Jerry Avins <jya@ieee.org> writes:> Randy Yates wrote: >> Jerry Avins <jya@ieee.org> writes: >>> 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.I agree with that, but that isn't the main point I'm trying to make. My point is that to fully describe some phenomena requires an "extended system," whatever the representation we choose to use for that extended system is. To recast your statement above in these terms, a specific representation may be convenient but not necessary---yes, I agree. But we need SOME representation of the extended system (e.g., the complex numbers) and CANNOT simply use the simple system (e.g., the reals) to accomplish certain tasks (e.g., represent all N roots of an Nth-order polynomial). -- % Randy Yates % "Midnight, on the water... %% Fuquay-Varina, NC % I saw... the ocean's daughter." %%% 919-577-9882 % 'Can't Get It Out Of My Head' %%%% <yates@ieee.org> % *El Dorado*, Electric Light Orchestra http://www.digitalsignallabs.com
Absolute Beginner - inverted signal?
Started by ●June 9, 2008
Reply by ●June 12, 20082008-06-12
Reply by ●June 12, 20082008-06-12
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: > >> Randy Yates wrote: >>> Jerry Avins <jya@ieee.org> writes: >>>> 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. > > I agree with that, but that isn't the main point I'm trying to make. > > My point is that to fully describe some phenomena requires an "extended > system," whatever the representation we choose to use for that extended > system is. To recast your statement above in these terms, a specific > representation may be convenient but not necessary---yes, I agree. But > we need SOME representation of the extended system (e.g., the complex > numbers) and CANNOT simply use the simple system (e.g., the reals) to > accomplish certain tasks (e.g., represent all N roots of an Nth-order > polynomial).I completely agree about roots. In electrical theory, though, we use complex numbers as ordered pairs, which is overkill (but we grow to think it fundamental). x+iy, R=jX; it is rare that extracting a root comes into play. Sin(r), cos(r) often serves where we use cos(r) + j*sin(r) for convenience. (I don't remember the formulas for the sums of sines and cosines. I use complex numbers to derive the relations when I need them. I'm all for convenience!) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●June 13, 20082008-06-13
Jerry Avins <jya@ieee.org> writes:> Randy Yates wrote: >> Jerry Avins <jya@ieee.org> writes: >> >>> Randy Yates wrote: >>>> Jerry Avins <jya@ieee.org> writes: >>>>> 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. >> >> I agree with that, but that isn't the main point I'm trying to make. >> >> My point is that to fully describe some phenomena requires an "extended >> system," whatever the representation we choose to use for that extended >> system is. To recast your statement above in these terms, a specific >> representation may be convenient but not necessary---yes, I agree. But >> we need SOME representation of the extended system (e.g., the complex >> numbers) and CANNOT simply use the simple system (e.g., the reals) to >> accomplish certain tasks (e.g., represent all N roots of an Nth-order >> polynomial). > > I completely agree about roots. In electrical theory, though, we use > complex numbers as ordered pairs, which is overkill (but we grow to > think it fundamental). x+iy, R=jX; it is rare that extracting a root > comes into play. Sin(r), cos(r) often serves where we use cos(r) + > j*sin(r) for convenience. (I don't remember the formulas for the sums > of sines and cosines. I use complex numbers to derive the relations > when I need them. I'm all for convenience!)Well once you agree there, it's the similar situation for negative frequencies, isn't it? If you're talking about a two-dimensional "wheel," then you need to specify whether it's turning CW or CCW. Now you don't HAVE to use negative numbers (that's just one representation) - you could use positive numbers plus the letters [CW | CCW] - but, somehow, you must represent the information, and any two representations are isomorphic. -- % 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 13, 20082008-06-13
On Mon, 09 Jun 2008 14:18:17 -0500, "BobTheDog" <andrew.capon@zen.co.uk> wrote:>Hi Guys, > >First please excuse my ignorance, I am just starting in DSP as a bit of a >hobby. > >I Have Richard Lyons "understanding dsp" here and am working my way >through the "Sampling Bandpass Signals" chapter. I understand the idea of >Aliases and aliasing in the sampled signal spectrum but in the diagrams for >the continuous signal spectrum there is always an inverse of the signal. So >say there is an amplitude of 1 and 2MHz then there is also have an >amplitude of 1 at -2MHz. > >This doesn't seem to be explained, probably as it is something basic that >I should already know. > >Could someone direct me to some information that would explain this to >me. > >Thanks for any help. > >AndyHello Andy, Forgive me for taking so long to reply to you. You have pointed out an explanatory "gap" in my Chapter 2 material. I'm so accustomed to showing signal spectra having negative-frequency components that I just went ahead and created figures such as Figure 2-4 through Figure 2-7. At the bottom of page 24 I very briefly mentioned that the concept of negative-frequency was going to be useful to us, and that the reader could go to Chapter 8 for more info on negative-frequency. But I now see that this 'super-brief' Chapter 2 mention of negative-frequency may well leave the reader in an "uncertain & uncomfortable" state of puzzlement. If I ever create a 3rd edition to my book, I'll definitely add a bit more Chapter 2 explanation of the reason for, and usefulness of, drawing signal spectra having negative-frequency components. So Andy, Chapter 3 shows how we obtain negative-frequency results when we use the DFT to perform spectrum analysis, and Chapter 8 gently, and thoroughly, describes how negative-frequencies are related to real and complex signals. By the way, please end me an E-mail and we'll figure a way for me to send the book's errata to you (if you're interested.) [-Rick-]
Reply by ●June 13, 20082008-06-13
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: > >> Randy Yates wrote: >>> Jerry Avins <jya@ieee.org> writes: >>> >>>> Randy Yates wrote: >>>>> Jerry Avins <jya@ieee.org> writes: >>>>>> 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. >>> I agree with that, but that isn't the main point I'm trying to make. >>> >>> My point is that to fully describe some phenomena requires an "extended >>> system," whatever the representation we choose to use for that extended >>> system is. To recast your statement above in these terms, a specific >>> representation may be convenient but not necessary---yes, I agree. But >>> we need SOME representation of the extended system (e.g., the complex >>> numbers) and CANNOT simply use the simple system (e.g., the reals) to >>> accomplish certain tasks (e.g., represent all N roots of an Nth-order >>> polynomial). >> I completely agree about roots. In electrical theory, though, we use >> complex numbers as ordered pairs, which is overkill (but we grow to >> think it fundamental). x+iy, R=jX; it is rare that extracting a root >> comes into play. Sin(r), cos(r) often serves where we use cos(r) + >> j*sin(r) for convenience. (I don't remember the formulas for the sums >> of sines and cosines. I use complex numbers to derive the relations >> when I need them. I'm all for convenience!) > > Well once you agree there, it's the similar situation for negative > frequencies, isn't it? If you're talking about a two-dimensional > "wheel," then you need to specify whether it's turning CW or CCW. Now > you don't HAVE to use negative numbers (that's just one representation) > - you could use positive numbers plus the letters [CW | CCW] - but, > somehow, you must represent the information, and any two representations > are isomorphic.CW and CCW are rather arbitrary and limited views of the world. The notion can't adequately describe which way the wheels of your car turn. Most of us would agree that all the wheels turn the same way when a car moves forward, yet the "standard" engineering nomenclature has the right wheels turning clockwise and the left ones, counterclockwise.* That's all just an amusing side issue. A sinusoid is not a wheel. Sin(-x) = -sin(x) and cos(-x) = cos(x). The former has an inversion. When I run sin(ft) through a unity-gain inverting amplifier, would you insist that sin(-ft) comes out? That wouldn't be wrong, but rather than deal with negative frequency or time marching backward, I prefer to write -sin(ft). I don't have a problem using negative frequency when it's appropriate. I *do* have a problem with letting the notion become so ingrained that it seems to be indispensable. Jerry _________________________________ * The standard describes the direction when looking into the shaft. A double-ended motor, such as those used on many grinders and polishers, turns both ways at once. -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
