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Initial value theorem:
Could you please also help me with
these queries related to initial value theorem? Thanks a lot.
Best wishes
PG
I'm finding it difficult to answer some of these questions because they reveal many misconceptions and needless probing. I don't mean that to say you shouldn't ask such questions when you feel the need, but sometimes the person trying to answer needs to point out that you may be trying to run before you can walk. This is a subject that takes time to learn. It's not overly difficult, but there are many many details that take more than a few days to absorb, and you may be trying to absorb too much, too fast.
So, I can't answer it all, but I can try to make a few points that might help.
Q1: Here my comment is that you need to be careful to ask "why" about definitions. The initial value theorem is stated in a particular way, and they tell you about that particular way. Asking "why is it so" is somewhat silly (we all do it, of course) because the answer is "that's how it is defined" . So, then we might ask, "why is it defined this way, and not another way". Usually the answer is either that it is defined in the most useful way and alternative definitions are less useful or are useless. Or, sometimes, the definition is chosen because that is the form we can actually say something about. Perhaps we would like to define something different, but the different way is unsolvable and the defined way is solvable. So, we either define the most useful way, or the way that allows us to say something useful.
In this case, I think the latter is a likely answer. We can say something useful if we define the initial value as t=0+, and we run into theoretical issues if we try other ways. The clear problem with t=0- is that the initial value at t=0- is always zero. So, how useful can that be?
Q2: Basically, just realize that short time scales match up with high frequencies and long time scales match up with low frequencies. But here we are just exploring mathematical relations. Follow the math to get the correct conclusions, and then just mentally note that t=∞ says something about s=0 and t=0 says something about s=∞. The detailed understanding will come with time. So, be patient.
Q3: This is really hard to respond to. Poles are not functions, they are locations where functions blow up. Poles and zeros can be related to functions in the sense that if we know the function is a ratio of two polynomials in s, then the poles and zeros essentially define the function. The text is merely stating that you don't need to worry about where the LT of the function blows up, which is defined by the pole locations. It simply isn't relevant to the initial value of the function, or the final value of sF(s).
Before we were talking about functions that the Laplace transform deals with, which is something different than poles. exp(st) can be thought of as a test function that we can apply to a system, and each value of s in the complex plane creates a different test function. Don't confuse this function with the function we are transforming, whether it be the input signal or the system's impulse response function. If these seems like a lot of detail and seems confusing, then I would say it is because you have not solved enough practical problems yet. Doing practical problems can more quickly help you get these concepts down.
When you are thinking about these things, use examples, and even use Matlab to explore.
For example, a unit step function u(t) has a value of 1 at t=0+. The transform is 1/s. Hence, the initial value theorem works because f(0+)=lim(s->∞) sF(s) becomes 1=s/s=1
For example, a unit ramp function has a value of 0 at t=0+. The transform is 1/s^2. Hence, the initial value theorem works because f(0+)=lim(s->∞)sF(s) becomes 0=lim(s->∞) 1/s=0
For example, an impulse function has a value of infinity at t=0+. The transform is 1. Hence, the initial value theorem works because f(0+)=lim(s->∞)sF(s) becomes ∞=lim(s->∞) s=∞