On Sun, 18 Oct 2009 17:18:42 -0700, Fully Half Baked wrote:
On Oct 18, 8:02 pm, Tim Wescott <t... at (no spam) seemywebsite.com> wrote:
On Sun, 18 Oct 2009 09:45:12 -0700, Fully Half Baked wrote:
On Oct 17, 8:05 pm, Tim Wescott <t... at (no spam) seemywebsite.com> wrote:
On Sat, 17 Oct 2009 12:30:43 -0500, fisico32 wrote:
A system is described by an equation that relates the output
function y(t) and the input function x(t). Both x(t) and y(t) are
functions of time. If the system is time invariant, it means that
the mechanisms of the equation are not time dependent (the
coefficients are constants).

Ex: y(t)=3*x(t)+ [x(t)]^2

This means that y depends only "implicitly" on time, but not
explicitly: y(x)= y(x(t)).
y(t) can be written as a function of time only however: Ex:
x(t)=2*t, then y(t)=3*(2t)+ [2t]^2

If the system is time variant, then the equation describing the
relation between input and output has time t variable appearing as
an explicit variable:

Ex: y(t)=5*t+x(t)

so y(t,x)=y(t, x(t)).
y(t) can be written only as a function of time t too. x(t)=2t,
then y(t)=5*t+2t

Q: If I was ONLY given the function y(t) and was asked if it is
the output of a time variant or invariant system, would I be able
to tell?

thanks
fisico32

Normal terminology in signals & systems is to call x(t) and y(t)
_signals_.  Yes, their values are functions of time, but you don't
really care about the "functional" part nearly as much as you care
about their behavior.  Think of them as continuous vectors that
"just are" more than as functions.

A more general way to describe a time varying system is to define
the system h as

y(t) = h(x(t), t).

In other words, h is some "thing" that acts on the input signal x(t)
to generate the output signal y(t).  The nice thing about the above
definition is that you can immediately shift the input and output
signals by some time t_s:

y(t - t_s) =? h(x(t - t_s), t).

If the above y and x _always_ match the non-shifted case for _all_
possible time shifts and _all_ possible input signals then the
system is time invariant.

Now, to answer your question:

No.  If you were given both the input and the output signals for all
time, you could _sometimes_ determine that the system was either
time varying or nonlinear.  I don't think you could conclusively
prove that the system was a linear time invariant system just from
one sample x(t) and it's resulting sample y(t), however.

--www.wescottdesign.com

Is it not true that if a system is nonlinear it's spectrum will have
changed which is easily measurable as long as the change isn't too
small to measure?

Any system can _change_ the spectrum of the input.  A linear system can
only change the amplitude of energy that was already in the input, a
time- varying system can only convolve the input spectrum with it's own
"time- varying-ness" spectrum and change the amplitude of energy that's
already in the input, and will do so in a way that obeys superposition..
 A nonlinear system can do any damn thing it pleases with the spectrum
of the input signal _and_ won't obey superposition.

But the OP is asking if he can look at the output signal _only_.  If he
really means what he says, that he's just been handed a signal and told
"here, this is the output of a system" then he has no clue about the
linearity or time invariance of the system.  If he's given the input
_and_ the output, then he may be able to say "that system isn't LTI",
but with just one input and one output signal I don't think he can say
_for sure_ that the system is indeed LTI, or if not if it is nonlinear
or time varying.

--www.wescottdesign.com

Ok maybe my idea of what is linear or not is a bit too simplistic. What
I mean by change the spectrum is new frequencies are introduced not just
changes in amplitude or shifts in time. Having said that fisico32 has
said "not only nonlinear systems generate new frequencies in the output
spectrum" but I don't know how that can happen if a system is said to be
linear unless you deliberately generate new frequencies and add them to
the output but then the overall effect is still nonlinear.

A time-varying system will add frequencies to the spectrum, because it
multiplies the signal by a time-varying parameter.  The effect on the
spectrum is to convolve the signal's spectrum with the time-varying
parameter's spectrum.

So a time varying system can't generate frequencies from _nothing_, but
it can certainly have more frequencies at the output than at the input.

--www.wescottdesign.com

Tim Wescott wrote:
On Mon, 19 Oct 2009 11:17:09 -0400, Jerry Avins wrote:

Tim Wescott wrote:

   ...

A time-varying system will add frequencies to the spectrum, because it
multiplies the signal by a time-varying parameter.  The effect on the
spectrum is to convolve the signal's spectrum with the time-varying
parameter's spectrum.

So a time varying system can't generate frequencies from _nothing_, but
it can certainly have more frequencies at the output than at the input.
Can a nonlinear system generate frequencies from _nothing_?

Define "nothing".

A time-varying system cannot generate an output signal with spectral  
components that are unrelated to the spectrum of the input signal.  A
nonlinear system can -- define the system

y = h(x, t)

as y = sin(w*t)

It's time-varying, it's nonlinear (it certainly doesn't obey
superposition!), and it generates an output signal from as close to
nothing as you can get.

You lost me. (That's not hard.) As far as I can see, you're setting up
sin(w*t) = h(x, t). Then the innards of h(x, t) are immaterial; y is
given. Where's the non-linearity? Are you describing an oscillator with
dummy input terminals? If so, we agree.

A balanced diode bridge is non-linear. No x emerges at all, only its
harmonics. It's so bad that the term "distortion" hardly applies.
Nevertheless, when no signal is applied, none emerges.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Oct 19, 2:28 am, Jerry Avins <j... at (no spam) ieee.org> wrote:

To continue in a simplistic vein, imagine an amplifier with time-varying
gain; the input is a single sinusoid of 2KHz, and the gain varies
sinusoidally at a frequency of 50 Hz. What is the output spectrum?

You got me there.

As a first approximation (which most of the time is all you're
bothered about with a bridge rectifier) a diode bridge is something
like abs(x). Is that a nonlinear function?

Maybe it's more like this
http://en.wikipedia.org/wiki/Nonlinear_system

Fully Half Baked wrote:
On Oct 19, 2:28 am, Jerry Avins <j... at (no spam) ieee.org> wrote:

   ...

To continue in a simplistic vein, imagine an amplifier with time-varying
gain; the input is a single sinusoid of 2KHz, and the gain varies
sinusoidally at a frequency of 50 Hz. What is the output spectrum?

   ...

You got me there.

A two KHz carrier modulated at 50 Hz. The (positive) spectrum consists
of three spikes; one each at 1950, 2000, and 2050 Hz.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Fully Half Baked wrote:

   ...

Maybe it's more like this
http://en.wikipedia.org/wiki/Nonlinear_system

More like in what way

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Oct 19, 5:26 pm, Jerry Avins <j... at (no spam) ieee.org> wrote:
Tim Wescott wrote:
On Mon, 19 Oct 2009 11:17:09 -0400, Jerry Avins wrote:

Tim Wescott wrote:

   ...

A time-varying system will add frequencies to the spectrum, because
it multiplies the signal by a time-varying parameter.  The effect
on the spectrum is to convolve the signal's spectrum with the
time-varying parameter's spectrum.

So a time varying system can't generate frequencies from _nothing_,
but it can certainly have more frequencies at the output than at
the input.
Can a nonlinear system generate frequencies from _nothing_?

Define "nothing".

A time-varying system cannot generate an output signal with spectral
components that are unrelated to the spectrum of the input signal.  A
nonlinear system can -- define the system

y = h(x, t)

as y = sin(w*t)

It's time-varying, it's nonlinear (it certainly doesn't obey
superposition!), and it generates an output signal from as close to
nothing as you can get.

You lost me. (That's not hard.) As far as I can see, you're setting up
sin(w*t) = h(x, t). Then the innards of h(x, t) are immaterial; y is
given. Where's the non-linearity? Are you describing an oscillator with
dummy input terminals? If so, we agree.

A balanced diode bridge is non-linear. No x emerges at all, only its
harmonics. It's so bad that the term "distortion" hardly applies.
Nevertheless, when no signal is applied, none emerges.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

As a first approximation (which most of the time is all you're bothered
about with a bridge rectifier) a diode bridge is something like abs(x).
Is that a nonlinear function?

On Oct 21, 8:12 pm, Jerry Avins <j... at (no spam) ieee.org> wrote:
Fully Half Baked wrote:
On Oct 19, 2:28 am, Jerry Avins <j... at (no spam) ieee.org> wrote:
...

To continue in a simplistic vein, imagine an amplifier with time-varying
gain; the input is a single sinusoid of 2KHz, and the gain varies
sinusoidally at a frequency of 50 Hz. What is the output spectrum?
...

You got me there.
A two KHz carrier modulated at 50 Hz. The (positive) spectrum consists
of three spikes; one each at 1950, 2000, and 2050 Hz.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

I meant you proved me wrong. What happened to 50Hz?