Fourier Transform Duality Rect and Sinc Functions

Explains how the square waveform, Rec(t), and the Sinc function are related via the Fourier Transform. They are extremely important functions in digital communications and electronics, as well as elsewhere.
Note that in this video I used the following definition of the Sinc function: sinc(x)=sin(x)/x
** Also note that I made a minor typo when I wrote out the answer to the integral on the first line of mathematics. The -1/jw term should be positive. The line below is correct though, so I didn't carry the error through - it was just a "typo".
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[@yellowboxster06]

Not sure what’s better than a “thumbs up” but that was an excellent presentation.

[@iain_explains]

Thanks for your nice comment. Glad you liked the video.

[@jefferytownsend7787]

Thank you for taking the time to do this.

[@iain_explains]

Glad it was helpful!

[@TheTanmaybishnoi]

wow u just explained so simply, this concept that i was having a very tough time understanding! Thanks!

[@iain_explains]

Glad it helped!

[@sevfx]

Am I missing something, or shouldn't the integral evaluate to -1/jw * (e^(-jwT) - e^(jwT)) so the exponents' signs switched? In this case it doesn't really matter, because the cos and sins cancel/add up nicely, but i got confused for a sec. Nice explanation of the duality, thanks for the vid :)

[@iain_explains]

Ah, yes, you're right. Thanks for letting me know. I must have got mixed up with the negatives as I was writing it out quickly. The following line is correct though.

[@jesussanchez-prieto3643]

Congratulations Iain, for your great explanation.

[@iain_explains]

Thanks. Glad you liked it.

[@goskamatala]

Thankyou for clearing out what some guy couldnt!

[@iain_explains]

Happy to help!

[@andreasmiller5448]

Very nicely done! You are an excellent teacher. Thank you.

[@iain_explains]

Glad it was helpful!

[@maksymkloka7819]

Theory is all nice and this video explained a lot. Is there a video or some resource where we can understand what this practically means. Lets say you take a square wave and put it through an ideal filter, what would the output look like. Would you be able to see the ringing in the time domain?

[@iain_explains]

I'm not sure what you're asking exactly, sorry. But for a practical example when rect and sinc are "traded off" see: "How are Data Rate and Bandwidth Related?" kzbin.info/www/bejne/kHO2p4CYhJWghrM

[@canocan5050]

Can you do a frequency shifted rect function by -f0 and has a width Delta f, which will be inverse transformed in the time domain using duality and not the integrals

[@Technocratoo7]

Plz do make a video on spectral leakage in dsp and Gibbs phenomenon..thanku in advance

[@iain_explains]

Thanks for the suggestions, I've added them to my "to do" list.

[@tripd4949]

Thank you sir, very clear explanation, and I appreciate it.

[@iain_explains]

Glad it was helpful!

[@martatxrress]

thank you sm

[@BlancoYT]

Can you do an example where x(t) is the sinc function and X(jw) is the rect function?

[@tim-701cca]

Thanks for sharing, nice explanation. I have something not sure. Is X(jw)a notation only? As in the video, you use X(jt) for time domain and jw for frequency. Why can’t we use X(w), X(t). Also, what does bandwidth means? Can you help to explain this term that I always heard but didn’t understand?

[@iain_explains]

Two videos to answer your questions: "Transform Notation" kzbin.info/www/bejne/r2mUgGywataheKc and "How are Data Rate and Bandwidth Related?" kzbin.info/www/bejne/kHO2p4CYhJWghrM

[@varunchakravarthy5844]

It's very understandable now thank you sir.

[@iain_explains]

You are most welcome

[@tomoechel3759]

Thanks for the video, super helpful. One question: Why exactly is the bandwidth infinite in the frequency domain (when having a rectangular shape in the time domain)?

[@iain_explains]

Well if you think of adding sinusoidal waveforms together (in the time domain) to form the square waveform in the time domain, you'd need to be adding in some extremely high frequency waveforms to generate in the sharp corners of the square. In fact, you would need infinitely high frequencies to generate those sharp transitions from high to low (which are infinitely sharp vertical transitions.)

[@tomoechel3759]

@@iain_explains Thanks for the quick reply - understood. Great content. :)

[@iain_explains]

Glad it's been helpful.

[@f1u1c1k.y1o1u]

​@@iain_explains ​​ is there any relationship between the sinc waveform that appears as the Fourier of Rect(x) versus the sinc waveform that appears in the infinite degree Lagrange polynomial basis? It doesn't make sense. Lagrange's sinc is formed by a polynomial of infinite degree, which is just an infinite product of equally spaced monomials, but the Fourier of the Rect(x) isn't a polynomial, it's a trig-polynomial (Of infinitely many complex exponential terms) So how do they both lead to sinc? What does it really mean?

[@koche9917]

Please help me to understand this. Is the "square waveform" just a square pulse ? What if the square is a periodical square waveform ( clock ) ? Then won't the frequency domain not a sinc function ?

[@iain_explains]

The "square waveform" is the waveform that I drew in the top left. This video should help to explain the clock waveform: "What is the Bandwidth of a Digital Clock Waveform?" kzbin.info/www/bejne/bKWTl4qDibxnaac

[@rishabhkumar1050]

hello sir I want to ask that is there any practical use of duality property in real life time?

[@iain_explains]

Yes, there are many. For example, in digital communications we use a sinc function/waveform in the time domain for pulse shaping, so naturally we want to know about its frequency domain properties. Since we know that the Fourier transform of a Rect function in the time domain is a sinc in the frequency domain, we can use the duality result to find the Fourier transform of the sinc pulse shaping filter. See this video for more details on pulse shaping: "Pulse Shaping and Square Root Raised Cosine" kzbin.info/www/bejne/h5abf4Suac6Ve5o

[@gajodharsingh9590]

2:02 ( sin wT )/wT = Sa(wT) and Sa(wT) = sinc (wT/pi) so ( sin wT )/wT = sinc(wT/pi) i am not understanding this

[@iain_explains]

You may have missed seeing my note in the description below the video. In this video I used the following definition of the sinc function: sinc(x)=sin(x)/x (Note that there are two different definitions of the sinc function: this one, and the other one where sinc(x)=sin(pi x)/(pi x) )

[@harrysvensson2610]

4:15 Calling that a bandpass filter, hmmm, I suppose a lowpass filter counts as a bandpass filter.

[@iain_explains]

You're right, it does! The terminology sometimes gets a bit relaxed when thinking about digital communications systems, for example wireless/mobile communications, where these low-pass (base-band) representations are up-converted by a carrier waveform, and hence the "low pass" baseband signals become "band pass" passband signals.

[@prathameshbhosale9968]

Sir you explained it very nice it helped me❤, but you look like Eric Garcetti 😅...

[@iain_explains]

Glad you liked the video.

[@zechenxiong6067]

You missed a minus sign in the exponentials

[@iain_explains]

Yes, I already made a note about that in the Details section below the video.

[@velmuruganr9321]

really Banks(thats minionese) a lot

[@iain_explains]

Glad it was helpful.