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?

# RF complex mixing

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. > > GeneHello 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 webpage:> > 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 negative sign in front of them. This is consistent mathematically. 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. > > -- > RichHi. 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.