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causal, non-causal system

Discussion in 'Mathematics and Physics' started by PG1995, Nov 29, 2012.

  1. PG1995

    PG1995 Active Member

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    Hi

    Could you please help me with these queries? The Q1 is given below. Thank you very much and please don't forget I'm new to this stuff.

    Q1:
    A causal system is one which responds only during or after an input signal is applied. Or, as the book says, "Any system for which the zero-state response occurs only during or after the time in which it is excited is called a causal system".

    But if I were to define it then I would say, a causal system is one whose output, y(t), does not depend on the future value of an input signal, x(t), or its some future output value. A causal system should always use present or previous value of an input signal and/or some previous output value. For instance, this equation is for a causal system, y(t)=x(t)+x(t-1). And this one is for a non-causal system, y(t)=x(t)+x(t+1)+y(t+1); you can see that the x(t+1) and y(t+1) expect future values. In other words, if you want to know the output's value at t=3 then you must first know its input signal's value at t=3 and its output value t=3. Such a system cannot implemented in real-time environment.

    Regards
    PG
     
    Last edited: Nov 29, 2012
  2. steveB

    steveB Well-Known Member Most Helpful Member

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    For Q4, you have entirely missed the point made. A sinusoidal signal that is turned on at t=0 is not a true periodic sine wave. A proper true sine wave exists for all time from negative infinity to positive infinity.

    This is a mathematical artifact. In the real world we can test systems with sine waves by turning the sine wave on and then waiting for the transients to decay to zero. If you wait long enough, then the test you do is a valid represetntation of a true sine wave that existed for all time.

    In other words, the signal sin(wt)*u(t) can be used (or at least approximated), but the signal sin(wt) does not exist in the real world.

    For Q3, your question does make sense. Actually, throwing a switch is essentially an input to the system. However, a separate question is whether the system that is "switched" is linear and/or time invariant. But, these properties are independent from the property of causality. Since you described a circuit that can be built, then it should be a causal system. Non-causal systems are not realizable.

    For Q2, I think you are correct. It's really a matter of definition, and I don't have the formal definition in front of me. However, what you said makes sensee.

    For Q1, there does not seem to be a question here, but what you say seems reasonable.
     
    Last edited: Nov 29, 2012
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  3. PG1995

    PG1995 Active Member

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    Thanks a lot, Steve.

    In a way this is new to me! I didn't think that throwing a switch is an input to the system. Let's use that definition from the book, "Any system for which the zero-state response occurs only during or after the time in which it is excited is called a causal system". The word of our concern here is, I think, "excited". I think in your view, an input to a system could be a signal or an operation such as throwing a switch, and therefore a system can be excited either by a signal or an operation such as turning on a switch. Please let me know if I have it correct. Thank you.

    Best wishes
    PG
     
    Last edited: Nov 29, 2012
  4. dave

    Dave New Member

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  5. MrAl

    MrAl Well-Known Member Most Helpful Member

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    "y(t)=x(t)+x(t+1)+y(t+1); you can see that the x(t+1) and y(t+1) expect future values"

    What about this which depends on a future input value:
    y(t)=A*x(t)+B*x(t+1)
    (note the missing y(t+1) term and included constants just to make it more realistic)

    Is this causal? You cant really tell without knowing more about what x is. If x is periodic we can implement this system in real life.
     
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  6. steveB

    steveB Well-Known Member Most Helpful Member

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    I think you are correct, but this definition from the book is troublesome to me also. If you activate a circuit by throwing a switch, you might want to say that you have reactivated the process of a natural response which is not a zero state response. To me, causal systems are more general than this restrictive view. We are talking about cause and effect and anything real is automatically causal based on our current understanding of phyiscs. If we specify something with non-causal properties, then we can't build it.

    If i walk up to a circuit with the intention of throwing a switch, and an instant before I actually throw it, the circuit discharges, what do I have? Is this a non-causal circuit that somehow prepredicted that I was going to throw the switch? No, of course not because a real system will not do this. There would need to be a causal reason for what I observe. Some possible reasons are as follows.

    1. I triggered a vibration which set off the switch.
    2. The switch just happened to malfunction as I was getting ready to throw it.
    3. My friend had a remote control and was playing a joke on me.
    4. The system had a detector to sense when a person got near and activated an autmatic turn on.


    Another strange thing about the definition is that I'm free to specify a system (on paper) that has a zero state response that occurs during or after the excitation, but I can then add to the definition that if the system gets stored with energy above a certain value for longer than a certain amount of time, it will discharge 12 hours before someone decides to apply a signal to it. Well, this is a non-causal system, even though it meets that definition of a causal system. The non-causal behavior does not happen when the system is in the zero state. Very confusing!
     
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  7. PG1995

    PG1995 Active Member

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    Thank you for the explanation. I get it now.

    Best wishes
    PG
     
  8. PG1995

    PG1995 Active Member

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    Hi

    Below are given two definitions of causality.

    Definition #1:
    A causal system is one whose output, y(t), does not depend on the future value of an input signal, x(t), or its some future output value. A causal system should always use present or previous value of an input signal and/or some previous output value.

    Definition #2:
    The mathematical definition of causality states that a system is causal if all output values, y[n0], depend only on input values x[n] for n<=n0 . Another way of saying this is the present output depends only on past and present input values.

    I think my query is not directly related to causality. I have read that every system is turned on at t=0. Is this time t=0 really a turn on time, or, is it the time reference for marking the moment when we started analyzing or noticing the output of system? If it really is the time when the system is turned on then 'every' practical system's output should be zero for t<0. Perhaps, in a very strict sense, it is true to say that a system is turned on at t=0 but I would say that in theory it's better to say that this refers to the moment when we started analyzing the system's output. For example, once the transients have died away we can assume that the system has been running since a long time which tends to negative infinity but t=0 marks the time when its output was started to being analyzed. I'm sorry that my query is once again confusing. I hope you can see my confusion as you mostly do. Thank you.

    Regards
    PG
     
  9. PG1995

    PG1995 Active Member

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    It might be possible that you have missed my previous post. Otherwise, it's okay, take you time. Thanks.
     
  10. steveB

    steveB Well-Known Member Most Helpful Member

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    I don't think there is any requirement to say that the system in turned on at t=0 or n=0. It is convenient to do so, but not required. Any signals (delta, step, ramp functions) are defined to start at zero but we can shift them as needed. For example u[n] can be written as u[n-k] to delay the start of the turn on by k time steps.

    Impulse responses always start at t=0 or n=0 (that doesn't mean they can't have zero values for t=0 or n=0) because we use the impulse function to define the input and the impulse function has values only at t=0 or n=0.
     
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