itsbhanu@yahoo.com (Bhanu Prakash Reddy) wrote in message news:<28192a4d.0307142216.4c6ee88@posting.google.com>...> Hi, > Can anyone explain the concept of Negative frequencies clearly. Do > they really exist?Hi Bhanu, You've gotten lots of responses. Allow me to give you my perspective, which addresses some of the comments others have made in this thread. To begin with, let me answer your question with a question: do integers really exist? Do real numbers really exist? Do complex numbers really exist? All of these questions may potentially be answered "no" depending on how philosophical you want to get. However, in studying math I have found a wonderful field which I think is rich in meaning. I think it applies here. That field is called "abstract algebra" or "modern algebra" - it deals with the concepts of groups, rings, and fields. You will get many many hits if you Google for some of these keywords. A "ring" is basically a set along with two operations, addition (+) and multiplication (*), that satisfy certain criteria. For example, there must be an identity element for addition and a different identity element for multiplication; There must be additive inverses; The operations must be associative; etc. A field is a ring in which the elements have multiplicative inverses as well. The common number systems you know about are all rings. For example, integers, real numbers, and complex numbers are all rings. There are many other not-so-obvious examples as well, such as the set of all 8x8 matrices under matrix addition and multiplication. Real numbers and complex numbers are fields as well (all fields are rings, but not all rings are fields). Now there's one more concept from abstract algebra that is germane - that of "isomorphism." If two rings are isomorphic to one another, then they are essentially the same mathematical beast. If not, then they're not. For example, the integers are not isomorphic to the reals, but the complexes are isomorphic to the set of all two-by-two invertible matrices of the form [a b; -b a]. OK. So the purpose of stating all this is this: I believe that the concept of a ring is so important that any system which can be classified as a ring is worthy to be considered "real." Further, I consider any two rings that are not isomorphic to one another distinct. For example, you cannot equate the real numbers to the complex numbers because they are not isomorphic. Now proceeding on this axiom, we can say that the complex numbers are "real." Therefore we can also say that the the quantity e^{i*2*pi*f} is "real," where f is any element of the real numbers. So a negative value for f is legitimate and "real." Also, this uniqueness of negative frequencies only comes out when interpreting complex numbers and not real numbers since reals are not isomorphic to the complex. Now we get to the bottom line, which Clay Turner already discussed. The basic difference between frequency when thinking in terms of real numbers versus complex numbers is the concept of dimension. In some sense, real numbers are one-dimensional while complex numbers are two-dimensional. So, as Clay illustrated, the sign of the frequency can be used to indicate the direction in the plane (clockwise or counterclockwise) that a rotating vector is traveling. So, in this sense, negative frequency is real because it matters in the complex numbers and complex numbers are real if we base our definition of "real" on rings. Further, this concept of negative frequencies being real is not due to real numbers but complex numbers since the real numbers are not isomorphic to the complex numbers. And that's my $0.02. --Randy
Negative Frequencies
Started by ●July 15, 2003
Reply by ●July 16, 20032003-07-16
Reply by ●July 16, 20032003-07-16
Tom: [snip]> > Peter > > Consultant > > Indialantic By-the-Sea, FL. > > That is interesting but does this not give some causality problems? For asignal> f(t) > its Fourier transform F(f(t)) is F(w) and F(f(-t)) = F(-w) or are yousaying> > F(-w) = F(f*(t)) ie the Fourier TF of the complex conjugate as you have a > complex time-domain signal.Therefore the answer would be yes if you havecomplex> signals in the time-domain and no otherwise?That maybe right... > > Tom[snip] Negative frequency is NOT negative time. Consider the argument of exp(phi) with phi = wt = 2*pi*f*t. phi can be negative if either of w or t are negative, but not both. Just because w is negative does not mean that t is negative and time is running backwards. Positive and negative frequencies have nothing whatsoever to do with causality and the direction of time, rather they simply have to do with direction of rotation. Counterclockwise rotation is a positive angular frequency and clockwise rotation is a negative angular frequency, nothing more nothing less. No mystery, no arm waving, just plain direction of rotation. When your car is moving forwards its' tires are rotating with a positive frequency when the car backs up the tires have a negative frequency. Just realize that it takes two co-ordinates [hence complex numbers or complex signals] to discern the direction of rotation. -- Peter Consultant Indialantic By-the-Sea, FL.
Reply by ●July 16, 20032003-07-16
Rune Allnor wrote:> > ...> My point was merely that the lower side band > appears because of the negative ferquency components of the baseband > representation are shifted by modulation as well. If you do a spectrum > analysis (positive frequencies only) at baseband and then of the modulated > signal, I have been told[*] that you find that the bandwidth of the > modulated signal is twice the bandwidth of the baseband signal.That's a possible viewpoint, and a productive one. It's not conclusive because it isn't the only one. AM modulation if a carrier by a single baseband cosine is defines by the equation f(t) = cos(w_c*t)*[1 + m*cos(w_m*t), where f(t) is the modulated waveform, w_c is the carrier frequency, w_m is the modulating frequency, and m is the modulation percentage. Trigonometric identities show that f(t) consists of the original carrier from the 1 in the bracket term, and two additional frequencies, w_c + w_m and w_c - w_m, each with amplitude m/2. The math in no way insists that w_c - w_m be construed as w_c + (-w_m), although that's not ruled out.> > The negative frequencies may very well be a mathematical abstraction, > but with a measurable manifestation. The only reason for the observable > doubling of bandwidth is the negative frequency components. Thus, it > is tempting to conclude that negative frequencies exist.No. The bandwidth doubling comes about because for every sideband frequency some amount above the carrier, there is another frequency equally far below. It is as reasonable to think of the lower sideband as positive frequencies subtracted from the carrier as it is to think of it as negative frequencies added to it. There is no question that the notions of negative frequencies is consistent, and that it can greatly simplify thinking about difficult cases. An example is the carrier frequency descending below the highest modulating frequency.>...> I don't know Wiechert and > Sommerfeldt's work, so I don't want to comment on that.Search on "harmonic" in http://www.amphilsoc.org/library/guides/ahqp/bios.htm> What I do know > is that the complex numbers and the complex exponentials, with their > negative and positive frequencies, are useful. I don't think it's a > problem with negative frequencies as such, but with the properties of time. > When analysisn wave propagation the poitive and negative wavenumbers (that > play the same role in the spatial Fourier transform as ferquency in the > temporal transform) relate to the direction waves travel. > > > Jerry > > > > P.S. Does -sin(at) imply -[sin(at)], sin(-a*t), or sin(a*-t)? Maybe, > > instead of being composed of (relatively) negative frequencies, the > > lower sideband runs in (relatively) negative time. Really! :-) > > Again, it's a problem with time, not the frequency concept as such. > What you say appear to make no sense, but only because you explicitly > talk about time. Start out with -sin(kx) and you have outlined the basis > for one of many clues that are used to identify various types of waves > in a seismic data record. > > Rune > > [*] The phrase "I have been told" is included because I haven'y actually > done that exercise myself, but that's what pop out of the maths. > I do put sufficiently trust in maths and physics to predict that's what's > going to happen.I make two claims: that negative frequencies can be dispensed with if one works hard enough (not that it is worth doing), and that showing that negative frequencies simplify analyses or promote our understanding of them doesn't serve to establish their reality. Is that heresy? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 16, 20032003-07-16
Glen Herrmannsfeldt wrote:> > "Jerry Avins" <jya@ieee.org> wrote in message > news:3F1473F2.47CF9A7D@ieee.org... > > > It is exactly the same quantity when "f" is positive and "t" is > > negative. How can you tell which is is the real way? > > Remember, though, that the universe does not conserve time reversal > symmetry. > > Close, but not exactly. > > -- glenBut the math knows nothing about that. a - b is indistinguishable from a + -b even when a and b have dimensions of time. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 16, 20032003-07-16
Peter Brackett wrote:>...> I have participated in the design and production of both digital signal > processors and analog signal processing systems which process complex > signals having both positive and negative frequencies and I can assure you > that they are quite practical "real" and often quite useful.Practical, yes. Useful, yes. But any calculation can be completed with extra effort without invoking them. I don't speak to whether they are real here, only to your claim that they are. It seems to me you have demonstrated only utility, not reality.>... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 16, 20032003-07-16
Rune Allnor wrote:> > ...> My point was merely that the lower side band > appears because of the negative ferquency components of the baseband > representation are shifted by modulation as well. If you do a spectrum > analysis (positive frequencies only) at baseband and then of the modulated > signal, I have been told[*] that you find that the bandwidth of the > modulated signal is twice the bandwidth of the baseband signal.That's a possible viewpoint, and a productive one. It's not conclusive because it isn't the only one. AM modulation of a carrier by a single baseband cosine is defined by the equation f(t) = cos(w_c*t)*[1 + m*cos(w_m*t), where f(t) is the modulated waveform, w_c is the carrier frequency, w_m is the modulating frequency, and m is the modulation percentage. Trigonometric identities show that f(t) consists of the original carrier from the 1 in the bracket term, and two additional frequencies, w_c + w_m and w_c - w_m, each with amplitude m/2. The math in no way insists that w_c - w_m be construed as w_c + -w_m, although that's not ruled out.> > The negative frequencies may very well be a mathematical abstraction, > but with a measurable manifestation. The only reason for the observable > doubling of bandwidth is the negative frequency components. Thus, it > is tempting to conclude that negative frequencies exist.No. The bandwidth doubling comes about because for every sideband frequency some amount above the carrier, there is another frequency equally far below. It is as reasonable to think of the lower sideband as positive frequencies subtracted from the carrier as it is to think of it as negative frequencies added to it. There is no question that the notions of negative frequencies is consistent, and that it can greatly simplify thinking about difficult cases. An example is the carrier frequency descending below the highest modulating frequency.>...> I don't know Wiechert and > Sommerfeldt's work, so I don't want to comment on that.Search on "harmonic" in http://www.amphilsoc.org/library/guides/ahqp/bios.htm> What I do know > is that the complex numbers and the complex exponentials, with their > negative and positive frequencies, are useful. I don't think it's a > problem with negative frequencies as such, but with the properties of time. > When analysisn wave propagation the poitive and negative wavenumbers (that > play the same role in the spatial Fourier transform as ferquency in the > temporal transform) relate to the direction waves travel. > > > Jerry > > > > P.S. Does -sin(at) imply -[sin(at)], sin(-a*t), or sin(a*-t)? Maybe, > > instead of being composed of (relatively) negative frequencies, the > > lower sideband runs in (relatively) negative time. Really! :-) > > Again, it's a problem with time, not the frequency concept as such. > What you say appear to make no sense, but only because you explicitly > talk about time. Start out with -sin(kx) and you have outlined the basis > for one of many clues that are used to identify various types of waves > in a seismic data record. > > Rune > > [*] The phrase "I have been told" is included because I haven'y actually > done that exercise myself, but that's what pop out of the maths. > I do put sufficiently trust in maths and physics to predict that's what's > going to happen.I make two claims: that negative frequencies can be dispensed with if one works hard enough (not that it is worth doing), and that showing that negative frequencies simplify analyses or promote our understanding of them doesn't serve to establish their reality. Is that heresy? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 17, 20032003-07-17
"Jerry Avins" <jya@ieee.org> wrote in message news:3F16006C.4E35B67@ieee.org...> Rune Allnor wrote: > > > > ... > > > My point was merely that the lower side band > > appears because of the negative ferquency components of the baseband > > representation are shifted by modulation as well. If you do a spectrum > > analysis (positive frequencies only) at baseband and then of themodulated> > signal, I have been told[*] that you find that the bandwidth of the > > modulated signal is twice the bandwidth of the baseband signal. > > That's a possible viewpoint, and a productive one. It's not conclusive > because it isn't the only one. AM modulation of a carrier by a single > baseband cosine is defined by the equation > > f(t) = cos(w_c*t)*[1 + m*cos(w_m*t), > > where f(t) is the modulated waveform, w_c is the carrier frequency, w_m > is the modulating frequency, and m is the modulation percentage. > Trigonometric identities show that f(t) consists of the original carrier > from the 1 in the bracket term, and two additional frequencies, > w_c + w_m and w_c - w_m, each with amplitude m/2. The math in no way > insists that w_c - w_m be construed as w_c + -w_m, although that's > not ruled out.Jerry, The point is addressed by: f(t) = cos(w_c*t)*[1 + m*cos(w_m*t) expressed using complex exponentials: f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2]*{1 + m*[exp(w_m*t)/2 + exp(-w_m*t)/2]}, Multiplying out: f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2] + m*{.... ...exp(w_c*t)*exp(w_m*t)/2 + exp(w_c*t)*exp(-w_m*t)/2 + exp(-w_c*t)*exp(w_m*t)/2 + exp(-w_c*t)*exp(w_m*t)/2} Collecting exponents in products of exponentials: f(t) = [exp(w_c*t)/2 + exp(-w_c*t)/2] + m*{.... ...exp[(w_c+w_m)*t]/4*exp[(w_c-w_m)*t]/4*+ exp[(-w_c+w_m)*t]/4+ exp[(-w_c-w_m)*t]/4} Collecting terms at or near positive and negative carrier frequencies: f(t) = exp(w_c*t)/2 + [exp(w_c+w_m)*t]/4 +[exp(w_c-w_m)*t]/4 + exp(-w_c*t)/2 + [exp(-w_c+w_m)*t]/4 +[exp(-w_c-w_m)*t]/4 So, there is a positive and a negative carrier component and there is a positive and negative sideband associated with each of those carrier components. Addition is an operation whether the specific addition ends up in subtraction - so that's not the point. For example, if we said exp-(+w_c*t) or exp(-w_c*t) it wouldn't matter. The function is the same. I really think this isn't about the specific operations being performed, it's about the function that results. Fred
Reply by ●July 17, 20032003-07-17
Jerry: [snip]> real here, only to your claim that they are. It seems to me you have > demonstrated only utility, not reality. > > > ... > > Jerry[snip] Hmmm... well, we shipped several of the systems to customers [DoD, Lincoln Labs], and they paid $MM's for the kit. They even looked at the rotating frequency displays with smiles on their faces during acceptance testing. Can't get much more real than that! BTW... in that complex analog system the highest frequencies in use were ~ 20 Hz and so you could actually see the positive and negative frequencies rotating in "real time" on the scope displays and watch the negative frequencies being attenuated into oblivion at the output of the single sided bandpass filters as you decreased the frequency down through zero Hz, neat stuff! -- Peter Consultant Indialantic By-the-Sea, FL.
Reply by ●July 17, 20032003-07-17
Jerry Avins <jya@ieee.org> wrote in message news:<3F16006C.4E35B67@ieee.org>... ...> I make two claims: that negative frequencies can be dispensed with if > one works hard enough (not that it is worth doing), and that showing > that negative frequencies simplify analyses or promote our understanding > of them doesn't serve to establish their reality. Is that heresy? > > JerryNo. I think it's realistic reasoning. You choose what tools you want to use. Either a "physically correct" view, that requires hard work but does not induce any controversial issues, or one that is easy to work with but comes at the expence of possible confusion over the "negative frequency" concept. It's a fair choice. Rune
Reply by ●July 17, 20032003-07-17
