The unit step function. u(t) is 0 for t < 0 and 1 for t > 1. It's value at t = 0 depends on who is defining it with 0, 0.5, and 1 being the usual options.What is u ?
What is the definition, mathematically, of a periodic function?Title says everything , I attached a pic in the post. M(t) is u(t+0.5)-u(t-0.5)
Something is missing. In my opinion, a steady DC voltage of V, or even v=0, does not qualify as periodic but would satisfy that equation.the definition is x(t+T) = x(t) where T is the period...
There are an infinite number of values of Tp that satisfy the requirement for a periodic signal.Hi,
A periodic signal satisfies this:
y(t+Tp)=y(t)
a few things to note here are:
1. y(t) is the amplitude.
2. t is the variable, which could be time or x for distance for example.
3. Tp is a constant, that describes the period.
#1 simply means the height of the signal at any given time.
#2 is of course the running variable that varies from some lower value to some higher value.
#3 is a constant which never changes and only one value of Tp is allowed for a given periodic signal.
Simply put, the periodic signal has a period Tp where the local pattern repeats indefinitely, and that period never changes, and there is only one value of Tp possible for any given signal.
DC is a special case that can be handled a couple of different ways -- and, like many special cases, can cause problems if we aren't careful. Strictly speaking a DC signal is periodic with any non-zero value being an acceptable choice for the period. By convention, the fundamental period is taken to be infinity making the fundamental frequency 0 Hz. But another convention is that the period of a periodic signal must be finite. We seem to have a contradiction but we really don't (yet). While the fundamental period is infinite, there are plenty of periods that are finite. Where we do run into a contradiction is with the convention that the fundamental period is the smallest positive period for which the definition is satisfied, but for a DC signal we pick the longest since there IS no smallest. A period of 0 trivially satisfies the definition as usually given for ANY signal, so the strict definition requires Tp to be strictly greater than zero.Something is missing. In my opinion, a steady DC voltage of V, or even v=0, does not qualify as periodic but would satisfy that equation.
I'm at a loss as to what is supposed to be seen from that.
Hi,There are an infinite number of values of Tp that satisfy the requirement for a periodic signal.
For instance:
v(t) = Asin(2πt/T)
The obvious choice for Tp is T. But it is NOT the only choice possible. Pick Tp = 10T.
Does
v(t+Tp) = v(t)?
Let's see.
v(t+Tp) = Asin(2π(t+10T)/T)
v(t+Tp) = Asin(2πt/T + 2π·10T/T)
v(t+Tp) = Asin(2πt/T + 2π·10)
v(t+Tp) = Asin(2πt/T)
Yep. 10T is a period of this function.
We have a special name for the SMALLEST value of Tp that satisfies the definition: We call it the fundamental period.
Hi,I can tell/state my student opinion and argumentation as why it might be considered as non-periodic without considering all the technical detail already mentioned because I think there is no need to go that deep into conventions and mathematical approaches, etc etc.
As a simple student, we usually place 3 dots before and after the representation of a non time-limited signal so that we know that signal extends to -infinity and +infinity. And when the representation of the signal don't include those 3 dots, the signal is limited in time, thus, we cannot say it's periodic in (all) time. At most we could eventually say it is periodic within a certain range of time included in the representation.
This could be one argument without mathematical definitions, conventions and detailed stuff like that.
Another argument could be that a periodic function repeats itself every T secs.
Even if that function is not limited in time, can you see it repeating anywhere in time if you plot it in your calculator? I couldn't. Yes, I plotted it in my calculator!
So if it doesn't repeat itself, it cannot be periodic.
Then the definition of what the period is needs to be such that "10T" doesn't satisfy it.Hi,
Yes you can acknowledge more than one repeat time, but 10T is not *the* period.
For example, if we ask the question:
"What is the period of that sine wave?"
we do not EVER answer "10T".
Hi again,Then the definition of what the period is needs to be such that "10T" doesn't satisfy it.