Hello Everyone, Can anyone point me the definition for a transversal equalizer? Some books only treat linear and adaptive equalizers, and some talk about transversal equalizers. It seems that all linear equalizer are transversal equalizers but I not convinced that this is true and I do not know if the opposite holds as well!? Very confused!! Pointers to references are also appreciated. Looking forward to hearing from you, Huke

# Definition for transversal equalizers

Started by ●February 20, 2005

Posted by ●February 20, 2005

"huke" <hugo.harada@gmail.com> wrote in message news:1108910094.822203.226820@z14g2000cwz.googlegroups.com...> Hello Everyone, > > Can anyone point me the definition for a transversal equalizer? Some > books only treat linear and adaptive equalizers, and some talk about > transversal equalizers. It seems that all linear equalizer are > transversal equalizers but I not convinced that this is true and I do > not know if the opposite holds as well!? Very confused!! > > Pointers to references are also appreciated. > > Looking forward to hearing from you, > > Huke >Before there were truly discrete, digital filters, there were continuous time or analog filters (such as delay lines) that were "tapped" at discrete points. The signal input appeared in delayed versions: it transversed in time (i.e. "went across") as you viewed or accessed the delay line taps at one instant of time. From an anlysis point of view, a transversal filter and a FIR filter can be viewed as the same thing. The frequency response is periodic in the inverse of the (assumed regular) tap separation in time. A transversal equalizer is a transversal filter (or FIR filter) that usually has adjustable weights / coefficients in the filter so that the equalization can be adjusted for best performance. In the latest IEEE Spectrum there is an article by or about Bernie Widrow who pioneered LMS adaptive filters. The article mentions Robert Lucky's work at Bell Labs that preceded on adaptive equalizers. There are many ways and many objective criterion for doing "adaptive"..... So, just substitute "FIR" for "transversal" and you should be OK. And, yes, usually linear - although modifying the coefficients very rapidly could cause the filter to become "nonlinear" during the change. Fred

Posted by ●February 20, 2005

Posted by ●February 21, 2005

"huke" <hugo.harada@gmail.com> writes:> Hello Everyone, > > Can anyone point me the definition for a transversal equalizer? Some > books only treat linear and adaptive equalizers, and some talk about > transversal equalizers. It seems that all linear equalizer are > transversal equalizers but I not convinced that this is true and I do > not know if the opposite holds as well!? Very confused!! > > Pointers to references are also appreciated. > > Looking forward to hearing from you, > > HukeHi Huke, Here is the way I would categorize these terms: Equalizers can be either adaptive or fixed. Within either of those categories, you can have the following subdivisions: linear non-linear | --------------------- | | transversal (FIR) non-transversal (IIR) See, e.g., Proakis' "Digital Communications" section "Linear Equalization." -- % Randy Yates % "The dreamer, the unwoken fool - %% Fuquay-Varina, NC % in dreams, no pain will kiss the brow..." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Eldorado Overture', *Eldorado*, ELO http://home.earthlink.net/~yatescr

Posted by ●February 21, 2005

On Mon, 21 Feb 2005 14:08:08 GMT, Randy Yates <yates@ieee.org> wrote:>"huke" <hugo.harada@gmail.com> writes: > >> Hello Everyone, >> >> Can anyone point me the definition for a transversal equalizer? Some >> books only treat linear and adaptive equalizers, and some talk about >> transversal equalizers. It seems that all linear equalizer are >> transversal equalizers but I not convinced that this is true and I do >> not know if the opposite holds as well!? Very confused!! >> >> Pointers to references are also appreciated. >> >> Looking forward to hearing from you, >> >> Huke > >Hi Huke, > >Here is the way I would categorize these terms: Equalizers can be >either adaptive or fixed. Within either of those categories, you >can have the following subdivisions: > > linear non-linear > | > --------------------- > | | >transversal (FIR) non-transversal (IIR) > >See, e.g., Proakis' "Digital Communications" section "LinearEqualization." Note that transversal implies FIR, but FIR does not imply transversal. Regards, Allan

Posted by ●February 21, 2005

"Allan Herriman" <allan.herriman.hates.spam@ctam.com.au.invalid> wrote in message news:dprj1198ics0h3co4tq6qs31h45vpo14q1@4ax.com...> On Mon, 21 Feb 2005 14:08:08 GMT, Randy Yates <yates@ieee.org> wrote: > >>"huke" <hugo.harada@gmail.com> writes: >> >>> Hello Everyone, >>> >>> Can anyone point me the definition for a transversal equalizer? Some >>> books only treat linear and adaptive equalizers, and some talk about >>> transversal equalizers. It seems that all linear equalizer are >>> transversal equalizers but I not convinced that this is true and I do >>> not know if the opposite holds as well!? Very confused!! >>> >>> Pointers to references are also appreciated. >>> >>> Looking forward to hearing from you, >>> >>> Huke >> >>Hi Huke, >> >>Here is the way I would categorize these terms: Equalizers can be >>either adaptive or fixed. Within either of those categories, you >>can have the following subdivisions: >> >> linear non-linear >> | >> --------------------- >> | | >>transversal (FIR) non-transversal (IIR) >> >>See, e.g., Proakis' "Digital Communications" section "Linear>>Equalization." > > > Note that transversal implies FIR, but FIR does not imply transversal.Alan, Can you explain further? I don't see a difference. For example, both have FIR so I agree that transversal implies FIR. So, there must be something that you attribute to FIR that somehow makes it non-transversal. The application of coefficients is on samples or a continuum (either one) that transeverses (goes across) time (or whatever the sample domain might be - such as space/distance). Whether the data is continuous or discrete samples doesn't change the nature of FIR. But, I hasten to acknowledge that *most* of the time we refer colloquially to "FIR" as a filter that operates on discrete (and quantized) samples. Fred

Posted by ●February 21, 2005

On Mon, 21 Feb 2005 10:14:44 -0800, "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote:> >"Allan Herriman" <allan.herriman.hates.spam@ctam.com.au.invalid>wrote in>message news:dprj1198ics0h3co4tq6qs31h45vpo14q1@4ax.com... >> On Mon, 21 Feb 2005 14:08:08 GMT, Randy Yates <yates@ieee.org>wrote:>> >>>"huke" <hugo.harada@gmail.com> writes: >>> >>>> Hello Everyone, >>>> >>>> Can anyone point me the definition for a transversal equalizer?Some>>>> books only treat linear and adaptive equalizers, and some talkabout>>>> transversal equalizers. It seems that all linear equalizer are >>>> transversal equalizers but I not convinced that this is true and Ido>>>> not know if the opposite holds as well!? Very confused!! >>>> >>>> Pointers to references are also appreciated. >>>> >>>> Looking forward to hearing from you, >>>> >>>> Huke >>> >>>Hi Huke, >>> >>>Here is the way I would categorize these terms: Equalizers can be >>>either adaptive or fixed. Within either of those categories, you >>>can have the following subdivisions: >>> >>> linear non-linear >>> | >>> --------------------- >>> | | >>>transversal (FIR) non-transversal (IIR) >>> >>>See, e.g., Proakis' "Digital Communications" section"Linear>>>Equalization." >> >> >> Note that transversal implies FIR, but FIR does not imply transversal. > >Alan, > >Can you explain further? I don't see a difference. > >For example, both have FIR so I agree that transversal implies FIR. > >So, there must be something that you attribute to FIR that somehow makes it >non-transversal. The application of coefficients is on samples or a >continuum (either one) that transeverses (goes across) time (or whatever the >sample domain might be - such as space/distance). > >Whether the data is continuous or discrete samples doesn't change the nature >of FIR. >But, I hasten to acknowledge that *most* of the time we refer colloquially >to "FIR" as a filter that operates on discrete (and quantized)samples. Hi Fred, I assume that 'transversal' is a property of a filter implementation, whereas 'FIR' is a property of the impulse response of a filter. I'm using your definition of transversal (from an earlier post in this thread), which I interpreted as meaning that the output would be formed by adding weighted time delayed copies of the input signal. To prove my point that FIR does not imply transversal, we need to find a filter that is both (1) FIR, and (2) not transversal. A boxcar averager (as used in a CIC) is an example of such a filter, as it has a recursive implementation. y[n] = y[n-1] + x[n] - x[n-k], for some constant k, and y[-1] = 0. This gives the same (finite) impulse response as this transversal filter: n y[n] = sum x[n] n-k+1 Regards, Allan

Posted by ●February 21, 2005

Allan Herriman <allan.herriman.hates.spam@ctam.com.au.invalid> writes:> [...] > To prove my point that FIR does not imply transversal, we need to find > a filter that is both (1) FIR, and (2) not transversal. > > A boxcar averager (as used in a CIC) is an example of such a filter, > as it has a recursive implementation. > > y[n] = y[n-1] + x[n] - x[n-k], for some constant k, and y[-1] = 0. > > This gives the same (finite) impulse response as this transversal > filter: > n > y[n] = sum x[n] > n-k+1This is not equivalent to the recursive implementation given above. I think you meant to write something like n y[n] = sum x[m]. m = n-k+1 However, I agree with your point. -- % Randy Yates % "Remember the good old 1980's, when %% Fuquay-Varina, NC % things were so uncomplicated?" %%% 919-577-9882 % 'Ticket To The Moon' %%%% <yates@ieee.org> % *Time*, Electric Light Orchestra http://home.earthlink.net/~yatescr

Posted by ●February 22, 2005

On Tue, 22 Feb 2005 03:34:13 GMT, Randy Yates <yates@ieee.org> wrote:>Allan Herriman <allan.herriman.hates.spam@ctam.com.au.invalid> writes: >> [...] >> To prove my point that FIR does not imply transversal, we need to find >> a filter that is both (1) FIR, and (2) not transversal. >> >> A boxcar averager (as used in a CIC) is an example of such a filter, >> as it has a recursive implementation. >> >> y[n] = y[n-1] + x[n] - x[n-k], for some constant k, and y[-1] = 0. >> >> This gives the same (finite) impulse response as this transversal >> filter: >> n >> y[n] = sum x[n] >> n-k+1 > >This is not equivalent to the recursive implementation given above. I >think you meant to write something like > > n > y[n] = sum x[m]. > m = n-k+1Yes indeed! My brain starts to fade around six in the morning, at which time the little pixies help me with my news postings. Regards, Allan

Posted by ●February 22, 2005

"Allan Herriman" <allan.herriman.hates.spam@ctam.com.au.invalid> wrote in message news:ehak119ie6q0tlp7k9gka22uub34mrtkvl@4ax.com...> On Mon, 21 Feb 2005 10:14:44 -0800, "Fred Marshall" > <fmarshallx@remove_the_x.acm.org> wrote: > >> >>"Allan Herriman"<allan.herriman.hates.spam@ctam.com.au.invalid> wrote in>>message news:dprj1198ics0h3co4tq6qs31h45vpo14q1@4ax.com... >>> On Mon, 21 Feb 2005 14:08:08 GMT, Randy Yates <yates@ieee.org>wrote:>>> >>>>"huke" <hugo.harada@gmail.com> writes: >>>> >>>>> Hello Everyone, >>>>> >>>>> Can anyone point me the definition for a transversal equalizer?Some>>>>> books only treat linear and adaptive equalizers, and some talkabout>>>>> transversal equalizers. It seems that all linear equalizer are >>>>> transversal equalizers but I not convinced that this is trueand I do>>>>> not know if the opposite holds as well!? Very confused!! >>>>> >>>>> Pointers to references are also appreciated. >>>>> >>>>> Looking forward to hearing from you, >>>>> >>>>> Huke >>>> >>>>Hi Huke, >>>> >>>>Here is the way I would categorize these terms: Equalizers can be >>>>either adaptive or fixed. Within either of those categories, you >>>>can have the following subdivisions: >>>> >>>> linear non-linear >>>> | >>>> --------------------- >>>> | | >>>>transversal (FIR) non-transversal (IIR) >>>> >>>>See, e.g., Proakis' "Digital Communications" section"Linear>>>>Equalization." >>> >>> >>> Note that transversal implies FIR, but FIR does not imply transversal. >> >>Alan, >> >>Can you explain further? I don't see a difference. >> >>For example, both have FIR so I agree that transversal implies FIR. >> >>So, there must be something that you attribute to FIR that somehow makes >>it >>non-transversal. The application of coefficients is on samples or a >>continuum (either one) that transeverses (goes across) time (or whatever >>the >>sample domain might be - such as space/distance). >> >>Whether the data is continuous or discrete samples doesn't change the >>nature >>of FIR. >>But, I hasten to acknowledge that *most* of the time we refer colloquially >>to "FIR" as a filter that operates on discrete (and quantized)samples.> > Hi Fred, > > I assume that 'transversal' is a property of a filter implementation, > whereas 'FIR' is a property of the impulse response of a filter. > > I'm using your definition of transversal (from an earlier post in this > thread), which I interpreted as meaning that the output would be > formed by adding weighted time delayed copies of the input signal. > > To prove my point that FIR does not imply transversal, we need to find > a filter that is both (1) FIR, and (2) not transversal. > > A boxcar averager (as used in a CIC) is an example of such a filter, > as it has a recursive implementation. > > y[n] = y[n-1] + x[n] - x[n-k], for some constant k, and y[-1] = 0. > > This gives the same (finite) impulse response as this transversal > filter: > n > y[n] = sum x[n] > n-k+1 >OK - if we include recursively implemented FIRs then they aren't transversal. But, that's a special case that only applies to a small subset of FIRs. Correct but limited. I like to think of the recursive implementation of FIRs as sort of a curiosity with limited application - no matter how cool and even valuable some of those implementations may be. Let's not have the tail wag the dog. What if we turn it around and say: "In some cases a FIR can be implemented recursively and then isn't a transversal filter. However, a FIR filter can *always* be implemented as a sum of delayed inputs and thus, can always be transversal" ?? In that sense, FIR can very reasonably imply transversal. Fred

Posted by ●February 24, 2005

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:0fednQVKW-v_DoPfRVn-og@centurytel.net...> > "Max Hauser" <maxREMOVE@THIStdl.com> wrote in message > news:111oecet5h9b407@corp.supernews.com... >> "Allan Herriman" >>> >> >> Neglecting novelty items like recursive FIR implementations, there are >> other famous nonrecursive structures that also realize Finite Impulse >> Responses but lack the transversal (tapped-delay) structure. A parallel >> set of delay lines of different lengths or delays, each weighted at their >> input or output, and finally summed, is such a structure. (It has been >> useful in some delay-line technologies that had constraints, for example >> certain CCD technologies circa the 1970s.) >> >> -- Max Hauser > > In the olden days when delay lines of any appreciable length were > expensive we used a relatively wideband torsional wire acoustic delay line > over the range 100kHz to 1MHz. The line wasn't tapped. We wanted to > delay signals up to 100kHz or so and create a FIR filter sort of response. > So, we used frequency domain multiplexing on the line and used the line > multiple times: > > output at baseband at t > first pass at 150kHz downtranslate a tap out at t+T > uptranslate for: > second pass at 300kHz, downtranslate a tap out at t+2T > upstranslate for: > third pass at 450kHz, downtranslate a tap out at t+3T > > End result, one delay element of length T for a FIR filter of length 4. > So, in a way, these were different length delay lines in parallel even > though the physical line was the same one. > Note that this isn't recursive because all the signal paths are > independent / not added - well they are added in the delay line but > separated in frequency so the end result is not recursive. > > FredWell, on second thought, it seems that this implementation was more serial than parallel. Fred

Posted by ●February 24, 2005

"Max Hauser" <maxREMOVE@THIStdl.com> wrote in message news:111oecet5h9b407@corp.supernews.com...> "Allan Herriman" >> > > Neglecting novelty items like recursive FIR implementations, there are > other famous nonrecursive structures that also realize Finite Impulse > Responses but lack the transversal (tapped-delay) structure. A parallel > set of delay lines of different lengths or delays, each weighted at their > input or output, and finally summed, is such a structure. (It has been > useful in some delay-line technologies that had constraints, for example > certain CCD technologies circa the 1970s.) > > -- Max HauserIn the olden days when delay lines of any appreciable length were expensive we used a relatively wideband torsional wire acoustic delay line over the range 100kHz to 1MHz. The line wasn't tapped. We wanted to delay signals up to 100kHz or so and create a FIR filter sort of response. So, we used frequency domain multiplexing on the line and used the line multiple times: output at baseband at t first pass at 150kHz downtranslate a tap out at t+T uptranslate for: second pass at 300kHz, downtranslate a tap out at t+2T upstranslate for: third pass at 450kHz, downtranslate a tap out at t+3T End result, one delay element of length T for a FIR filter of length 4. So, in a way, these were different length delay lines in parallel even though the physical line was the same one. Note that this isn't recursive because all the signal paths are independent / not added - well they are added in the delay line but separated in frequency so the end result is not recursive. Fred

Posted by ●February 24, 2005

"Max Hauser" <maxREMOVE@THIStdl.com> wrote in message news:111oecet5h9b407@corp.supernews.com...> "Allan Herriman" >> >.......................A parallel set of delay lines of different lengths >or delays, each weighted at their input or output, and finally summed, is >such a structure. (It has been useful in some delay-line technologies that >had constraints, for example certain CCD technologies circa the 1970s.) > > -- Max HauserMax, A parallel set of delay lines *is* transversal..... Transversal doesn't say anything about "serial" only about "across" which the delay lines of different length accomplish. Fred

Posted by ●February 24, 2005

Hello Steve, Max, Allan, Fred, Randy, Jerry, Thank you for all the feedback. The answer could not be clearer! Regards, Hugo

Posted by ●February 24, 2005

Jerry Avins wrote:> Allan Herriman wrote: > >> On Wed, 23 Feb 2005 00:52:57 -0800, "Max Hauser" >> <maxREMOVE@THIStdl.com> wrote: >> >> >>> "Fred Marshall" innews:a9ednW1bu_E7jYTfRVn-iA@centurytel.net...>>> >>>> ... >>>> Before there were truly discrete, digital filters, there were >>>> continuous time or analog filters (such as delay lines) that were >>>> "tapped" at discrete points. >>> >>> >>> Not just _before_ digital filers by the way, but also after. Here >>> are some modern, monolithic continuous-time FIR examples. (Click on >>> "Images" for the figures.) >>> >>> http://tinyurl.com/548jv >>> >>> (Remember: Not all FIR filters are transversal. Not all FIR >>> filters are digital. Not all non-digital FIR filters are even >>> discrete-time.) >> >> >> >> >> A SAW (Surface Acoustic Wave) filter is another good example of a >> continuous time transversal filter. >> >> The surface acoustic waves are launched by "interdigitaltransducers">> which are just interleaved electrodes deposited on the surface of the >> quartz. The gain (& hence the weight of that 'tap') is proportional >> to the overlap between the electrodes. >> >> http://koigakubo.hitachi.co.jp/~cs/cd/eng/technical/device/saw/ >> >> Regards, >> Allan > > > SAWs are frequency selective depending on the spacing of the > electrodes relative to the speed of the wave. If a pulse is launched > down the substrate and under a set of electrodes with progressively > decreasing spacing, a chirp is produced. When that chirp is applied to > to the variable-pitch electrode of an identical device, pulses emerge > from its other electrode. Even if the chirps overlap in time, the > pulses can be distinct. There are several very useful applications. > > JerryMany pulse radars used to use that technique for generating and recompressing their chirps. It offers a somewhat limited dynamic range, though, as all SAW devices do. Bulk waves and other side effects of the generation of the surface wave are a key cause. Another is the fact that things don't stop abruptly at the edges of the inter-digital transducers. SAW devices can generally be characterised as reproducable, predictable, stable, but never really great performers. This is getting away from the original issue. The type of SAW device Allan referred to is an exact analogue of a conventional DSP FIR filter, and the same design methods apply to both. The only difference is, the DSP version doesn't require fiddle factors to compensate for the effects around the edges of the transducers. Regards, Steve

Posted by ●February 23, 2005

Allan Herriman wrote:> On Wed, 23 Feb 2005 00:52:57 -0800, "Max Hauser" > <maxREMOVE@THIStdl.com> wrote: > > >>"Fred Marshall" in news:a9ednW1bu_E7jYTfRVn-iA@centurytel.net... >> >>>... >>>Before there were truly discrete, digital filters, there werecontinuous>>>time or analog filters (such as delay lines) that were"tapped" at>>>discrete points. >> >>Not just _before_ digital filers by the way, but also after. Here are some>>modern, monolithic continuous-time FIR examples. (Click on"Images" for the>>figures.) >> >>http://tinyurl.com/548jv >> >>(Remember: Not all FIR filters are transversal. Not all FIR filters are >>digital. Not all non-digital FIR filters are even discrete-time.) > > > > A SAW (Surface Acoustic Wave) filter is another good example of a > continuous time transversal filter. > > The surface acoustic waves are launched by "interdigitaltransducers"> which are just interleaved electrodes deposited on the surface of the > quartz. The gain (& hence the weight of that 'tap') is proportional > to the overlap between the electrodes. > > http://koigakubo.hitachi.co.jp/~cs/cd/eng/technical/device/saw/ > > Regards, > AllanSAWs are frequency selective depending on the spacing of the electrodes relative to the speed of the wave. If a pulse is launched down the substrate and under a set of electrodes with progressively decreasing spacing, a chirp is produced. When that chirp is applied to to the variable-pitch electrode of an identical device, pulses emerge from its other electrode. Even if the chirps overlap in time, the pulses can be distinct. There are several very useful applications. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������

Posted by ●February 23, 2005

On Wed, 23 Feb 2005 00:52:57 -0800, "Max Hauser" <maxREMOVE@THIStdl.com> wrote:>"Fred Marshall" in news:a9ednW1bu_E7jYTfRVn-iA@centurytel.net... >> ... >> Before there were truly discrete, digital filters, there were continuous >> time or analog filters (such as delay lines) that were "tapped"at>> discrete points. > >Not just _before_ digital filers by the way, but also after. Here are some >modern, monolithic continuous-time FIR examples. (Click on "Images"for the>figures.) > >http://tinyurl.com/548jv > >(Remember: Not all FIR filters are transversal. Not all FIR filters are >digital. Not all non-digital FIR filters are even discrete-time.)A SAW (Surface Acoustic Wave) filter is another good example of a continuous time transversal filter. The surface acoustic waves are launched by "interdigital transducers" which are just interleaved electrodes deposited on the surface of the quartz. The gain (& hence the weight of that 'tap') is proportional to the overlap between the electrodes. http://koigakubo.hitachi.co.jp/~cs/cd/eng/technical/device/saw/ Regards, Allan

Posted by ●February 23, 2005

"Fred Marshall" in news:a9ednW1bu_E7jYTfRVn-iA@centurytel.net...> ... > Before there were truly discrete, digital filters, there were continuous > time or analog filters (such as delay lines) that were "tapped" at > discrete points.Not just _before_ digital filers by the way, but also after. Here are some modern, monolithic continuous-time FIR examples. (Click on "Images" for the figures.) http://tinyurl.com/548jv (Remember: Not all FIR filters are transversal. Not all FIR filters are digital. Not all non-digital FIR filters are even discrete-time.) -------- "To converse at the distance of the Indes by means of sympathetic contrivances may be as natural to future times as to us is a literary correspondance." -- Joseph Glanvill, 1661 [as quoted by Scott Dorsey, news:4723@pyr.gatech.edu, rec.audio, 1988]

Posted by ●February 23, 2005

"Allan Herriman"> > I assume that 'transversal' is a property of a filter implementation, > whereas 'FIR' is a property of the impulse response of a filter. >I thought that a good summary, and in accord with what much of the literature has used in the past 40 years or so. Neglecting novelty items like recursive FIR implementations, there are other famous nonrecursive structures that also realize Finite Impulse Responses but lack the transversal (tapped-delay) structure. A parallel set of delay lines of different lengths or delays, each weighted at their input or output, and finally summed, is such a structure. (It has been useful in some delay-line technologies that had constraints, for example certain CCD technologies circa the 1970s.) -- Max Hauser

Posted by ●February 22, 2005

Fred said:>>OK - if we include recursively implemented FIRs then they aren't >>transversal. >>But, that's a special case that only applies to a small subset of FIRs. >>Correct but limited. I like to think of the recursive implementation of >>FIRs as sort of a curiosity with limited application - no matter how cool >>and even valuable some of those implementations may be. Let's not have >>the >>tail wag the dog. >> >>What if we turn it around and say: >>"In some cases a FIR can be implemented recursively and then isn't a >>transversal filter. However, a FIR filter can *always* be implemented as >>a >>sum of delayed inputs and thus, can always be transversal" ?? >>In that sense, FIR can very reasonably imply transversal. >Allan said:> They never should have stopped teaching logic in schools. > > AllanOh geez, now I have to construct a syllogism for testing.... ?? I agree with what you said. Thanks for the clarification - it took me a while. Fred