How do Complex Numbers relate to Real Signals?

Explains the link between sinusoidal signals (in the "real world") and complex numbers (in the "maths world").
One point to note is that I have used "j" for the complex variable, instead of "i" (which is more commonly used). This is because I am an electrical engineer, and we use "i" for electrical current, so we use "j" for the complex variable to avoid confusing ourselves (except then of course it can be confusing for physicists and mathematicians! ... sorry about that, but it is just the way it is.)
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Related videos: (see www.iaincolling...)
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[@navjotsingh2251]

If you make more videos like this about complex analysis you will sure become more popular, there aren’t that many good videos about the topic so you have great potential!!!! Excellent video btw.

[@iain_explains]

Glad you found it helpful. Thanks for the suggestion on the topic of complex analysis, I'll give it some thought.

[@PradeepReddy-v9q]

There are enough students in the world who are willing to learn. There are just Not enough Good Teachers. Thanks for being one !

[@iain_explains]

I'm glad you like the explanations in my videos.

[@seshansesha7645]

The world needs teachers like you....Thank you professor for keeping it soo simple to understand.

[@iain_explains]

Thanks for your nice comment. I'm glad you are finding the videos helpful.

[@Deutche6928]

After 6 years studying this topic, I finally get a full understanding of it, I hace no words, Thank you so much!

[@iain_explains]

I'm so glad it helped!

[@pardhasaradhi816]

I am in love with your lectures. Thanks prof Iain. In the world full of seekers, you are a giver. You are in a great position with busy schedules and research, and still contributing your valuable time here. Thanks

[@iain_explains]

Thanks so much. I'm really glad you like my videos.

[@DigitalMirrorComputing]

I have a video coming up that involves complex numbers and hence, I was doing a bit of research on how to explain the topic as it is... well... "complex" :) I have to say that your calming voice, sheer love for what you do (which comes through the video) and absolutely flawless tutoring skills are just beautiful to watch! Not to mention, incredibly inspiring. No fancy animations, just pure skill and a high level of engagement. Thank you for inspiring us and... mate, 100% subscribed!

[@iain_explains]

Thanks for your very nice comment. I'm glad you found the video helpful.

[@dmitrikazantsev3692]

Thanks Iain! When I watched your other video, I asked myself a question, and next suggested video was exactly about this question of mine. When I watched that suggested video, I asked myself another question, and next suggested video was again to the point - this one. When I watched this video, I asked myself something again, and in 2 min was able to find your another video about explaining exactly that. So when I'll finish it, I expect I will find something again exactly targeting my future question. This is speechless. I envy your students.

[@iain_explains]

Thanks for your nice comment. I'm glad you like all the videos. Have you seen my webpage, that has a fully categorised listing of my videos on different topics? iaincollings.com

[@ilgazmore]

Wow just wow. I am an electronics and communication student and i have been dealing with the problem of keeping mind all that formulas in the lectures like communication I or electromagnetic waves I-II. Now i understand and i dont have to. Thank you so much

[@iain_explains]

I'm glad it's clicked into place for you.

[@clems6989]

I have been in engineering and used this system for many years. This is the simplest explanation I have ever seen, Great Job !!

[@iain_explains]

Glad you liked it.

[@swat_katz_tbone]

Really top notch content presented in such an intuitive way. My best wishes for your future videos. I just realized that sin wave is just a plot of projection of a point on the y axis in the complex plane over time

[@iain_explains]

Glad you like the videos. And it's nice to know that the explanations are helping you to understand these fundamental concepts. It's the reason I'm making the videos, so it's great to hear.

[@kyrospapa6001]

amazing video, i have never thought of observing a movement in complex plane, into 2 different movements, one in real axis and one in imaginary axis, just amazing video

[@iain_explains]

I'm so glad it was helpful!

[@chrismcmahon7957]

Finally i understand why the real world application is what it is! years of a-level and uni study didn't explain this in a way i understood. Thanks so much for finally making it click!

[@iain_explains]

I'm really glad to hear that. Glad you've found it helpful.

[@FinalDestinationAllah]

Your videos Lecture kicking me to work in Analog domain at the age of 40. wonderful and lovely people like you make this planet beautiful. grateful to you so much.

[@iain_explains]

That's great to hear. Thanks for your nice comment.

[@se2702]

Such a good and fresh way to look at this. Brilliant again Iain!

[@iain_explains]

I'm so glad you like my videos. It's great to hear.

[@efazsafwan7804]

Hi, just wanted to say thanks for the video. It was very helpful. I've seen an alternating voltage vcos(wt+phi) being equated to Ve^j(phi).e^jwt or something along those lines. My question is, the expression on the left hand side of the equation is a cosine function, yet it is mathematically being represented in the complex exponential form (e^jtheta) where the imaginary part is being ignored? How does the math work? Thanks.

[@iain_explains]

The formula you're talking about is: cos(wt) = 0.5e^(jwt) + 0.5e^(-jwt) In words, this means cos(wt) can be written as the sum of two rotating complex numbers (rotating as w increases), that both start at the point 1+j0 (ie. on the real axis). The first complex number is rotating counter-clockwise and the second is rotating clockwise. When you add the two together, the imaginary components cancel each other out, and all that is left is the real component, which is a cos wave on (or along) the real axis. I hope this helps.

[@stevenjalter6386]

@@iain_explains Thank you for this explanation. However, apart from the mathematics I am puzzled as to *why* you introduce the second, negative (CW) rotation in the first place. What is the reason or purpose behind doing this?

[@_SeaH0rse]

Thank you SO much for this. Really. I've spent years now trying to understand this relationship.

[@iain_explains]

Glad it was helpful!

[@SeanAM2023]

I wish I had attended your class earlier. It's just GREAT!!!

[@iain_explains]

I'm really glad you've finding the videos helpful. Thanks for the nice comments.

[@miftahmahmood1198]

Your teaching method is simply wonderful and awe-inspiring. The best video I saw regarding this concept. Thanks a lot. May God bless you.

[@iain_explains]

Thanks for your nice comment. You are most welcome. Glad you like the videos.

[@SotirakisPeklivanas]

I appreciate this lecture. It finally cleared up the complex relationship. I cannot help thinking that the maths can work just as well without the complex number. Are you able to expand on this thought?

[@iain_explains]

Certainly everything could be done with "real" numbers, but then it would be necessary to deal with a lot of expressions involving sin and cos functions, which is cumbersome and also the relationship between amplitude and phase would not be clearly visible.

[@Ykotb08]

amazing explanation and really simple, thank you! Just hope the sound was a bit higher because I used maximum volume and it is not that high.

[@iain_explains]

Glad you liked the video. Sorry about the volume. Since I made the video I've bought a new microphone.

[@alexhunter4800]

Best content for a quick lunch break! Thank you 💚

[@iain_explains]

I'm so glad you like the videos.

[@tanner1985]

So enlightening with that calm approach, thanks!

[@iain_explains]

Glad you enjoyed it!

[@zuhalormanoglu5091]

I feel enlightened! Thank you for the simple and intuitive explanations. Everything makes perfect sense now.

[@iain_explains]

Glad it was helpful!

[@cyberphox1]

Great nuggets of info, well presented

[@iain_explains]

Glad you liked it

[@simayiya4421]

This is the coolist and simplist way to explaine theses curves. Many thanks sir.

[@iain_explains]

Glad you liked it!

[@sagarrawat7203]

I didn't get it why youtube don't vedio like yours 😢. I shows useless vedios. I have got your channel when using vpn for different vedios. Lucky! have found ur channel ❤❤

[@mnada72]

What a magnificent insight ! Thank you

[@iain_explains]

Glad it was helpful!

[@loveen3186]

Your teaching is extraordinary - thank you so much for your help!

[@iain_explains]

Glad it was helpful!

[@Electrologia]

This video is a life saver. Thanks a lot! for such a great explanation!

[@iain_explains]

Glad it was helpful!

[@riasadbari1872]

You are brilliant at explaining complex (ignore the pun) concepts. Surprised that you have so few subscribers

[@iain_explains]

Thanks! 😃

[@tanmoybhakat7912]

I face problems to connects the exp(ix) with cosx and sinx in real world view point Now it is clear Thanks a lot sir ☺️

[@iain_explains]

I'm glad to hear that my video helped.

[@siddharthjoshi6811]

Sir you have really nice content, you are quite underrated here on youtube, I hope to see alot more interesting videos.

[@iain_explains]

Thanks, I'm glad you've found the videos useful.

[@ellentuane4068]

What an excellent video! It has a great insight about real world use of complex signal

[@iain_explains]

Glad you liked it!

[@zuzukouzina-original]

I remember that we used complex numbers to describe frequency and phase shift in one word. You can than easily combine 2 frequencies with different phase shifts by doing simple calculations.

[@iain_explains]

Yes, that's right. Here's my video on that topic: "Why are Complex Numbers written with Exponentials?" kzbin.info/www/bejne/eaqYeoSkd9V9paM

[@vichetduk9218]

Hi, I really enjoy the video but i have a few questions. why do you define theata moving in a negative direction to create the second equation? when you add the two equations, you get back the cos function which is a real function/signal which is defined at the beginning. I am feel, it is going around and around and confusing! or Is it the goal that the cos function can be represented by complex functon?

[@iain_explains]

Yes, exactly. By representing the cos function in terms of complex exponentials, it allows calculations to be done directly according to the rules of complex number mathematics, which is very powerful and convenient.

[@binhnguyenquoc3249]

great vid, but what I don't understand is why our signal is represented as a point moving in circle? If a physical signal (like a current in a circuit) is represented as a point moving in circle (which I don't understand why we do that), then I can understand the need of representing signal as a complex mathematic formula, but isn't a physical signal (in this case a current in a circuit)can be sufficiently represented as a sinusoidal wave, then why do we even need all this complex representation. Thank you

[@iain_explains]

Hopefully this video will provide more insights: "Visualising Complex Numbers with an Example" kzbin.info/www/bejne/nonPZqiOa76mnpI

[@nahopyga]

Mind blowing explanation. I always found it difficult to conceptualize.

[@iain_explains]

Glad it was helpful!

[@leocmen]

Have no words to thank you

[@iain_explains]

I'm glad you liked the video.

[@hieutrung2883]

wonderful explain that i ever seen in my life. Thanks prof so much.

[@iain_explains]

Thanks for your nice comment. I'm so glad the video was helpful.

[@Titurel]

Thanks so much for making this so clear and easy!

[@iain_explains]

Glad it was helpful!

[@no-de3lg]

I have question im wanting to understand the basics to know how mri system generate rf pulse and what confuses me that the rf generator I know nothing about sine wave or sine function please could you explain to me why they plot everything related to sine wave why they are important how do waves travels Also how is complex wave formed is it just sum of other waves and if two waves with different frequency was emited do they sum up to form a complex wave like if i recorded my voice it will not be single frequency will be weird shaped wave what is this

[@iain_explains]

Hi, I think the best suggestion would be to watch my videos on the Fourier Transform at www.iaincollings.com/signals-and-systems

[@GowthamanPrabhu]

Why do we use complex number on the y-axis to model the waves? Why can't we do the same using normal numbers?

[@iain_explains]

The axes represent orthogonal basis functions. You need some way to differentiate between the two orthogonal functions. Hopefully this will help: "Visualising Complex Numbers with an Example" kzbin.info/www/bejne/nonPZqiOa76mnpI

[@kpk331]

@5:17 What is the logical justification for saying that, that point is e^iTheta?

[@iain_explains]

This video will hopefully answer you question: "Why are Complex Numbers written with Exponentials?" kzbin.info/www/bejne/eaqYeoSkd9V9paM

[@rickjesse3763]

Very helpful and clear explanation. thank you.

[@iain_explains]

Glad it was helpful!

[@ahmetserdr2920]

Can you explain numerical controllerd oscillator?

[@iain_explains]

Thanks for the suggestion. I've put it on the "to do" list (but I should warn you, it's a long list).

[@huyenpham4261]

I spent so much time with complex numbers in math during my bachelor degree, but I did not understand its application. Thanks for your great video.

[@iain_explains]

Glad it helped!

[@baranpekkolaystudent820]

I love you man

[@oke4631]

you made teh sybject justa bit more comprehensible thank you

[@iain_explains]

Glad it helped!

[@chivo2468]

if e'(j*tetha) = 1 and is represented as a hypothenuse in the Re-Img diagram, then why isn't the first expression e'(j*tetha) = sqrt ( cos'2 (tetha) + sen'2 (tetha) ) ? Theorics always drawing aces from the sleeve without explaining...

[@iain_explains]

It's not represented as a "hypothenuse". It's represented exactly as it is. e^(j*theta) is a complex number. It is not equal to 1. It's magnitude equals 1. It only equals 1 for the following values of theta: ... -4pi, -2pi, 0, 2pi, 4pi, , ... Try plugging an angle into cos(theta)+j*sin(theta) and you will find that the resulting complex number lies on the unit circle (for any theta). This is the definition of e^(j*theta).

[@sagarrawat7203]

Thank you sir ❤. I always think how can real signal can be imaginary. But now it's clear. On doubt i have is why the use complex no. ? They can use vectors instead of this. Please! explain sir.

[@iain_explains]

It is possible to do lots of analysis using the complex numbers in exponential form, that you can't do in vector form.

[@sagarrawat7203]

@@iain_explains thank u sir ❤️.

[@PurplePeopleEater96]

Thank you. But it's not clear: 1. Why use complex numbers and not just cos and sin, real numbers in both axes. 2. Why add two signals going in opposite directions on the same circle.

[@iain_explains]

The two axes are orthogonal, and the complex number is a convenient way of representing these two components in a single "number". Yes, you could keep them seperate, but you would still need to ensure the mathematical relationships between the two are maintained when using them in calculations. Perhaps this video will help: "Visualising Complex Numbers with an Example" kzbin.info/www/bejne/nonPZqiOa76mnpI

[@nicolasperez4292]

what happens if you plot something in the time domain that has an imaginary component? In your example you plotted cos(theta), but what if you had to plot something like (j)(cos(theta))?

[@iain_explains]

Perhaps you haven't quite grasped the concept. "Imaginary" signals do not exist. They are just real signals with a phase shift compared to the "zero phase" cos wave. You can plot complex valued functions either by plotting their amplitudes and phases, or by plotting their "real" component and their "imaginary" component. In this video, I plotted the "real" component with the horizontal plot (sin(theta)), and the "imaginary" component with the vertical plot (cos(theta)).

[@nicolasperez4292]

@@iain_explains I see, thank you for that. Let me rephrase my question a bit, because I wasn't quite clear: When someone writes down a signal using euler's formula (as shown in this video) are they referring only to the 'real' component? Additionally, what is the point/benefit of representing a signal using euler's formula?

[@iain_explains]

The complex exponential is a way of writing down a complex number in terms of its amplitude and its angle from the positive real axis. If the angle is a function of time (eg. if theta = omega x time ) then the complex number rotates around a circle, as time increases. If you add it to another complex number that has the same amplitude, but the negative phase, then the imaginary components of the overall complex number will cancel each other out, and the overall complex number will just be real-valued. If you plot that as a function of time, it will be a real valued cos wave. Hope this makes sense.

[@therealspixycat]

Excellent explained!!! Thank you very much!

[@iain_explains]

Glad it was helpful!

[@therealspixycat]

@@iain_explains the e^j.w.t is just a notation only?

[@iain_explains]

It's not just notation. The exponential function has the properties that make the definition exp(j a) = cos(a) + j sin(a) make sense.

[@therealspixycat]

@@iain_explains that part I don't understand. The rest is clear. Has that something to do with the circumference of the point from the original 0,0 coordinate to its position when turned to angle theta?

[@iain_explains]

Here's a video I just made of this topic. "Why are Complex Numbers written with Exponentials?" kzbin.info/www/bejne/eaqYeoSkd9V9paM

[@MahmutKaraarslanStudent]

I appreciate you sir

[@iain_explains]

I'm glad you like the videos.

[@ser7_ha]

can u give me example , where we use imaginary part in our real live , please ?

[@iain_explains]

This video might help: "Visualising Complex Numbers with an Example" kzbin.info/www/bejne/nonPZqiOa76mnpI

[@ibrahimallafi1460]

helpful video, sir

[@iain_explains]

Glad you found it useful.

[@vikasgupta-hl9sq]

Very helpful

[@iain_explains]

Glad to hear that

[@hotshotsongohan2898]

Is this used in radio?

[@iain_explains]

Absolutely, yes. You might like to watch this: "Amplitude Modulation AM Radio Signal Transmission Explained" kzbin.info/www/bejne/Y4G6mGBmoJeiodU

[@hotshotsongohan2898]

@@iain_explains thx prof.

[@honestman276]

Thanks. From Bangladesh.

[@iain_explains]

Glad you found it useful.

[@bhanusriboddeda1187]

Nice vedio sir Thanks

[@iain_explains]

Glad you found it useful.

[@RisteardB2007]

Thank you so much

[@iain_explains]

You're most welcome. I'm glad it was helpful.

[@AMNHM]

Big thanks!

[@iain_explains]

Glad you liked it.

[@mohamedabdalla7547]

thank you sir

[@iain_explains]

Glad you found it useful.

[@LL-ue3ek]

You did a good job explaining the geometry and the math, and thank you for that. But, I still don't understand how you are able to just jump in in the middle of the derivation, and out of nowhere, simply force the vertical axis to be "Imaginary". Intuitively, the vertical axis is visible, touchable, measurable and physical; it's everything but "imaginary"! What's imaginary about it? which aspect? in what way? I know you can simply call it "imaginary" and go from there to do the rest of the math, but why? What is the underlying reason for saying the vertical axis is "imaginary" or "complex" while it actually can't be more real? What is the hidden mathematical nature that the vertical axis posses which the horizontal axis doesn't, that sets it apart from the horizontal axis (which represents the "real") ? If you can arbitrarily assign "imaginary" or "real" status to either axis like you did, what's wrong if I assign the x-axis to be imaginary instead?

[@iain_explains]

There's nothing that makes it "imaginary" except that it is an orthogonal basis vector/function, and prior to its invention in the early 1700's, everything was on a single Real line, so it was necessary to "imagine" a two-dimensional number. These videos might help with intuition: "Why are Complex Numbers written with Exponentials?" kzbin.info/www/bejne/eaqYeoSkd9V9paM and "Visualising Complex Numbers with an Example" kzbin.info/www/bejne/nonPZqiOa76mnpI . And this video discusses an application in digital communications: "Is the Imaginary Part of QAM Real?" kzbin.info/www/bejne/bJLWdaewlrudmLM

[@MrPabloguida]

It seems a good video but the volume is too low.

[@ridhaturki1907]

Thanks much 😊😊😊

[@iain_explains]

Most welcome 😊

[@amahbubul85]

I understood the link to cos and sin. But this could be done using any two axis, no? Why an real imaginary axis was needed?

[@amahbubul85]

Is to make square of costheta + square of sintheta = 1?

[@iain_explains]

Any two _orthogonal_ axes, yes. But then you could represent them in terms of a simple rotation from the real and imaginary axis. So why not use the real and imaginary, for which a vast toolbox of mathematical results have been developed.

[@GowthamanPrabhu]

Why are you using a complex number (imaginary i) to model the waves mathematically? Why can't we use just normal numbers. What have we achieved using complex?

[@iain_explains]

When you use the complex representation it makes a lot of mathematical calculations easier, compared to having to deal with trigonometric identities of cos(.) and sin(.). Hopefully this will help: "Why are Complex Numbers written with Exponentials?" kzbin.info/www/bejne/eaqYeoSkd9V9paM

[@wilsongomes3360]

Thanks

[@iain_explains]

You're welcome.

[@manfredbogner9799]

Sehr gut

[@GhilasBabahamed]

Thank you so much.

[@iain_explains]

Glad it helped!

[@rahulteen1780]

Sir, I liked the class but i want to know why you are looking side ways?

[@johncgibson4720]

I don't see the advantage of using imaginary number on the Y-axis, instead of just using the plain old y unit vector. And I don't see the advantage of using the euler exponent.

[@iain_explains]

This video might help: "Visualising Complex Numbers with an Example" kzbin.info/www/bejne/nonPZqiOa76mnpI

[@awaisahmadsiddiqi6505]

in real life, we normally dont have a "periodic" signal. how to relate that now?

[@iain_explains]

Good point, but you may be surprised to find how many systems have an underlying periodic signal component - at least during the time that the system is in operation. For example, in AM radio, the voice/music signal (non-periodic) from the microphone in the studio is multiplied by a periodic sinusoid at the carrier frequency of the radio station in order to be able to transmit the signal over the air from an antenna, without interfering with other radio stations that are "modulated" at other carrier frequencies (eg. at 702 KHz for the "702 ABC Sydney" radio station). Check out this video on the channel for more information on this: kzbin.info/www/bejne/Y4G6mGBmoJeiodU More generally, it has been shown that ANY signal can be constructed/represented by a summation of sinusoids at different frequencies - even a square on/off digital signal. This maybe sounds a bit hard to believe at first, but this mathematics underpins the analysis of all time-varying signals. See this video on the channel for more details: kzbin.info/www/bejne/boeZeZxjoLVse6c

[@hassanelgogry1908]

ترجمة الفيديو

[@srabrownn]

I still cant understand why we have to use imaginary number? Cant we replace it with a normal number?

[@iain_explains]

Certainly everything can be done without using complex numbers, but the calculations are much more complicated. Complex numbers enable us to do many mathematical calculations very efficiently. For example, see: "Why are Complex Numbers written with Exponentials?" kzbin.info/www/bejne/eaqYeoSkd9V9paM and "Visualising Complex Numbers with an Example" kzbin.info/www/bejne/nonPZqiOa76mnpI