Fred Marshall wrote:> > "Martin Eisenberg" <martin.eisenbergNOS@PAMudo.edu> wrote in > message news:1076894877.106551@ostenberg.wh.uni-dortmund.de... >> Bob Cain wrote: >> >> > I understand that the magnitude of the frequency response of >> > an FIR filter is the intersection of a tube going down >> > through the unit circle with the curvaceous surface created >> > by the placement of the zeros of its roots. >> >> With the surface defined by the modulus of the complex value >> assigned to each point in the z-plane, yes. Since few are able >> to imagine objects of the four spatial dimensions the actual >> situation calls for, something's got to give. Fred threw out >> the real direction of the s-plane to make room for his spirals; > > Well, I don't think I "threw it out", I rather thought that I'd > superimposed...What I meant is this: The quantities s and H(s) are distinct, so I don't think you can superimpose some of their components -- any more than you can superimpose the x and y axes of a real->real graph -- without losing something. In fact, you didn't actually *use* the direction in question as Re(s) in your post, only as Re(H(s)). Consider the spirals about both the jw line and the -1+jw line. Now that Re(s) has come into play, there's no way to avoid their potential crossing because space of perception lacks another offset possibility. In other words, anywhere we can put that "-1", something already exists. Whether that's "throwing out a dimension" or "superposition" seems moot. Martin -- Please help refine my English usage! -= Send your critique by email. =- Quidquid latine dictum sit, altum viditur.
FIR roots and frequency response
Started by ●February 13, 2004
Reply by ●February 16, 20042004-02-16
Reply by ●February 16, 20042004-02-16
"Martin Eisenberg" <martin.eisenbergNOS@PAMudo.edu> wrote in message news:1076926298.516596@ostenberg.wh.uni-dortmund.de...> Fred Marshall wrote: > > > > Well, I don't think I "threw it out", I rather thought that I'd > > superimposed... > > What I meant is this: The quantities s and H(s) are distinct, so > I don't think you can superimpose some of their components -- any > more than you can superimpose the x and y axes of a real->real > graph -- without losing something. In fact, you didn't actually *use* > the direction in question as Re(s) in your post, only as Re(H(s)). > Consider the spirals about both the jw line and the -1+jw line. Now > that Re(s) has come into play, there's no way to avoid their > potential crossing because space of perception lacks another offset > possibility. In other words, anywhere we can put that "-1", something > already exists. Whether that's "throwing out a dimension" or > "superposition" seems moot. >Martin, Hmmmm... Well, there are many perspectives that work. For one thing, I could do the same thing without thinking "superposition" and just create a new 3-D figure. Otherwise, I think of it as plotting two curves or data points on the same set of axes - thus superposition - both of them are there. So, the axes do double duty to display different (and maybe related) things on the same "chart". In this case, as you point out, we have "s=x+jw" and we have "H(jw)=A(jw)e^jwp" ... something like that. So, if we share the "w" axis for both, then we have to figure out how to display H(jw). The choice I made was to show the real part aligned with the "x" axis and the imaginary part in the new third dimension. This means that zero phase shift vectors point to the right in "x" and that 90 degree phase shift vectors point up out of the x/w plane, etc. Just a 3-D plot of two things superimposed on that plot. Your analogy of real>real plots (which is really the same thing) might be demonstrated like this: (it doesn't mix up the x and y axes at all) There is one function of x and y - so we can plot it in an x/y plane. There is another function of x and y and z - so we can plot it by adding a z axis to the plane above and making a 3-D plot without changing any of the aspects of the original x/y plot. Of course, it's going to be that the function of x,y,z can lie on or cross the x,y plane (or not). That's the idea. Tying them together where there are common elements (in this case, "w") seems useful. Fred
Reply by ●February 16, 20042004-02-16
Fred Marshall wrote:> Hmmmm... Well, there are many perspectives that work.Indeed. After reading your post I think the glass was half empty to me and half full to you. Martin -- The power of accurate observation is commonly called cynicism by those who don't have it. --George Bernard Shaw
Reply by ●February 17, 20042004-02-17
Ronald H. Nicholson Jr. wrote:> > The location of both the poles and the zeros is important. Imagine a > crank handle at the location of every pole or zero on the complex > plane.Wonderful visualization aid, Ronald. Many thanks. I'm still left wondering what it all means, however. All I can see is that what is left in the sum of the integrators after a single revolution is a minimum, zero, when the zeros are inside the circle but the signifigance of that remains obscure to me in a physical sense. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
Reply by ●February 17, 20042004-02-17
Fred Marshall wrote:> > Now, I can imagine there might be another mapping or assignment of > dimensions that could work more along the lines you've stated (BTW, where > did you get this?) but I'm hard pressed to see how a simple projection onto > a plane or a tube (respectively) can yield magnitude - because magnitude > isn't represented here anywhere.I dunno, I just thunk it up. As should be obvious my experience has not been with traditional filter design at all or with feedback systems so my feeling for the complex plane representation never gelled. Actually, I've always thought, for some reason I can't remember, perhaps an analogy I was given that I totally misunderstood, that if you plotted the magnitude of the transfer function (I failed to state the critical word "magnitude" in my first post) as a surface above the plane with the poles and zeros deforming it appropriately that the magnitude function we usually think about and plot linearly in a frequency response plot was the intersection of that surface with a cylinder perpindicular to the complex plane passing through the unit circle. In the s plane it would be the intersection with a plane perpindicular to it passing through the imaginary axis. From what I'm reading in the thread I'm still not sure whether that is right or dead wrong. :-) Now I am trying to understand how the phase function varies as you traverse that circle. I do understand Ronald's explanation and see that (since I'm only thinking now about FIR's) the placement of the zeros inside or outside determine whether the integral of the total of the contributions of all the zeros can only be a multiple of 2*pi if they are all inside but I'm still not making the connection as to why that would cause the energy in an impulse response to all bunch up at the start when it's minimum phase and what is really being minimized if I look at just the usual (unwraped) phase plot (assuming it even shows up there.) Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
Reply by ●February 17, 20042004-02-17
"Ronald H. Nicholson Jr." <rhn@mauve.rahul.net> wrote in message news:c0jn3f$6dn$1@blue.rahul.net...> In article <c0holl016fs@enews1.newsguy.com>, > > The location of both the poles and the zeros is important.The way I learned to visualize this was to imagine that the s-plane was covered with a thin rubber sheet. Then, imagine that a pencil were stood on end, under the sheet, like a tent pole at each pole location. Then, imagine that a thumbtack is set to hold the sheet down to the plane wherever there's a zero. It's helped now and then! Fred
Reply by ●February 17, 20042004-02-17
robert bristow-johnson wrote:> >>I'm trying to visualize how reflecting a zero from outside >>to inside the unit circle causes the phase function to >>become closer to zero everywhere. > > > it's a lot easier visualizing it on the s-plane for continuous-time LTI > systemsOk. I do basically understand the mapping between them.>> Actually I'm trying to >>understand just what it is that's minimized in minimum phase >>filters. If that's not it, please enlighten me. > > > oh, it is it. this is much easier to see for the s-plane. given a > particular magnitude response that can be attained with poles and zeros in > the s-plane, the only "constellations" of poles and zeros that will give you > *that* particular magnitude response (let's not consider different constant > gain factors) is the set of constellations that have the poles and zeros in > some position or in reflections about the jw axis. of course poles must be > in the left-half plane so they can't be reflected to the other side of the > jw axis. but zeros can.Understood. I'm only thinking about all-zero systems right now anyway.> but if they are reflected from the left-hand plane > to the right-hand plane, you can see that their angle will always be greater > than 90 degrees whereas they were always less than 90 degrees before. so to > get the smallest set of angles on those zeros, you gotta have them all in > the left-half plane.Doh! Pretty simple, huh? <Bob blushes>> > does that do it, Bob?At last. Thanks for the patience. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
Reply by ●February 17, 20042004-02-17
Bob Cain wrote: ...> Now I am trying to understand how the phase function varies as you > traverse that circle. I do understand Ronald's explanation and see that > (since I'm only thinking now about FIR's) the placement of the zeros > inside or outside determine whether the integral of the total of the > contributions of all the zeros can only be a multiple of 2*pi if they > are all inside but I'm still not making the connection as to why that > would cause the energy in an impulse response to all bunch up at the > start when it's minimum phase and what is really being minimized if I > look at just the usual (unwraped) phase plot (assuming it even shows up > there.)I'm out in a limb here, waving my arms rather than doing math, but here goes <deep breath>: Adding phase shift implies adding or subtracting time. In real life, subtracting isn't possible, so extra phase shift means extra delay. I seems that simple to me. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 17, 20042004-02-17
"Bob Cain" <arcane@arcanemethods.com> wrote in message news:c0s60402rrh@enews3.newsguy.com...> Fred Marshall wrote: > > > > > > Now, I can imagine there might be another mapping or assignment of > > dimensions that could work more along the lines you've stated (BTW,where> > did you get this?) but I'm hard pressed to see how a simple projectiononto> > a plane or a tube (respectively) can yield magnitude - because magnitude > > isn't represented here anywhere. >..............>that if you plotted the magnitude of the > transfer function (I failed to state the critical word > "magnitude" in my first post) as a surface above the plane > with the poles and zeros deforming it appropriately that the > magnitude function we usually think about and plot linearly > in a frequency response plot was the intersection of that > surface with a cylinder perpindicular to the complex plane > passing through the unit circle. In the s plane it would be > the intersection with a plane perpendicular to it passing > through the imaginary axis.Bob, Oh, well then..... you already *have* the magnitude surface for all values of s or z. So, sure, a perpendicular plane or cylinder will simply be cut by the magnitude surface at those values on the jw axis in s or on the unit circle in z. Fred
Reply by ●February 17, 20042004-02-17
In article <40321b52$0$3084$61fed72c@news.rcn.com>, Jerry Avins <jya@ieee.org> wrote:>Bob Cain wrote: > ... >> Now I am trying to understand how the phase function varies as you >> traverse that circle. I do understand Ronald's explanation and see that >> (since I'm only thinking now about FIR's) the placement of the zeros >> inside or outside determine whether the integral of the total of the >> contributions of all the zeros can only be a multiple of 2*pi if they >> are all inside but I'm still not making the connection as to why that >> would cause the energy in an impulse response to all bunch up at the >> start when it's minimum phase and what is really being minimized if I >> look at just the usual (unwraped) phase plot (assuming it even shows up >> there.) > >I'm out in a limb here, waving my arms rather than doing math, but here >goes <deep breath>: > >Adding phase shift implies adding or subtracting time. In real life, >subtracting isn't possible, so extra phase shift means extra delay.Here's another "hand-wavey" one: Consider the unit delay. That's a zero at the origin. It's phase response will spiral by 2*pi per trip around the unit cylindar. The phase response of an N unit delay line will spiral by 2*pi*N. Wave hands: The more any arbitrary frequency response in the z plane "resembles" one of these spirals, the more the time response will "resemble" a delay line of N taps. A minimum phase filter is the one closest to "resembling" a zero-delay delay line. IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
