Hello Forum, this is a picky question: in the Fourier Transform pair the forward transform needs to have the complex exponential with positive OR negative sign. The inverse transform is correct as long as it has the opposite sign. Why? Why does the sign not matter as long as they are opposite in the two transforms? Next, there could be a 1/sqrt(2pi) factor in both,1/2pi in front of just one, or the other. In some books the factor is not even present. The answer has to do with energy I guess. Energy is the integral of the Int[x(t)x*(t)] dt = Int[F(w)F*(w)]dw..... thanks fisico30
Fourier transform normalization
Started by ●February 7, 2009
Reply by ●February 7, 20092009-02-07
On Feb 7, 9:42�am, "fisico30" <marcoscipio...@gmail.com> wrote:> Hello Forum, > > �this is a picky question: in the Fourier Transform pair the forward > transform needs to have the complex exponential with positive OR negative > sign. The inverse transform is correct as long as it has the opposite > sign. > Why? Why does the sign not matter as long as they are opposite in the two > transforms?it's because, while +j and -j are not the same (imaginary) number (they are not zero and they are negatives of each other), there is *no* qualitative difference, in fact *no* other difference between the two. the only other descriptive feature is one they share: they both square to be -1. as long as you're consistent, +j and -j are completely interchangable in mathematical equations.> Next, there could be a 1/sqrt(2pi) factor in both,1/2pi in front of just > one, or the other. In some books the factor is not even present. > The answer has to do with energy I guess.the answer has more to do with scaling of the "t" axis and the "f" or "w" axis. this definition of the FT has no 1/(2pi) scaling factor nor 1/sqrt (2pi) outside of any integrals (but there is a "df" instead or "dw"). +inf x(t) = integral{ X(f) e^(+j*2*pi*f*t) df} -inf +inf X(f) = integral{ x(t) e^(-j*2*pi*f*t) dt} -inf there are many, many advantages to using this definition, all largely revolving around the fact that the duality theorem is quite simplified, being that the forward and inverse transform are functionally identical (remember "+j" and "-j" are really the same imaginary thing, even if they are negatives of each other). the theorems for duality, convolution, energy (Parseval's), shifting in t or f, sinc() vs. rect(), x(0) and X(0) are all greatly simplified and easier to remember. when i'm using the common "omega" definitions, i never remember whatever scaling factor i have to put wherever to make the theorems correct. you *do* have to remember that instead of seeing "w" you see "2*pi*f" and that derivatives or integrals have the 2*pi factor in them (but the advantage is that it's exactly the same no matter which way you're going; from t to f or from f to t).> Energy is the integral of the �Int[x(t)x*(t)] dt = Int[F(w)F*(w)]dw.....no, since you're using radian frequency "w", those two energy integral are not equal. you need a 1/(2pi) in there on one or the other. that's the disadvantage to the "w" definition. you forget those necessary scaling factors. r b-j
Reply by ●February 8, 20092009-02-08
fisico30 wrote:> this is a picky question: in the Fourier Transform pair the forward > transform needs to have the complex exponential with positive OR negative > sign. The inverse transform is correct as long as it has the opposite > sign. > Why? Why does the sign not matter as long as they are opposite in the two > transforms?You turn screws one direction to screw them in, the opposite to remove them. It doesn't matter, left or right handed screw, in either case you turn the opposite direction to remove as insert. -- glen
Reply by ●February 8, 20092009-02-08
On 7 Feb, 15:42, "fisico30" <marcoscipio...@gmail.com> wrote:> Hello Forum, > > �this is a picky question: in the Fourier Transform pair the forward > transform needs to have the complex exponential with positive OR negative > sign. The inverse transform is correct as long as it has the opposite > sign. > Why? Why does the sign not matter as long as they are opposite in the two > transforms?Careful! Poking too deep into that sort of questions is dangerous! There might be faster ways to turn mad, but those would likely involve consuming 'funny' mushrooms or injecting illegal psychopharmaca. Or both. Just accept that this is how the maths works, and leave the question at that. Rune
Reply by ●February 8, 20092009-02-08
On Feb 7, 8:42�am, "fisico30" <marcoscipio...@gmail.com> asked:> �this is a picky question: in the Fourier Transform pair the forward > transform needs to have the complex exponential with positive OR negative > sign. The inverse transform is correct as long as it has the opposite > sign. > Why? Why does the sign not matter as long as they are opposite in the two > transforms?The sign should not matter as long as the forward and inverse transforms use opposite signs, as several people have pointed out already. The reason that the negative sign is used in the forward transform comes from the tradition of Fourier series. We want to express a periodic signal x(t) as a sum of complex exponential functions, and many people feel that an expression of the form x(t) = sum c_n exp(j n2pi t/T) is slightly more "natural" than x(t) = sum c_n exp(-j n2pi t/T)). If we wish to have this level of comfort, then we are forced to define c_n = int_0^T x(t) exp(-j n2pi t/T) and this carries over to the Fourier transform. Denizens of this group are, of course, far more fearless since they routinely define the z- transform as A(z) = sum a_n z^{-n} and have no compunction about mixing positive and negative indices, but that is a discussion for a different thread. Hope this helps --Dilip Sarwate
Reply by ●February 8, 20092009-02-08
On Feb 8, 8:38�am, "dvsarw...@yahoo.com" <dvsarw...@gmail.com> wrote:> c_n = int_0^T x(t) exp(-j n2pi t/T)Oooops, that expression is missing a 1/T factor.... See also the nice discussion by Robert Bristow-Johnson re factors and why one should avoid them.
Reply by ●February 8, 20092009-02-08
On 8 Feb, 15:38, "dvsarw...@yahoo.com" <dvsarw...@gmail.com> wrote:> On Feb 7, 8:42�am, "fisico30" <marcoscipio...@gmail.com> asked: > > > �this is a picky question: in the Fourier Transform pair the forward > > transform needs to have the complex exponential with positive OR negative > > sign. The inverse transform is correct as long as it has the opposite > > sign. > > Why? Why does the sign not matter as long as they are opposite in the two > > transforms? > > The sign should not matter as long as the forward and inverse > transforms > use opposite signs,This points out the fact that the sign does not matter. It does not answer the OP's question, which is *why* it doesn't matter. Rune
Reply by ●February 8, 20092009-02-08
Rune Allnor wrote:> On 7 Feb, 15:42, "fisico30" <marcoscipio...@gmail.com> wrote: >> Hello Forum, >> >> this is a picky question: in the Fourier Transform pair the forward >> transform needs to have the complex exponential with positive OR negative >> sign. The inverse transform is correct as long as it has the opposite >> sign. >> Why? Why does the sign not matter as long as they are opposite in the two >> transforms? > > Careful! Poking too deep into that sort of questions is dangerous! > There might be faster ways to turn mad, but those would likely > involve consuming 'funny' mushrooms or injecting illegal > psychopharmaca. Or both. > > Just accept that this is how the maths works, and leave the question > at that.Just accepting the math is a copout. How can you have confidence in it if it seems to make no sense? When I was first taught long division, I asked the teacher why the procedure worked. She said to multiply the answer out and I would see that it works. I said that I knew *that* it worked; I wanted to know *how* it worked. She didn't even understand that that was a real question. Is that your idea of appropriate education? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 8, 20092009-02-08
On 8 Feb, 16:01, Jerry Avins <j...@ieee.org> wrote:> Rune Allnor wrote: > > On 7 Feb, 15:42, "fisico30" <marcoscipio...@gmail.com> wrote: > >> Hello Forum, > > >> �this is a picky question: in the Fourier Transform pair the forward > >> transform needs to have the complex exponential with positive OR negative > >> sign. The inverse transform is correct as long as it has the opposite > >> sign. > >> Why? Why does the sign not matter as long as they are opposite in the two > >> transforms? > > > Careful! Poking too deep into that sort of questions is dangerous! > > There might be faster ways to turn mad, but those would likely > > involve consuming 'funny' mushrooms or injecting illegal > > psychopharmaca. Or both. > > > Just accept that this is how the maths works, and leave the question > > at that. > > Just accepting the math is a copout. How can you have confidence in it > if it seems to make no sense? > > When I was first taught long division, I asked the teacher why the > procedure worked. She said to multiply the answer out and I would see > that it works. I said that I knew *that* it worked; I wanted to know > *how* it worked. She didn't even understand that that was a real > question. Is that your idea of appropriate education?These are three questions: 1) What's the procedure? (what the teacher had taught you) 2) How does it work? (your question to the teacher) 3) Why does it work? (The OPs question) Question 1 is the basis of all education: Teach procedures as a first approach. On the advanced level one can apporach the fundamental "how" - presumably based on algebra or number theory, in your case. The "why" is a completely different cup of tea: Why does math work at all? Because of interference or design from a divine entity? Because of coincidence? Because of a quirk of the Human mind? I'm generally known as the "mad scientist", more often than not going against the general opinions, so I prefer to leave those questions alone. Rune
Reply by ●February 8, 20092009-02-08
On 2009-02-08 11:01:41 -0400, Jerry Avins <jya@ieee.org> said:> Rune Allnor wrote: >> On 7 Feb, 15:42, "fisico30" <marcoscipio...@gmail.com> wrote: >>> Hello Forum, >>> >>> this is a picky question: in the Fourier Transform pair the forward >>> transform needs to have the complex exponential with positive OR negative >>> sign. The inverse transform is correct as long as it has the opposite >>> sign. >>> Why? Why does the sign not matter as long as they are opposite in the two >>> transforms? >> >> Careful! Poking too deep into that sort of questions is dangerous! >> There might be faster ways to turn mad, but those would likely >> involve consuming 'funny' mushrooms or injecting illegal >> psychopharmaca. Or both. >> >> Just accept that this is how the maths works, and leave the question >> at that. > > Just accepting the math is a copout. How can you have confidence in it > if it seems to make no sense? > > When I was first taught long division, I asked the teacher why the > procedure worked. She said to multiply the answer out and I would see > that it works. I said that I knew *that* it worked; I wanted to know > *how* it worked. She didn't even understand that that was a real > question. Is that your idea of appropriate education? > > JerryIf you do the FT (or IFT) twice you get a time reversed sequence. So somewhere along the way you will need to reverse the time. The next question is why the time gets reversed. Walk throught the repeated FT and watch very carefully how the orthogonality of the basis functions works. You will either be asleep from boredom or will understand the details of the ortogonality relationships.
