Probably the weirdest function I encountered as an engineering student 

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?

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?”

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).

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?

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

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.

I’m waiting for an April Fools sarcastic math explanation

I'm just taking a class on signal theory, and can confirm this. Honestly, it's amazing the power of this function

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

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

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)

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.

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).

In engineering it's known as the 'unut impulse function' which is pretty much the same thing

Temba, his arms wide.

Convolution of impulse, then the walls feel

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.

literally took my signals and systems finals yesterday lmao

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.

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

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

It will if you study mechanical or electrical engineering at least

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.

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.

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.