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

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?  

Thank you, not sure why I got confused....
So, I guess the impulse response h , in the general case, could be written
as

h [delta(t-t0), t]

If the system is time invariant, then h[ delta(t-t0) ] there is no time dependance t.

So, for example, if the function h contains both t and t_0 (the location
of the delta), in such a way that we ca always pair t and t_0 in a
subtraction, then the system is time invariant:

h(delta(t-t_0)) =  3+(t-t_0)^2. This system would be time
invariant.......

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

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?

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

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.

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.

On 10/19/2009 10:26 AM, glen herrmannsfeldt wrote:

There is in non-linear optics something called a phase conjugate mirror.

It reflects a signal (light beam), reversed in time from the original.
(It has to be large enough to do that.) One use for them is in
fiber optics systems with dispersion: If you put a phase conjugate
mirror in the middle of a long optical fiber, the second half will
undo the dispersion for the first half.
(snip)

That sounds like it'd fit in the causal/non-causal thread really well.

Jerry Avins<jya at (no spam) ieee.org> wrote:
Tim Wescott wrote:
(snip)

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_?

Isn't that what we call an oscillator?

There is in non-linear optics something called a phase conjugate mirror.

It reflects a signal (light beam), reversed in time from the original.
(It has to be large enough to do that.) One use for them is in
fiber optics systems with dispersion: If you put a phase conjugate
mirror in the middle of a long optical fiber, the second half will
undo the dispersion for the first half.

If you have a phase conjugate mirror with gain (they are active
devices needing input power), it gets very interesting if you
hold a shiny object nearby. (The reflected wavefront arrives
back at the same phase and higher amplitude than when it left,
just in time for a new reflection.)

-- glen

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.

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.

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?

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