Hello! Does anyone have references on (lowpass) filters with maximally flat magnitude at DC and minimum phase? A first look only turned up linear- phase and reduced-delay designs. I'm particularly interested in FIR right now, but also IIR. Thanks, Martin
Max-flat min-phase filters
Started by ●May 19, 2011
Reply by ●May 19, 20112011-05-19
On 05/19/2011 09:42 AM, mcode wrote:> Hello! > > Does anyone have references on (lowpass) filters with maximally flat > magnitude at DC and minimum phase? A first look only turned up linear- > phase and reduced-delay designs. I'm particularly interested in FIR > right now, but also IIR.In the continuous-time world that's a Bessel filter. I don't know if there's an equivalent IIR sampled-time filter. In FIR, a filter with a Gaussian shape may get you 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 ●May 19, 20112011-05-19
On 19 Mai, 19:00, Tim Wescott <t...@seemywebsite.com> wrote:> In FIR, a filter with a Gaussian shape may get you that.Thanks Tim, but I don't see how that's either flat or mp? Martin
Reply by ●May 19, 20112011-05-19
On May 19, 12:42�pm, mcode <mcodes...@gmx.net> wrote:> Hello! > > Does anyone have references on (lowpass) filters with maximally flat > magnitude at DC and minimum phase? A first look only turned up linear- > phase and reduced-delay designs. I'm particularly interested in FIR > right now, but also IIR. > > Thanks, > MartinMaximally flat magnitude at DC results in Butterworth filters. Maximally flat phase response at DC results in Bessel (Thompson) filters. There should be a ton of info on these - check wiki. IHTH, Clay
Reply by ●May 19, 20112011-05-19
On May 19, 12:42�pm, mcode <mcodes...@gmx.net> wrote:> Hello! > > Does anyone have references on (lowpass) filters with maximally flat > magnitude at DC and minimum phase? A first look only turned up linear- > phase and reduced-delay designs. I'm particularly interested in FIR > right now, but also IIR. > > Thanks, > MartinIf you have a linear-phase FIR filter that works for you, you can transform it into a minimum-phase filter by moving all of the zeros outside the unit circle to their reciprocals (i.e. if you have a zero at z = z1, move it to z = 1/z1; there may be a conjugation required in there too, my memory is fuzzy). The result will be an FIR with the same magnitude response (so it should still be maximally flat) and a minimum-phase response. Jason
Reply by ●May 19, 20112011-05-19
mcode wrote:> Hello! > > Does anyone have references on (lowpass) filters with maximally flat > magnitude at DC and minimum phase?That's a classic definition of a Butterworth filter.> A first look only turned up linear- > phase and reduced-delay designs. I'm particularly interested in FIR > right now, but also IIR.You can certainly truncate a Butterworth impulse response at some length to make a FIR; however making a minimum phase FIR is a strange idea. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●May 19, 20112011-05-19
On 5/19/2011 12:44 PM, Vladimir Vassilevsky wrote:> > > mcode wrote: > >> Hello! >> >> Does anyone have references on (lowpass) filters with maximally flat >> magnitude at DC and minimum phase? > > That's a classic definition of a Butterworth filter. > >> A first look only turned up linear- >> phase and reduced-delay designs. I'm particularly interested in FIR >> right now, but also IIR. > > You can certainly truncate a Butterworth impulse response at some length > to make a FIR; however making a minimum phase FIR is a strange idea. > > Vladimir Vassilevsky > DSP and Mixed Signal Design Consultant > http://www.abvolt.comNot so strange. Hermann and Schussler came up with a design approach for doing it. And I was able to get a Remez exchange -based algorithm to do it for minimax stopbands. They go like this: H-S said (I paraphrase): Find a linear phase filter that meets the design criteria but with all-positive stopbands. One way to do this is to design a regular filter with single zeros and then add a constant equal to the peak of the stopband. Another way to do this is to slightly modify the Remez exchange algorithm so that the stopband objective function changes at each step - making the stopband objective function equal to half of the peak stopband error. In the first method the stopbands have to *all* have equal ripple peaks. In the second method, each stopband weight can be different and each stopband can be weighted within. Since this is a linear phase filter, each zero not on the unit circle appears in a quad of zeros (one outside the unit circle at Z, one inside the unit circle at 1/Z) and those two paired with their brothers having the negative of the imaginary parts. Take the square root of the filter transfer function. This is done by selecting half the roots (in pairs of course) and throwing out all the zeros outside the unit circle and one from each pair on the unit circle. This results in a minimum phase filter that's still equiripple. I suppose the same trick can be done on a maximally flat filter. The one thing to note is that the stopband ripples increase as the sqrt. So if they are 10^-6 they become 10^-3. This means that the "prototype" filter initially found has to be twice as good (in dB) in the stopbands than the final filter will be. I guess this method is also known as spectral factorization. Eric Jacobsen has a method that uses P-M for nonlinear phase FIR filters using P-M that's pretty cool: http://www.dspguru.com/dsp/tricks/using-parks-mcclellan-to-design-non-linear-phase-fir-filter Maybe this one is easier to use. Now, isn't "minimum-delay" synonymous with "minimum phase"? Fred
Reply by ●May 20, 20112011-05-20
Fred Marshall wrote:> On 5/19/2011 12:44 PM, Vladimir Vassilevsky wrote: > >> >> >> You can certainly truncate a Butterworth impulse response at some length >> to make a FIR; however making a minimum phase FIR is a strange idea.> Not so strange. Hermann and Schussler came up with a design approach > for doing it. And I was able to get a Remez exchange -based algorithm > to do it for minimax stopbands."How" is no question. The question is why would anyone need a minimum phase FIR.> Eric Jacobsen has a method that uses P-M for nonlinear phase FIR filters > using P-M that's pretty cool:Cool as a demo but as design method it isn't optimal in any sense.> Now, isn't "minimum-delay" synonymous with "minimum phase"?Not quite. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●May 20, 20112011-05-20
On May 20, 5:57�am, Vladimir Vassilevsky <nos...@nowhere.com> wrote:> Fred Marshall wrote:> > Now, isn't "minimum-delay" synonymous with "minimum phase"? > > Not quite.Define 'delay'. Not at all that easy if the phase response is nonlinear. 'Minimum phase' is well-defined in terms of the phase response; 'Minimum delay' is not. Rune
Reply by ●May 20, 20112011-05-20
On May 20, 12:58�am, Rune Allnor <all...@tele.ntnu.no> wrote:> On May 20, 5:57�am, Vladimir Vassilevsky <nos...@nowhere.com> wrote: > > > Fred Marshall wrote: > > > Now, isn't "minimum-delay" synonymous with "minimum phase"? > > > Not quite. > > Define 'delay'. Not at all that easy if the phase > response is nonlinear. 'Minimum phase' is well-defined > in terms of the phase response; 'Minimum delay' is not.Instead of "minimum delay" (which, having precise meaning is subject to haggling) I use "prompt" to indicate an early peak in the impulse response. Jerry -- Engineering is the art of making what you want from things you can get.
