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## Re: explaining i/q

 From: Fabian Schwartau Subject: Re: explaining i/q Date: Tue, 3 Nov 2020 12:40:24 +0100 User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:68.0) Gecko/20100101 Thunderbird/68.10.0

```Hi Kristoff,

As some people already pointed out here, it is always good to give
multiple views for the concept of complex numbers, as different minds
work differently and some explanation may push the right buttons for
someone, but not for someone else. Especially for such a complex (pun
intended) and quite theoretical topic. So, here is an other view which I
like to use when I teach the concept to students and it is based on a
frequency point of view:
If you have a purley real valued signal (like all signals in reality
are) they have a mirrored spectrum. So the positive frequencies are the
same as the negative ones (except for inverse phase values). This is
logical, as there is no real intuitive concept of negative frequencies
in reality. Let's assume a signal at (and around) a carrier frequency
fc. This will show up as two "hills" of signals around fc and -fc in the
spectrum. One definition of complex signals is now that I use a
(complex) mixer to mix the signal from the positive frequencies down to
zero, so that fc is now at zero frequency (in time domain I multiply by
exp(-j*2*pi*fc), which is the one-sided version of a cosine). -fc is now
at -2fc. Then I use an ideal low pass filter to remove the part at -2fc
and I am left with a signal around zero frequency, which is not
symmetrical any more (in general). As the original real-values signal is
symmetrical I do not remove any information with the low pass filter, so
the original signal is reconstructable by mixing it back up to fc and
taking the real part of the complex result to get the real-values
two-sided spectrum again (and I have to multiply the amplitudes by 2 to
compensate the "lost" energy due to the low pass filtering and taking
only the real part of the result).
Hope that is clear, but when your audience does not come from an
electrical engineering point of view it might be even more confusing to
introduce the concept of a spectrum - although it is a vital part of
signal processing and you will have to introduce it at somepoint anyway.

Best regards,
Fabian

Am 03.11.20 um 00:02 schrieb Kristoff:
> Hi all,
>
> I was watching the webinar of Heather on GNU Radio today, when it came
> to me that one of the most difficult part doing a presentation of GNU
> Radio is the data-types, .. and especially these 'complex numbers'.
> The problem, or at least for me, is that when you mention 'GNU Radio
> complex numbers', you also have to mention iq-signals, which is a topic
> that is very difficult to explain in 10 seconds to an audience who has
> never seen anything about i/q sampling before.
>
>
> I have been thinking on how to explain the concept of I/Q signalling in
> just a few lines, e.g. in the context of -say- a workshop on GR.
>
>
> I have this idea in my head:
>
> Statement:
> 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).
>
>
> To make this visual:
> Take half a sine-wave and plot it out for every 45 degrees.
> This gives you 5 points: 0 (0 degrees), sqrt(2)/2 (45 degrees), 1 (90
> degrees), sqrt(2)/2 (135 degrees) and 0 (180 degrees).
>
> Now look at the 2nd and the 4th point (45 degrees and 135 degrees), if
> you sample this using only real/float values, these two points are
> exactly the same (sqrt(2)/2). Just based on these values by themselves
> (i.e. remove all other points from the graph), there is no way you can
> tell that at the first point (45 degrees) the graph was going up, while
> at the 135-degrees point the graph was going down.
>
> So, ... what i/q sampling does, is for every point "x", it not only
> provide the value of the graph at that point of time, but also
> information of the slope of the graphs at that time.
>
>
> This also explains while i/q sampling is done by not just taking the
> value of a signal at point "t", but also at 1/4 period later (which is
> the information you need to determine the 'slope' of that graph at point
> 't')
>
>
> So, ... is this statement correct?
>
> If it is more-or-less correct and it can help provide a basic mental
> image of the concept of i/q sampling, I would be more then happy! :-)
>
>
>
>
> 73
> kristoff - ON1ARF
>
>
>
>

```

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