# RF complex mixing

Started by July 10, 2018
```Hello everyone. Could anyone explain please the next issues

we know that there are no negative frequencies, so during real rf mixing the"-" frequencies fold back to positive F axis. so why during complex RF mixing the "-" Frequencies exist? I think that I understand that the complex signal can have "-" frequencies, but if we look at x(t)*cos  and x(t)*-sin terms as ar real rf mixing process each. the result will not contain "-" frequencies, but if we look on it as complex one - it will.

why?
```
```On 11.07.2018 1:49, udmitri85@gmail.com wrote:
> Hello everyone. Could anyone explain please the next issues
>
> we know that there are no negative frequencies, so during real rf mixing the"-" frequencies fold back to positive F axis. so why during complex RF mixing the "-" Frequencies exist? I think that I understand that the complex signal can have "-" frequencies, but if we look at x(t)*cos  and x(t)*-sin terms as ar real rf mixing process each. the result will not contain "-" frequencies, but if we look on it as complex one - it will.
>
> why?
>

If I've understood your question correctly -- because a real RF signal
contains two copies of the complex signal -- one centered at the carrier
frequency, and the mirror image of it centered at minus the carrier
frequency.

That topic is explained in excellent detail in Richard Lyons' book,
"Understanding Digital Signal Processing", which is very insightful.
Chapter 9, IIRC.

Gene

```
```On Tuesday, July 10, 2018 at 3:49:20 PM UTC-7, Dmitri wrote:
> Hello everyone. Could anyone explain please the next issues
>
> we know that there are no negative frequencies, so during real rf mixing the"-" frequencies fold back to positive F axis. so why during complex RF mixing the "-" Frequencies exist? I think that I understand that the complex signal can have "-" frequencies, but if we look at x(t)*cos  and x(t)*-sin terms as ar real rf mixing process each. the result will not contain "-" frequencies, but if we look on it as complex one - it will.
>
> why?

Hello Dmitri.
Complex-valued signals and complex mixing are puzzling ideas when you first encounter them. Don't worry, it's that way with almost everyone. When you have nothing better to do, please have a look at the blog at:

https://www.dsprelated.com/showarticle/192.php

If you have any questions regarding that blog, you're welcome to send me an e-mail at: R-dot_Lyons-at-ieee-dot-org.
```
```On Wednesday, July 11, 2018 at 5:25:25 AM UTC-7, Gene Filatov wrote:

>
> If I've understood your question correctly -- because a real RF signal
> contains two copies of the complex signal -- one centered at the carrier
> frequency, and the mirror image of it centered at minus the carrier
> frequency.
>
> That topic is explained in excellent detail in Richard Lyons' book,
> "Understanding Digital Signal Processing", which is very insightful.
> Chapter 9, IIRC.
>
> Gene

Hello Gene Filatov.
Thanks for the "plug" for my "Understanding DSP" book.
If you have a copy of that book you might want to have a look at the following web page:

https://www.dsprelated.com/showarticle/1094.php
```
```On 12.07.2018 9:06, Richard (Rick) Lyons wrote:
> On Wednesday, July 11, 2018 at 5:25:25 AM UTC-7, Gene Filatov wrote:
>
>>
>> If I've understood your question correctly -- because a real RF signal
>> contains two copies of the complex signal -- one centered at the carrier
>> frequency, and the mirror image of it centered at minus the carrier
>> frequency.
>>
>> That topic is explained in excellent detail in Richard Lyons' book,
>> "Understanding Digital Signal Processing", which is very insightful.
>> Chapter 9, IIRC.
>>
>> Gene
>
> Hello Gene Filatov.
> Thanks for the "plug" for my "Understanding DSP" book.
> If you have a copy of that book you might want to have a look at the following web page:
>
> https://www.dsprelated.com/showarticle/1094.php
>

Hello Richard,

Thanks for providing the link! I've updated my old copy of the errata.

Also, I apologize to the OP for providing the wrong reference --
actually it's chapter 8 ("Quadrature signals") he might be interested in.

```
```On Tue, 10 Jul 2018 15:49:16 -0700 (PDT), udmitri85@gmail.com wrote:

>Hello everyone. Could anyone explain please the next issues
>
>we know that there are no negative frequencies, so during real rf mixing th=
>e"-" frequencies fold back to positive F axis. so why during complex RF mix=
>ing the "-" Frequencies exist? I think that I understand that the complex s=
>ignal can have "-" frequencies, but if we look at x(t)*cos  and x(t)*-sin t=
>erms as ar real rf mixing process each. the result will not contain "-" fre=
>quencies, but if we look on it as complex one - it will.
>
>why?

Think of the hands on a clock.   The second hand rotates around the
clock face at 1/60 rotations per second, the minute hand at 1/360
rotations/sec, and the hour hand at 1/4320 per second.    Since the
clock hands rotate, you could imagine running the clock backwards so
that the hand rotate counter-clockwise instead of clockwise.   In
order to distinguish the backwards, ccw, rotations of the clock at the
same frequencies from the forward, clock rotations, it's easy to put a

The same is true with complex signals, which can be represented as
vectors rotating around the origin in the complex plane.   They can
rotate forwards or backwards, and we distinguish the rotational
direction by the sign of the frequency.

Since real-valued signals don't have this property, as they occupy
only one dimension of the complex plane, there is no distinction
between positive and negative frequencies.   They cannot be
distinguished from each other.  Plot a sine wave vs time and it is the
same regardless of +/- frequency.   Plot a sine wave with another sine
wave of the same frequency in quadrature, and the direction of
rotation is unambiguous.

In addition to Rick's excellent book and articles, I'll offer this
one, which is related and might help visualise the issues.   Skip the
radio architecture parts if they're not interesting.

https://www.dsprelated.com/showarticle/51.php

```
```On July 10, Dmitri wrote:
> we know that there are no negative frequencies, so during real rf
> mixing the"-" frequencies fold back to positive F axis.

Watch a film of spoked wagon wheels.  At a certain speed,
the wheels appear to rotate backwards.  It's a consequence
of aliasing, due to the low sample rate of the human eye.

That's a  negative frequency.

--
Rich

```
```RichD  <r_delaney2001@yahoo.com> wrote:

>Watch a film of spoked wagon wheels.  At a certain speed,
>the wheels appear to rotate backwards.  It's a consequence
>of aliasing, due to the low sample rate of the human eye.
>
>That's a  negative frequency.

I'm going to disagree with this analogy, as it is an aliasing
effect and the original question applies to continuous time
signals.

Steve
```
```udmitri85@gmail.com writes:

> Hello everyone. Could anyone explain please the next issues
>
> we know that there are no negative frequencies,

I don't know that. As others have shown (I especially liked Jacobsen's
view), negative frequencies are valid. I also view negative frequency
as a valid and useful concept.

> so during real rf
> mixing the"-" frequencies fold back to positive F axis. so why during
> complex RF mixing the "-" Frequencies exist? I think that I understand
> that the complex signal can have "-" frequencies, but if we look at
> x(t)*cos and x(t)*-sin terms as ar real rf mixing process each. the
> result will not contain "-" frequencies, but if we look on it as
> complex one - it will.

I would not say "the negative frequencies fold back to the positive F
axis." I can be shown that the spectrum of a real signal is Hermitian
symmetric. That means that once you have one side (either positive or
negative) of a real signal's spectrum, the signal is fully described and
there is no further information in the other side (negative or positive,
respectfully) of the spectrum.

With complex signals, this is not the case. You must consider both sides
of the spectrum to fully "know" a complex signal in the frequency
domain.

So to summarize, I would say there are negative frequencies in the
spectrum of a signal whether the signal is real or complex. In the case
of a real signal, the negative frequencies don't provide any more
information about the signal once the positive frequencies are known
and can thus be ignored.
--
Randy Yates, DSP/Embedded Firmware Developer
Digital Signal Labs
http://www.digitalsignallabs.com
```
```On Thursday, July 12, 2018 at 11:13:45 AM UTC-7, RichD wrote:
> On July 10, Dmitri wrote:
> > we know that there are no negative frequencies, so during real rf
> > mixing the"-" frequencies fold back to positive F axis.
>
> Watch a film of spoked wagon wheels.  At a certain speed,
> the wheels appear to rotate backwards.  It's a consequence
> of aliasing, due to the low sample rate of the human eye.
>
> That's a  negative frequency.
>
> --
> Rich

Hi.
I always thought the wagon wheels spokes appeared to rotate backwards because the old movie cameras took a series of periodic snapshots. I.E., 24 individual snapshots each second. As such, the video sample rate was 24 samples/second.
```