Re: getting freqz from fft (complex numbers)


Andor wrote:
> Rune Allnor wrote:
> > * This is the L0 or L1 norm of the spectrum, I never remember the
> > correct term.
>
> It should be easy to remember the correct term because an L0 norm
> doesn't exist.

OK. It's been a while since I read about these kinds of things.
The L2 norm was the intersting one, anyway.

> > The nice thing about norms is that if
> > ||x|+|y||>||x|+|z||
> > in one norm, it is also true in all other norms. So the
> > computationally
> > cheap norms can be used to find the extrema, and the usual
> > L2 norm can be used only where one wants to know the details.
>
> That won't work, Rune. Consider the two numbers z1=1.0, z2=.9 exp(i
> pi/4). Then
>
> ||z1||_L2 > ||z2||_L2
>
> but
>
> ||z1||_L1 < ||z2||_L1

Ouch. It seems you are right. If so, it means the choise of norm
changes the topology of a normed space? At frist glance I find that
very hard to believe... but then, maybe not.

If I was right in my statement yesterday, it would mean that all
optimization problems needed to be solved only in the least squares
sense, since that solution would be optimal in all other senses, too.
That's clearly not the case, so I was wrong yesterday.

Thanks, Andor, for the correction.

Rune

.

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