  • I have now fixed much of the damage. While the presentation is probably superior the way it is now, thanks to Jorge, I would request that unless you understand the details of the section marked "caveats", you refrain from further mangling this article. Loisel 06:58, 29 Mar 2004 (UTC)

• Loisel, I don't claim to be an expert in math and signal processing, but I don think I am illiterate either. Note that a signal was *defined* in the article to be a *function* from [0,1] to C; that L^2, as defined, is therefore a set of *functions*; and that the S_0 operator merely evaluates f at specific points. Thus S_0 f is *always* well-defined, not only for L^2 signals or continuous signals, but for *any* function f defined on [0,1], even e.g. the is-rational predicate.
   
  Presumably what you meant is that S_0 is not *continuous*: which is true, and indeed relevant for sampling theory and practice -- but not to explain aliasing, as far as I can see. So perhaps this observation should go to the signal sampling page?
   
  I also don't understand the point of introducing the S_1 operator in this page, since it is neither a mathematically correct way to avoid aliasing, nor what is done in practice (where one typically uses a Gaussian-like sampling kernel, for physical limitations and mathematical reasons). Again, perhaps the S_1 operator should go to the sampling or anti-aliasing page, as a pedagogical example to introduce the concept of general convolution sampling?Jorge Stolfi 13:31, 29 Mar 2004 (UTC)
  • Or perhaps S_1 could be moved to a later section, titled e.g. "Aliasing under convolution sampling" or some such?Jorge Stolfi 14:07, 29 Mar 2004 (UTC)

L^2 is usually defined as a space of functions. However, for technical reasons, if two functions f and g in L^2 agree everywhere but on a small set (for instance, if they disagree at a single point) then it is not possible to distinguish f and g in L^2. This is because L^2 is in fact the space specified in the article modulo a certain equivalence relation. If f and g in L^2 agree everywhere except on a set E of measure 0, then f and g are indistinguishable in L^2. This is because ||f-g||=0. In order for || || to be a norm, we have to guarantee that ||f||=0 only when f=0. This particular piece of information is useful, but belongs in the Lp spaces article; at best, a reference to the relevant bit of that article should be inserted.

This nuance is often explained at some point when initially doing L^p spaces, never to be mentioned again (this is for instance how Folland and Rudin do it.) While it may be good to explain this technicality somehow, my opinion is that it belongs in the L^p spaces article, not this one. If you beg to differ, go ahead and make the change; however, you may then have to explain some measure theory so that people understand when functions in L^2 are indistinguishable.

Also worth mentioning, if one wishes to dig into the details, is that the evaluation map x->f(x), while not defined for all x \in R, is defined for all x \in E where R\E is a set of zero measure and E is called the Lebesgue set of f. Unfortunately, E changes with f, and so the "Lebesgue set of L^2" is empty. Still, the evaluation map, for a fixed x and with f varying in some subspace of V of L^2 whose Lebesgue sets contain x, is not continuous in f \in V.

The S_1 operator is just another linear map from L^2 to C^n, like S_0. However, it is better than S_0 since it is actually defined on L^2 (this refers to the paragraph I just wrote, which explains why S_0 isn't clearly defined on L^2.) It is also a better filtering method than point sampling (the Fourier transform of the point sampling method fails to decay, but the uniform averaging method has a Fourier transform which decays like sinx/x (or 1/x if you prefer) which is not very good but better than nothing. A gaussian filter has a Fourier transform that decays like a gaussian, which is much better. Of course, a sinc filter has a Fourier transform that looks like a square wave, which is often considered perfect. All the filtering methods I just mentionned do help with anti-aliasing, even S_1. Loisel 06:52, 8 Apr 2004 (UTC)

Sound example

The sound example needs more explanation: what is the sampling rate, what exactly is meant by "bandlimited", and what should the listener pay attention to. Jorge Stolfi 18:43, 24 Mar 2004 (UTC)

• It is great now! Jorge Stolfi 13:53, 29 Mar 2004 (UTC)
  • Thanks to your suggestion. Glad you like the updated version. -- Tlotoxl 17:00, 29 Mar 2004 (UTC)

The Nyquist criterion is in fact simplistic...see below

Having specifically said that the Nyquist condition is simplistic, I don't think the following theory adequately explains why this is the case. I mean, it may explain it, but it doesn't specifically summarize why it is that the Nyquist criterion is therefore simplistic. I don't think readers should have to dig so much, only to find vague statements under caveats -- Tlotoxl 10:15, 31 Mar 2004 (UTC)

I added something at the end. Do you like it? Loisel 07:24, 8 Apr 2004 (UTC)