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Re: explaining i/q

From: David Hagood
Subject: Re: explaining i/q
Date: Mon, 2 Nov 2020 18:07:09 -0600
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The main difference between sampling with reals ('floats') and i/q sampling with complex numbers, is that the latter does not only provide theĀ  instantaneous value (voltage) of a signal at a certain point of time, but also includes phase information (i.e. the slope of the signal at that point).

The phase is not the slope of the signal - the slope of the signal is the first derivative with respect to time of the signal.

The phase is something completely different - it's the angle of the signal. So, what does "angle" mean?

Imagine a sine wave - it's zero at zero degrees, 1 at ninety degrees, 0 at 180 degrees, and -1 at 270 degrees. OK, so if I tell you the signal is 1, you know the phase angle is 90, because that's the only place sin(x) = 1. But what if I tell you it's at 0 - is the angle 0, or 180? There's literally no way to tell. But, what if I also give you the cosine of the angle? Cosine is 1 at zero, zero at ninety, -1 at 180, and zero again at 270. If I tell you that sin(x) is 1, and cos(x) is zero, the angle HAS to be 90. I and Q, or in phase and quadrature, are sine and cosine (or rather, usually cosine and sine, due to the way math works). Getting rid of that ambiguity about what the angle of the signal does all sorts of nice things when doing the math - it allows you to get rid of the negative frequency part of a signal (or more accurately, it allows you to unambiguously represent a negative frequency vs. a positive frequency).

Another way to visualize it - you may have seen the GIF of the ballet dancer silhouette , which can appear to be spinning left-to-right or right-to-left, just depending upon how you choose to interpret it. That's because you only have one part of the signal - you have the X part, or the in-phase part, but you don't have the Y part or quadrature part. Now, if you had both parts - if you saw not only the image along the Y axis (you see x and z), but you also saw the image along the X axis (seeing y and z - seeing the figure from the left or right side), then you would immediately know which way she was spinning.

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