Quaternions for filters

"Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." ? Lord Kelvin,

1892. (from the Wikipedia entry on Quaternions)

If you're attracted to quaternions because of the weirdo math -- back away slowly until you think no one is looking at you any more, then run like hell. Or get a PhD in math, and have fun. In general filtering problems they're totally useless -- complex numbers model signal processing tasks quite well; there's absolutely nothing that quaternions can add to that particular mix.

Don't even try to use them until you thoroughly understand them, the consequences of using them, and the implications of the fact that they are not commutative. I have used them, in a Kalman filter for a navigation application, and for that application they were the least of all the various evils that are available for keeping track of 3D rotations. But in and of themselves, quaternions offer nothing of use unless you're working in those self-same 3D rotations, and because of their character their use is exacting, tedious, mind-bending, and esoteric.

So -- they're fun stuff if you're a university mathematician and can use them to attract grant money, they're highly useful (in a sump-pump-to-a- sewer-worker sort of way) if you're doing 3D graphics or navigational solutions, and a useless curiosity otherwise.

--
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

In addition to the other Tim's comments :), I would add: there isn't much (anything?) you can do with [hyper]complex numbers that matrices of sufficient order cannot. For instance, linear algebra on matrices of the form, IIRC, [ a b] z = a + bi --> [-b a]

are indistiguishable from complex numbers (including commutativity and division -- this matrix is nonsingular (= inverse exists) for (a = b) != 0).

The complex number e^(i*theta) is useful in rotations; the corresponding matrix is of course: [ cos theta sin theta] [-sin theta cos theta] Corresponding rotations in higher dimensions can be formed by incorporating this submatrix into a higher dimension's identity matrix. IIRC, this, in turn, is indistinguishable from quaternions, when using a 4x4 matrix with a

3x3 rotation, which looks something like: [ trig trig trig 0 ] [ trig trig trig 0 ] [ trig trig trig 0 ] [ 0 0 0 1 ] (I don't remember or care what the exact trigonometry is..) IIRC, the equivalent of a three-dimensional rotation holds w constant (in quaternion-space), hence the 'extra' row and column.

High dimension matrices are very helpful to discrete signal processing. Multiplication is numerical convolution, so filtering and frequency analysis are relatively simple. A set of samples of length N looks like a vector; if you multiply the vector, entry by entry, by different (sampled) sine waves, representing the Nth sample of the sine wave by the j'th column of a matrix, and the order of sine wave (fundamental or i'th harmonic) as the row, simply multiplying this vector of data by the matrix is a Fourier transform. And I have no proof for it, but evidently, and apparently, this matrix is self-inverse (that is to say, F{F{f(t)}} = f(t) -- the [normalized] Fourier transform "un-does" itself; likewise, denoting the Fourier transform matrix as F, F*F = 1, the NxN identity matrix). So you don't need to generate anything funny to go from frequency to time domain.

You'd never do this practically, of course; an FFT is O(N*log N) compexity, while (unoptimized) matrix multiplication is at least O(N^2), and usually O(N^3). But it's handy for small things, very easy on theory (linear algebra is very powerful, you can write a simple equation which actually encapsulates billions of computations), and a good way to get your feet wet in Matlab/Octave (whose fundamental data type is the matrix).

Tim

--
Deep Friar: a very philosophical monk.
Website: http://webpages.charter.net/dawill/tmoranwms

"Jeffery Tomas"  wrote in message 
news:jgg7vk$98a$1@dont-email.me...
> We know that most filters have amplitude and phase characteristics that 
> are represented by a complex function. Does anyone know if Quaternions 
> offer anything more than the complex numbers for filters, Fourier 
> transform, etc?
>

There seems to be some real geniuses hear that probably know everything about Quaternions without ever even passing math 101(not necessarily you specifically, but the others. No wonder I had them on ignore)

The fact is that Quaternions are useful in some areas that allow one to do more than complex numbers. Quaternions are very useful in 3D fractals.

1. Quaternions are an extension of the complex numbers. These fools that think extension are useless should be saying the same thing about complex numbers too, since, after all, they are just an extension of the reals... oh, they need to say it about rationales, reals, matrices, integrals, and just about everything else in mathematics.

1. Quaternions are a subset of matrices. They have additional properties/constraints that make them behave certain ways. You do point this out but it's better to think of quaternions in there own right... even if you don't like them much. This is important and what makes them useful. We could talk in terms of matrices... but if you want to advocate that then you should be doing it for the complex numbers too.

2. You are right for the most part about rotations. What makes the Fourier transform work seems to be the rotational aspect. If you try any arbitrary pseudo-Eulerian kernel in the Fourier transform you end up with another Fourier like transform(one that can at least be written as a FT). This made me wonder how Quaternions would work. They would transform a real signal into, effectively, R^4. We know that the FT transforms R to R^2. I wonder what 2 extra dimensions would yield? Again, see the 3D fractals.

(I don't know if it's commonly mentioned about these but any slice(intersection with a plane) of such a fractal yields a 2D view that is some view of the complex version.)

Most likely what a quaternion would offer is 2 additional phase relations. What these relations from an extended FT would represent physically would be what I am interested in.

I guess you will be even less keen on Clifford Algebras then which are a superset of all the common number systems and some others.

I know some physicists who are recasting physics in that notation with the occasional new insight occurring along the way.

(not for the mathematically faint hearted - may cause headaches)

--
Regards,
Martin Brown

OK, I forgot the tie-in with theoretical physics. Oops. Apparently quaternions describe rotations of particles involved in quantum electrodynamics fairly well, and octonions work to describe quantum chromodynamics.

So if you're in love with quaternions you can go for a math _or_ a physics PhD.

--
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

Oh. You're trolling. I see.

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

Oh. You're trolling. I see.

-------

Na, just adding another loser to my ignore list... have a nice day!

Do you seriously think these guys care about anything useful? As long as they have a calculator that can compute sqrt's and pi they can do all the math they need(which isn't much). I would doubt they know what an Algebra is much less a Clifford Algebra.

There responses demonstrate they have no clue about the power of generalizations. This alone shows there lack of intelligence.

Your use of "there" for "their" doesn't convey a good impression of yours.

I'm not going to play your game of who has the smartest 3rd grade education... when you graduate 4th grade give me a call and maybe we can then see who is smarter.

You will not win.

Same here.

I know... you can't beat stupid! I guess you can revel in your ignorance!