Hi again,

Well a function is periodic if:
y(t+Tp)=y(t)

but the period is Tp, so we have a secondary expression:
y(t+n*Tp)=y(t)

where 'n' is the number of periods, however Tp is still the actual period.

I guess that is why we say the period of sin(2*pi*f*t) is 1/f, yet we can still have "periods" (plural) of duration n/f.

To follow along with your calculation just to throw a little more math into it, if i start with y=sin(2*pi*f*t) and replace t with n/f (n an integer) i get:
y=sin(2*pi*n)

which is zero for any integer 'n'.

I also assume we are talking about simpler functions too here.

I'm not sure I'm following the point here. Yes, y=sin(2πn) is zero for any integer 'n'. So what? The replacement you did only applies at t = 0 and sin(0) = 0, so we expect the function to be zero at any integer multiple of anything that can be called a period away from t = 0.

If we want the definition to have a preferred period, then we just need to incorporate that into the definition (something that is often done). At the same time, we need to patch up a bigger hole because the definition as you've given it here (and which is often the way it is given) means that EVERY function is periodic since if Tp = 0, every function satisfies the definition. Both of these are easy to deal with and the definition of periodicity is often found along the lines of the following:

A function is periodic if and only if a strictly positive value Tp exists such that

y(t+n*Tp)=y(t)

for every value 't' and every positive integer 'n'. The fundamental period of y(t) is the smallest value of Tp that satisfies this constraint.

This definition still has a weak spot for a DC signal since while there are infinitely many values of Tp that satisfy the constraint, the fundamental period is undefined but in the limit would be infinity. So to patch that up (which is done far less frequently) the definition would need to deal with this case explicitly, which when it is done, usually either simply excludes DC signals from consideration or defines the frequency of such a signal to be zero (by using a limiting process as a clearly periodic signal is reduced to a DC signal by letting the period grow to infinity such that y(t) = y(0) for any finite value of t). No matter what you do, the question of the periodicity of a DC signal is going to involve a kludge.

 

I'm not sure I'm following the point here. Yes, y=sin(2πn) is zero for any integer 'n'. So what? The replacement you did only applies at t = 0 and sin(0) = 0, so we expect the function to be zero at any integer multiple of anything that can be called a period away from t = 0.

If we want the definition to have a preferred period, then we just need to incorporate that into the definition (something that is often done). At the same time, we need to patch up a bigger hole because the definition as you've given it here (and which is often the way it is given) means that EVERY function is periodic since if Tp = 0, every function satisfies the definition. Both of these are easy to deal with and the definition of periodicity is often found along the lines of the following:

A function is periodic if and only if a strictly positive value Tp exists such that

y(t+n*Tp)=y(t)

for every value 't' and every positive integer 'n'. The fundamental period of y(t) is the smallest value of Tp that satisfies this constraint.

This definition still has a weak spot for a DC signal since while there are infinitely many values of Tp that satisfy the constraint, the fundamental period is undefined but in the limit would be infinity. So to patch that up (which is done far less frequently) the definition would need to deal with this case explicitly, which when it is done, usually either simply excludes DC signals from consideration or defines the frequency of such a signal to be zero (by using a limiting process as a clearly periodic signal is reduced to a DC signal by letting the period grow to infinity such that y(t) = y(0) for any finite value of t). No matter what you do, the question of the periodicity of a DC signal is going to involve a kludge.

Hello again,

You stated that:
"A function is periodic if and only if a strictly positive value Tp exists such that y(t+n*Tp)=y(t)"

but we dont really have to specify it that way to test for periodicity, we only have to use y(t+Tp)=y(t).
The little 'n' comes in when we want to know the period.
So the form without the 'n' tests for periodicity,and the form with the 'n' tests for the actual period.
To say it another way, one form tests to see if the function is periodic, and the other actually finds the period.

Not sure if that changes much here however when we talk about DC. I think more to the point of DC is how we actually do a circuit when we have to involve a DC source, in the most general way. When we have a DC source in a circuit where any kind of frequency would be involved, we have to "turn on" that DC source by specifying it as a step input of amplitude A in:
V=A*u(t)

but i guess there could be other views on this and that's why i start to think it depends on the context of the math.

Ohms Law:
I=E/R

obviously here we are thinking of E being on forever, and never having a starting point, so it started at minus infinity and never ends. So this gives us two views of what DC really is in the analytical sense.

Wolfram defines a constant non zero signal as being periodic having ANY period.

There is an interesting point to be made though in the context of a Fourier Series. That is, the DC component has it's own formula that is different from the frequency components. Why would the DC component calculation be so different if it was so much the same as the AC components.

I guess there is no single answer. If the DC signal is introduced as a step input then it is clearly not periodic, but if it is introduced as starting from minus infinity then it might be. The actual application and the choice of the analyst would come into play here.

The only other thing i can think of right now is that can DC ever exist from minus infinity. really. If we view time as starting at the big bang, then any DC signal that we say started at minus infinity must have started at the time somewhere around the time of the big bang. Do we consider that as having always existed or as a step change.

 

Hello again,

You stated that:
"A function is periodic if and only if a strictly positive value Tp exists such that y(t+n*Tp)=y(t)"

but we dont really have to specify it that way to test for periodicity, we only have to use y(t+Tp)=y(t).
The little 'n' comes in when we want to know the period.
So the form without the 'n' tests for periodicity,and the form with the 'n' tests for the actual period.
To say it another way, one form tests to see if the function is periodic, and the other actually finds the period.

Fine. Is

y(t) = m·t + b

periodic?

Since you say that we don't need to specify that Tp needs to be strictly positive, this function must be periodic because I can choose Tp = 0 and

y(t + 0) = m·(t+0) + b = m·t + b = y(t)

Thus it satisfies the test for periodicity.

The definition for periodicity MUST require that Tp be non-zero, otherwise EVERY function is trivially periodic.

Adding in the 'n' does NOTHING to find the actual period. ANY multiple of the fundamental period will still satisfy the periodicity constraint whether you include the 'n' or not. If you want to find the fundamental period, then you need to specify either that Tp must be strictly positive and that the fundamental period is the smallest period satisfying the periodicity constraint, or you must specify that Tp be non-zero and that the fundamental period is the absolute value of the Tp having the smallest magnitude.

 

Fine. Is

y(t) = m·t + b

periodic?

Since you say that we don't need to specify that Tp needs to be strictly positive, this function must be periodic because I can choose Tp = 0 and

y(t + 0) = m·(t+0) + b = m·t + b = y(t)

Thus it satisfies the test for periodicity.

The definition for periodicity MUST require that Tp be non-zero, otherwise EVERY function is trivially periodic.

Adding in the 'n' does NOTHING to find the actual period. ANY multiple of the fundamental period will still satisfy the periodicity constraint whether you include the 'n' or not. If you want to find the fundamental period, then you need to specify either that Tp must be strictly positive and that the fundamental period is the smallest period satisfying the periodicity constraint, or you must specify that Tp be non-zero and that the fundamental period is the absolute value of the Tp having the smallest magnitude.

Hi,

Oh so you want to say that Tp has to be non zero? That's fine with me i guess as i just assumed it was.
I guess the definitions we find in books assume it is non zero simply because it is there i guess, but it is interesting that they dont seem to mention that. Maybe they assume we have basic common sense, however in this field we usually see things more precisely defined.

 

Hi,

Oh so you want to say that Tp has to be non zero? That's fine with me i guess as i just assumed it was.

It's a reasonable assumption to make and one that nearly anyone will make intuitively. But it IS an assumption and definitions should never rely on assumptions.

I guess the definitions we find in books assume it is non zero simply because it is there i guess, but it is interesting that they dont seem to mention that. Maybe they assume we have basic common sense, however in this field we usually see things more precisely defined.

Exactly. Humans are pretty good at filling in the gaps in this kind of stuff and doing so correctly more often than not. Even when we mention all the fine points initially, we often stop doing so after that because we assume everyone understands them. But then we get sloppy and stop mentioning them even to someone that's just seeing it for the first time. Textbook authors, particularly of lower level books, are notoriously sloppy. Part of this is true sloppiness on their part and part of it is not wanting to cloud the message with "unnecessary" detail. Midlevel and higher physics and math texts seem to be the most conscientious about covering all the subtle details. Of course, then there's legal documents -- they frequently go into excruciating detail precisely because they don't want any way for anyone to interpret them differently than how it was intended, and yet they still are because even fairly simple interactions between humans in the real world are enormously more complicated than quantum physics ever dreamed of being.

 

It's a reasonable assumption to make and one that nearly anyone will make intuitively. But it IS an assumption and definitions should never rely on assumptions.



Exactly. Humans are pretty good at filling in the gaps in this kind of stuff and doing so correctly more often than not. Even when we mention all the fine points initially, we often stop doing so after that because we assume everyone understands them. But then we get sloppy and stop mentioning them even to someone that's just seeing it for the first time. Textbook authors, particularly of lower level books, are notoriously sloppy. Part of this is true sloppiness on their part and part of it is not wanting to cloud the message with "unnecessary" detail. Midlevel and higher physics and math texts seem to be the most conscientious about covering all the subtle details. Of course, then there's legal documents -- they frequently go into excruciating detail precisely because they don't want any way for anyone to interpret them differently than how it was intended, and yet they still are because even fairly simple interactions between humans in the real world are enormously more complicated than quantum physics ever dreamed of being.

Hi,

Yes, and when new people come into the field they dont have the experience to understand what is a reasonable assumption, so they have to rely on a set of well written assumptions.