On Sun, May 19, 2013 at 10:18:35AM +0200, Jörn Nettingsmeier wrote: > On 05/02/2013 01:26 AM, Richard Dobson wrote: > > > I have always understood it to mean that the behaviour is not dependent > > upon ~when~ the signal is injected. Thus, a plain delay is TI because > > everything is always delayed the same way; while a modulated effect such > > as a flanger (maybe using a variable delay) is not TI as exactly what > > comes out depends in the time something goes in. > > ... > > for practical purposes, i guess fons' definition is more useful, > because then the term "LTI" system is strictly limited to something > that can be fully described with an impulse response.
Despite what I wrote before, I tend to agree with Richard. If I interpret his formulation correctly, a system is time-invariant iff, when the output for input x(t) is y(t), then the output for x(t + T) is y(t + T), for any T. It's actually quite difficult to formulate a stronger version *unless* you assume that the system is linear as well. A linear time-invariant (LTI) system is fully defined by an impulse response, or by a transfer function in the frequency domain. Now consider three cases: 1. A filter, 2. A tremolo effect, 3. A compressor. The filter is LTI, while the tremolo and compressor are not. Do they fail to be LTI because they are not linear, or because they are not time-invariant ? The tremolo fails Richard's TI criterion. But it *is* linear in a very strong sense: for any a(t) and b(t) Tremolo (a(t) + b(t)) == Tremolo (a(t)) + Tremolo (b(t)). The compressor is time-invariant according to Richard's criterion. But it isn't linear in the way the tremolo is. It could be said to be linear 'at any instant', assuming attack and release times are non-zero. But that is a somewhat problematic definition of linearity, since apart from trivial cases (pure gain) linear processes depend on the input's or output's history, and are not defined by some relation at a single instant. So it seems that a stronger definition of TI is not necesssary. Ciao, -- FA A world of exhaustive, reliable metadata would be an utopia. It's also a pipe-dream, founded on self-delusion, nerd hubris and hysterically inflated market opportunities. (Cory Doctorow) _______________________________________________ Sursound mailing list Sursound@music.vt.edu https://mail.music.vt.edu/mailman/listinfo/sursound
