Why are Complex Numbers written with Exponentials?

Iain Explains Signals, Systems, and Digital Comms

Подписаться 65 тыс.

Просмотров 8 тыс.

50% 1

Explains how complex numbers can be written in the form r.e^(i theta). This is a useful representation because it makes it easy to multiply complex numbers and perform other mathematical manipulations.
Related videos: (see: iaincollings.com)
• How do complex numbers relate to real signals?: • How do Complex Numbers...
• Visualising Complex Numbers with an Example • Visualising Complex Nu...
• What is negative frequency?: • What is Negative Frequ...
• Sampling Bandlimited Signals: Why are the Samples "Complex"? • Sampling Bandlimited S...
• How are Complex Baseband Digital Signals Transmitted? • How are Complex Baseba...
For a full list of Videos and Summary Sheets, goto: www.iaincolling...
* One point to note is that I have used "i" for the complex variable in this video. Sometimes electrical engineers use "j" instead. This is because electrical engineers use "i" for electrical current, so they use "j" for the complex variable to avoid confusing themselves (except then of course it can be confusing for physicists and mathematicians! ... that's just the way it is.)

Опубликовано:

30 сен 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 65

@yoda11235 Год назад

It's been 40 years since I did anything with signal processing. This is a great refresher.

@iain_explains Год назад

I'm so glad you liked the video. Have you seen that I have a webpage that lists all the other videos on the channel grouped in topics? iaincollings.com

@zhou6486 2 года назад

Fascinating topic and concise explanations. Many thanks, professor.

@iain_explains 2 года назад

Glad you liked it!

@johndunn5272 Год назад

You have explained the missing link between the use of mathematics and representing signals with a robust consistent and truth-ing representation

@fifaham Год назад

It is due to the property [ ( d(e^x) / dx) = e^x ] and this happens only when e = 2.7183... Good presentation - The Universe knew that but apparently we human discovered that by accidental calculations. There are many shocking irrational constants which we human discovered them that unleash amazing realities and irrational numbers deserve volumes of books to talk about. Thank you for the video.

@iain_explains Год назад

Yes, indeed, the universe really is amazing! Glad you liked the video.

@hamishmckay7386 7 месяцев назад

By far the best explanation of e and complex numbers on the internet

@iain_explains 7 месяцев назад

I'm so glad you liked it.

@minhhoanggia 2 года назад

Nice explanation. I used to use Taylor expansion to explain, but it is too mathematical.

@iain_explains 2 года назад

Yes, and in my opinion infinite series expansions are not intuitive, so any explanations that are based on them, tend to leave the student wondering if it's a bit of maths "magic".

@kpk331 Год назад

Agree. However that explanation is really strong, once you understand how Taylor expansion is formulated. We also get the feeling that the world we live in and things and phenomena we see around are really mathematical!

@john2.7183 Год назад

Very nice explanation. I am reviewing things that have forgotten and never fully understood.. This is excellent.

@iain_explains Год назад

Glad you liked it

@MrFab0117 2 года назад

Very good and intuitive explanation. As always 👍🙏

@iain_explains 2 года назад

Glad yo liked it.

@kevinzhang4221 7 месяцев назад

nice and thank you, from a Chinese ALEVEL teacher:)

@iain_explains 7 месяцев назад

I'm glad you liked it.

@LeBooth 2 года назад

I got an A in my Digital Communications class because of you. This is great stuff.

@iain_explains 2 года назад

That's so great to hear. Thanks for letting me know. I'm glad my videos helped.

@zeroxdan 6 месяцев назад

This literally blew my mind haha thank you!

@iain_explains 6 месяцев назад

I'm glad it helped. I hope your mind has recovered from being blown. 🤣

@kpk331 Год назад

I think a direct simpler explanation would be: First, have a clear idea of what exactly are 'e' and 'e^x', i. e, exponential growth etc. where both 'e' and 'x' are real. i. e, firmly remaining in the real domain itself. Then consider the expansion of e^x as a series, all terms real. Similarly see the expansions of sin(x) and cos(x) as well. Now just substitute 'ix' in place of 'x' in the expansion of e^x. Apply the result i x i = -1. Now we can see there the series of cos(x) and sin(x) connected in the form cos(x) + i sin (x) showing that e^ix = cos(x) + i sin (x). This is the simplest explanation. Otherwise how could you figure out how a number can be raised to an imaginary power?!. Imaginary is after all imaginary!!

@iain_explains Год назад

Well yes, the first part of the approach you suggest is the same as how I started my video (ie. showing what "e" is, as a function), and then the second part of your suggestion is a common approach, typically used by mathematicians (who like to work with infinite summations, eg. the series expansion you suggest). I find that for most people though, it is more intuitive to work/think/understand things in the _finite_ world, and not have to resort to techniques that rely on infinite arguments. I guess it's just that different people find different things "intuitive".

@kpk331 8 месяцев назад

@@iain_explains Thx. However I think what is implied in "finite world" is, things are real and conceivable. Imaginary numbers, although good to represent rotatory movement, wave motion etc., are not of this world! They are the result of our laws related to basic arithmetic operations. There is something incomprehensible about them. Isn't it?

@devduttapandey 6 месяцев назад

I absolutely agree with your approach simple and yet so powerful , how about making a video on constellation diagram , i did not find any @@iain_explains

@power-max 2 года назад

I was playing with 'e', 'j', and pi the other day to understand how the classic exponential notation works. I know that i^0=0, i^1=i, i^2=-1, and i^3=-i and circles back around. I found on the calculators it also works for decimal numbers and ends up being a vector that rotates with a frequency of 0.25Hz around the unit circle when you substitute 'x' or 't' for the exponent. Does this have any practical value and why isn't this used as a notation more often? Seems more convenient than having to deal with 2*pi radians for angular frequency all the time. 9:34 Damn, that is useful! Will keep that in mind!

@iain_explains 2 года назад

Sorry, I'm not sure what you're asking exactly. I'm glad you found the video useful though.

@geze2004 Год назад

Euler could not have said it better.

@iain_explains Год назад

That's nice of you to say.

@dmitrikazantsev3692 Год назад

Thank you, I like your videos

@iain_explains Год назад

Glad you like them!

@wssz112 Год назад

i in fact never thought about it....

@iain_explains Год назад

Well, you learn something new every day. 😁

@alfonsosimobas4391 2 года назад

BRAVO!!!! really understandable!!! many years ago, I would had liked to have this class :)

@iain_explains 2 года назад

Glad you liked it. And glad it reminded you of many years ago 😁

@clarklau1292 2 года назад

Hi, professor. I'm recently reading a paper about underwater ofdm, there is a perfomance metrics BLER, but there is no explanation on how to calculated it. So my question is how BLER is calculated. Is there a theoretical way or just by counting the number of fails blocks on receiving end ? Thank you!

@iain_explains 2 года назад

It depends on how they define a "block", and what sort of error correction coding/decoding they are using. It also depends if there is correlation between the bit errors. It is difficult to get analytical results for this. Generally people just use/develop simulators and count the number of blocks that have errors in them.

@priyadarshimukherjee9571 2 года назад

In my opinion, if we just state the words "De Moivre's Theorem", it will be enough to understand. Moreover, regarding the circular explanation of the exponential, this is especially helpfull when the concept of negative frequency is being explained to someone for the first time, i.e. the direction of rotation.

@iain_explains 2 года назад

It sounds like you're already very familiar with the topic. Not everyone is.

@王一蒙-s5v 2 года назад

Thanks for all videos professor, really a helpful channel!! And could you please make a video about MIMO codebook?

@iain_explains 2 года назад

You'll need to be a bit more specific about what you mean by "MIMO codebook". Have you seen my videos on MIMO that are already on the channel? www.iaincollings.com/digital-communications#h.3443rkjmzp5v

@thanhucnguyen6903 10 месяцев назад

You are amazing sir!

@iain_explains 10 месяцев назад

Thanks, I'm so glad you like the videos.

@khalifi2100 2 года назад

Thanks for the explanation in a simple and intuitive way. In usual mathematicians use Taylor series for the exponential to end up (After rearranging terms) to a two Taylor series one for sine and the other for cosine. All roads lead to Rome.

@iain_explains 2 года назад

Yes, thanks for your comment. To me, derivatives (gradients) are more intuitive and practical compared to infinite series.