Jerry Avins wrote:> Kari Pesonen wrote: > >>"Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message news:28192a4d.0307142216.4c6ee88@posting.google.com... >> >>>Hi, >>>Can anyone explain the concept of Negative frequencies clearly. Do >>>they really exist? >> >>About 30 years ago I wondered the same question (do negative frequencies >>exist?) >>Here you have one example from acoustics: >>Let's assume a sound wave propagating from a source at speed c to certain >>direction. We sample this sound wave by a microphone moving at speed >>2c to the same direction. What we get? The exact time domain wave form >>but the signal represented in time reversed. >>We can define the instantaneous frequency as the phase change in time >>unit (f = d phi/dt). In the microphone signal this change is negative >>when compared with the original signal. The frequency is negative by >>definition. >> >>In real life we can easily listen and analyze sounds "by negative frequencies" >>playing recorded signals backwards. Although the signal played forwards >>and backwards have same spectrum they sounds differently (at least when >>listened by humans) >> >>Kari Pesonen > > > There is linear or rotary motion involved in playing a tape or record > backward, so direction reversal makes sense. What about the loudspeaker > cone? Does that vibrate backwards?In active noise control, a speaker can be used to absorb sound energy.> > The examples above are a sort of time reversal. Does that really negate > the frequencies? If so, how? > > Jerry
Negative Frequencies
Started by ●July 15, 2003
Reply by ●July 21, 20032003-07-21
Reply by ●July 21, 20032003-07-21
Rune Allnor wrote:> Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms?What if I say that there are as many axioms as natural numbers? bye, -- piergiorgio
Reply by ●July 21, 20032003-07-21
Jerry Avins wrote:> Maybe I said it wrong. What I mean is that for every a, there is one and > only one cos(a), and that both a and cos(a) are scalars.For each a there is one cos(a), but not only one. Clearly cos(0) = 1 and cos(2pi) = 1, so there are two a (I should I write? as?), but one cos... bye, -- piergiorgio
Reply by ●July 21, 20032003-07-21
"Piergiorgio Sartor" <piergiorgio.sartor@nexgo.REMOVE.THIS.de> wrote in message news:2drtu-f56.ln1@lazy.lzy...> Vanamali wrote: > > > I am in Jerry's camp on this issue. Sometimes I get the feeling that > > there are people who think that exp(jwt) is more fundamental than sine > > and cosine. I don't think this is correct because one cannot define > > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > > building block its definition should not depend on sine and cosine. > > There is also an other argument: it cannot be > more basic, since "j" is more... complex... > > In fact "j" was introduced in order to solve > sqrt(-1), so it extends the basic set of R > and thus cannot be a building block.But sin(x)=(exp(j x)-exp(-j x))/(2j) which looks more complex than exp(j x) -- glen
Reply by ●July 21, 20032003-07-21
Glen Herrmannsfeldt wrote:> But sin(x)=(exp(j x)-exp(-j x))/(2j) which looks more complex than exp(j x)Let's say you're not allowed to use "j"... :-) bye, -- piergiorgio
Reply by ●July 21, 20032003-07-21
On Sun, 20 Jul 2003 22:01:51 -0700, Rune Allnor wrote:> Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms?This can probably explain things better than I can... http://planetmath.org/encyclopedia/Axiom.html -- Matthew Donadio (m.p.donadio@ieee.org)
Reply by ●July 21, 20032003-07-21
"Piergiorgio Sartor" <piergiorgio.sartor@nexgo.REMOVE.THIS.de> wrote in message news:4outu-676.ln1@lazy.lzy...> Glen Herrmannsfeldt wrote: > > > But sin(x)=(exp(j x)-exp(-j x))/(2j) which looks more complex than exp(jx)> > Let's say you're not allowed to use "j"... :-)OK, use i then. -- glen
Reply by ●July 21, 20032003-07-21
Piergiorgio Sartor wrote:> > Jerry Avins wrote: > > > Maybe I said it wrong. What I mean is that for every a, there is one and > > only one cos(a), and that both a and cos(a) are scalars. > > For each a there is one cos(a), but not only one. > > Clearly cos(0) = 1 and cos(2pi) = 1, so there are > two a (I should I write? as?), but one cos... > > bye, > > -- > > piergiorgioThe point please? I wrote -- anyway, intended to write, that for any a, there is one and only one cos(a). Do you contradict that? Jerry P.S. "Contradict" means "speak against". When writing, should we use "contrascribe"? -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 21, 20032003-07-21
Stan Pawlukiewicz wrote:> > Jerry Avins wrote: > > Kari Pesonen wrote: > > > >>"Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message news:28192a4d.0307142216.4c6ee88@posting.google.com... > >> > >>>Hi, > >>>Can anyone explain the concept of Negative frequencies clearly. Do > >>>they really exist? > >> > >>About 30 years ago I wondered the same question (do negative frequencies > >>exist?) > >>Here you have one example from acoustics: > >>Let's assume a sound wave propagating from a source at speed c to certain > >>direction. We sample this sound wave by a microphone moving at speed > >>2c to the same direction. What we get? The exact time domain wave form > >>but the signal represented in time reversed. > >>We can define the instantaneous frequency as the phase change in time > >>unit (f = d phi/dt). In the microphone signal this change is negative > >>when compared with the original signal. The frequency is negative by > >>definition. > >> > >>In real life we can easily listen and analyze sounds "by negative frequencies" > >>playing recorded signals backwards. Although the signal played forwards > >>and backwards have same spectrum they sounds differently (at least when > >>listened by humans) > >> > >>Kari Pesonen > > > > > > There is linear or rotary motion involved in playing a tape or record > > backward, so direction reversal makes sense. What about the loudspeaker > > cone? Does that vibrate backwards? > > In active noise control, a speaker can be used to absorb sound energy. > > > > > The examples above are a sort of time reversal. Does that really negate > > the frequencies? If so, how? > > > > JerryI see. Would you say that the speaker emits negative frequencies that cancel the positive ones by combining with them? An interesting idea! Come to think of it, no. That would imply that a negative frequency is a positive one with a 180� phase reversal. So what _did_ you mean? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 21, 20032003-07-21
Vanamali wrote:> > Jerry Avins <jya@ieee.org> wrote in message news:<3F1418BA.822C806@ieee.org>... > > > No amount of math will reach a conclusion here, because the issue is not > > math, but philosophy. What is clear is that a Fourier transform with > > sines and cosines doesn't use negative frequencies in the analysis. > > > > Calculating with complex exponentials entails using negative > > frequencies. That doesn't confirm the existence negative frequencies or > > of complex exponentials. It simplifies manipulations while extending the > > repertoire of necessary concepts. > > I am in Jerry's camp on this issue. Sometimes I get the feeling that > there are people who think that exp(jwt) is more fundamental than sine > and cosine. I don't think this is correct because one cannot define > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > building block its definition should not depend on sine and cosine.I don't think what I'm about to say is concretely related to what you're getting at here, but it is prompted by your ideas, Vanamali. I get the impression that Jerry (and others?) think that there is nothing really gained in using the complex over the reals. I vehemently disagree. Jerry's argument has been, over and over, that whatever you can do with the complex, you can do with the reals - it just may take a few more operations. It's precisely those "few more operations" that make the complex (along with the operations of addition and multiplication) a significantly different mathematical beast. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr
