One method of creating a quadrature signal is to pass an input signal through two parallel allpass networks, where the outputs of the two networks differ in phase by 90 degrees. Each allpass network is typically high-order to obtain reasonable accuracy in terms of holding the phase difference over frequency. One advantage of this technique compared with the more usual anti-symmetric Hilbert filter is that the number of multiplies is smaller for a given accuracy (note that the Hilbert filter is perfect for phase accuracy, but suffers from amplitude errors and becomes very long if you need a frequency range that spans a large portion of the band from 0 to PI, whereas the allpass filters are perfect for amplitude accuracy but cannot hold the phase difference at frequencies near DC and PI). The problem I have is designing the allpass filter networks to hold the 90-degree phase difference. In the past I have seen people apply non-linear optimizers to the problem. I am trying to solve it with Matlab fminsearch, but am having trouble getting a starting point that is close enough that it converges to the desired 90-degree phase difference. Has anyone seen any papers on this?. Thanks! Bob Adams

# constant 90-degree phase shift using parallel allpass filter networks

Started by ●July 10, 2008

Reply by ●July 11, 20082008-07-11

Robert Adams wrote:> One method of creating a quadrature signal is to pass an input signal > through two parallel allpass networks, where the outputs of the two > networks differ in phase by 90 degrees. Each allpass network is > typically high-order to obtain reasonable accuracy in terms of holding > the phase difference over frequency. One advantage of this technique > compared with the more usual anti-symmetric Hilbert filter is that the > number of multiplies is smaller for a given accuracy (note that the > Hilbert filter is perfect for phase accuracy, but suffers from > amplitude errors and becomes very long if you need a frequency range > that spans a large portion of the band from 0 to PI, whereas the > allpass filters are perfect for amplitude accuracy but cannot hold the > phase difference at frequencies near DC and PI). > > The problem I have is designing the allpass filter networks to hold > the 90-degree phase difference. In the past I have seen people apply > non-linear optimizers to the problem. I am trying to solve it with > Matlab fminsearch, but am having trouble getting a starting point that > is close enough that it converges to the desired 90-degree phase > difference. > > Has anyone seen any papers on this?.Hi Bob You could try FDLS for this (at least to get sensible starting values for the coefficients). Using FDLS for phase compensation was discussed in the thread "Allpass filter with specific phase response Options" by the FDLS inventor Greg Berchin. I'm curious: the two allpass networks you want are supposed to have a phase shift of 90� with respect to each other. However, does the output of each network have a constant phase shift? Regards, Andor

Reply by ●July 11, 20082008-07-11

"Robert Adams" <robert.adams@analog.com> wrote in message news:d0b3aa98-a31e-48a1-87ba-93cc24adabb9@r66g2000hsg.googlegroups.com...> The problem I have is designing the allpass filter networks to hold > the 90-degree phase difference. In the past I have seen people apply > non-linear optimizers to the problem. I am trying to solve it with > Matlab fminsearch, but am having trouble getting a starting point that > is close enough that it converges to the desired 90-degree phase > difference. > > Has anyone seen any papers on this?.IIRC the 90 degree IIR networks are designed analytically by lowpass transformation. The standard approximations like Chebyshev, etc. can be used. Vladimir Vassilevsky DSP and Mixed Signal Consultant www.abvolt.com

Reply by ●July 11, 20082008-07-11

On Jul 11, 2:48�am, Andor <andor.bari...@gmail.com> wrote:> Robert Adams wrote: > > One method of creating a quadrature signal is to pass an input signal > > through two parallel allpass networks, where the outputs of the two > > networks differ in phase by 90 degrees. Each allpass network is > > typically high-order to obtain reasonable accuracy in terms of holding > > the phase difference over frequency. One advantage of this technique > > compared with the more usual anti-symmetric Hilbert filter is that the > > number of multiplies is smaller for a given accuracy (note that the > > Hilbert filter is perfect for phase accuracy, but suffers from > > amplitude errors and becomes very long if you need a frequency range > > that spans a large portion of the band from 0 to PI, whereas the > > allpass filters are perfect for amplitude accuracy but cannot hold the > > phase difference at frequencies near DC and PI). > > > The problem I have is designing the allpass filter networks to hold > > the 90-degree phase difference. In the past I have seen people apply > > non-linear optimizers to the problem. I am trying to solve it with > > Matlab fminsearch, but am having trouble getting a starting point that > > is close enough that it converges to the desired 90-degree phase > > difference. > > > Has anyone seen any papers on this?. > > Hi Bob > > You could try FDLS for this (at least to get sensible starting values > for the coefficients). Using FDLS for phase compensation was discussed > in the thread "Allpass filter with specific phase response Options" by > the FDLS inventor Greg Berchin. > > I'm curious: the two allpass networks you want are supposed to have a > phase shift of 90� with respect to each other. However, does the > output of each network have a constant phase shift? > > Regards, > Andor- Hide quoted text - > > - Show quoted text -No, it does not. Each allpass network will display increasing phase shift with frequency, but if you were to plot them on the same graph the difference in phase would be 90 degreees. Bob

Reply by ●July 11, 20082008-07-11

On 11 Jul., 12:08, Robert Adams <robert.ad...@analog.com> wrote:> On Jul 11, 2:48�am, Andor <andor.bari...@gmail.com> wrote: > > > > > > > Robert Adams wrote: > > > One method of creating a quadrature signal is to pass an input signal > > > through two parallel allpass networks, where the outputs of the two > > > networks differ in phase by 90 degrees. Each allpass network is > > > typically high-order to obtain reasonable accuracy in terms of holding > > > the phase difference over frequency. One advantage of this technique > > > compared with the more usual anti-symmetric Hilbert filter is that the > > > number of multiplies is smaller for a given accuracy (note that the > > > Hilbert filter is perfect for phase accuracy, but suffers from > > > amplitude errors and becomes very long if you need a frequency range > > > that spans a large portion of the band from 0 to PI, whereas the > > > allpass filters are perfect for amplitude accuracy but cannot hold the > > > phase difference at frequencies near DC and PI). > > > > The problem I have is designing the allpass filter networks to hold > > > the 90-degree phase difference. In the past I have seen people apply > > > non-linear optimizers to the problem. I am trying to solve it with > > > Matlab fminsearch, but am having trouble getting a starting point that > > > is close enough that it converges to the desired 90-degree phase > > > difference. > > > > Has anyone seen any papers on this?. > > > Hi Bob > > > You could try FDLS for this (at least to get sensible starting values > > for the coefficients). Using FDLS for phase compensation was discussed > > in the thread "Allpass filter with specific phase response Options" by > > the FDLS inventor Greg Berchin. > > > I'm curious: the two allpass networks you want are supposed to have a > > phase shift of 90� with respect to each other. However, does the > > output of each network have a constant phase shift? > > > Regards, > > Andor- Hide quoted text - > > > - Show quoted text - > > No, it does not. > Each allpass network will display increasing phase shift with > frequency, but if you were to plot them on the same graph the > difference in phase would be 90 degreees.Ah, ok. So you generate a quadrature pair, but neither of the outputs is equal to the input ...

Reply by ●July 11, 20082008-07-11

On Jul 11, 6:12�am, Andor <andor.bari...@gmail.com> wrote:> On 11 Jul., 12:08, Robert Adams <robert.ad...@analog.com> wrote: > > > > > > > On Jul 11, 2:48�am, Andor <andor.bari...@gmail.com> wrote: > > > > Robert Adams wrote: > > > > One method of creating a quadrature signal is to pass an input signal > > > > through two parallel allpass networks, where the outputs of the two > > > > networks differ in phase by 90 degrees. Each allpass network is > > > > typically high-order to obtain reasonable accuracy in terms of holding > > > > the phase difference over frequency. One advantage of this technique > > > > compared with the more usual anti-symmetric Hilbert filter is that the > > > > number of multiplies is smaller for a given accuracy (note that the > > > > Hilbert filter is perfect for phase accuracy, but suffers from > > > > amplitude errors and becomes very long if you need a frequency range > > > > that spans a large portion of the band from 0 to PI, whereas the > > > > allpass filters are perfect for amplitude accuracy but cannot hold the > > > > phase difference at frequencies near DC and PI). > > > > > The problem I have is designing the allpass filter networks to hold > > > > the 90-degree phase difference. In the past I have seen people apply > > > > non-linear optimizers to the problem. I am trying to solve it with > > > > Matlab fminsearch, but am having trouble getting a starting point that > > > > is close enough that it converges to the desired 90-degree phase > > > > difference. > > > > > Has anyone seen any papers on this?. > > > > Hi Bob > > > > You could try FDLS for this (at least to get sensible starting values > > > for the coefficients). Using FDLS for phase compensation was discussed > > > in the thread "Allpass filter with specific phase response Options" by > > > the FDLS inventor Greg Berchin. > > > > I'm curious: the two allpass networks you want are supposed to have a > > > phase shift of 90� with respect to each other. However, does the > > > output of each network have a constant phase shift? > > > > Regards, > > > Andor- Hide quoted text - > > > > - Show quoted text - > > > No, it does not. > > Each allpass network will display increasing phase shift with > > frequency, but if you were to plot them on the same graph the > > difference in phase would be 90 degreees. > > Ah, ok. So you generate a quadrature pair, but neither of the outputs > is equal to the input ...- Hide quoted text - > > - Show quoted text -Yes, that's right. Bob

Reply by ●July 11, 20082008-07-11

On Jul 11, 2:48�am, Andor <andor.bari...@gmail.com> wrote:> You could try FDLS for this (at least to get sensible starting values > for the coefficients). Using FDLS for phase compensation was > discussed in the thread "Allpass filter with specific phase response > Options" by the FDLS inventor Greg Berchin.I experimented a little bit with FDLS this morning, and with a "reasonable" filter size I can get 0dB��dB and 90��1� from approximately Fs/40 to just under Fs/2. Bob, would that satisfy your performance requirements? Greg Berchin

Reply by ●July 11, 20082008-07-11

On Jul 11, 7:55�am, Greg Berchin <gberc...@sentientscience.com> wrote:> On Jul 11, 2:48�am, Andor <andor.bari...@gmail.com> wrote: > > > You could try FDLS for this (at least to get sensible starting values > > for the coefficients). Using FDLS for phase compensation was > > discussed in the thread "Allpass filter with specific phase response > > Options" by the FDLS inventor Greg Berchin. > > I experimented a little bit with FDLS this morning, and with a > "reasonable" filter size I can get 0dB��dB and 90��1� from > approximately Fs/40 to just under Fs/2. �Bob, would that satisfy your > performance requirements? > > Greg BerchinGreg Thanks for running this through FDLS. I actually need much better accuracy. In the end I am using this for making a frequency shifter, so quadrature phase error will result in leakage into the "Lower sideband" (sorry, my past history as a ham radio operator causes me to use strange terminology!). These leakage components should be < -80 dB. I haven't gone through the math but I assume roughly that this corresponds to a phase error on the order of 90 degrees/10000. I am not that familiar with FDLS, perhaps you can give me an overview? Bob

Reply by ●July 11, 20082008-07-11

On Jul 11, 5:03�am, "Vladimir Vassilevsky" <antispam_bo...@hotmail.com> wrote:> "Robert Adams" <robert.ad...@analog.com> wrote in message > > news:d0b3aa98-a31e-48a1-87ba-93cc24adabb9@r66g2000hsg.googlegroups.com... > > > The problem I have is designing the allpass filter networks to hold > > the 90-degree phase difference. In the past I have seen people apply > > non-linear optimizers to the problem. I am trying to solve it with > > Matlab fminsearch, but am having trouble getting a starting point that > > is close enough that it converges to the desired 90-degree phase > > difference. > > > Has anyone seen any papers on this?. > > IIRC the 90 degree IIR networks are designed analytically by lowpass > transformation. The standard approximations like Chebyshev, etc. can be > used. > > Vladimir Vassilevsky > DSP and Mixed Signal Consultantwww.abvolt.comCan you elaborate a little more, or provide a reference? This sounds interesting. Bob

Reply by ●July 11, 20082008-07-11

On Jul 11, 7:55�am, Greg Berchin <gberc...@sentientscience.com> wrote:> On Jul 11, 2:48�am, Andor <andor.bari...@gmail.com> wrote: > > > You could try FDLS for this (at least to get sensible starting values > > for the coefficients). Using FDLS for phase compensation was > > discussed in the thread "Allpass filter with specific phase response > > Options" by the FDLS inventor Greg Berchin. > > I experimented a little bit with FDLS this morning, and with a > "reasonable" filter size I can get 0dB��dB and 90��1� from > approximately Fs/40 to just under Fs/2. �Bob, would that satisfy your > performance requirements? > > Greg BerchinGreg Never mind, I just read your paper and realized that FDLS stands for frequency-domain least-squares; I thought at first it was the name of a program that was commercially available. I also note that you mention amplitude and phase accuracy, so I assume the FDLS method cannot be constrained to produce allpass filters (otherwise the amplitude error would be zero). Bob