# Is u(t+0.5)-u(t-0.5) non periodic ? Why ?

#### AlexMak

Joined Jan 2, 2018
32
Title says everything , I attached a pic in the post. M(t) is u(t+0.5)-u(t-0.5)

#### wayneh

Joined Sep 9, 2010
16,398

#### WBahn

Joined Mar 31, 2012
25,904
What is u ?
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.

• wayneh

#### WBahn

Joined Mar 31, 2012
25,904
Title says everything , I attached a pic in the post. M(t) is u(t+0.5)-u(t-0.5)
What is the definition, mathematically, of a periodic function?

Does M(t) satisfy this definition?

#### AlexMak

Joined Jan 2, 2018
32
the definition is x(t+T) = x(t) where T is the period... How can I see if it satisfies the definition ? I mean, I can see there is only one period in there but im still not sure . It might be a trick.

#### WBahn

Joined Mar 31, 2012
25,904
If someone were to claim that it is periodic, what would the period have to be?

Most people require that the period of a periodic signal be finite.

#### AlexMak

Joined Jan 2, 2018
32
So it is non periodic because it doesnt have a visible or stated number of periods right ? Is there any other reasons ?

#### wayneh

Joined Sep 9, 2010
16,398
the definition is x(t+T) = x(t) where T is the period...
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.

#### MrAl

Joined Jun 17, 2014
7,659
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. Note this does not include DC because DC has no single value of Tp that satisfies the relationship above as any value of Tp satisfies the relationship for a DC signal.

The given signal is a uint step followed sometime later by a negative unit step, so it only changes two times for all time. For a signal to be periodic, it has to change many times over the course of all time. In fact, it has to change an infinite number of times.

There are also other types of periodic functions however that do not satisfy the relationship above, but they are usually noted as being special. A simple example is:
y(t+Tp)=-y(t)

#### WBahn

Joined Mar 31, 2012
25,904
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.
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.

#### WBahn

Joined Mar 31, 2012
25,904
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.
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.

It would be convenient just to declare that DC signals are aperiodic, but that can get you into trouble in some cases, too. It's somewhat akin to whether 1 is a prime number or not. In some instances it would be very convenient if were, but in others it is very inconvenient. In that case, however, he have a strongly universal convention that it is not and, in those cases where it would be convenient if it were, we just have to specify something like "for one and all prime numbers". Since there is no similar universal convention regarding the periodicity of a DC signal, it is best to always consider it as a special case and include or exclude it explicitly where appropriate.

#### Bordodynov

Joined May 20, 2015
2,623

#### WBahn

Joined Mar 31, 2012
25,904
I'm at a loss as to what is supposed to be seen from that.

What is the point of waveform Y? How does it address the TS's question?

I'm not saying it doesn't, I'm just saying that I'm not seeing it.

#### Bordodynov

Joined May 20, 2015
2,623

#### MrAl

Joined Jun 17, 2014
7,659
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,

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".

This reason for this i think is because there is no application that would benefit from this 'definition' of the period.
In other words, whatever we can do with 10 periods, we can do with just one, and if we do find a use for 10 periods, we dont default to calling THAT the 'period'.

However i do see that this puts a damper on the description i gave on "DC". If we have a repeat pattern with integer repeat time, then that is still an infinite number despite the fact that it is related by an integer and not a general ratio of two integers. So where does that leave us?

We end up back at square 1 where we have to define what a pattern is. This in turn leads me to think that something that NEVER changes (DC voltage) can not be said to have a pattern because it never changes. Everything else we see that has a pattern has at least two different values and that is how we can see a pattern at all.

THAT, in turn, leads me to the additional interesting view that there is no such thing as DC. This is an extreme view however, but when you think about it, we almost always think of some starting point for an experiment, where that experiment started some time after the big bang.
In the practical sense this makes sense too because when we have any circuit, we always have some point in time that came after the big bang that is the point in time where we turned the circuit 'on'. That is, applied power, and before that there was no signal at all. After we turn on we may get what we call "DC" and what we call "AC", but even that DC had to have a starting point, and that would have been a step signal. The step signal, no matter where it started from, at one time was zero and after turn on it became non zero, so we see that we had only ONE change, and it never repeats that change. A periodic signal always has more than one change over time.
I realize that sometimes we think of DC as having existed for all time, since minus infinity, but i dont think we ever use that in an actual calculation unless we allow a bunch of assumptions such as no time dependent components.

So really here the more interesting question i think was that whether or not DC was periodic. I dont believe it is periodic for two reasons now:
1. It never changes and therefore can have no pattern (view from -infinity).
2. There's no such thing as a DC signal because at some point the circuit had zero input and it must be turned on at some point to get that DC, and that's not DC really, that's a step signal.

Feel free to elaborate. Possibly show an example of where sin(wt) has a period of 1/(10*f) instead of 1/f, or rather, an application where this does any good to define it that way. I know one example where defining the period as 1/2 the ORIGINAL period has some benefit, but it leads to a result that seems to have no benefit.
More to the point, your thoughts on DC and the 'period'.

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
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.

#### MrAl

Joined Jun 17, 2014
7,659
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.
Hi,

Well the books always state y(t+Tp)=y(t) as the criterion so we were looking at that.
Then someone asked the question about a DC signal, which is a constant and since K1=y(t+Tp)=y(t) it seems plausible that DC would be a periodic signal. But then if that is true, a DC signal for example is 0v, and is that periodic? So the most probable answer is that it depends on the context.

2vdc might be considered periodic if it fits into a theory that considers all periodic functions and the results work out ok, but 2vdc might not be considered periodic if it fits into a different theory that considers all periodic functions but the results do not work out ok.

0vdc on the other hand may fit into both theories, but it probably does not serve any useful purpose other than saying that a given theory works at 0vdc also so we can state the domain as 0<=x<infinity instead of 0<x<infinity.

The math analysis is there so that we have a test for periodicity that anyone can use to test their signal. If we get too casual then problems could come up. Math is used in order to be more precise; state something in more exact terms.

Some of these problems we talk about sometimes border on the pure philosophical because there are different ways to interpret things sometimes and the different interpretations could lead to different conclusions and not all the interpretations were considered in the original work because the authors dont see the need to explain every tiny variation that might spark up a curiosity that stems from a more or less absurd interpretation.

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#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Hello once more!

I didn't want to say that what have been discussed here is irrelevant! I just wanted to make my point of view that is a very simple and rudimentar way of saying things at "student way".

Psy

#### WBahn

Joined Mar 31, 2012
25,904
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".
Then the definition of what the period is needs to be such that "10T" doesn't satisfy it.

#### MrAl

Joined Jun 17, 2014
7,659
Then the definition of what the period is needs to be such that "10T" doesn't satisfy it.
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.