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zero-forcing equalizer statement in Sklar

Started by Randy Yates October 11, 2013
On Monday, October 14, 2013 6:15:59 AM UTC-7, Randy Yates wrote:

> 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. >
Perhaps this would have been more clear if I had written: This can come from a known training sequence, or in the blind case, from the channel providing enough correct values of the transmitted data that the adaptation can converge. The eye pattern -is- improved, but it won't converge in the blind case starting from no correct knowledge of the transmitted data. Convergence of the adaptation of the coefficients requires that you know most of the transmitted data correctly. That is what you get from the channel itself when the eye is open. When you adapt the coefficients from the channel data with the eye open, you are using knowledge of the transmitted data. When the eye is open, the channel itself is giving you that knowledge of the transmitted data. That's the limitation: ZFE only cleans up an eye that is already open, it doesn't adapt to open up the eye from scratch. Dale B. Dalrymple
On Mon, 14 Oct 2013 13:09:19 -0400, Randy Yates wrote:

> "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...
That's one that I hadn't thought of. You can't easily look at a Bode plot and call out the settling time of a system. You can't always look at at Bode plot and know that a filter won't ring -- you can often tell if it will be bad, but you can't always tell when it will damp easily. Conversely, you can't easily go from a step response to a Bode plot (in fact, you can't easily do this numerically if there's much noise). I know there's more, but the examples aren't just rolling off my fingertips -- sorry. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
>"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...) >
It's not an assumption. When the channel includes the filtering by equalizer itself, its length will always be greater than the equalizer time span. Suppose channel length (including all Tx, Rx filterings and actual impulse response of the transmission medium) is M and equalizer taps are N, the total length of the equivalent channel is M+N-1, while the equalizer has only N tunable parameters, hence there's always some residual ISI. Why calculating distortion including filtering by equalizer. Because it's at the output of the equalizer where we want to see the close to ideal signal.
>> 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 >
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>"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. >
I hope my last post answers this as there is no assumption here. ISI memory will always be longer than the finite length ZF equalizer, as it's the output of the equalizer where ISI is being monitored.
>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?
I didn't understand what exactly you are referring to here.
>-- >Randy Yates >Digital Signal Labs >http://www.digitalsignallabs.com >
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>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
When I read Sklar, I found that there are some errors in the book here and there according to my understanding, so if you believe something to be doubtful there, try other sources. Unfortunately, there is no single book/reference which is very good about synchronization and equalization as a complete system, although for synchronization only, both Meyr and Mengali's books are great.
> >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 >
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>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.
Correct. That's why we can't equalize a channel whose input in frequency domain is not sufficiently dense with not very different magnitudes. To 'sample' the channel, ideally we should have an input signal with unity values on all frequencies. I think that's why channel sounding is done via PN sequences which are uncorrelated with different shifts in time domain (making the matrix invertible) while exhibiting good flat spectral properties. In Sklar's equation, you will see that if input was a delta function, then x matrix would be identity there, which is directly related to this concept. Even if the input is OK, and the channel is really bad at some frequencies, then inverting x will give large compensating values on those frequencies, which will enhance the noise. That's why ZF is only good for mild channels.
>-- >Randy Yates >Digital Signal Labs >http://www.digitalsignallabs.com >
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