Rune Allnor wrote:> On 12 Jun, 14:28, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote: >> On Wed, 11 Jun 2008 08:58:47 -0700 (PDT), Mark <makol...@yahoo.com> >> wrote: > >>> us old analog guys have learned that perspective using old fashioned >>> heterodyne spectrum analyzers. If you tune below the zero, you can >>> actually see the negative frequencies, mirror images of the positive >>> ones...sure enough they are there... :-) >>> Mark >> Hi Mark, >> Neat. I think I tried that decades ago, but I forget >> what it was that I saw on the old fashioned spectrum >> analyzer. It seems like you've verified that Steve Smith >> was not just "making up" this whole notion of negative >> frequencies. > > What about AM? What's the point of SSB AM without > the concept of 'negative frequency'? No need to > play tricks with the novices minds by 'tuning the > spectrum analyzer to -5 Hz' when a trivial mixer > will show the effect. Once you see the output > of that analog mixer with an extra sideband below > the carrier, you understand that something 'weird' > is going on. Negative frequencies are there to help > you explain what is happening and you avoid the > hurdle of trying tho understand what it means to > 'tune a reciever to negative frequency'.2*sin(a)*cos(b) = sin(a+b) + sin(a-b). I see sum and difference frequencies, but no negative ones. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Absolute Beginner - inverted signal?
Started by ●June 9, 2008
Reply by ●June 12, 20082008-06-12
Reply by ●June 12, 20082008-06-12
Jerry Avins <jya@ieee.org> writes:> [...] > There's no need to assume negative frequencies for explaining what one > sees on the spectrum analyzer, but as with exponential forms for sine > and cosine, it is a convenient thing to do. Elevating a convenient > construct to an immutable reality can occasionally lead one far > astray.Ah, Jerry. Sheesh, we've talked about this how many times now? :) This time I think I might have just the right concept that integrates our two perspectives. As with complex numbers, the concept exists because the "simple version" cannot represent fully what's going on. You either lose information (if you ignore it) or you start adding words (or, in the standard cases, mathematical notation) to describe them. So, e.g., just by the fact that you can't fully specify the frequency of e^{j*2*pi*f} without allowing f to be negative, means that there is something more than just "positive frequencies" going on. Same thing with complex numbers. Whether you call them "complex numbers" or "things that must use two real values and special ways of adding and multiplying," they're still something that different plain old real numbers and the standard associated arithmetic operations. Maybe I'm making an error in interpreting you, but it has seemed for many years now that you are in denial that these types of "extended systems" really do exist. -- % Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven. %% Fuquay-Varina, NC % She love the way Puccini lays down a tune, and %%% 919-577-9882 % Verdi's always creepin' from her room." %%%% <yates@ieee.org> % "Rockaria", *A New World Record*, ELO http://www.digitalsignallabs.com
Reply by ●June 12, 20082008-06-12
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: >> [...] >> There's no need to assume negative frequencies for explaining what one >> sees on the spectrum analyzer, but as with exponential forms for sine >> and cosine, it is a convenient thing to do. Elevating a convenient >> construct to an immutable reality can occasionally lead one far >> astray. > > Ah, Jerry. Sheesh, we've talked about this how many times now? :) This > time I think I might have just the right concept that integrates our two > perspectives. > > As with complex numbers, the concept exists because the "simple version" > cannot represent fully what's going on. You either lose information (if > you ignore it) or you start adding words (or, in the standard cases, > mathematical notation) to describe them.So far, we fully agree (I think).> So, e.g., just by the fact that you can't fully specify the frequency of > e^{j*2*pi*f} without allowing f to be negative, means that there is > something more than just "positive frequencies" going on.Not so. I can't fully or even partially describe sin(2*pi*f*t) in terms of positive exponentials alone. I needn't construe e^-{j*2*pi*f*t} as involving negative frequency. If I choose to be even more ornery, I can construe it as involving negative time. The exact equivalence of several representations implies that no one of them is fundamental.> Same thing > with complex numbers. Whether you call them "complex numbers" or "things > that must use two real values and special ways of adding and > multiplying," they're still something that different plain old real > numbers and the standard associated arithmetic operations. > > Maybe I'm making an error in interpreting you, but it has seemed for > many years now that you are in denial that these types of "extended > systems" really do exist.Existence is a slippery matter. Those extended systems certainly exist in my mind. My stand is that while they are convenient notations, they are not reality in themselves. Music exists without staves. While some can hear a symphony simply by reading a score, the score is not the music. Need I amplify? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●June 12, 20082008-06-12
Jerry Avins <jya@ieee.org> writes:> [...]> Those extended systems certainly exist in my mind. My stand is that > while they are convenient notations, they are not reality in > themselves.I'm not saying that any particular representation is the system, but rather that the fact a system requires extending means the extended system is no longer the simple system, i.e., the extended system exists as a new system distinct from the simple system, and thus the extended system is real. But of course the representation of a system is not real, just as the sequence "1, 2, 3, ..." that is arranged as ascii bytes on your harddrive is not the natural numbers (and yes, staves of music are not the music). -- % Randy Yates % "Rollin' and riding and slippin' and %% Fuquay-Varina, NC % sliding, it's magic." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Living' Thing', *A New World Record*, ELO http://www.digitalsignallabs.com
Reply by ●June 12, 20082008-06-12
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: >> [...] > >> Those extended systems certainly exist in my mind. My stand is that >> while they are convenient notations, they are not reality in >> themselves. > > I'm not saying that any particular representation is the system, but > rather that the fact a system requires extending means the extended > system is no longer the simple system, i.e., the extended system exists > as a new system distinct from the simple system, and thus the extended > system is real.If by extension you mean the inclusion of imaginary numbers, that isn't required. Read J.C.Maxwell's treatment of electromagnetic waves in space. (A Dover reprint of "Treatise" is available.) It has no imaginary numbers, just a frightful mess of three-dimensional sets of integral equations. Give me vector analysis please, but don't tell me that imaginary numbers and negative time are inherent. They can both be dispensed with at some cost.> But of course the representation of a system is not real, just as the > sequence "1, 2, 3, ..." that is arranged as ascii bytes on your > harddrive is not the natural numbers (and yes, staves of music are not > the music).I think we're saying the same thing. Only the emphasis is different. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●June 12, 20082008-06-12
Jerry Avins <jya@ieee.org> writes:> [...] > If by extension you mean the inclusion of imaginary numbers, that > isn't required.No, that is not what I mean. Complex numbers are a representation. What I mean is that there is some system that is different than the plain old real number system along with the standard arithmetic operations, i.e., the field R(+,*) (to use a little abstract algebra), and that without it, you cannot accomplish certain things (like represent all N roots of any Nth-order polynomial with coefficients in R(+,*)).> I think we're saying the same thing.I don't think we are, Jerry. -- % Randy Yates % "Remember the good old 1980's, when %% Fuquay-Varina, NC % things were so uncomplicated?" %%% 919-577-9882 % 'Ticket To The Moon' %%%% <yates@ieee.org> % *Time*, Electric Light Orchestra http://www.digitalsignallabs.com
Reply by ●June 12, 20082008-06-12
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: >> [...] >> If by extension you mean the inclusion of imaginary numbers, that >> isn't required. > > No, that is not what I mean. Complex numbers are a representation. > > What I mean is that there is some system that is different than the > plain old real number system along with the standard arithmetic > operations, i.e., the field R(+,*) (to use a little abstract algebra), > and that without it, you cannot accomplish certain things (like > represent all N roots of any Nth-order polynomial with coefficients in > R(+,*)).Yes; for that, one needs complex numbers. Complex numbers close arithmetic. By "close", I mean that every operation can be performed on any number. Negative numbers close subtraction, fractions close division. Imaginary numbers make possible the extraction of roots.>> I think we're saying the same thing.>> I don't think we are, Jerry.We agree, I think, that neither imaginary numbers nor negative time are needed to describe sinusoids of arbitrary phase. We agree that it is convenient to use them. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●June 12, 20082008-06-12
glen herrmannsfeldt wrote:> Jerry Avins wrote: > (snip) > >> There's no need to assume negative frequencies for explaining what one >> sees on the spectrum analyzer, but as with exponential forms for sine >> and cosine, it is a convenient thing to do. Elevating a convenient >> construct to an immutable reality can occasionally lead one far astray. > > I would say somewhere in between. Not immutable, but more than > just a convenience. Using phasors and complex numbers to represent > phase shifted sin or cos is convenient. (Voltage and current are > still really real.)"More than just a convenience" and "is convenient". I think I missed the point you made.> and in another post Jerry wrote: > > > 2*sin(a)*cos(b) = sin(a+b) + sin(a-b). I see sum and difference > > frequencies, but no negative ones. > > Yes, but if a<b (and you add a t such that they are frequencies) > then you have negative frequency with no complex exponential > in sight. > > It isn't just a side effect of the exponential transform.Good point. But you still don't need to tune your receiver to a negative frequency to pick up sin(a-b)t signal. It's alias shows up (with appropriate phase). Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●June 12, 20082008-06-12
Jerry Avins wrote: (snip)> There's no need to assume negative frequencies for explaining what one > sees on the spectrum analyzer, but as with exponential forms for sine > and cosine, it is a convenient thing to do. Elevating a convenient > construct to an immutable reality can occasionally lead one far astray.I would say somewhere in between. Not immutable, but more than just a convenience. Using phasors and complex numbers to represent phase shifted sin or cos is convenient. (Voltage and current are still really real.) and in another post Jerry wrote: > 2*sin(a)*cos(b) = sin(a+b) + sin(a-b). I see sum and difference > frequencies, but no negative ones. Yes, but if a<b (and you add a t such that they are frequencies) then you have negative frequency with no complex exponential in sight. It isn't just a side effect of the exponential transform. -- glen
Reply by ●June 12, 20082008-06-12
Jerry Avins wrote:> glen herrmannsfeldt wrote: >> Jerry Avins wrote: >> (snip) >> >>> There's no need to assume negative frequencies for explaining what >>> one sees on the spectrum analyzer, but as with exponential forms for >>> sine and cosine, it is a convenient thing to do. Elevating a >>> convenient construct to an immutable reality can occasionally lead >>> one far astray. >> >> I would say somewhere in between. Not immutable, but more than >> just a convenience. Using phasors and complex numbers to represent >> phase shifted sin or cos is convenient. (Voltage and current are >> still really real.) > > "More than just a convenience" and "is convenient". I think I missed the > point you made. > >> and in another post Jerry wrote: >> >> > 2*sin(a)*cos(b) = sin(a+b) + sin(a-b). I see sum and difference >> > frequencies, but no negative ones. >> >> Yes, but if a<b (and you add a t such that they are frequencies) >> then you have negative frequency with no complex exponential >> in sight. >> >> It isn't just a side effect of the exponential transform. > > Good point. But you still don't need to tune your receiver to a negative > frequency to pick up sin(a-b)t signal. It's alias shows up (with > appropriate phase).Recall that sin(a-b)t = -sin(b-a)t. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
