# Definition for transversal equalizers

Started by February 20, 2005
```Hello Everyone,

Can anyone point me the definition for a transversal equalizer? Some
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

```
```"huke" <hugo.harada@gmail.com> wrote in message
> 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

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

```
```Hi Fred,

Thank you very much for your valuable input.

I really appreciated it.

Hugo

```
```"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."
--
%  Randy Yates                  % "The dreamer, the unwoken fool -
%% Fuquay-Varina, NC            %  in dreams, no pain will kiss the brow..."
%%% 919-577-9882                %
```
```On Mon, 21 Feb 2005 14:08:08 GMT, Randy Yates <yates@ieee.org> wrote:

>
>> 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.

Regards,
Allan
```
```"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:
>
>>
>>> 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

```
```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:
>>
>>>
>>>> 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.

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
```
```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+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
```
```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+1

Yes 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
```
```"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:
>>>
>>>>
>>>>> 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.
>
> 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

```