Viewing Frequency using transforms

Discussion in 'Math' started by justin2014, Nov 26, 2014.

  1. justin2014

    Thread Starter New Member

    Nov 26, 2014
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    why always the complex value part holds the frequency and the real part is utilized for damping effect.But I suppose that there's no damping in fourier transform due to the sense that it just contain an complex valued part i.e e−jωt.Why couldn't the real part(eαt) in Laplace transform contain information about frequency?Am I wrong with the concept I have stated?Could anyone help me.
     
  2. Papabravo

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    The real part contains no information on frequency because resistance has no dependence on frequency. Reactance on the other hand is frequency dependent. In the expression of a complex quantity, the real part defines the radius of a circle. The imaginary part controls the rotational velocity of a vector whose length is defined by the real part.
     
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  3. studiot

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    Clean and simple explanation.
     
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  4. justin2014

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    Thanks for your answer.I think you really know about the complex and real part in a transformation like laplace or fourier.So could you tell me what really the real and complex parts in a transformation like fourier or laplace means in simple terms.When you say 'resistance' and 'reactance' I can't find resistance eventhough I think reactance comes with the term $j\omega$
     
  5. Papabravo

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    The transforms 'like Fourier and Laplace' don't really care about the physical world. They are mathematical constructs that aid in the solution of problems with physical significance which, would otherwise be hard to solve. In the case of the Laplace transform we can solve a differential equation into an algebraic equation. Finding the roots of a polynomial equation rests on techniques that have been know for centuries.

    Don't know why you are having trouble with definitions, try these #1 google hits
    Resistance:
    Reactance
     
  6. studiot

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    Just a side note that Fourier Transforms are quite different from Fast Fourier Transforms, for when you come to study FFT's, which have considerable use in many branches of engineering.

    Fourier, Laplace, Hankel, Mellin and other Transforms are a formal analytical techniques for solving analytical equations in mathematics.

    As noted they can be used to solve differential equations to obtain analytical formulae.

    The Fast Fourier Transform is a numerical method to process data (measurements) and obtain/contruct a numerical model of the physical quantities of interest.
     
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  7. justin2014

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    Nov 26, 2014
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    Thanks for yor answer.I would like to just know what the real and complex parts in Fourier transforms and Laplace transform mean in context of signal processing.I think you started with previous answer with the explanation of it but could you just make it more simpler.Also the answer you told today is kind of mathematical approach that doesn't relate anything to signal processing eventhough mathematical concepts are used in signal processing.Could you help me.
     
  8. Papabravo

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    It is a common fallacy that complex things can be made simpler and simpler if only our powers of expression were good enough. I do not subscribe to this fanciful notion. Complex numbers may seem like an odd way to keep track of things, but their power comes from the ability track multiple aspects of a signal. People get hung up on the nature of the imaginary unit and its definition as the square root of -1. That is perhaps the least useful way of looking at the imaginary unit. I could tell you that j, the imaginary unit, was a rotation operator, but that might or might not be a simpler concept. I could tell you that a complex quantity has both a Cartesian and a polar representation. Again that might or might not be simpler. Because I am not a mind reader and cannot look into you mind or your thought processes; you might have to help me to help you by making clear what you do and do not understand.

    1. Are you comfortable with complex arithmetic in Cartesian form? Do you know how to add, subtract, multiply, and divide numbers of the form a + jb?
    2. Are you familiar with complex arithmetic in polar form? Do you know how to add, subtract, multiply, and divide numbers of the form M∠θ
    3. Can you convert one form into another?
    4. If we got rid of the imaginary unit j, and used ordered pairs to represent complex numbers would that be helpful?
    5. Are you familiar with Euler's identity? [​IMG]
    This would be a good start for further discussion.
    http://en.wikipedia.org/wiki/Euler's_identity
     
    Last edited: Dec 17, 2014
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  9. studiot

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    You are asking why use imaginary numbers in the exponentials.

    This is simply because {e^{it}} is oscillatory and {e^{t}} is not

    This can be seen from Euler's formulae connecting the complex exponential to the

    {e^{it}} = \cos t + \sin t

    circular sine and cosine functions which are oscillatory

    compared to the real exponential to the hyperbolic sine and cosine which are not oscillatory.

    {e^t} = \cosh t + \sinh t


    This is why the exponential solution to the wave equation is complex.
     
    Last edited: Nov 29, 2014
  10. justin2014

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    Nov 26, 2014
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    Thanks.That's really a good answer.But let me say I know all these 5 points that you have mentioned.Yes of course I know Euler's identity makes a circle of unit radius.But what still confuses me is these questions:
    1. Why do we take a unit circle for analyzing signals?
    2. How can a circle be manipulated to analyze a signal?(I haven't never seen a signal like a circle)
    3. Why doesn't the real part holds information about frequency while the complex part does?(It might be as you said that "the complex part could also be used in polar representation".But how is frequency related to polar representation?)
    Could you please help me in answering these questions/
     
  11. studiot

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    Justin, although I seem to be talking to myself here,

    What does a circle do that a hyperbola or parabola does not?

    answer: it loops back on itself.

    And what is frequency?

    answer: how often something loops back on itself.

    You cannot construct something that loops back on itself on the real axis alone, in fact you cannot construct it on a single axis you need at least two.
    We need the complex solutions to give us a two dimensional representation.

    Why do we take the unit circle?

    answer: we don't always, but the unit circle is the simplest closed curve and its radius can easily be multiplied by a constant to give any size we wish.
     
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  12. Papabravo

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    1. The circle is the ideal geometric shape for analyzing things which are periodic. Whatever amount of time it takes for a point to move around the circle it always returns to it's initial location. At any point on the circle it's position is determined by the cosine and sine of the angle from the x axis.
    2. Sure you have. It is called an analog clock. The minute hand makes one revolution of 2π radians in 1 hour. The hour hand makes 1 revolution of 2π radians in 12 hours.
    3. Frequency is related to the polar representation by telling you how many times per second a vector centered at the origin goes around the circle. It has dimensions of (1/sec) and is thus an angular velocity since radians are dimensionless. The real part of a complex exponential has nothing whatever to do with the angular velocity of that vector and thus no effect or information on frequency. The only thing it can effect is the radial dimension or magnitude when expressed in polar form.
    Maybe these animations will be helpful
    http://resonanceswavesandfields.blogspot.com/2007/08/phasors.html#basic-phasor-1
     
    Last edited: Nov 29, 2014
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  13. justin2014

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    Nov 26, 2014
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    Thanks for your answer.But could you tell me why we can't represent a loop on real axis because I think you can represent the equation of a circle on real axis eventhough it is possible on complex axis too.
     
  14. justin2014

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    Nov 26, 2014
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    Thanks for your answer.The animation seems to be helpful but since it's frequency is in decimals it's bit hard to grasp.I'll try looking for other animations in the web.
    1. Why Fourier transform only holds for periodic signals or does it holds for non periodic waves too?
    2. As you told:"The real part of a complex exponential has nothing whatever to do with the angular velocity",why can't we take angular frequency on real axis because angular frequency is too an real quantity isn't it?
    3. Also why didn't we take a sin or cos signal which is also periodic to analyze a signal instead of a circle.
    Could you help me in answering these questions.
     
    Last edited: Dec 1, 2014
  15. Papabravo

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    I'm sorry, but I just cannot find the words to help you understand. Maybe someone else can find a way to express these concepts in a way you can understand.
     
  16. studiot

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    Can you?

    What is the difference between the y axis and the jx axis?

    It's simple, if you plot a loop of any shape on a sheet of paper you have two axes, the x axis and the y axis.

    This means that your formulae need to reflect that as a function of two variables ( x and y) not one (just x).

    So what exponentials do you propose incorporating y?

    The beauty of the complex approach is that you only have one axis, the x (or in your case the t) axis.
    To obtain the second axis you rotate it 90 degrees by applying the complex operator j.
    In fact you can obtain any rotation angle you choose by multiplying this by a constant.

    The fact that the complex approach introduces a rotation has already been mentioned.

    And the whole point of the rotation is the link to frequency, that you are seeking.

    On two real axes omega is just a real constant and has no meaning as an angular frequency.
    But allow a single axis to become a rotating arm and the constant omega takes on a new significance.

    Remember frequency refers to something ( a process) that repeats itself after a fixed interval of time, again and again indefinitely.

    I only entered this thread to add value, I thought papabravo was doing so well, I'm sorry you are having trouble with his excellent explanations, perhaps you should read them several times?
    Certainly old hands will tell you that you often have to do this when studying new material, in order to understand it properly.
    There's no guarantee you will 'get it' first time.

    As regards to your specific questions

    1) You appear to misunderstand Fourier Transforms and periodic waveforms.
    Periodic waveforms are defined to extend from minus infinity to plus infinity, ie they have to repeat indefinitely backwards and forwards.
    Strictly, Fourier Series refer to periodic waveforms, although some tricks may be employed to go from say zero to infinity.
    The Fourier Integral was developed to extend FS to non periodic waveforms - That is to represent a function over a single finite interval I. Note that the function may or may not repeat over that range.
    Fourier Transforms are a a formalised development of the Fourier Integral for the same purpose.

    2)I have explained that in relation to rotation and one versus two variables.

    3)Papabravos links showed the relationship between a circle and trigonometric functions.

    Further we do use trigonometric representation, they are called Fourier Series.
    However the requirements are that the functions they represent repeat continually from minus infinity to plus infinity.
     
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  17. Papabravo

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    @studiot -- Thanks man, I just did not have it in me to make the effort on this one.
     
  18. MrAl

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    Jun 17, 2014
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    Hi,

    Just a quick note here:

    e^(i*t)=cos(t)+i*sin(t)

    and

    e^(-i*t)=cos(t)-i*sin(t)

    where 'i' is the imaginary operator. Note that it appears as a factor of sin(t) in this relationship.
     
  19. justin2014

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    Nov 26, 2014
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    @studiot:Thanks.But still I have doubts regarding complex representation for angular frequency.They are:
    1. Do you meant to say that for plotting angular frequency on real axis you would need to rotate a vector to the same axis?
    2. But if we rotate in rotate an vector in the real axis by 90 degree wouldn't it rotate towards 90 degree eventhough it is on the y axis?
    3. Are you saying that it's better to represent it on the complex axis since there is only one variable but in real axis we have to consider two variables?Also whether this rotation by j vector is possible only on unit circle?
    Could you help me in answering these questions that I have raised here.
     
  20. studiot

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    Only if you pay some attention to what I, and others have said to you.

    We have assumed that the only things you haven't understood are embodied in the questions you have asked.

    But I think it runs deeper than that.

    Is it a language difficulty, is English your first language?

    Edit: Please note this is not meant as a criticism, I'm simply trying to find a way to help.
     
    Last edited: Dec 17, 2014
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