Can someone explain me IQ?

[Awesome14]

I will watch Dave's Alan's video. I mentioned I became skeptical because the concepts presented could be merely mental exercises. For example, negative frequency. Is that merely a mathematical construct? Also, what real physical property is represented by Q?

Mathematicians can make sense out of negative frequency, negative resistance, or a negative square root. I just try not to think about it too much. It's no less real. But if you read the entire paper you cited in the OP, I/Q is a just mathematical construct. A good one, too.

[Doc Daneeka]

I had some similar arguments once after someone asked me to explain what harmonics are, then 'do they really exist'

A simple analogy could be: you have 10 objects in a pile on the table. I say it is two groups of objects one of 7 and one of 3. Most people wouldn't have a problem with that. Do these two groups really exist? what does it matter if for instance you only care about how many there are? thinking of the pile as two groups combined works out the same as one big pile, or three or ten individual, whatever.

I and Q are similar: You have a sinusoidal signal, a reference point in time called 0, now you say what amplitudes of cosine and sine when added together give me that signal? The answer is I and Q. Do I and Q actually exist? Maybe you are physically combining an I and Q to get your signal. maybe not. The thing is it does not matter, the result s of anything you do to the signal are the same if you consider it as 'just a signal' or a sum of two components I and Q, which is why we use I and Q (it simplifies calculations for instance)

You just have two choices for your model (important to know there is a difference between our abstract models and physical reality) - one signal or I and Q components. I and Q don't exist any more (or less) than 'a sinusoidal function of t'

[rs20]

^ Spot on. I had said in an earlier message that an LO is required to make the terms I and Q make sense (I = in phase with LO, Q = in quadrature with LO). But indeed, as Doc Daneeka points out, all you have to do is define a particular point in time as your t=0 point, and you may, if you choose, thereby define I and Q to be the sine and cosine components (where sin means sin(omega*t) and cos means cos(omega*t), which only makes unambiguous sense if you have a defined, absolute t=0 reference point).

[hamster_nz]

What I was wondering was, if I take my power supply and simply turn the knob up and down to create a sine wave, then what is Q?

If you are creating a 1Hz signal then you are creating an equal amount of 1Hz and -1Hz.

Mathematically cos(x) and cos(-x) are always equal, but when you add sin(x) to sin(-x) you always get zero, so your totaled Q value is zero.

Wrong, this is totally nonsensical. What you're saying is that any sinewave I encounter in nature is somehow "fundamentally" a cosine and definitely not a sine. What the?

I'll take you through it slowly....

Make yourself comfortable with the symmetry in the sin() and cos() functions: sin(x) = -sin(-x) and cos(x) = cos(-x) (see https://brilliant.org/wiki/symmetry-in-trigonometric-graphs/)

Say you have a cosine wave, with an amplitude of 2:

  f(x) = 2cos(x).

You can make it a complex, but with a zero imaginary component.
  f(x) = 2cos(x) + i*0

You can decompose that into two halves:
- one a positive frequency (phase advancing with x)
      g(x) = cos(x)+i*sin(x)
- one with negative frequency (phase decreasing with x)
      h(x) = cos(-x)-i*-sin(-x)
(note that because of the symmetries these are almost the same functions, but with opposite signs on the second imaginary term so they cancel each other out).

That gives you an equivalent function of:
 
  f(x) = 2cos(x) = g(x) + h(x) = cos(x)+i*sin(x) + cos(-x)+i*sin(-x)

And if you want to check then you can use the symmetries to reduce everything back to f(x) = 2cos(x).
    f(x) = cos(x)+i*sin(x) + cos(-x)+i*sin(-x)
    f(x) = cos(x)+cos(-x) + i(sin(x)+sin(-x))
    f(x) = 2cos(x) + i*0

So far so good.

What have we just done?

We have split the cos(x) into two complex functions (g(x) and h(x), that when added give you the original, real-valued, signal f(x). If you graph each of these halves on the complex plain, you have two points spinning around (0,0), one spinning, clockwise, the other counter clockwise. However, if you add them together and graph them, you just get a point that moves to along the real axis, tracing out 2cos(x).

Note that both halves start at 1+i*0 when x=0 - but of course you already knew that  because they are half of the 2cos() function.

What about Sines?

For f(x) = 2sin(x) you can decompose it into a slightly more confusing, less clear halves:

   g(x) =  sin( x)  + i * cos(x)
   h(x) = -sin(-x)  - i * cos(-x)

Once again see how that uses the symmetries of  sin(x) = -sin(-x) and cos(x) = cos(-x), with the imaginary terms cancelling each other out.

One is a function of the 'positive' frequency, the other the 'negative' frequency, which when added together give the original completely real-valued function.

When viewed on the complex plane these are also two points rotating around (0,0) in different directions, but unlike the cos() wave (where both are at 1+0i when x=0), one point is at 0+1i, the other point at 0-1i when x = 0.


Why is this useful?
It then becomes pretty visually intuitive why multiplying a baseband signal with a carrier gives the "sum+difference" side bands - the g(x) half gives you the upper side band, the h(x) half gives you the lower side band.
It also becomes pretty clear what is going on when generate / demodulate USB and LSB signals.
It also becomes clear that multiplying by a I+Q carrier is pretty much 'moving the zero point' on the frequency spectrum, and you just need special care for what happens to those pesky 'negative' frequencies.

So what is the point to all this?
It is just as valid, and sometimes helpful to view any 'real' signal (such as somebody tracing a sine wave with a PSU) as the sum of positive and negative frequencies that have an imaginary component. I sure makes a lot of SDR math / DSP stuff more intuitive and easier to grok.

This is just like how it is sometime helpful to view a real signal as a summation of sine and cosine waves.

I mean really, is that 3 minute song on the radio music really an nearly infinite group of sine+cosine waves of different phases and amplitudes?

[rs20]

What I was wondering was, if I take my power supply and simply turn the knob up and down to create a sine wave, then what is Q?

If you are creating a 1Hz signal then you are creating an equal amount of 1Hz and -1Hz.

Mathematically cos(x) and cos(-x) are always equal, but when you add sin(x) to sin(-x) you always get zero, so your totaled Q value is zero.

Wrong, this is totally nonsensical. What you're saying is that any sinewave I encounter in nature is somehow "fundamentally" a cosine and definitely not a sine. What the?

I'll take you through it slowly....

...

Yep, I agree with everything you wrote in your second message. Still calling out the total nonsensicality of your original message where you said that a hand-made sinusoid definitively has "a totalled Q value of zero" without specifiying any absolute time base or LO.