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==song==
Guest
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 Posted:
Wed Jan 05, 2005 2:41 am Post subject:
Definition of Signal Energy |
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The signal energy in decrete-time is defined as
Energy = sigma(|x(n)|^2)
I understand why it is squared. To prevent the energy from cancelling
out by negative values of x(n).
But I still don't get why absolute value is required here.
Will it be positive value after squared?
I have heard it is used if the signal is complex valued.
Anyone give me a specific example where absolute sign is necessary.
Thanks.
==song== |
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glen herrmannsfeldt
Guest
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| Posted:
Wed Jan 05, 2005 3:32 am Post subject:
Re: Definition of Signal Energy |
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==song== wrote:
| Quote: | The signal energy in decrete-time is defined as
Energy = sigma(|x(n)|^2)
I understand why it is squared. To prevent the energy from cancelling
out by negative values of x(n).
|
Well, if it was just to prevent the negatives from cancelling
then absolute value would do it. The physics requires it
to be squared. For electrical signals, consider a voltage
across a resistor. The current is V/R, power is IV or V**2/R.
Energy is power*time if constant, the integral of power*dt
if not, and the sum of power * delta t if discrete.
| Quote: | But I still don't get why absolute value is required here.
Will it be positive value after squared?
|
For a complex number, x+iy, the square is x**2-y**2+2ixy.
|x+iy|**2, for a variety of mathematical reasons which you
can find in any book on complex analysis, is x**2+y**2
and the right answer to this question.
-- glen |
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Guest
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| Posted:
Fri Jan 07, 2005 3:42 am Post subject:
Re: Definition of Signal Energy |
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Thanks Glen,
It is truely helpful to me to understand more about the term of signal
energy.
However, some portions are still unclear.
For electrical signal, the power at a certain time is 1/R*V^2(t) as
you mentioned.
So the energy(work) will be integral of the power in a period of time.
I can see the square of voltage in this equation. But, what if energy
for an arbitrary signal instead of electrical signal?.
The other way to describe the absolute value is that;
"The distance from that point to zero(origin)"
It clearly explain to me that the absolute value of a complex number
or signal is _positive and real number_.
Threrefore, the absolute in the equation is truly used for preventing
from cancelling the energy out. Is it correct understanding?
Thanks.
==song==
On Tue, 04 Jan 2005 14:32:17 -0800, glen herrmannsfeldt
<gah@ugcs.caltech.edu> wrote:
| Quote: |
==song== wrote:
The signal energy in decrete-time is defined as
Energy = sigma(|x(n)|^2)
I understand why it is squared. To prevent the energy from cancelling
out by negative values of x(n).
Well, if it was just to prevent the negatives from cancelling
then absolute value would do it. The physics requires it
to be squared. For electrical signals, consider a voltage
across a resistor. The current is V/R, power is IV or V**2/R.
Energy is power*time if constant, the integral of power*dt
if not, and the sum of power * delta t if discrete.
But I still don't get why absolute value is required here.
Will it be positive value after squared?
For a complex number, x+iy, the square is x**2-y**2+2ixy.
|x+iy|**2, for a variety of mathematical reasons which you
can find in any book on complex analysis, is x**2+y**2
and the right answer to this question.
-- glen |
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glen herrmannsfeldt
Guest
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| Posted:
Fri Jan 07, 2005 4:28 am Post subject:
Re: Definition of Signal Energy |
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song@cogeco.ca wrote:
(snip)
| Quote: | For electrical signal, the power at a certain time is 1/R*V^2(t) as
you mentioned.
So the energy(work) will be integral of the power in a period of time.
I can see the square of voltage in this equation. But, what if energy
for an arbitrary signal instead of electrical signal?.
|
Mechanical energy is force times distance, integral if force isn't
constant. Force is the time derivative of momentum, distance
is velocity times time, F=dp/dt=d(mv)/dt
dE = F dx=d(mv)/dt v dt or m v dv
Kinetic energy is the integral of m v dv, or (1/2) m v**2.
| Quote: | The other way to describe the absolute value is that;
"The distance from that point to zero(origin)"
It clearly explain to me that the absolute value of a complex number
or signal is _positive and real number_.
Threrefore, the absolute in the equation is truly used for preventing
from cancelling the energy out. Is it correct understanding?
|
Consider the potential energy of a mass on a spring in two
dimensional space with the spring tied to the origin. The distance
from the origin is x**2 + y**2, the energy is 1/2 k (x**2 + y**2).
That is, the sum of the x and y energies, there is no xy term.
If the position is described using a complex number x+iy then
the distance is sqrt(x**2+y**2), the definition of the absolute
value of the complex number.
If x=A cos wt and y=A sin wt then the energy is
(1/2) k A**2 (sin(wt)**2 + cos(wt)**2) = (1/2) k A**2.
-- glen |
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Guest
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| Posted:
Fri Jan 07, 2005 8:00 am Post subject:
Re: Definition of Signal Energy |
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On Thu, 06 Jan 2005 15:28:35 -0800, glen herrmannsfeldt
<gah@ugcs.caltech.edu> wrote:
| Quote: |
song@cogeco.ca wrote:
(snip)
For electrical signal, the power at a certain time is 1/R*V^2(t) as
you mentioned.
So the energy(work) will be integral of the power in a period of time.
I can see the square of voltage in this equation. But, what if energy
for an arbitrary signal instead of electrical signal?.
Mechanical energy is force times distance, integral if force isn't
constant. Force is the time derivative of momentum, distance
is velocity times time, F=dp/dt=d(mv)/dt
dE = F dx=d(mv)/dt v dt or m v dv
Kinetic energy is the integral of m v dv, or (1/2) m v**2.
The other way to describe the absolute value is that;
"The distance from that point to zero(origin)"
It clearly explain to me that the absolute value of a complex number
or signal is _positive and real number_.
Threrefore, the absolute in the equation is truly used for preventing
from cancelling the energy out. Is it correct understanding?
Consider the potential energy of a mass on a spring in two
dimensional space with the spring tied to the origin. The distance
from the origin is x**2 + y**2, the energy is 1/2 k (x**2 + y**2).
That is, the sum of the x and y energies, there is no xy term.
If the position is described using a complex number x+iy then
the distance is sqrt(x**2+y**2), the definition of the absolute
value of the complex number.
If x=A cos wt and y=A sin wt then the energy is
(1/2) k A**2 (sin(wt)**2 + cos(wt)**2) = (1/2) k A**2.
-- glen
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Sorry, I lost here what you try to tell me.
What I have heard is that the absolute is meaningless for real-valued
signal.
I want to know why absolute value is necessary for complex vauled
signal.
Thanks.
==song== |
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glen herrmannsfeldt
Guest
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| Posted:
Fri Jan 07, 2005 8:01 am Post subject:
Re: Definition of Signal Energy |
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song@cogeco.ca wrote:
(snip)
| Quote: | Sorry, I lost here what you try to tell me.
What I have heard is that the absolute is meaningless for real-valued
signal.
I want to know why absolute value is necessary for complex vauled
signal.
|
Well, what is wanted is x**2+y**2, in a two dimensional
real case as I described, or in the case of sin(wt)
and cos(wt). If you represent x and y as a complex
number, then the square of the magnitude of that number
is x**2+y**2, but the square of the complex number is
x**2-y**2+2ixy.
Absolute is fairly useless for complex numbers, too,
but magnitude squared is useful.
Mathematically, it usually comes out conjugate of variable
time variable, that is, (x-iy) * (x+iy), which you can also
verify is x**2+y**2. Magnitude squared is a little easier
to write.
-- glen |
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Clay S. Turner
Guest
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| Posted:
Fri Jan 07, 2005 9:22 pm Post subject:
Re: Definition of Signal Energy |
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"glen herrmannsfeldt" <gah@ugcs.caltech.edu> wrote in message
news:X4pDd.28273$3m6.9042@attbi_s51...
| Quote: | Mathematically, it usually comes out conjugate of variable
time variable, that is, (x-iy) * (x+iy), which you can also
verify is x**2+y**2. Magnitude squared is a little easier
to write.
|
Hello Glen and others,
Here you have hit on the crux of the matter. While (z conj)* z and |z|^2 are
equivalent, most like to think of magnitude squared as this is more
intuitive for real valued quantities.
As for the OP, in Physics we like for the energy to be a scalar quantity. So
in the case of kinetic energy, we can express it as 0.5*m*(v dot v) where v
is the velocity vector and v dot v is the inner product whose value is
|v|^2. So in all of this business of using scalars, complex numbers or
vectors, the energy needs to be defined in a way that reduces these things
down to a scalar involving a magnitude squared.
Clay |
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