Just to inscribe this on the wall in cyberspace: There are _four basic kinds_ of Fourier transforms*: The continuous Fourier transform, AKA _the_ Fourier transform: A continuous-time signal of infinite extent is transformed by X(w) = \int_{-\infty}^{\infty} x(t) * e^{j*w*t} dt At times (as when finding power spectra) this may be a limit, taking the average over time, rather than finding the transform itself. The continuous Fourier transform of a cyclical signal, AKA the Fourier series: A continuous-time signal of period t0 is transformed by X_n = 1/t0 * \int_x1^\{x1+x0} x(t) * e^{j*t*2*\pi*n/t0} dt (I'm not sure what to call this one; usually one sees it's descendant, the z transform): A discrete-time signal, x_n of infinite extent is transformed by X(w) = \sum_{-\infty}^{\infty} x_n e^{j*w*n} This coughs up a transform that is continuous in frequency, but cyclic over 2*pi The Fourier transform of a discrete-time _cyclic_ signal, x_n with period N. AKA the Discrete Fourier Transform: X_k = \sum_{n=n0}^{n0+N} x_n e^{j*n*2*\pi*k/N} Each of these transforms is very similar to the others, yet each one has some _fundamental differences_, not least of which is the types of inputs and outputs. In general, note that: --> continuous time implies infinite frequency extent. --> discrete time implies cyclic frequency extent. Since the Fourier transform and its inverse are duals, that means that the two above rules, well, rule, and the following two are mere corollaries: --> continuous frequency implies infinite time extent --> discrete frequency implies cyclic time extent From this you get the four possible combinations: continuous-time infinite --> continuous-frequency infinite. continuous-time cyclic --> discrete-frequency infinite. discrete-time infinite --> continuous-frequency cyclic. discrete-time cyclic --> discrete-frequency, cyclic. I'm pulling this from memory, but you can find it in books with titles like "Signals and Systems"; the one on my shelf is by Oppenheim, Willsky & Young. There. I'll put down my can of spray paint now. * Forget multi-dimensional ones for the moment -- they can be made up of combinations of these four types. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html

# Appendix A: Types of Fourier Transforms

Started by ●January 10, 2011

Reply by ●January 10, 20112011-01-10

Tim Wescott <tim@seemywebsite.com> wrote: (snip)> * Forget multi-dimensional ones for the moment -- they can be made up of > combinations of these four types.The Fourier transform(s) are separable in rectangular coordinates, but in other coordinate systems you need different transforms. There is, for example, in cylindrical coordinates a Fourier-Bessel transform. -- glen

Reply by ●January 10, 20112011-01-10

a couple of suggestions: i would express together both the forward transform and the inverse. for the continuous F.T., i might suggest expressing both the angular- frequency version ("omega") and the cyclical-frequency version ("f") which is sometimes called the "unitary" definition. maybe make a relationship between F.T. and L.T. for continuous-time and DTFT and Z.T. for discrete-time. maybe the best thing to do (since ASCII math sorta sucks) is to plop up a page to point to. maybe the right combination of Wikipedia articles (and if the articles don't say it just the way you want it, then edit the damn article). On Jan 10, 3:00�pm, Tim Wescott <t...@seemywebsite.com> wrote:> Just to inscribe this on the wall in cyberspace: > > There are _four basic kinds_ of Fourier transforms*: > > The continuous Fourier transform, AKA _the_ Fourier transform: > > A continuous-time signal of infinite extent is transformed by > > X(w) = \int_{-\infty}^{\infty} x(t) * e^{j*w*t} dt > > At times (as when finding power spectra) this may be a limit, taking the > average over time, rather than finding the transform itself. > > The continuous Fourier transform of a cyclical signal, AKA the Fourier > series: > > A continuous-time signal of period t0 is transformed by > > X_n = 1/t0 * \int_x1^\{x1+x0} x(t) * e^{j*t*2*\pi*n/t0} dt > > (I'm not sure what to call this one; usually one sees it's descendant, > the z transform):i think they call that the DTFT, "Discrete-Time Fourier Transform" which represents the values on the unit circle of the Z Transform. r b-j

Reply by ●January 10, 20112011-01-10

On 01/10/2011 12:45 PM, robert bristow-johnson wrote:> > a couple of suggestions: > > i would express together both the forward transform and the inverse. > > for the continuous F.T., i might suggest expressing both the angular- > frequency version ("omega") and the cyclical-frequency version ("f") > which is sometimes called the "unitary" definition. > > maybe make a relationship between F.T. and L.T. for continuous-time > and DTFT and Z.T. for discrete-time. > > maybe the best thing to do (since ASCII math sorta sucks) is to plop > up a page to point to. maybe the right combination of Wikipedia > articles (and if the articles don't say it just the way you want it, > then edit the damn article). >And to think that I've been trying to think of a subject for a new article, and the light just wasn't going on.> > On Jan 10, 3:00 pm, Tim Wescott<t...@seemywebsite.com> wrote: >> Just to inscribe this on the wall in cyberspace: >> >> There are _four basic kinds_ of Fourier transforms*: >> >> The continuous Fourier transform, AKA _the_ Fourier transform: >> >> A continuous-time signal of infinite extent is transformed by >> >> X(w) = \int_{-\infty}^{\infty} x(t) * e^{j*w*t} dt >> >> At times (as when finding power spectra) this may be a limit, taking the >> average over time, rather than finding the transform itself. >> >> The continuous Fourier transform of a cyclical signal, AKA the Fourier >> series: >> >> A continuous-time signal of period t0 is transformed by >> >> X_n = 1/t0 * \int_x1^\{x1+x0} x(t) * e^{j*t*2*\pi*n/t0} dt >> >> (I'm not sure what to call this one; usually one sees it's descendant, >> the z transform): > > i think they call that the DTFT, "Discrete-Time Fourier Transform" > which represents the values on the unit circle of the Z Transform.So it is -- somewhere in the mussed and dusty corners of my brain I already knew that. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html

Reply by ●January 10, 20112011-01-10

On Jan 10, 4:00�pm, Tim Wescott <t...@seemywebsite.com> wrote:> On 01/10/2011 12:45 PM, robert bristow-johnson wrote: > > > > > a couple of suggestions: > > > i would express together both the forward transform and the inverse. > > > for the continuous F.T., i might suggest expressing both the angular- > > frequency version ("omega") and the cyclical-frequency version ("f") > > which is sometimes called the "unitary" definition. > > > maybe make a relationship between F.T. and L.T. for continuous-time > > and DTFT and Z.T. for discrete-time. > > > maybe the best thing to do (since ASCII math sorta sucks) is to plop > > up a page to point to. �maybe the right combination of Wikipedia > > articles (and if the articles don't say it just the way you want it, > > then edit the damn article). > > And to think that I've been trying to think of a subject for a new > article, and the light just wasn't going on.would this be an article you put on your website, or would it appear somewhere else? lemme know if you want a collaborator. it may be a rehashing of O&W (and now "Buck", i think), but there is a nice straight-forward concept flow in the pedagogy from the tenets of Linear System Theory to all of these transforms. 1. Superposition and Linearity 2. Time Invariance 3. Causal and non-causal systems 4. Convolution 5. Sinusoids and exponentials (Euler's theorem) 6. e^(jwt) as eigenfunction for LTI (transfer functions) magnitude 7. maybe an optional chapter on phasors 8. Fourier Series (with implications regarding LTI) 9. Continuous Fourier Transform (extend the F.S. period to infinity) 10. Laplace Transform as a generalization of C.F.T. (implication to transfer function concept) 11. Sampling theorem and reconstruction (periodicity in freq domain from sampling in time) 12. DTFT as CFT of dirac-sampled signal 13. ZT as LT of dirac-sampled signal (or as the generalization of DTFT) 14. DFT as DFS (Discrete Fourier Series) DFT as a special case of the DTFT 15. FFT as a means of calculating the DFT note that there is no mention of applications. "just the concepts, ma'am." r b-j

Reply by ●January 10, 20112011-01-10

On 01/10/2011 02:10 PM, robert bristow-johnson wrote:> On Jan 10, 4:00 pm, Tim Wescott<t...@seemywebsite.com> wrote: >> On 01/10/2011 12:45 PM, robert bristow-johnson wrote: >> >> >> >>> a couple of suggestions: >> >>> i would express together both the forward transform and the inverse. >> >>> for the continuous F.T., i might suggest expressing both the angular- >>> frequency version ("omega") and the cyclical-frequency version ("f") >>> which is sometimes called the "unitary" definition. >> >>> maybe make a relationship between F.T. and L.T. for continuous-time >>> and DTFT and Z.T. for discrete-time. >> >>> maybe the best thing to do (since ASCII math sorta sucks) is to plop >>> up a page to point to. maybe the right combination of Wikipedia >>> articles (and if the articles don't say it just the way you want it, >>> then edit the damn article). >> >> And to think that I've been trying to think of a subject for a new >> article, and the light just wasn't going on. > > would this be an article you put on your website, or would it appear > somewhere else? > > lemme know if you want a collaborator. it may be a rehashing of O&W > (and now "Buck", i think), but there is a nice straight-forward > concept flow in the pedagogy from the tenets of Linear System Theory > to all of these transforms. > > 1. Superposition and Linearity > 2. Time Invariance > 3. Causal and non-causal systems > 4. Convolution > > 5. Sinusoids and exponentials (Euler's theorem) > 6. e^(jwt) as eigenfunction for LTI (transfer functions) > magnitude > 7. maybe an optional chapter on phasors > > 8. Fourier Series (with implications regarding LTI) > 9. Continuous Fourier Transform (extend the F.S. period to infinity) > 10. Laplace Transform as a generalization of C.F.T. > (implication to transfer function concept) > > 11. Sampling theorem and reconstruction > (periodicity in freq domain from sampling in time) > 12. DTFT as CFT of dirac-sampled signal > 13. ZT as LT of dirac-sampled signal > (or as the generalization of DTFT) > > 14. DFT as DFS (Discrete Fourier Series) > DFT as a special case of the DTFT > > 15. FFT as a means of calculating the DFT > > note that there is no mention of applications. > "just the concepts, ma'am."Hmm. Email sent. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html

Reply by ●January 10, 20112011-01-10

On Jan 10, 3:00�pm, Tim Wescott <t...@seemywebsite.com> wrote:> Just to inscribe this on the wall in cyberspace: > > There are _four basic kinds_ of Fourier transforms*: > > The continuous Fourier transform, AKA _the_ Fourier transform: > > A continuous-time signal of infinite extent is transformed by > > X(w) = \int_{-\infty}^{\infty} x(t) * e^{j*w*t} dt > > At times (as when finding power spectra) this may be a limit, taking the > average over time, rather than finding the transform itself. > > The continuous Fourier transform of a cyclical signal, AKA the Fourier > series: > > A continuous-time signal of period t0 is transformed by > > X_n = 1/t0 * \int_x1^\{x1+x0} x(t) * e^{j*t*2*\pi*n/t0} dt > > (I'm not sure what to call this one; usually one sees it's descendant, > the z transform): > > A discrete-time signal, x_n of infinite extent is transformed by > > X(w) = \sum_{-\infty}^{\infty} x_n e^{j*w*n} > > This coughs up a transform that is continuous in frequency, but cyclic > over 2*pi > > The Fourier transform of a discrete-time _cyclic_ signal, x_n with > period N. �AKA the Discrete Fourier Transform: > > X_k = \sum_{n=n0}^{n0+N} x_n e^{j*n*2*\pi*k/N} > > Each of these transforms is very similar to the others, yet each one has > some _fundamental differences_, not least of which is the types of > inputs and outputs. �In general, note that: > > --> continuous time implies infinite frequency extent. > --> discrete time implies cyclic frequency extent. > > Since the Fourier transform and its inverse are duals, that means that > the two above rules, well, rule, and the following two are mere corollaries: > > --> continuous frequency implies infinite time extent > --> discrete frequency implies cyclic time extent > > �From this you get the four possible combinations: > > continuous-time infinite --> continuous-frequency infinite. > continuous-time cyclic --> discrete-frequency infinite. > discrete-time infinite --> continuous-frequency cyclic. > discrete-time cyclic --> discrete-frequency, cyclic. > > I'm pulling this from memory, but you can find it in books with titles > like "Signals and Systems"; the one on my shelf is by Oppenheim, Willsky > & Young. > > There. �I'll put down my can of spray paint now. > > * Forget multi-dimensional ones for the moment -- they can be made up of > combinations of these four types. > > -- > > Tim Wescott > Wescott Design Serviceshttp://www.wescottdesign.com > > Do you need to implement control loops in software? > "Applied Control Theory for Embedded Systems" was written for you. > See details athttp://www.wescottdesign.com/actfes/actfes.htmlhttp://www.dspguide.com/ch8/1.htm

Reply by ●January 10, 20112011-01-10

On 01/10/2011 03:09 PM, brent wrote:> On Jan 10, 3:00 pm, Tim Wescott<t...@seemywebsite.com> wrote: >> Just to inscribe this on the wall in cyberspace: >> >> There are _four basic kinds_ of Fourier transforms*: >> >> The continuous Fourier transform, AKA _the_ Fourier transform: >> >> A continuous-time signal of infinite extent is transformed by >> >> X(w) = \int_{-\infty}^{\infty} x(t) * e^{j*w*t} dt >> >> At times (as when finding power spectra) this may be a limit, taking the >> average over time, rather than finding the transform itself. >> >> The continuous Fourier transform of a cyclical signal, AKA the Fourier >> series: >> >> A continuous-time signal of period t0 is transformed by >> >> X_n = 1/t0 * \int_x1^\{x1+x0} x(t) * e^{j*t*2*\pi*n/t0} dt >> >> (I'm not sure what to call this one; usually one sees it's descendant, >> the z transform): >> >> A discrete-time signal, x_n of infinite extent is transformed by >> >> X(w) = \sum_{-\infty}^{\infty} x_n e^{j*w*n} >> >> This coughs up a transform that is continuous in frequency, but cyclic >> over 2*pi >> >> The Fourier transform of a discrete-time _cyclic_ signal, x_n with >> period N. AKA the Discrete Fourier Transform: >> >> X_k = \sum_{n=n0}^{n0+N} x_n e^{j*n*2*\pi*k/N} >> >> Each of these transforms is very similar to the others, yet each one has >> some _fundamental differences_, not least of which is the types of >> inputs and outputs. In general, note that: >> >> --> continuous time implies infinite frequency extent. >> --> discrete time implies cyclic frequency extent. >> >> Since the Fourier transform and its inverse are duals, that means that >> the two above rules, well, rule, and the following two are mere corollaries: >> >> --> continuous frequency implies infinite time extent >> --> discrete frequency implies cyclic time extent >> >> From this you get the four possible combinations: >> >> continuous-time infinite --> continuous-frequency infinite. >> continuous-time cyclic --> discrete-frequency infinite. >> discrete-time infinite --> continuous-frequency cyclic. >> discrete-time cyclic --> discrete-frequency, cyclic. >> >> I'm pulling this from memory, but you can find it in books with titles >> like "Signals and Systems"; the one on my shelf is by Oppenheim, Willsky >> & Young. >> >> There. I'll put down my can of spray paint now. >> >> * Forget multi-dimensional ones for the moment -- they can be made up of >> combinations of these four types. >> > > http://www.dspguide.com/ch8/1.htmYup. Something like that. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html

Reply by ●January 10, 20112011-01-10

On Jan 10, 3:00�pm, Tim Wescott <t...@seemywebsite.com> wrote:> Just to inscribe this on the wall in cyberspace: > > There are _four basic kinds_ of Fourier transforms*: > > The continuous Fourier transform, AKA _the_ Fourier transform: > > A continuous-time signal of infinite extent is transformed by > > X(w) = \int_{-\infty}^{\infty} x(t) * e^{j*w*t} dt > > At times (as when finding power spectra) this may be a limit, taking the > average over time, rather than finding the transform itself. > > The continuous Fourier transform of a cyclical signal, AKA the Fourier > series: > > A continuous-time signal of period t0 is transformed by > > X_n = 1/t0 * \int_x1^\{x1+x0} x(t) * e^{j*t*2*\pi*n/t0} dt > > (I'm not sure what to call this one; usually one sees it's descendant, > the z transform): > > A discrete-time signal, x_n of infinite extent is transformed by > > X(w) = \sum_{-\infty}^{\infty} x_n e^{j*w*n} > > This coughs up a transform that is continuous in frequency, but cyclic > over 2*pi > > The Fourier transform of a discrete-time _cyclic_ signal, x_n with > period N. �AKA the Discrete Fourier Transform: > > X_k = \sum_{n=n0}^{n0+N} x_n e^{j*n*2*\pi*k/N} > > Each of these transforms is very similar to the others, yet each one has > some _fundamental differences_, not least of which is the types of > inputs and outputs. �In general, note that: > > --> continuous time implies infinite frequency extent. > --> discrete time implies cyclic frequency extent. > > Since the Fourier transform and its inverse are duals, that means that > the two above rules, well, rule, and the following two are mere corollaries: > > --> continuous frequency implies infinite time extent > --> discrete frequency implies cyclic time extent > > �From this you get the four possible combinations: > > continuous-time infinite --> continuous-frequency infinite. > continuous-time cyclic --> discrete-frequency infinite. > discrete-time infinite --> continuous-frequency cyclic. > discrete-time cyclic --> discrete-frequency, cyclic. > > I'm pulling this from memory, but you can find it in books with titles > like "Signals and Systems"; the one on my shelf is by Oppenheim, Willsky > & Young. > > There. �I'll put down my can of spray paint now. > > * Forget multi-dimensional ones for the moment -- they can be made up of > combinations of these four types. > > -- > > Tim Wescott > Wescott Design Serviceshttp://www.wescottdesign.com > > Do you need to implement control loops in software? > "Applied Control Theory for Embedded Systems" was written for you. > See details athttp://www.wescottdesign.com/actfes/actfes.htmlhttp://12000.org/my_notes/transforms/transforms.png Hope this helps. Greg

Reply by ●January 10, 20112011-01-10

On Jan 10, 6:39�pm, Greg Heath <he...@alumni.brown.edu> wrote:> > http://12000.org/my_notes/transforms/transforms.png > > Hope this helps.nice chart. one old issue that gets me into fights here is that i disagree with the mathematical factuality of this statement: "Finite x[n] length N, zero elsewhere". it's oft repeated, you see it in print, and it's wrong. r b-j