Glad you found it helpful.

Thanks for your nice comment. Glad you like the videos.

Thanks for your nice comment. I'm so glad you like the channel.

Glad you found it useful.

He really does!

still does haha

I'm so glad the video helped to clarify things for you.

Glad it was helpful!

Thanks. I'm glad you liked it.

Thanks.

Glad you like them!

Glad it was helpful!

Glad it was helpful!

Glad it was helpful!

Have you seen the list of videos on my webpage? iaincollings.com Let me know if there are other specific topics you'd like.

Yes, that's right.

Perhaps you haven't quite understood the points that are being made in the video, sorry. In reality, all signals only have +ve spectrum. There is no "negative" spectrum. No signals oscillate with negative frequencies. It's just that engineers often choose to represent the "negative rotating phases" of a signal by showing its effect as a "negative" frequency.

Ah yes, great question. I should have thought to mention something about that in the video. Anyway, one reason shows up when we start to look at sampling, and another good example is when we look at modulating signals onto carrier waveforms. In both of these examples, the "baseband"/"low frequencies" (both positive and negative) get shifted up to higher frequencies. I'm not sure if it clear from that description, but check out these two videos where you'll see that happening: Sampling: kzfaq.info/get/bejne/d8mlhNF6vc--YnU.html and Modulating: kzfaq.info/get/bejne/Y7aHmpBh0JjTm6c.html

You're welcome.

Yes, you're right, it is possible to write the Fourier transform without needing the concept of negative frequencies, however it is most commonly done in a way that does induce the negative frequency concept - as illustrated at the 1:25min mark of the video.

Thanks. Glad you liked it.

One example is sin(w_1 t). This can be written 1/(2j) [ e^(j w_1 t) - e^(-j w_1 t) ], which you can see gives complex valued delta functions at w_1 and -w_1 in the Fourier transform (due to the 1/(2j) term that multiplies both of the complex exponentials). So, actually the only signals that do _not_ have complex valued Fourier transforms, are ones that only contain cos(wt) components (with no phase offsets - since as soon as you include a phase offset, you are getting a sin(wt) component).

NOMA separates users in the "code domain", and SDMA separates users in the "spatial domain". You can use them together. For example, you could implement SDMA at a base station, by using sectorized antennas, and then you could implement NOMA in each sector.

Glad it helped

Thank you! Cheers!

Sounds fascinating, but you'd have to ask a physicist.

I'm so glad you liked it.

@@iain_explains Once I get a job I will pay you, for helping everyone to make understand these concepts..💕💕

That would be very nice. Good luck in your job search.

sin(wt) is the same as cos(wt), except that the points are rotated 90 degrees around the unit circle, due to the 1/j factor in the complex exponential representation of sin. See this video for more details: "How do Complex Numbers relate to Real Signals?" kzfaq.info/get/bejne/irKHeJZpnbC7d6c.html

I dont think so. Fundamental topics are much more important. His purpose is not to prepare you for exam.

Thanks for your recommendation. I'm glad you like the videos.

Speed/velocity is not the same as Frequency. In your example, "Frequency" is the number of rotations per second. So it's a positive number. It doesn't include the direction of the spin.

@@iain_explains But the number of cycles per second (Hertz) is also a positive number; does not include direction either. I specifically said angular (rotational) speed. Rotational speed seems to be very similar to frequency. And angular speed is obtained by multiplying by 2π. One rotation is equivalent to one full cycle. One rotation per second should be equivalent to one cycle per second. Why do you say they are not the same?

Speed is measured in distance per unit time (eg. metres/sec), or angle per unit time (eg. radians/sec). Frequency is measured in "number of times something happens" per unit time. They're different things.

@@iain_explains Sorry, but I still don't get you fully. What about the term, "revolutions per second"? It has got nothing to do with distance, and the angle is always 2*pi radians. We are mainly counting the number of complete revolutions. I think I should have said angular frequency (or rotational frequency) instead of angular speed?

OK, yes, I guess you can say that. Probably even better to say "revolutions per second in a clockwise direction", so you can say counter-clockwise is "negative".

Sure. But the additional 90 deg aspect would have potentially confused some people, if I hadn't at least done it with cosine first. Then the video would have been longer. I've got another video that covers the phase aspect: "Is Phase important in the Fourier Transform?" kzfaq.info/get/bejne/jd92gpmqp7eUc5s.html

Sorry, I don't know what you mean by "interference frequency". Perhaps this video might help: "How do Complex Numbers relate to Real Signals?" kzfaq.info/get/bejne/irKHeJZpnbC7d6c.html

Yes, it has that effect. Although the aim is not to "get rid" of the imaginary part. The aim is to be able to represent real signals in the general complex baseband form.

@@iain_explains that is right, so what do we have? We had single vector on positive frequency that represents 1 harmonics with its own amplitude, frequency and phase shift(using vector length and angle). But the problem is that we actually have 2 vectors, representing this harmonic, not only 1. So the second vector has coordinates such as conjugated complex number of the first complex number. The vectors have the same length and the same angle, if we consider clockwise rotating then clockwise angle is positive in that rotating ofc. But at the same time clockwise rotating is negative frequency, that doesn't really have much physical explanation. 2 vectors with equal length and equal angle represent 1 harmonic. So we need negative frequency for completeness of the solution? What do you think? Sorry for bothering you. Thx for the video, helped me a lot)

I think the point is that negative frequency doesn't really exist. There is no physical phenomena that oscillates with a negative number of periods per second. To most people, "frequency" is like temperature measured in Kelvin. There is an absolute zero, and negative just doesn't make sense physically. But when people come across the Fourier Transform for the first time, and see a plot that shows both positive and negative values for frequency, then it can be confusing. However, if you understand that the plot is just part of the representation of a complex exponential, and that the negative portion of the frequency plot simply represents a negative direction of rotation in the complex plane, then it all falls into place. In terms of your question, you may find it helpful to read the Wikipedia page for the Fourier Transform, and go down to the heading: "Sine and cosine transforms". It explains that the FT can be written in terms of Sin and Cosine functions, instead of the complex exponential. Again though, it needs two functions at the same frequency, which is mirrored by the two functions (positive and negative frequency components) in the complex exponential form.

@@iain_explains ty

No. Notice that all the plots I drew in the video go forwards in time. In summary, the cos waveform can be constructed from two complex exponential components. One travels in a positive direction around the unit circle (as time increases) at a rate w_0, the other travels in a negative direction around the unit circle (as time increases) also at a rate w_0. On a Frequency plot, we show the positive directions on the right hand side, and the negative directions on the left hand side (which makes it look like a negative frequency). It's really a positive frequency travelling in the negative direction around the unit circle. It might help to watch the video I made on the relationship between complex numbers and the real world: kzfaq.info/get/bejne/irKHeJZpnbC7d6c.html

@@iain_explains Great , thank you very much !

I'm not sure what your question is, sorry. Maybe you're asking why I said "height"? OK, technically, the "height" of a delta function is infinity, but since we can't draw infinite height, traditionally we draw a delta function with a height equal to the area under the delta function, which in this case is pi. For more details of the delta function, see the video: "Delta Function Explained" kzfaq.info/get/bejne/ot-ilNSlsr3Yd58.html

@@iain_explains Many thanks for your response. Yes it was the pi term that confused me. The problem is me not remembering my uni maths properly. I think mostly we left it out as it is a constant term as this simplifies the look of things. Also I think we referred to the 'weight' of the delta functions rather than 'height'. Sorry should have phrased my question better. Liked and subsribed.

Glad you found it helpful.

Thank you

It does. When k is negative, it corresponds to negative frequencies. See: "Fourier Series and Eigen Functions of LTI Systems" kzfaq.info/get/bejne/nbihZq5lv7evmWw.html