Probably the weirdest function I encountered as an engineering student
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- เผยแพร่เมื่อ 13 มิ.ย. 2024
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Dirac rolls "worst function ever"
Asked to leave mathematics
The other neat thing about the dirac delta is that it can be presented as a smooth function. It's actually kind of mathematically brilliant
Have you never heard of the derivate of the delta function?
As a mathematician I am obligated by law to point out that the Dirac delta function isn't actually a function, but a distribution (or measure).
Function in the sense of vertical line test, agree. Function in the sense of mapping, disagree.
but isn't the distribution defined as a function? a bell curve distribution has its own function too, in the sense y = f(x)
@@erikhicks6184 Sure, you can define a function to something like the extended real numbers, or the reals with some extra element called ∞, but that still doesn't immediately clear up what the integral of that function should be. Moreover, if you define multiplication for this new element called ∞ as a*∞ = ∞ for all nonzero a, (or maybe you allow for a sign), then the integral still wouldn't have the desired property for that. It really isn't a function.
As a european engineering graduate I learned it as the Dirac Impulse.
As an engineer, I am obligated by law to say, “who cares?”
The Dirac Delta function can also be thought of as the Normal distribution with mean and variance both at 0
That's pretty cool
In fact that can be made rigorous! The nerd way would be to say that as both mean and variance tend to zero, the Gaussian distribution converges weakly to the Dirac delta distribution!
Or you could say this is the one way that a normal distribution is also a uniform distribution.
Also, sick pfp! Always good to see a fellow fractal enjoyer!
That's actually the most rigorous definition of the delta if you're putting it as a function -- delta (t) = lim (s->0) e^-(t/s)^2
@@NathanSimonGottemer Except you are missing the 1/sqrt(s) factor such that the pointwise limit doesn't exist at 0 😂. If you pick the right topology though, yes absolutely. It's also not the most rigorous, because any rigorous definition is equally rigorous. Personally, I prefer to define it as a distribution because once you know the set-up, it's the easiest to make sense of (for me).
The Dirac Delta function is what you get when you allow engineering students to do pure math 😂
I forgot that zach made educational videos 😅
Don't worry, you're not alone. We all remember when he uploads a video, then forget the next day 😂.
I pretty much only watch his comedy skits in Zach Star Himself. I have to watch countries after WW2 video once every day.
Same
@@zachstarits been awhile lol
Are you willing and able to watch his educational videos?
The convolution is the way in music the reverb is calculated. They make the shortest and loudest possible hit inside a church or a concert hall and record the response. This gives everything is needed to matematically simulate that exact space and allow us to imagine any instrument playing there with that exact ambient. Fascinating.
Or you can just record the transfer function directly with a chirp
In one of my differential equations assignments, we were given four impulse responses from the four cardinal directions around a microphone (L+R channels rather), which could be a clap, a balloon popping, anything like that. We were tasked with recording something and then manipulating it with convolutions using a few lines of MATLAB to change our sound to appear as if it were coming from different places around your head.
The assignments in that course were pretty neat sometimes. Plus we got the obligatory "how to make rudimentary autotune" explanation. I looked back at my assignment and I have no idea why I did this, but my MATLAB file was named oriAndTheBlindForest.m lmao
As an engineer in dynamics and vibration, when we study a system, we need to know how it responds freq by freq. To do this, we have two solutions :
- make the system vibrate at specific frequencies and sweep the frequency to the max value and see at each moment how it responds,
- make a "bang" test : use a hammer, smash the object, and see the response. Then with convolution and Fourier transform, we get back to the response frequency by frequency
The latter of the two sounds much more fun! Until your nightmares of performing convolution for your differential equations class come back to haunt you...
can you point in a direction that goes deeper in the latter and fun solution ? THANK you in advanced .
Finally, a scientific answer to why percussive maintenance is so effective
that 2nd is called a wide-band signal.
I remember being amazed by this when I first saw a video of it being done both ways (though they didn't use a hammer - just played a very short click!). It has economic advantages in that the first method needed an anechoic chamber; but with a click, it was all over before any echo could return.
after seing a bunch of zach's videos on other channel its hard to dismiss ironic notes from his voice
I’m waiting for an April Fools sarcastic math explanation
"Air resistance is accounted for" what is this blasphemy this is not a world I want to live in
That’s until you get to the Discrete time Fourier transform and then they’re all Dirac functions
I'm just taking a class on signal theory, and can confirm this. Honestly, it's amazing the power of this function
Laplace transforms and Engineering Dynamics at play
Mad mathematicians incoming: "It's not a function! It's a Distribution."
3..2..1..
Damn right
B-but it's!
Im a physicist but i must admit that it was my first thought aswell upon seeing this video
a distribution is just a probability measure and a probability measure is just a function. no mathematician will complain
@@bestgrill9647 if youre saying that you didnt have a professor rant about how distributions are not functions for 10 minutes
In math we call the "impulse response" a Green's function. We integrate the Green's function, and that's the convolution.
It's a distribution, not a function 🤓. You need somewhat complicated math to derive the delta distribution cleanly.
In physics you typically use it to describe mass or charge density for an infinitly small particle.
Also the step function is also known as the Heaviside or Theta function.
It's a generalized function. Using them you can find derivatives of common functions.
For example, the derivative of |x| is sign(x). And the derivative of sign(x) is 2*delta(x). It's strange that 2*infinity is not the same as just infinity, but that's correct
I love direc delta because remembering that the inverse laplace of a constant is that constant multiplied by the direc delta function gave me 20 extra points on a Circuits 2 quiz
I want to mention that your rectangular formation of Dirac Delta function can be fudged to provide any value to the integral
Eg.
From -a -> a we have an area of n
This means our height would need to be n/2a
Take a -> ∞ and you get the same resuly, just with an area of n
I've been self-studying math for about 2 years now. Currently, I'm working through a DiffEq textbook and covered the Dirac Delta Function a couple of chapters ago, along with convolution. This video was great because it allowed me to prove to myself that I did actually learn it and was able to follow along and even preemptively guess the next topic. Thanks!
Mathematicians: wow such a strange function
Programmers: it's just if(x==0){return INT_MAX;} else {return 0;}
Realistically, since everything programmers deal with is discrete, they would be dealing with the discrete analog of the Dirac delta function, which is the kronecker delta function.
d[n]=1, n=0
d[n]=0, everything else.
@@markgross9582which is just logical not
@@U20E0 what do you mean it’s logical not? Are you talking about how Boolean vars in most languages consider 0 false and every other number true?
@@markgross9582That combined with the fact that in most languages true and false are just 1 and 0 with a taped-on moustache.
@@markgross9582 in programming, there is a distinction between inverting a boolean value (logical not) and flipping all bits of a binary representation of a number (bitwise not)
TBF the Laplace Transform is still useful here because it turns out that convolution gets turned into multiplication in the frequency domain and also the FT and LT of the delta function are both 1
It works for solving no homogeneous differential equations, but strictly (formally) speaking it is not a function. Mathematicians had to create a new theory to formalize those weird things Physicists were using; it is called distribution theory, some call it generalized function: because the formal definition of function does not include it.
In our university we had a laser that had the power of 10^14 watts for 10^-12 seconds.
Holy shit
@@dielaughing73 The total energy is just 100 joules, but as it is released in such a tiny amount of time, it can turn the air into plasma.
@@skyscraperfan that's friggin awesome
I remember the university I did my PhD at had a similar laser. I didn't ever get to play around with it for anything but I remember a few stories of what it did to the surfaces of different materials. Putting little craters into titanium blocks and such.
I want to add that in mathematics this dirach delta he defined does not exist, infact the condition that the integral is 1 is impossible for a function 0 everywhere except for one point. Indeed in mathematics we use the dirach delta a lot but without this condition. Still this is very useful in physics as explained in this video. This makes this function even more wonderfull.
In math everything you define clearly exists.
@@robegatt yeah, but the dirac distribution is not a "well-defined function", it's as ill-defined as "defining" f(x) as a function that's always negative, but its integral is positive.
@@robegatt In math it's not a function, it's a distribution. It does not satisfy the requirements of a function.
@@username8644 technically is a limit of the definition of a function, but since it fits with differential calculus, which is based on the concept of limit, it all goes well.
Yeah, the Dirac Delta "Function" is a misnomer. Still very useful. If you want to be a mathematician about it, there are a lot of ways to define it. A method accessible to a intro integral calculus course would be: Define a sequence of functions d_n with the property:
1) integral from -oo -> oo of each d_n = 1
2) d_n >= 0 for all x.
3) For any integrable function f, lim n-> oo of integral from -oo -> oo of f * d_n = f(0).
Then we abuse notation and say any time d(x) is inside the integral, we really mean to take the limit as n -> oo of that integral, where we replace d(x) with d_n(x).
Lots of sequences of functions satisfy this property, one is the one Zach gave. There are also "easier" ways to define the dirac delta, but require further math (measure theory and Lebegue integration is the most common way).
While several comments have already noted it's not a function (it's a distribution as well as a measure), it's worth knowing the true functions whose distributional limit is the Dirac delta are called nascent delta functions, in case you want to look up the rigorous details.
As a computer engineering major, the delta function is something that still amazes me. The concept of an impulse response blew my mind when I first learned it; seeing its applications in things like filter design, digital signal processing, and even control systems. Also the fact that convolution in the time domain maps to multiplication in the frequency domain is something that still captivates me to this day.
They should call it the punch function
In engineering it's known as the 'unut impulse function' which is pretty much the same thing
When I learned about this in diff eq, I was so blown away by the brilliance. To be able to mathematically express impulse is just so genius because it ends up setting a system in motion but multiplying it by 1. Just brilliant.
Holy shit he’s back
Nifty. The Dirac Delta also allows an analytical form of sampling which leads to all the DSP stuff.
I'm willing to learn more from Zach's engineering channel and enjoy these videos equally much, but sadly as a highschooler I am not able to understand most of the topics and content :(
i've been doing a project on fourier transforms and i only realised last week that the frequency spike FT of a single sine wave is modelled with a dirac delta function as well. I first read about them in a QM book where a 3D delta function describes a point particle. Very versatile piece of maths!
Dirac and Jalad at tinagra. When the walls fell.
Temba, his arms wide.
Convolution of impulse, then the walls feel
You just casually gave the best intuitive definition of convolution
I love the explanation on convolution, never seen it explained more intuitively
Probably one of the best videos for intuition about signals and systems that I have ever seen
Thank you for making a video on the Dirac Delta function, I have studying it for some time and I hope this video will help me understand it better.
Dirac functions were definitely the craziest thing my differential equations course covered.
For me it was the Laplace Transform. I wasn't exposed to delta functions until later in my physics classes.
@@ThePrimeMetric Laplace transforms are so cool. Higher level mathematics are truly a marvel.
@@ShadowSlayer1441 In my opinion, Fourier Transforms are even cooler. To be honest I haven't really used Laplace transforms since my first ODE class. I don't know what their applications are outside of solving differential equations but Fourier transforms seem to do the trick just as well. Their pretty similar, Laplace transforms are just the real-valued analog I guess, but I haven't seen them used for anything besides solving differential equations.
I've used Fourier transforms in many classes though and even used it for some physics research. My favorite applications for them is Fraunhofer diffraction from Optics and using them to parametrize any curve or surface.
I was actually wrong here. I probably knew this at some point and forgot but the frequencies of the Laplace transform can take on complex values. So the Fourier Transform is actually a special case of the Laplace transform. A Fourier transform decomposes a function into sinusoids and the Laplace transform decomposes functions into exponentials and sinusoids. So they each have their own strengths and weakness. Laplace transforms are in general probably better for solving differential equations because they are more stable with exponential growth or decay.
i started off thinking of this as just the derivative of the unit step
Just came across the dirac delta recently in Quantum Mechanics. It used to be a pretty strange function to me but the application in Quantum beautiful. Truly a function by mathematicians, for mathematicians
I was once a TA for this subject in college. One helpful analogy that students loved was the Taco Shop or the Furniture Store. At either, ingredients or raw materials (alluding to the input curves) go into the Shop or Store (System to Convolute with) and each produced nachos, tacos, or burritos or a chair, table, or shelf (alluding to the output). The output would “take the shape/presentation” of the directive at the Shop/Store at that moment. The analogy may not be 100% accurate, but oh how fondly I remember teaching Convolution and seeing how students began to understand what it all meant.
2:21 Everyone who's taken AP Physics before has unknowingly been using this function the entire time.
Amazingly great video. Didn't realize I was also going to finally understand impluse function, convolution and the LTI control response. I wish my control class professor had said it this way.
Great job on this one Zack!
Love the educational content ❤..you have a gift of explaining
your timing is impeccable seeing I got an exam on this in 3 hours ❤
literally took my signals and systems finals yesterday lmao
Currently learning this as an ee student and it definitely confused me at first
Elite timing
I hate that I saw the thumbnail and went “Dirac delta function!”
**Diff EQ flashbacks set in**
Zach being able to teach so seriously, and yet willing to teach unseriously, is such a blessing
1:20 that was unexpected but glad to see Zach star himself on Zach star channel.
I'd say that it isn't built into the dirac delta that it can predict the response of a system to any impulse, but instead that that's what convolution does. Convolution multiplies all points of time of an impulse with the response the system has after the time delay since that impulse, and sums/ integrates over the time the impulse has acted. In reality, the concept of convolution is not so complex, although it becomes a complex calculation requiring computers in order to be done in a reasonable ammount of time. I also love how it intersects with the topic of Fourier transforms, in that we can use Fourier transforms to compute a convolution.
huh... So that's what my controls professor was on about.
I like to think that Dirac delta function is the laplacian of the fundamental solution to the laplace equation.
While the functional and exact formulations of these kinds of fomulas and tools are extremely interesting. I have to point out that the most useful part of this is how wasy it is to plug them into computers and numerically calculate stuff with them. There will always be an error sure. But the fact that you can plug a whatever record of an impulse response and numerically convolve it with whatever signal to obtain the behaviour of the system is invaluable for simulation and signal processing.
Woah this was really cool. CUrrently studying for my differential equations final and was cool to see how the dirac delta and step function are related
In calculus 4 ( differential equations), when I saw the Dirac delta I was really asking myself if it was from the actual dirac, because he is basically a half god and lived so close to us in time. Could not believe we were gonna use something from him in an engineering course.
I learned this for both electromagnetism and laplace transformations. Beautiful function, I really like how it behaves so neatly despite such an unorthodox definition .
I am curious though, what is the co-domain of the function? As far as I know, infinity isnt a number and is not an element in the set of real numbers but the approximated functions leading up to the dirac delta do have a co-domain of R.
Great video btw, these really help me understand what I am doing in my physics class to a deeper level.
To be mathematically precise, dirac delta function does not make sense as a function, but as a distribution. In fancy math language qe say its defined as a continious linear functional from the space of smooth, compactly supported functions topologized with an inductive limit topology, but in human language you can think of it as something that only makes sense under the integral sign multipled by a continious function.
So this is what the impulse response means in guitar amp cabinets.
Bruh where was this video when i needed it. I just finished my linear systems and signals class today 😭. Good intuition
The delta function also has a derivative. A good place to learn how to make sense of such "functions" (which are distributions, not functions) is Lieb & Loss's analysis book.
Thank you for explaining this better than any of my professors!
I hope you are not breathin the air that relates to that particular 'air resistance'. The impulse response of that spring-mass system has no oscillation at all, so it must be sitting in oil or something ;-)
I'm a matematicians so I have to say that. Delta dicara is a distribution of funtion with has 1 if x=a and 0 x=/=a. Area over this funcition is 0. However measure od any set with have element a is 1.
I just did dynamics, system modelling, and control systems and I never noticed this lol
Thank you! I'm reading a book and was confused by this today and of course you read my mind from the future and upload this
Taking me back to my college electrical engineering undergraduate days in the early eighties at UC Berkeley. Thanks.
God damn, I am absolutely blessed by the timing of this video. I have an exam in "Systems and transforms" math course in 4 weeks.
you could construe a function with an arbitrary area k by saying y=k/2a when x=0, then take the limit to say the Dirac delta function has area k under the graph.
point is, the area is undefined, and the function being "at infinity" is meaningless for real valued functions.
I kinda skimmed through, but wanted to mention, the reason the dirac delta might not make sense as a function, is because it is sometimes used in place of a density, but a point mass has no sensible density arguably. But if we are integrating against distributions, it totally can still make sense as a measure, with respects to a Lebesgue integral. Measures just give you the amount of stuff in a set, so for a continuous distribution you have the riemann integral over a region as usual, but for a dirac, you just get all the mass if the measured set contains the point mass and none of the mass if the measured set does not contain the point mass.
I was so confused for the first few seconds like “hold on, this voice isn’t meant to be on a math education vid” 😭
Dirac delta function only makes sense as limit of sequence of functions. It could also be the limit of a gaussian with the standard deviation going to zero
It also makes sense as probability measure supported on a point, as the "evaluation" distribution or as hyperfunction with representative 1/z, and i guess it has many more equivalent definitions
@@massipiero2974 the interpretation of the probability measure is that it's the distribution of a variable that can only be one value
It's a distribution, not a function per se. You can define a function in the intermediate limits.
A little correction: technicaly the Dirac's delta is not a function but (Schwartz) distribution
Zach's in his signal processing era right as I finish 2 courses on it.
Now that's something I'm proud to know too much about
Space vs time perspective. No space = eternity, no time = infinite plane, a film slide.
The title is like " This cow is the weirdest human I've ever met".
Dirac delta is also used in quantum mechanics
I was a physics student, and the function seemed totally obvious and not weird at all; you can litearlly integrate and pick out values since you times any function by 0 every except at x = a, where the integral is = f(a).
int_a_b; f (x) d(x-a) dx = f(a)
I remember having a Digital Signal Processing course at university. We recorded a clap, and used convolution between clap recording and any other sound. This was effectively applying a filter to the sound and making it sound like it was recorded in the room where clap was recorded.
I wonder if these things have a connection
They do, that technique is called Impulse Response Reverb, and uses exactly this principle!
Well yes. The equations modeling sound are linear time invariant, so the you essentially convoluted a general input with the impulse response.
@@markgross9582 convolved
i just learned convolution like 2 days ago in my differential equations class, nice to see it might actually come up in my engineering degree again
It will if you study mechanical or electrical engineering at least
The convolution looks a lot like a crosscorrelation with time shifted function?
nice theory you got there 😮
Dirac delta is not a function, it is a linear functional on the space of test functions.
Well done. Would it be correct to say that a Green's function is a sort of generalization of the idea of an impulse response when applied to partial differential equations? I found Green's functions always kinda scary until I looked at them from this angle. If so, it would be cool to see a follow up video on Green's functions.
I just started learning about Green's functions so I'm not an expert but I would say Green's functions and impulse responses are one in the same, the Green's function is just mathematically exact. The Green's function is the exact output you get if the input is some shifted delta function. If you can model your system as a linear differential equation of the form Ly(x)=f(x), where L is a linear differential operator, you can define LG(x,x')=delta(x-x') and solve for G using the form of L and the boundary conditions. The hammer banging method, or whatever you want to call it, I believe is just a more empirical way of getting the approximate impulse response. After all you can't actually apply a delta function of force on something. You can get close though by hitting something very hard over a small area and contact time. An engineer probably has less of a reason for finding the exact impulse response (or Green's function) because:
1.) they are using a idealized model (simplified differential equation) to model a more complicated system and there are greater sources of error involved
or
2.) The system their dealing with is so complicated they don't have a differential equation that models it.
If you can get an approximate impulse response you don't really need to know what your differential equation is, all you need to know is the input or driving function. Then you can take the convolution of these two functions to get the response to the driving function.
Am I misunderstanding, or is the “flip, slide along, and integrate the product” a convolution? Edit: oh my god I commented this literally the second before “this is known…as convolution”
IT'S A DISTRIBUTION.
Thinking of the impulse function as the reflexivity after re-settlement of a pole flip.
The pole flips, decides to correct the flip which is incomplete since the flip is trying to confiscate or conquer the instability of a new territory or experience, ..., so, it does correct itself, but now has to account for and carry the instability which it did not conquer, now taking its' uninterrupted time to present itself.
The presence of this "abberation" has to be presented clearly and cleanly, and so its' reflexivity accumulates after settlement in a virtual capacitor, which suddenly materializes at a fixed point on the time line, and so is such realized.
In other words, time and space displacement chacterized by returning back to zero, leaves the impulse function as a memory of the previous now unknown event, other then, it happened.
The sliding thining box does slide in virtual space, and does have a fixed consistency time to present itself, before expiring in validity, ...however, the measure is not infinite, it is just long enough to be recognized by the system it is now part of., and absorbed by, and cannot be measured without disturbance, and risk of determination.
An after thought: It's the displacement spectra of a forgotten event that you know happened, but in itself does not carry enough info to explain what exactly it was.
Not that sure whether it needs to be shown on the upper y-axis, as half could be below, and one being chosen just for the convenience of not disturbing any further measures.
Aww, those times at university when I studied cybnernetics, control and regulators.
So the Dirac function is just a specific impulse applied to a system?
8:33, when taking derivative of the output side, the given result is considered to be for "t > 0" right? If not, how does one derivate the output? Thanks in advance
Why is there a need to flip the impulse response?
Playing around with:
((-8)(2abs(x) - abs(x-0.5) - abs(x+0.5) - (-8)(2abs(x) - abs(x+0.25) - abs(x-0.25))) / 4
You could set the height to an infinite amount, then Subtract basically all of it. This would leave a platform with slope of infinity, width of 1, and height of whatever you want. I'm too lazy to explain further, or to simplify, so... just trust me bro.
Zach always makes me want to study a certain topic on my own 😂.
By the way Zach, I highly recommend you get the textbook called “The Physics of Energy”. If an apocalypse occurs, that’s the book we need to restart civilization.
Heh. I only knew of its use in quantum mechanics. Thanks, man!
Sir you told before you will make more videos on how to take physics classes from computer like you couldn't find relativity there but didn't make till now. Please make a video on it😢😢😢
The ultimate function: x=0
the guy who found out what 1 was made this video... wow
Please cover laplace transform and fourier transforms too. This is where my eyes glazed over in my EE classes and I said fck it I'm switching to CS.
I have actually had a class on signal theory as part of my CS major 😅 Although I guess it wasn't _too_ in-depth on Laplace and Fourier
@@ef-tee I just didn't get convolution integrals. The class I was in the prof was just throwing around integrals as if this just justifies how the transforms work, no derivation or trying to solve the integrals. If I were to do EE again I would focus more on application than theory.
This is the first time i understood what people nean when they say math is beautiful
What if Washington and Lincoln encountered a Dirac delta function?