Systems Knowledge Question

...

it's never been tautological. and it turns out to not be entirely true (but it can be shown to be true for scalars that are real and rational, but it's still not tautological).

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

I think the practicality is that it is tautological in a broad sense. I was going to respond to Tim's initial post with some of the points that have since come out, mainly that for the most part superposition and linearity nearly define each other. In the majority of cases, and even with complex-valued signals, analysis of the linearity of operations is verifiable by confiming superposition (and scaling if one wishes).

The fact that there is an exception doesn't mean, to me, anyway, that it doesn't hold, it just means that one should be aware of the exception. We do this all the time for other things so I don't think it's a stretch to say "yes" to Tim's question in a general sense. It's like saying an analog signal is really analog and continuously variable, even though we know deep down that everything is quantized at the quantum physics level. It's not helpful in most conversations (or teaching) to bring the exception up in every discussion, or throw a definition out, because it may ultimately create more confusion.

Even recognized non-linear operations, done with non-linear devices, like mixing, can be treated as linear if held within a region of support where the behavior is linear. Likewise an amplifier can be driven into saturation, but that doesn't mean that linear analysis isn't completely appropriate if that's the operating mode of interest.

So, yes, there's an exception. It's an exception that's good to know and perhaps interesting or fun or useful to bring up when the case warrants, but generally speaking I think the broad answer to Tim's question is 'yes'. There may be a footnote but there usually is with almost all of this stuff.

The more time goes by the more I think that almost any assertion with a mathematical argument behind it is more likely to be opinion than fact. ;)

Eric Jacobsen

In comp.dsp Tim Wescott wrote: (snip)

Complex conjugate is a strange operator. What, for example, is its derivative?

-- glen

...

I don't think we have any exceptions here.

Thomas Richter commented on both the suggested conjugational and irrational exceptions. Dilip explained that the suggested complex domain exceptions come from operations that are not linear in the complex domain in a 2009 post:

**** *Nov 24 2009, 7:10 am * *Complex conjugation is not a linear operation over *the complex field. For linearity to hold, it must be *that L(cX) =3D cL(X) for all complex numbers c, which *is not true. Complex conjugation *is* a linear operation *if we construe complex numbers as constituting a *two-dimensional space over the real field. The mapping *(a,b) --> (a, -b) is a linear map from R^2 to R^2, while *the mapping X --> X* is not a linear map from C to C. * *Hope this helps * *Dilip Sarwate *****

Applying consistent usage of "complex" to both "operations" and "linearity" removes the exceptions.

Dale B. Dalrymple

in which direction?

linearity ==> superposition

or

superposition ==> linearity ?

the whole issue of the thread (well, half of it, TI was the other) is whether

superposition linearity

i used to think "yes", now it's "no". (but it's "yes" for any real and practical system that satisfies superposition.)

i can't decode that.

the simple question is: are there systems (or operations) that always satisfy superposition that do not satisfy linearity? since linearity is widely defined to mean superposition *and* homogeneous in scaling then, as best as my logic can tell, the question is equivalent to: are there systems (or operations) that always satisfy superposition that do not satisfy homogeneity in scaling?

i had previous thought the answer to both is "no" (with some hand-waving regarding the irrationals), but, if complex scalars and signals part of the picture, the answer is "yes" and you prove it by use of an exception that you can identify.

conj{}, Re{}, and Im{} are exceptions because, i suspect, they're not analytic functions. not "analytic" in the sense of Hilbert, but in the sense of Cauchy-Riemann. (because the derivative of imag w.r.t. imag does not equal the derivative of real w.r.t. real.) i suspect that this is the reason those complex functions do not satisfy scaling (even though they do satisfy superposition) and then are not linear operators (even though they do satisfy superposition). [thanks, glen.]

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

(with=20

the=20

The denominator of a fraction does not exist in and by itself, a more correct question is what is the derivative of the fraction?

?-)

a

of

=20

=20

Well of course, both the Fourier transform and the Laplace transform are linear and consistently defined for linear systems only.

(snip, someone wrote)

(then I wrote)

If you write f(x)=conj(x), and try to compute df/dx...

Yes, the denominator doesn't exist by itself, but you can take the derivative of 1/x, or a/x, or even the partial derivative of y/x.

-- glen

In order for a complex function u(x,y)+i*v(x,y) to have a derivative, the derivative has to be the same in all directions in the complex plane, which leads to the the Cauchy-Riemann condition, i.e.

partial u/partial x = partial v/partial y

and

partial v/partial x = - partial u/partial y.

Complex conjugate doesn't satisfy them, so it has no derivative.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510
845-480-2058

hobbs at electrooptical dot net
http://electrooptical.net

(with

the

assertion.

Whoosh. I heard that one get by me. I have forgotten too much math.

?8-/