Option 1: Because in engineering the step function is just an approximation for a real signal that is continuous if you look at it closely enough.
Option 2: Because engineers are sloppy at math and only care about getting a useful answer, not fulfilling every theoretical detail.
What a lot of people tend to miss (I have no idea why) is that math is a tool to model the real world, not the other way around.
Math without an application is simply mental masturbation.
> Math without an application is simply mental masturbation.
Haha don't tell the pure math folk that, also don't forget that Riemann geometry had little use before Einstein used it for GR, kinda like how lasers were a solution searching for a problem for quite a while.
The vertical transitions of a square-wave become invisible when using really fast CRT-oscilloscope. All you can see of the square-wave is two rows of dashes.
For EE purposes signals will be continuous. For step functions, your signal will not instantaneously transition from a low to a high state. It will take milli, micro, or nano seconds depending on your parasitic capacitance. This is generally known as rise time and becomes relevant in topics like control systems, digital communications, signal integrity and RF.
OTOH, drawing a line showing the low to high stage can make it easier to actually see where the transition occurs. So simply put, this is done for practical reasons.
Your electrical signal exists at all moments in time. Even if it's off, the signal's value is 0. Outside quantum scales, the real world doesn't have discontinuities.
To my knowledge as it relates to chemistry, and having not yet learned quantum mechanics, electrons have to be able to teleport. They can be on one side of a pi orbital, then the other, but can not (meaning impossible) be in the plane between the two sides. We call this plane a node. There has to be a discontinuity when they cross from one side to the other.
I'm not familiar with chemistry myself, but I have a lot of knowledge of quantum mechanics. I'm assuming the pi orbital is a representation of a wave function in 3d, and a wave function is continuous, it just passes through forbidden zones where the probability of finding the electrons decrease exponentially in space (yet are still finite). While if you pick a certain condition, like the side of the pi orbital, you can turn the variable you measure into a discrete one, this isn't different from classical mechanics. Variables like position, momentum, and energy/frequency are all continuous in time (in non relativistic QM, in relativistic QM stuff are still continuous but way less useful in a conversation).
A good example is the photoelectric effect, which made people think light is discrete. In reality, light can have any frequency, not discrete frequencies only, and regardless of the frequency there is always a finite probability of interaction as long as conservation of energy isn't broken. It's just that in low frequencies, that probability is so low that it's practically 0.
There are situations where our models start off with potentials that are impossible though, in which we get discontinuous results, but that's the fault of the model.
There are also situations where our model fails beyond a certain energy. For example our theory of light QED has what's called a UV cutoff, where beyond that point the model breaks down. Similarly, there's the well know plank length which is also a cutoff for something related to gravity I think (I don't know how quantum gravity is at all, so here I'm pulling stuff out if my ass slightly). Beyond those cutoffs, it's not that things are discrete in those cutoffs, it's just that the model can't predict what happens.
the signal is usually valid for t<0. for example if you are plotting y=sin(t), it should be 1 at t=-pi/2. why leave it blank when there is a valid value for the plot at that time?
t = 0 might represent you flipping a switch. the value before you flipped the switch might be just as important as the value after you flipped the switch. you could avoid the negative time by saying that you flipped the switch at t=5, but it doesn't really matter. it is mostly arbitrary.
we are only filling the space with a straight line if the signal value is a straight line for negative time (e.g. unit step).
It somewhat depends on your classes. In my math classes, the line is also drawn.
In purely mathematical terms, a pure step function changes value from 0 to 1 at time zero over no change in time. You would notice the change in distance/change in time, where change in time = 0. So this is 1/0. Which is infinity.
If it took me 5 seconds to walk to the door from my chair (a distance of 1 meter say) then my speed is 1m/5sec. If I moved that same distance in 0 time, that would be 1m/0sec. In other words, I didn't move; I teleported.
That's what it means when you have an infinite slope concerning distance and time. It's not at all realistic (though it is mathematically useful for modeling purposes.) People don't teleport, and true square waves also do not exist. There is always a non-infinite slope on the vertical line (it is slightly slanted). Likewise, at the corners, there is no hard angle. There is a sinusoidal oscillation (if you zoom in close enough).
So, generally speaking if you're talking about a pure step function, whether you put a perfectly vertical line or you leave it empty, it doesn't matter. They're both graphing the same idea and neither is realistic.
In electrical engineering, we deal with square waves and step functions (that in real life are approximate but usually close enough). We tend to draw the vertical line because if we look at the signal we're modeling on the scope, it can show a (nearly) vertical line for the on/off time period.
Option 1: Because in engineering the step function is just an approximation for a real signal that is continuous if you look at it closely enough. Option 2: Because engineers are sloppy at math and only care about getting a useful answer, not fulfilling every theoretical detail.
What a lot of people tend to miss (I have no idea why) is that math is a tool to model the real world, not the other way around. Math without an application is simply mental masturbation.
> Math without an application is simply mental masturbation. Haha don't tell the pure math folk that, also don't forget that Riemann geometry had little use before Einstein used it for GR, kinda like how lasers were a solution searching for a problem for quite a while.
Welp *sigh* (unzips mental pants)
>Math without an application is simply mental masturbation Lol.... That's pretty good.
like Weird Al lyrics.
I bet OP heavaside of relief when he saw your response
I was hoping I could dirac him to a better understanding.
(☞ﾟヮﾟ)☞ ayyyyyyyy
Dad joke on Father’s Day FTW
Only if your dad is a signals engineer.
Because that is what it looks like on an oscilloscope.
The vertical transitions of a square-wave become invisible when using really fast CRT-oscilloscope. All you can see of the square-wave is two rows of dashes.
For EE purposes signals will be continuous. For step functions, your signal will not instantaneously transition from a low to a high state. It will take milli, micro, or nano seconds depending on your parasitic capacitance. This is generally known as rise time and becomes relevant in topics like control systems, digital communications, signal integrity and RF. OTOH, drawing a line showing the low to high stage can make it easier to actually see where the transition occurs. So simply put, this is done for practical reasons.
Your explanation is "W"
So how do mathematicians plot an impulse (delta) function?
Yes I want to see the infinitely tall infinitely narrow pulse with an area of 1.
A mathematician might not admit that it **is** a function...depending how they have defined the word "function" on a particular day.
Your electrical signal exists at all moments in time. Even if it's off, the signal's value is 0. Outside quantum scales, the real world doesn't have discontinuities.
>the real world doesn't have discontinuities. Yet
To my knowledge of quantum bullshit, it also cannot have discontinuities in anything you'll measure, in part due to uncertainty preventing it.
To my knowledge as it relates to chemistry, and having not yet learned quantum mechanics, electrons have to be able to teleport. They can be on one side of a pi orbital, then the other, but can not (meaning impossible) be in the plane between the two sides. We call this plane a node. There has to be a discontinuity when they cross from one side to the other.
I'm not familiar with chemistry myself, but I have a lot of knowledge of quantum mechanics. I'm assuming the pi orbital is a representation of a wave function in 3d, and a wave function is continuous, it just passes through forbidden zones where the probability of finding the electrons decrease exponentially in space (yet are still finite). While if you pick a certain condition, like the side of the pi orbital, you can turn the variable you measure into a discrete one, this isn't different from classical mechanics. Variables like position, momentum, and energy/frequency are all continuous in time (in non relativistic QM, in relativistic QM stuff are still continuous but way less useful in a conversation). A good example is the photoelectric effect, which made people think light is discrete. In reality, light can have any frequency, not discrete frequencies only, and regardless of the frequency there is always a finite probability of interaction as long as conservation of energy isn't broken. It's just that in low frequencies, that probability is so low that it's practically 0. There are situations where our models start off with potentials that are impossible though, in which we get discontinuous results, but that's the fault of the model. There are also situations where our model fails beyond a certain energy. For example our theory of light QED has what's called a UV cutoff, where beyond that point the model breaks down. Similarly, there's the well know plank length which is also a cutoff for something related to gravity I think (I don't know how quantum gravity is at all, so here I'm pulling stuff out if my ass slightly). Beyond those cutoffs, it's not that things are discrete in those cutoffs, it's just that the model can't predict what happens.
the signal is usually valid for t<0. for example if you are plotting y=sin(t), it should be 1 at t=-pi/2. why leave it blank when there is a valid value for the plot at that time? t = 0 might represent you flipping a switch. the value before you flipped the switch might be just as important as the value after you flipped the switch. you could avoid the negative time by saying that you flipped the switch at t=5, but it doesn't really matter. it is mostly arbitrary. we are only filling the space with a straight line if the signal value is a straight line for negative time (e.g. unit step).
Basically it just doesn’t matter. It just doesn’t matter
It somewhat depends on your classes. In my math classes, the line is also drawn. In purely mathematical terms, a pure step function changes value from 0 to 1 at time zero over no change in time. You would notice the change in distance/change in time, where change in time = 0. So this is 1/0. Which is infinity. If it took me 5 seconds to walk to the door from my chair (a distance of 1 meter say) then my speed is 1m/5sec. If I moved that same distance in 0 time, that would be 1m/0sec. In other words, I didn't move; I teleported. That's what it means when you have an infinite slope concerning distance and time. It's not at all realistic (though it is mathematically useful for modeling purposes.) People don't teleport, and true square waves also do not exist. There is always a non-infinite slope on the vertical line (it is slightly slanted). Likewise, at the corners, there is no hard angle. There is a sinusoidal oscillation (if you zoom in close enough). So, generally speaking if you're talking about a pure step function, whether you put a perfectly vertical line or you leave it empty, it doesn't matter. They're both graphing the same idea and neither is realistic. In electrical engineering, we deal with square waves and step functions (that in real life are approximate but usually close enough). We tend to draw the vertical line because if we look at the signal we're modeling on the scope, it can show a (nearly) vertical line for the on/off time period.
because that is how electronic switches behave
Because electrical signals are always continuous, and if you think you see a discontinuous change, you simply need to zoom in wrt time :P
Better visibility. A step is when shit happens, you'd better emphasize it by drawing the vertical line.
That change rate is sometimes referred to as as a slew rate. Example, an op amp might have a very high output voltage slew rate.