Negative Frequency, Imaginary Numbers and the Complex Conjugate

[@karthikeyanm1986]

This is excellent! I have always wondered about negative frequency but never was able to understand. You explained it so beautifully! Also decomposing the signal into cosine and since components made me realise these concepts more clearly for the first time, I never even thought this decomposition existed!!! Everything is so intuitively explained, you are a fantastic teacher ❤

[@quaternion-pi]

Your book clearly explains these concepts that have always puzzled me. Thanks.

[@MarkNewmanEducation]

Thank you. It makes me SO happy to hear it. These are things that always puzzled me too. The book is a result of my own journey to understanding the Fourier Series.

[@vmmehta]

@@MarkNewmanEducation can you pls tell me which book have you published

[@EpsilonZRho]

Mark, I want to thank you for the effort and production quality you put into your videos - your animations are top-notch. While they haven't transformed me into an overnight mathematician superstar, they have significantly aided in giving me intuition about a topic I knew practically nothing of. Personally, I now consider you to be among the top youtube educational channels, which places you among esteemed company. I hope for everyone's sake that you keep making these high-quality videos, and that whatever reward you gain from it makes it worth your while. I just purchased your book in part to support your efforts, with its reasonable price being another thing I want to commend you for. Keep up the great work!

[@MarkNewmanEducation]

Thank you for your kind words. I LOVE making these videos as I found these concepts so hard to understand when I was a student and I get a real kick when people write back to me saying that I have helped them understand things I always found so difficult. I'm currently looking around for funding so I can spend my days just making videos. In the meantime, my second book: "How the Fourier Transform Works" is on the way. I'm busy writing it as we speak.

[@EpsilonZRho]

@@MarkNewmanEducation I'm looking forwards to your new book! You are a top-notch educator!

[@JohnyMad1988]

Saw two videos. Immediatly bkught the book. Freat explanations, to the point od the practitioner. I was rustt after some years and theae videos helped a lot.

[@MarkNewmanEducation]

Amazing. Really glad the videos were useful to you. There is a second book out now too which continues from where the first book ended. Here is a link to the series on Amazon: www.amazon.com/dp/B0BSJJ69Z1?binding=paperback&ref=dbs_dp_rwt_sb_pc_tpbk

[@aaron6807]

Sir, thank you so much. Your videos are great for giving intuition of complicated subjects

[@dosgos]

Perfect timing Mark! I was just studying the interesting concept of negative frequencies. Keep up the fantastic videos.

[@MarkNewmanEducation]

Oh, that's amazing! The concept of negative frequency baffled me for years and no-one seemed to worry why all the FT spectra were drawing with negative frequency. I'm happy to finally plug that knowledge gap. Really glad to have helped. Please share with anyone who you feel it might help. I'm already hard at work on the next video.

[@ec11368]

Thank you very much, Mark. You have corrected many misconceptions about the Fourier transformation. I understood many things confusing me about the Fourier transformation for more than six years. Many thanks.

[@MarkNewmanEducation]

Really glad to have helped. Please share and help me help more people.

[@bjrcomputacion]

I absolutely loved the explanation of the "real life" implications of negative frequencies, I work with audio synthesis and see those sidebands all the time, but I never thought of them as related to this concept and even to the spacing between radio signal bands, It seems so natural now. Thank you!

[@MarkNewmanEducation]

You're most welcome. I learned about single-side-band modulation years ago, but never understood it, or why it happened. The same is true of imaginary numbers. Only when I began working with these concepts professionally did the two come together for me in a practical way.

[@bertronixtv]

Thank you. You really made the abstracted concept a really chewable one. Awesome!

[@MarkNewmanEducation]

Amazing. It's nice to know that I have achieved what I set out to do. As Einstien once said: “Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.” Our job as teachers is to find that language.

[@bertronixtv]

@@MarkNewmanEducation Thanks for sharing. You are awesome. More to come 👍.

[@georgethomas8138]

Excellent! gave me insight and understanding I didn't have

[@MarkNewmanEducation]

Really happy to have helped. Please share so I can help others too.

[@petrparizek9945]

Let me add something to this. There's that thing called the Hartley transform. For a start, the Hartley kernel is neither a sine wave nor a cosine wave. It's phase-shifted exactly midway between these two. Now you might wonder: "What the hell does he need this strange phase shift for?" Well, if you play this thing backwards, what happens? Surprise surprise, it's the same as if you start a quarter-period ahead. This means that you can get any possible phase shift just by changing the ratio of the two amplitudes (i.e. the one of the positive-frequency Hartley sinusoid compared to that of the negative-frequency Hartley sinusoid). Which in turn tells me that this way, I can represent any possible signal that I could ever think of, with the added advantage that I don't need complex numbers this time. By which I mean, there's even a 1-dimensional way of representing negative frequencies and this representation still has its use and its meaning. Without negative frequencies, my signals would be similarly limited as if I sticked just to sine waves without cosine waves or vice versa.

[@MarkNewmanEducation]

Fascinating! Thanks so much. I must research this further.

[@amoghnelavigi]

Hi, can you give me some links or any sources, where I can understand this concept of negative frequency in more depth. Even after watching the video, I still have a lot of questions regarding its physical interpretations, real world applications etc. Can we generate signals with negative frequencies with an oscilloscope?

[@petrparizek9945]

@@amoghnelavigi It's the same thing as reversing the thing in question. If your periodic waveform has a frequency of 100 Hz and you play it backwards, then the frequency of your reversed signal is -100 Hz.

[@eshabyalcn2747]

Hi Mark, firstly thank you so much for excellent explanation. I just don't understand one thing. In your previous videos which about of Fourier Transform, you say "complex number result of Fourier Transform is contains either phase and amplitude information". The amplitude of real and imaginary component's of Fourier transform result also cover the phase information. During the performing Inverse Fourier Transform, we added negative frequency and positive frequency sinusoids as a result imaginary components are cancelled each other. In this situation, the magnitude of reconstructed sinusoid or real component of the reconstructed signal is 2 times bigger than real component of negative and positive frequency's ( actually this is eliminated by 1/2 which is out of integral). However, the imaginary part is 0. This situation, we get a real cosine wave and 0 phase degree. This is absolutely unacceptable because one properity of wave is dissapered. Please if I'm wrong you say this and explain me that mechanism.

[@danielbeirao6071]

Congratulations! Well explained!

[@MarkNewmanEducation]

Thank you. I've always found these concepts really difficult to understand. These videos are the result of how I finally understood it myself.

[@Albert_C]

Why does the baseband frequency of your voice contain negative frequency? Or is that the FFT results of the time domain voice signal?

[@MarkNewmanEducation]

That's the really weird thing. This negative frequency is a physical phenomenon. I have no way, beyond the ideas I expressed in the video, to intuitively explain why it is so. It just is! This may suggest that imaginary numbers are not so imaginary after all; that they too are a physical phenomenon. It's not just an artifact of the FFT. Certainly in physics they are now beginning to think that the imaginary dimension, as Decartes called it, is a real thing, not just a trick of the maths. Mind blowing, I know.

[@CombFilter-vo4gk]

There is no negative frequency in the baseband signal. The negative frequency band in the modulated signal is purely a result of the modulation process. Put simply: a modulating signal of Cos(ω1t to ω2t) multiplying a carrier signal of Cos(ωct) will produce: 1/2Cos(ωct + (ω1t to ω2t)) + 1/2Cos(ωct - (ω1t to ω2t)).

[@Jedlmind]

I cannot thank you enough.

[@deroberallmann9844]

Very good Video Mark ! Could you also make such a grate one on Discrete Cosine Transforms? Kind Regards

[@MarkNewmanEducation]

Thanks for the idea. I've added it to my list of future videos. However, I need to understand the DCT myself first. Will research it.

[@JohnyMad1988]

God bless you

[@fhqjklasdf]

I don't get the fact that there is a negative value of "Amplitude"....Can you explain about this, please? I think its about canceling out imaginary term, but isn't "Amplitude" absolute value? Thank you!

[@longluo1788]

Could you make a video explain the details of 2D Fourier transform? I am a bit confused about the MRI about the phase and sptial frequency. Thank you mark.

[@MarkNewmanEducation]

You are not the first to ask for this interestingly. This is something I will have to research. Thanks for the suggestion.

[@Mad_Speed]

I’ve watched a lot of stuff about imaginary or negative frequencies and while I know they exist and what they mean, I still haven’t the foggiest idea what they are and why they exist in signal processing or what they sound like when shifted upwards

[@caca-bt8et]

Dear Newman, initially thanks for your amazing videos you are a great instructor. I have a question to you why don't you upload these videos on Udemy. You can access more people with it.

[@MarkNewmanEducation]

Thank you for your kind words and suggestion. Udemy is indeed one of the directions I am considering. However, I need enough material for a whole course. I feel I am not quite there yet. I have many ideas though waiting to be turned into videos.

[@voidisyinyangvoidisyinyang885]

can you make a video on noncommutative phase of negative frequency as nonlocality? thanks

[@ZenoDiac]

Wow, new video!

[@bayestraat]

Could you explain what is a Hamming window and why is it applied with FFT?

[@MarkNewmanEducation]

kzbin.infoxoE2rnwFROs

[@vmmehta]

Can you please make an intuitive video on Reciprocal Space as a Fourier Transform of Real Space ?

[@MarkNewmanEducation]

Ooooh. I'd have to research that. Thanks for the idea. I've added it to the list.

[@gadxxxx]

It seems impossible ro buy your book at the moment. Amazon says it's unavailable. They only have a Kindle edition. I don't use Kindle. Any news on when it will be available in paperback.

[@MarkNewmanEducation]

Where are you located? It is already available in paperback, but due to the Amazon print on demand limitations, they only have the paperback available in certain countries. However, it should be possible to order in from a different Amazon store that does support the print on demand service.

[@gadxxxx]

@@MarkNewmanEducation I;m in Melbourne Australia. I've checked Amazon Australia and Amazon U.S. No luck there. Do you know of one where It's available? I don't know the price so I don't know if I can afford to buy it. 🙂

[@yosoylibre]

Excelent!

[@MarkNewmanEducation]

Thank you.

[@CombFilter-vo4gk]

I am afraid you have mis-represented the dual sideband of the modulated carrier as being an example of negative frequency. All of the transformations you have described are easily shown mathematically without resorting to complex numbers. This indicates that negative frequency is a trick of the maths.

[@abdeljalilpr2033]

Please watch Othman ibn Farouk's videos