I'm reading about the zero-forcing equalizer solution for ISI in [sklar], p. 155, where he says, For such an equalizer with finite length, the peak distortion is guaranteed to be minimized only if the eye pattern is initially open. Huh? I thought the whole idea of ZFE is to open up the eye pattern! This makes no sense to me. Perhaps what they are trying to say is that, if the "memory" in the ISI extends beyond the equalizer length, then the zero-forcing solution is not going to minimize the distortion. Clarifications welcome. --Randy @BOOK{sklar, title = "{Digital Communications}", author = "{Bernard~Sklar}", publisher = "Prentice Hall P T R", edition = "second", year = "2001"} -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com

# zero-forcing equalizer statement in Sklar

Started by ●October 11, 2013

Reply by ●October 12, 20132013-10-12

On Fri, 11 Oct 2013 22:38:49 -0400, Randy Yates wrote:> I'm reading about the zero-forcing equalizer solution for ISI in > [sklar], p. 155, where he says, > > For such an equalizer with finite length, the peak distortion is > guaranteed to be minimized only if the eye pattern is initially open. > > Huh? I thought the whole idea of ZFE is to open up the eye pattern! This > makes no sense to me. > > Perhaps what they are trying to say is that, if the "memory" in the ISI > extends beyond the equalizer length, then the zero-forcing solution is > not going to minimize the distortion. > > Clarifications welcome. > > --Randy > > @BOOK{sklar, > title = "{Digital Communications}", > author = "{Bernard~Sklar}", > publisher = "Prentice Hall P T R", > edition = "second", > year = "2001"}I don't know a whole bunch about the topic, but if by "peak distortion" he means the obvious -- that being the worst-case distortion anywhere in relation to the bit -- then I don't see how ZFE directly fixes that. Somehow it sounds more like a rule of thumb than a strict mathematical treatment. My (very sparse) understanding of ZFE has it defined in frequency-domain terms, while the eye pattern and the peak distortion are both time-domain phenomena. In my experience it's not always easy to draw strict parallels between the frequency domain and time domain. Did he offer any proof, or is that statement just tossed out for you to eat whole, without salt? -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com

Reply by ●October 12, 20132013-10-12

On Friday, October 11, 2013 7:38:49 PM UTC-7, Randy Yates wrote:> I'm reading about the zero-forcing equalizer solution for ISI in > [sklar], p. 155, where he says, > > For such an equalizer with finite length, the peak distortion is > guaranteed to be minimized only if the eye pattern is initially open. > > Huh? I thought the whole idea of ZFE is to open up the eye pattern! > This makes no sense to me. > ...> Clarifications welcome. > --RandyAdaptation of the ZF equalizer requires knowledge of the channel data. This can come from a known training sequence, or in the blind case, from the detector providing enough correct detections that the adaptation can converge. The eye pattern -is- improved, but it won't converge in the blind case starting from nothing. I think that the "peak distortion" refers to the amount of ISI, not a time waveform. Dale B. Dalrymple

Reply by ●October 14, 20132013-10-14

>On Friday, October 11, 2013 7:38:49 PM UTC-7, Randy Yates wrote: >> I'm reading about the zero-forcing equalizer solution for ISI in >> [sklar], p. 155, where he says, >>=20 >> For such an equalizer with finite length, the peak distortion is >> guaranteed to be minimized only if the eye pattern is initially open. >>=20 >> Huh? I thought the whole idea of ZFE is to open up the eye pattern! >> This makes no sense to me.=20 >> ... > >> Clarifications welcome. >> --Randy >Randy, you are right about a confusion present there. Actually eye pattern is not as much as important as compared to *wehre* this pattern is being monitored. Peak distortion is defined as sum of absolute values of non-zero indexed fininte number of channel taps (including the equalizer filtering). Now since there will always be some residual ISI in this case (equalizer cannot correct a 'channel' longer than its own length in general), the question is to find the optimum coefficients in this finite length case. LMS type solution to find the optimum can work here, but whether ZF will find the optimum solution here or not, is not known. The only condition in which optimum solution is given by ZF is that the eye is open *prior* to the equalizer, which means peak distotrtion is less than unity. Only in this case, ZF (middle tap = 1, rest 0) gives the optimum solution. _____________________________ Posted through www.DSPRelated.com

Reply by ●October 14, 20132013-10-14

>On Fri, 11 Oct 2013 22:38:49 -0400, Randy Yates wrote: > >> I'm reading about the zero-forcing equalizer solution for ISI in >> [sklar], p. 155, where he says, >> >> For such an equalizer with finite length, the peak distortion is >> guaranteed to be minimized only if the eye pattern is initially open. >> >> Huh? I thought the whole idea of ZFE is to open up the eye pattern!This>> makes no sense to me. >> >> Perhaps what they are trying to say is that, if the "memory" in the ISI >> extends beyond the equalizer length, then the zero-forcing solution is >> not going to minimize the distortion. >> >> Clarifications welcome. >> >> --Randy >> >> @BOOK{sklar, >> title = "{Digital Communications}", >> author = "{Bernard~Sklar}", >> publisher = "Prentice Hall P T R", >> edition = "second", >> year = "2001"} > >I don't know a whole bunch about the topic, but if by "peak distortion" >he means the obvious -- that being the worst-case distortion anywhere in >relation to the bit -- then I don't see how ZFE directly fixes that. > >Somehow it sounds more like a rule of thumb than a strict mathematical >treatment. My (very sparse) understanding of ZFE has it defined in >frequency-domain terms, while the eye pattern and the peak distortion are>both time-domain phenomena. In my experience it's not always easy to >draw strict parallels between the frequency domain and time domain. > >Did he offer any proof, or is that statement just tossed out for you to >eat whole, without salt? > >-- >Tim Wescott >Control system and signal processing consulting >www.wescottdesign.com >"In my experience it's not always easy to draw strict parallels between the frequency domain and time domain.". Tim, can you please explain and cite some example/s regarding this sentence. It's important for me because I always try to picture everything in both domains. Thanks. _____________________________ Posted through www.DSPRelated.com

Reply by ●October 14, 20132013-10-14

"commsignal" <58672@dsprelated> writes:>>On Friday, October 11, 2013 7:38:49 PM UTC-7, Randy Yates wrote: >>> I'm reading about the zero-forcing equalizer solution for ISI in >>> [sklar], p. 155, where he says, >>>=20 >>> For such an equalizer with finite length, the peak distortion is >>> guaranteed to be minimized only if the eye pattern is initially open. >>>=20 >>> Huh? I thought the whole idea of ZFE is to open up the eye pattern! >>> This makes no sense to me.=20 >>> ... >> >>> Clarifications welcome. >>> --Randy >> > > Randy, you are right about a confusion present there. Actually eye pattern > is not as much as important as compared to *wehre* this pattern is being > monitored.I presumed that in this context we are assuming perfect timing.> Peak distortion is defined as sum of absolute values of non-zero indexed > fininte number of channel taps (including the equalizer filtering).Well that's good to know.> Now since there will always be some residual ISI in this case > (equalizer cannot correct a 'channel' longer than its own length in > general), the question is to find the optimum coefficients in this > finite length case. LMS type solution to find the optimum can work > here, but whether ZF will find the optimum solution here or not, is > not known.So what you're saying is that Sklar's statement is meant to be taken under the assumption that the equalizer length is NOT great enough to cover the ISI memory? If so, that makes a lot more sense - thanks much! (It would have been nice for Sklar to be more explicit about that...)> The only condition in which optimum solution is given by ZF is that the eye > is open *prior* to the equalizer, which means peak distotrtion is less than > unity. Only in this case, ZF (middle tap = 1, rest 0) gives the optimum > solution.You seem to be implying now with this last paragraph that, even if the equalizer IS long enough to handle the ISI memory, we are still not guaranteed an optimum solution with the ZFE. Is that correct? If so, why not, assuming we have samples that are a) noiseless and b) generated by a random sequence. And aren't we being a little loose here with the word "open"? An eye pattern isn't either "open" or "not open," is it? -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com

Reply by ●October 14, 20132013-10-14

Randy Yates <yates@digitalsignallabs.com> writes:> "commsignal" <58672@dsprelated> writes: > >>>On Friday, October 11, 2013 7:38:49 PM UTC-7, Randy Yates wrote: >>>> I'm reading about the zero-forcing equalizer solution for ISI in >>>> [sklar], p. 155, where he says, >>>>=20 >>>> For such an equalizer with finite length, the peak distortion is >>>> guaranteed to be minimized only if the eye pattern is initially open. >>>>=20 >>>> Huh? I thought the whole idea of ZFE is to open up the eye pattern! >>>> This makes no sense to me.=20 >>>> ... >>> >>>> Clarifications welcome. >>>> --Randy >>> >> >> Randy, you are right about a confusion present there. Actually eye pattern >> is not as much as important as compared to *wehre* this pattern is being >> monitored. > > I presumed that in this context we are assuming perfect timing. > >> Peak distortion is defined as sum of absolute values of non-zero indexed >> fininte number of channel taps (including the equalizer filtering). > > Well that's good to know. > >> Now since there will always be some residual ISI in this case >> (equalizer cannot correct a 'channel' longer than its own length in >> general), the question is to find the optimum coefficients in this >> finite length case. LMS type solution to find the optimum can work >> here, but whether ZF will find the optimum solution here or not, is >> not known. > > So what you're saying is that Sklar's statement is meant to be taken > under the assumption that the equalizer length is NOT great enough to > cover the ISI memory? If so, that makes a lot more sense - thanks much! > (It would have been nice for Sklar to be more explicit about that...) > >> The only condition in which optimum solution is given by ZF is that the eye >> is open *prior* to the equalizer, which means peak distotrtion is less than >> unity. Only in this case, ZF (middle tap = 1, rest 0) gives the optimum >> solution. > > You seem to be implying now with this last paragraph that, even if the > equalizer IS long enough to handle the ISI memory, we are still not > guaranteed an optimum solution with the ZFE. Is that correct? If so, why > not, assuming we have samples that are a) noiseless and b) generated by > a random sequence. > > And aren't we being a little loose here with the word "open"? An eye > pattern isn't either "open" or "not open," is it?Also, and this may be directly related, how do we even know that the x matrix is non-singular? The ZFE assumes an inverse for x exists. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com

Reply by ●October 14, 20132013-10-14

dbd <dbd%ieee.org@gtempaccount.com> writes:> On Friday, October 11, 2013 7:38:49 PM UTC-7, Randy Yates wrote: >> I'm reading about the zero-forcing equalizer solution for ISI in >> [sklar], p. 155, where he says, >> >> For such an equalizer with finite length, the peak distortion is >> guaranteed to be minimized only if the eye pattern is initially open. >> >> Huh? I thought the whole idea of ZFE is to open up the eye pattern! >> This makes no sense to me. >> ... > >> Clarifications welcome. >> --Randy > > Adaptation of the ZF equalizer requires knowledge of the channel data.Hi Dale, Thanks for your response. By "channel data" did you mean the transmitted data? If so, then the ZFE solution formulated by Sklar does not anywhere require knowledge of the channel data.> This can come from a known training sequence, or in the blind case, > from the detector providing enough correct detections that the > adaptation can converge. The eye pattern -is- improved, but it won't > converge in the blind case starting from nothing.> I think that the "peak distortion" refers to the amount of ISI, not a > time waveform.Right, that seems to be the case. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com

Reply by ●October 14, 20132013-10-14

PS: Sklar definitely has at least one minor typo in that same section: The last sentence on p.155 begins with The sample of greatest magnitude contributing to ISI equals 0.0428, and the sum of all Then p.156 begins with The values of the equalized pulse samples... It could just be that the initial "The" on p.156 shouldn't be capitalized, but it makes me wonder if there are other errors in the area of the text. --Randy Randy Yates <yates@digitalsignallabs.com> writes:> I'm reading about the zero-forcing equalizer solution for ISI in > [sklar], p. 155, where he says, > > For such an equalizer with finite length, the peak distortion is > guaranteed to be minimized only if the eye pattern is initially open. > > Huh? I thought the whole idea of ZFE is to open up the eye pattern! > This makes no sense to me. > > Perhaps what they are trying to say is that, if the "memory" in the ISI > extends beyond the equalizer length, then the zero-forcing solution is > not going to minimize the distortion. > > Clarifications welcome. > > --Randy > > @BOOK{sklar, > title = "{Digital Communications}", > author = "{Bernard~Sklar}", > publisher = "Prentice Hall P T R", > edition = "second", > year = "2001"}-- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com

Reply by ●October 14, 20132013-10-14

"commsignal" <58672@dsprelated> writes:>>On Fri, 11 Oct 2013 22:38:49 -0400, Randy Yates wrote: >> >>> I'm reading about the zero-forcing equalizer solution for ISI in >>> [sklar], p. 155, where he says, >>> >>> For such an equalizer with finite length, the peak distortion is >>> guaranteed to be minimized only if the eye pattern is initially open. >>> >>> Huh? I thought the whole idea of ZFE is to open up the eye pattern! > This >>> makes no sense to me. >>> >>> Perhaps what they are trying to say is that, if the "memory" in the ISI >>> extends beyond the equalizer length, then the zero-forcing solution is >>> not going to minimize the distortion. >>> >>> Clarifications welcome. >>> >>> --Randy >>> >>> @BOOK{sklar, >>> title = "{Digital Communications}", >>> author = "{Bernard~Sklar}", >>> publisher = "Prentice Hall P T R", >>> edition = "second", >>> year = "2001"} >> >>I don't know a whole bunch about the topic, but if by "peak distortion" >>he means the obvious -- that being the worst-case distortion anywhere in >>relation to the bit -- then I don't see how ZFE directly fixes that. >> >>Somehow it sounds more like a rule of thumb than a strict mathematical >>treatment. My (very sparse) understanding of ZFE has it defined in >>frequency-domain terms, while the eye pattern and the peak distortion are > >>both time-domain phenomena. In my experience it's not always easy to >>draw strict parallels between the frequency domain and time domain. >> >>Did he offer any proof, or is that statement just tossed out for you to >>eat whole, without salt? >> >>-- >>Tim Wescott >>Control system and signal processing consulting >>www.wescottdesign.com >> > > "In my experience it's not always easy to draw strict parallels between the > frequency domain and time domain.". Tim, can you please explain and cite > some example/s regarding this sentence. It's important for me because I > always try to picture everything in both domains. Thanks.I'm with Tim - it's often (but not always) hard to relate characteristics in one domain to characterstics in the other. For example, try determining whether or not an impulse response is minimum phase. On the other hand, it's trivial given the Laplace (or z-) transform. But then you weren't asking me... -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com