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MrAl said:An impulse that is not unit can be thought of as a unit impulse with a gain. If you apply a unit step to a circuit and get 3.2v out at t=1 second, then instead you apply a step of 'amplitude' 2 you should expect to see the output rise to 6.4v at t=1 second. So it's as if the 'gain' had increased.
@Steve: I have corrected Q2. Please help me with it. You can also see my attempt at the solution of Q5.
For Q2, I would answer by saying that since the impulse function is best defined and understood in terms of the limiting process of a function with an integral as the characteristic relation, the derivative that has impulses in it is best understood if you think about integrating it.
MrAl said:An impulse that is not unit can be thought of as a unit impulse with a gain. If you apply a unit step to a circuit and get 3.2v out at t=1 second, then instead you apply a step of 'amplitude' 2 you should expect to see the output rise to 6.4v at t=1 second. So it's as if the 'gain' had increased.
Could you please elaborate on what you said above? Thanks.
PG1995 said:What do these results (1)δ(t), (3)δ(t) and 8δ(t-3) tell us about the derivative at discontinuity point of u(t), 3u(t) and g(t)?
An impulse that is not unit can be thought of as a unit impulse with a gain.
What do these results (1)δ(t), (3)δ(t) and 8δ(t-3) tell us about the derivative at discontinuity point of u(t), 3u(t) and g(t)?
Let me repeat what I said above. Personally, I think of an impulse as a value extractor. It extracts a value of a continuously varying function at some particular instant of time. A unit impulse will extract the value ... I was referring to Dirac delta function.
Now there is a related beast called Kronecker delta function for discrete time functions. ...Theoretically speaking, is it used to extract value of discrete time functions?
Hi
Some of the information on Kronecker delta function can be found here although for this discussion you don't need to refer the linked thread.
Let me repeat what I said above. Personally, I think of an impulse as a value extractor. It extracts a value of a continuously varying function at some particular instant of time. A unit impulse will extract the value as it is but an impulse which is not unit would amplify the extracted value; if it's >1 then it will amplify and if it's <1 then it will de-amplify. In this context I was referring to Dirac delta function.
Now there is a related beast called Kronecker delta function for discrete time functions. First of all, I don't get where it's used. Theoretically speaking, is it used to extract value of discrete time functions? If not, then where it's used. For continuous functions, the Dirac delta is used where it samples the value of continuous varying function. Please help me with this. Thank you.
@MrAl: What you say above is correct that when different things come together and work as a system then we get full understanding and appreciate their usefulness.This is true that studying about different topics in isolation makes things really difficult. But that's the way things are taught and this is how 'formal' education goes, at least where I live. Using your analogy. I have a window and I know it will be used someday. I need to know its properties before I see its real use. I can't wait for the day when I actually see it being used/fixed in some house. They are going to ask me about its properties, size, dimensions, etc. in an exam. It's possible they will ask me about different kinds of windows and it's quite possible they will ask me to show them how it's fixed in a frame although I haven't seen it being fixed. This world is a cruel place!
Regards
PG
In order to help speed up your comprehension of the answer to this question, I would like you first to express mathematically how the Dirac delta acts as a value extractor. I think your description is correct, but this needs to be more than words. Please show how this works.
I would say you are correct and once you show how the Dirac delta function is used to achieve "value extraction", it will be easy for us to show you now the Kronecker delta does the same for discrete time functions.