Orthogonal Basis Functions in the Fourier Transform

Explains how the Fourier Transform equation is in fact a projection of the time domain signal onto a set of orthogonal basis functions (the complex sinusoids).

* Note that in the final expression, I really should have put an extra set of brackets around the argument of each of the two cos functions. ie. inside the final integral it should be 1/2 cos((ω1 + ωb) t) + 1/2 cos((ω1 - ωb) t)

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[HitanshuPatel-xz]

to my understanding, it seems that the fourier transform of a function is the B-coordinate of a function multiplied by infinity, and the inverse fourier transform just maps the B-coordinates back into the time domain, correct? Also, it is seems that the integral of cosine(wt) doesnt actually equal 0, it just never exceeds 1/w. Are you using that to imply that two cosine functions of different frequency are orthogonal because when you divide it by the inner product of a cosine function with itself, you get 0?

Also, this was an amazing video. I am glad I was able to find a video that contains the beginnings of an abstract proof of fourier transforms. Most videos on fourier transforms and signals only teach you how to perform calculations.

[vtrandal]

Very nicely done. In the first half of the video you mention the dot product is analogous to what’s happening in “function space” when referring to the Fourier transform. Is this “function space” a Hilbert space? I’m not trying to quiz or match wits with you. I honestly don’t know and would like to know. Maybe it’s not an important or well formed question I’m asking. Anyway, I don’t expect a reply to every question. I may answer my own question eventually.

[drsimons]

Confused about how the last expression is written, you mean "1/2 cos((ω1 + ωb) * t)" rather than "1/2 cos(ω1 + ωb) * t" correct?

[JohnSmith-oksn]

I have no idea what you've said, but it's got to be awesome.

Today is not the day for me, I guess.

(*I gave you a thumbs up, too)

[ARMAN]

Why didn't you considered the imaginary part?

[manjumanl]

Ive been searching for this channel
Thanks .

[crewyard]

You simply answered all the questions on my mind, what a hero! I am grateful sir! In an academic desert that is competition focused where students are afraid of thinking of the questions on their mind because of their lack of fault tolerance and where rigor and intuition are inhibited because of these reasons, you are an oasis.

[jamesb]

Good video! Quick question.

Initially, you defined the dot product as the method of projecting, but later on you defined it as integral(x1*x2). Why were you able to do this? Just a guess, but is it related to the fact that a dot product = sum of individual components of x1 & x2 multiplied by each other, and an integral kind of works as multiplying individual components of each function together (with respect to time)?

[hongkyulee]

Wow... This video is so very I am so lucky because I am watching your nice explanation :D Thank you sir :D

[bobbaberson]

I got a bit confused @6:00. Afaik, we have two basis, one sin and the other cos. Then how is that if x is a pure cos, it is still orthogonal to cos? I was expecting sin to be orthogonal to cos...

[maxheadrom]

You mean to say the Fourier Transform maps the time domain into a infinite dimensional function space?

Awesome videos, dude!!! Thanks!

[badalsoren]

sir is there any specific reason to take the power of exponential negative I mean why it can't be exp^(jwt)

[tiffanykolesar]

Hi there! What does it mean practically if basis functions are orthogonal, but negatively correlated?

[torstenhendrich]

In the video before, the Fourier formula is shown with positive sign in the exponent. This video starts with the formula having a negative sign. This change is not explained, or?

[kalyansunkara]

Sir, can u start MATLAB series for Communcation systems also?