Hi, I am a junior undergrad studying a course on DSP. We have recently been introduced to linear phase sytems. It seems that we would very much like the LTI systems we design to have a zero/linear phase which we can account for appropriately. In other words, a non-linear phase response is highly undesirable in system design. Secondly, We have studied that an impulse response that exhibits Even or Odd symmetry can be shown to display a Generalized Linear Phase Response. Now, consider any arbitrary impulse response h[n], which may have any phase response (in general, non linear). Now it is trivial to decompose this into a sum of h1[n](even) + h2[n](odd). It also follows that these will be linear phase systems. So, we are able to represent an impulse response with arbitrary phase as a sum of two linear phase systems ! (which seemed to strike me as counter-intuitive). However, it is also appealing to see whether this observation can help us with system design [as a LTI system can be broken into two parallel branches - both of which shall be linear phase]. Is the above argument correct ? Can it be used to achieve better system analysis ? Any feedback/ideas on this would be highly educative. Thank you
Linear Phase Systems
Started by ●February 28, 2009
Reply by ●February 28, 20092009-02-28
noisysignal wrote:> > Hi, > > I am a junior undergrad studying a course on DSP. We have recently been > introduced to linear phase sytems. It seems that we would very much like > the LTI systems we design to have a zero/linear phase which we can account > for appropriately. In other words, a non-linear phase response is highly > undesirable in system design. > > Secondly, We have studied that an impulse response that exhibits Even or > Odd symmetry can be shown to display a Generalized Linear Phase Response. > > Now, consider any arbitrary impulse response h[n], which may have any > phase response (in general, non linear). Now it is trivial to decompose > this into a sum of h1[n](even) + h2[n](odd). It also follows that these > will be linear phase systems. > > So, we are able to represent an impulse response with arbitrary phase as a > sum of two linear phase systems ! (which seemed to strike me as > counter-intuitive).Why? How is that different than splitting the frequency domain into real and imaginary components.> However, it is also appealing to see whether this > observation can help us with system design [as a LTI system can be broken > into two parallel branches - both of which shall be linear phase].> > Is the above argument correct ?I don't see what is an argument? If it is required to combine the 2 parallel branches the result won't be linear phase (unless one of the branches contains nothing). -jim> Can it be used to achieve better system analysis ? Any feedback/ideas on > this would be highly educative. > > Thank you
Reply by ●February 28, 20092009-02-28
noisysignal wrote:> Hi, > > I am a junior undergrad studying a course on DSP. We have recently been > introduced to linear phase sytems. It seems that we would very much like > the LTI systems we design to have a zero/linear phase which we can account > for appropriately. In other words, a non-linear phase response is highly > undesirable in system design.I don't know about "highly". Engineering involves compromise, and linear phase usually is accompanied by less-than-minimum delay. Often, reducing delay is more important than maintaining linear phase.> Secondly, We have studied that an impulse response that exhibits Even or > Odd symmetry can be shown to display a Generalized Linear Phase Response. > > Now, consider any arbitrary impulse response h[n], which may have any > phase response (in general, non linear). Now it is trivial to decompose > this into a sum of h1[n](even) + h2[n](odd). It also follows that these > will be linear phase systems.If you mean that the individual decompositions will be linear phase, that is true by definition. If you assume that the sum of two arbitrary linear-phase impulse responses also has linear phase, you are mistaken.> So, we are able to represent an impulse response with arbitrary phase as a > sum of two linear phase systems ! (which seemed to strike me as > counter-intuitive). However, it is also appealing to see whether this > observation can help us with system design [as a LTI system can be broken > into two parallel branches - both of which shall be linear phase].Counter intuitive how? Is it counterintuitive to learn that any real function can be expressed as the sum of positive and negative parts, or that a complex function can be expressed as the sum of real and imaginary parts? What guides your intuition?> Is the above argument correct ?Yes. So what?> Can it be used to achieve better system analysis ? Any feedback/ideas on > this would be highly educative. > > Thank youJerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 28, 20092009-02-28
If you add the outputs of the two linear phase systems, the result can be anything. I give you an advice: use less of pompous words like "highly desirable", "in general", "system analysis" and such. Those words only make you look more stupid then you actually are. VLV noisysignal wrote:> Hi, > > I am a junior undergrad studying a course on DSP. We have recently been > introduced to linear phase sytems. It seems that we would very much like > the LTI systems we design to have a zero/linear phase which we can account > for appropriately. In other words, a non-linear phase response is highly > undesirable in system design. > > Secondly, We have studied that an impulse response that exhibits Even or > Odd symmetry can be shown to display a Generalized Linear Phase Response. > > Now, consider any arbitrary impulse response h[n], which may have any > phase response (in general, non linear). Now it is trivial to decompose > this into a sum of h1[n](even) + h2[n](odd). It also follows that these > will be linear phase systems. > > So, we are able to represent an impulse response with arbitrary phase as a > sum of two linear phase systems ! (which seemed to strike me as > counter-intuitive). However, it is also appealing to see whether this > observation can help us with system design [as a LTI system can be broken > into two parallel branches - both of which shall be linear phase]. > > Is the above argument correct ? > Can it be used to achieve better system analysis ? Any feedback/ideas on > this would be highly educative. > > Thank you > >
Reply by ●February 28, 20092009-02-28
On Sat, 28 Feb 2009 19:56:12 -0500, Jerry Avins wrote:> noisysignal wrote: >> Hi, >> >> I am a junior undergrad studying a course on DSP. We have recently been >> introduced to linear phase sytems. It seems that we would very much >> like the LTI systems we design to have a zero/linear phase which we can >> account for appropriately. In other words, a non-linear phase response >> is highly undesirable in system design. > > I don't know about "highly". Engineering involves compromise, and linear > phase usually is accompanied by less-than-minimum delay. Often, reducing > delay is more important than maintaining linear phase.(snip) Specifically, non-minimum phase elements in a feedback loop have exceedingly deleterious effects on stability and error sensitivity. It's a rare case when a designer would intentionally add a non-minimum-phase element -- much less something as high-delay as a linear phase element -- into a feedback loop. -- http://www.wescottdesign.com
Reply by ●February 28, 20092009-02-28
On Sat, 28 Feb 2009 16:29:37 -0600, "noisysignal" <kartikv6@gmail.com> wrote:>Hi, > >I am a junior undergrad studying a course on DSP. We have recently been >introduced to linear phase sytems. It seems that we would very much like >the LTI systems we design to have a zero/linear phase which we can account >for appropriately. In other words, a non-linear phase response is highly >undesirable in system design.As Jerry mentioned, even if "a non-linear phase response is highly undesirable", a lot of the time it isn't so bad to live with it. Often it is sufficient to have linear phase across a region of operation, and care much less about what happens elsewhere in the response range.>Secondly, We have studied that an impulse response that exhibits Even or >Odd symmetry can be shown to display a Generalized Linear Phase Response. > >Now, consider any arbitrary impulse response h[n], which may have any >phase response (in general, non linear). Now it is trivial to decompose >this into a sum of h1[n](even) + h2[n](odd). It also follows that these >will be linear phase systems.>So, we are able to represent an impulse response with arbitrary phase as a >sum of two linear phase systems ! (which seemed to strike me as >counter-intuitive). However, it is also appealing to see whether this >observation can help us with system design [as a LTI system can be broken >into two parallel branches - both of which shall be linear phase].I actually couldn't sort out exaclty what you were saying here until a couple of the other folks responded. Note that when two vectors are added together with tightly-controlled phases, the phase of the summed vector depends a LOT on the relative magnitudes of the vectors as well. So the phase response of the input isn't the only thing you need to know to control the phase response of the output.>Is the above argument correct ? >Can it be used to achieve better system analysis ? Any feedback/ideas on >this would be highly educative.As I just mentioned I think you missed a significant piece of the puzzle, which is that the magnitude responses have a lot to do with the output phase when two vectors are added together. Also, FWIW, thanks for identifying yourself as a student and the details of the context of your question (e.g., junior undergrad). I think it was good that you also outlined your thought and asked "is this right"? Doing homework on-line isn't tolerated well here, but asking questions so that you can learn and do your own homework is encouraged. Sometimes the line gets blurred, but I think your post was a good example of the right way to do it. Eric Jacobsen Minister of Algorithms Abineau Communications http://www.ericjacobsen.org Blog: http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php
Reply by ●February 28, 20092009-02-28
Vladimir Vassilevsky <antispam_bogus@hotmail.com> writes:> If you add the outputs of the two linear phase systems, the result can > be anything. > > I give you an advice: use less of pompous words like "highly > desirable", "in general", "system analysis" and such. Those words only > make you look more stupid then you actually are. > > VLVFor all your technical brilliance, Vladimir, you're an idiot when it comes to human interaction. This person is attempting to use the terminology he is being taught and perform the difficult task of assimilating these notions and synthesizing understanding. I'd say from this post he's one of the most promising students I've noticed on comp.dsp in years. He reminds me of Julius Kusuma several years back. --Randy> > > noisysignal wrote: >> Hi, >> >> I am a junior undergrad studying a course on DSP. We have recently been >> introduced to linear phase sytems. It seems that we would very much like >> the LTI systems we design to have a zero/linear phase which we can account >> for appropriately. In other words, a non-linear phase response is highly >> undesirable in system design. >> >> Secondly, We have studied that an impulse response that exhibits Even or >> Odd symmetry can be shown to display a Generalized Linear Phase >> Response. >> >> Now, consider any arbitrary impulse response h[n], which may have any >> phase response (in general, non linear). Now it is trivial to decompose >> this into a sum of h1[n](even) + h2[n](odd). It also follows that these >> will be linear phase systems. >> >> So, we are able to represent an impulse response with arbitrary phase as a >> sum of two linear phase systems ! (which seemed to strike me as >> counter-intuitive). However, it is also appealing to see whether this >> observation can help us with system design [as a LTI system can be broken >> into two parallel branches - both of which shall be linear phase]. >> >> Is the above argument correct ? >> Can it be used to achieve better system analysis ? Any feedback/ideas on >> this would be highly educative. >> >> Thank you >> >>-- % 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://www.digitalsignallabs.com
Reply by ●February 28, 20092009-02-28
On Feb 28, 5:29�pm, "noisysignal" <karti...@gmail.com> wrote:> > I am a junior undergrad studying a course on DSP. We have recently been > introduced to linear phase sytems. It seems that we would very much like > the LTI systems we design to have a zero/linear phase which we can account > for appropriately. In other words, a non-linear phase response is highly > undesirable in system design.it depends on what it is. it's not always undesirable.> Secondly, We have studied that an impulse response that exhibits Even or > Odd symmetry can be shown to display a Generalized Linear Phase Response. > > Now, consider any arbitrary impulse response h[n], which may have any > phase response (in general, non linear). Now it is trivial to decompose > this into a sum of h1[n](even) + h2[n](odd). It also follows that these > will be linear phase systems.but will they be causal? causality might not simply be desirable, it might be all you have.> So, we are able to represent an impulse response with arbitrary phase as a > sum of two linear phase systems ! (which seemed to strike me as > counter-intuitive). However, it is also appealing to see whether this > observation can help us with system design [as a LTI system can be broken > into two parallel branches - both of which shall be linear phase]. > > Is the above argument correct ?if causality is not a problem.> Can it be used to achieve better system analysis ? Any feedback/ideas on > this would be highly educative.there was a semester a while ago where i was (as an adjuct, i wasn't regular faculty) teaching a Systems and Signals course, using the text of the same title by Oppenhiem and Wilsky. they did some formal introduction (along with some formal notation, that i can't remember) about breaking functions into odd and even components. i didn't think it was particularly useful, but there are some theorems that apply. even and real functions have completely real Fourier Transforms. odd (and real) functions have completely imaginary FTs. when you fiddle around with causal functions, Hilbert Transforms, and the "analytic signal", remembering these even/odd properties have some utility. otherwise, i wouldn't carve out too much real estate in your brain for it. there's a lot of other things to learn. r b-j
Reply by ●February 28, 20092009-02-28
Tim Wescott wrote:> On Sat, 28 Feb 2009 19:56:12 -0500, Jerry Avins wrote: > >> noisysignal wrote: >>> Hi, >>> >>> I am a junior undergrad studying a course on DSP. We have recently been >>> introduced to linear phase sytems. It seems that we would very much >>> like the LTI systems we design to have a zero/linear phase which we can >>> account for appropriately. In other words, a non-linear phase response >>> is highly undesirable in system design. >> I don't know about "highly". Engineering involves compromise, and linear >> phase usually is accompanied by less-than-minimum delay. Often, reducing >> delay is more important than maintaining linear phase. > > (snip) > > Specifically, non-minimum phase elements in a feedback loop have > exceedingly deleterious effects on stability and error sensitivity. It's > a rare case when a designer would intentionally add a non-minimum-phase > element -- much less something as high-delay as a linear phase element -- > into a feedback loop.Well, as I wrote, engineering involves compromise. Sometimes, the simplicity of nearly-minimum phase works out better in digital systems. The OP may not realize that minimum delay and minimum phase go together. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●March 1, 20092009-03-01
On Feb 28, 5:29�pm, "noisysignal" <karti...@gmail.com> wrote:> Hi, > > I am a junior undergrad studying a course on DSP. We have recently been > introduced to linear phase sytems. It seems that we would very much like > the LTI systems we design to have a zero/linear phase which we can account > for appropriately. In other words, a non-linear phase response is highly > undesirable in system design. > > Secondly, We have studied that an impulse response that exhibits Even or > Odd symmetry can be shown to display a Generalized Linear Phase Response. > > Now, consider any arbitrary impulse response h[n], which may have any > phase response (in general, non linear). Now it is trivial to decompose > this into a sum of h1[n](even) + h2[n](odd). It also follows that these > will be linear phase systems. > > So, we are able to represent an impulse response with arbitrary phase as a > sum of two linear phase systems ! (which seemed to strike me as > counter-intuitive). However, it is also appealing to see whether this > observation can help us with system design [as a LTI system can be broken > into two parallel branches - both of which shall be linear phase]. > > Is the above argument correct ? > Can it be used to achieve better system analysis ? Any feedback/ideas on > this would be highly educative. > > Thank youGreat post! The argument you gave is absolutely right. But, the real problem is how can we achieve overall linear phase. Say, you decompose the original filter into two linear phase filters in parallel, one with group delay a1, the other with group delay a2. One way you might think of to achieve overall linear phase is to add delays |a1-a2| (suppose this is an integer) to one of the branch. Now it seemed that the problem is solved and you end up with a linear phase filter with group delay max(a1, a2). However, there is a catch. As you added a delay to one of the branch, the magnitude response will change accordingly, which is not what you want. I have not had the chance to design any filters used in real world. But, my guess is you probably would use linear phase filter at the first place. (Someone please correct me if I am wrong.)
