# Wiener solution for equalization

Started by March 26, 2007
```In order to find the Wiener solution for equalization purpose, do I need
to provide delay to the desired signal as in the case of LMS algorithm?
Will the result of the Wiener solution become better if I provide some
delay to the desired signal?

Thanks.
```
```On Mar 26, 9:05 am, "Richard_K" <ngy...@hotmail.com> wrote:
> In order to find the Wiener solution for equalization purpose, do I need
> to provide delay to the desired signal as in the case of LMS algorithm?
> Will the result of the Wiener solution become better if I provide some
> delay to the desired signal?
>
> Thanks.

In most cases, yes.  By how much, that is not clear.

Most of the things you know about the LMS is the same
for the Wiener filter.  Both minimize the square-error,
or some estimate of it.  LMS does it iteratively, and
Wiener does it in one-step, after obtaining an estimate
of the second-order statistics (correlations) that are
relevant.

Hope that helps.
Julius

```
```On Mar 27, 2:05 am, "Richard_K" <ngy...@hotmail.com> wrote:
> In order to find the Wiener solution for equalization purpose, do I need
> to provide delay to the desired signal as in the case of LMS algorithm?
> Will the result of the Wiener solution become better if I provide some
> delay to the desired signal?
>
> Thanks.

When you add delay you turn a filter into a smoother so it normally
improves things.

H.S-

```
```On Mar 26, 10:59 am, "julius" <juli...@gmail.com> wrote:
> On Mar 26, 9:05 am, "Richard_K" <ngy...@hotmail.com> wrote:
>
> > In order to find the Wiener solution for equalization purpose, do I need
> > to provide delay to the desired signal as in the case of LMS algorithm?
> > Will the result of the Wiener solution become better if I provide some
> > delay to the desired signal?
>
> > Thanks.
>
> In most cases, yes.  By how much, that is not clear.
>
> Most of the things you know about the LMS is the same
> for the Wiener filter.  Both minimize the square-error,
> or some estimate of it.  LMS does it iteratively, and
> Wiener does it in one-step, after obtaining an estimate
> of the second-order statistics (correlations) that are
> relevant.
>
> Hope that helps.
> Julius

If the signals are second-order stationary (and they should be for the
Wiener filter, since it is not adaptive), then adding a delay doesn't
really make sense. The system of equations that define the Wiener
filter tap weights only contain statistical values (the
autocorrelation matrix of the input signal and the cross-correlation
vector between the input signal vector and the desired response); for
any system that you would apply a Wiener filter to, these should be
shift-invariant. Therefore, adding a delay in the Wiener case wouldn't
have any effect on the statistical characterizations of the system,
resulting in the same tap-weight vector.

However, the LMS filter uses the desired response sequence as one of
its inputs, so the delay is important. You need to delay the desired
response by roughly the same amount as the LMS filter delays the
input, plus any other systems that might be cascaded (for instance,
for a zero-forcing equalizer, you need to account for any delay
imposed by the channel). If you don't delay the desired signal, then
you'll essentially get garbage out.

Jason

```
```On Mar 27, 7:15 am, cincy...@gmail.com wrote:

> If the signals are second-order stationary (and they should be for the
> Wiener filter, since it is not adaptive), then adding a delay doesn't
> really make sense. The system of equations that define the Wiener
> filter tap weights only contain statistical values (the
> autocorrelation matrix of the input signal and the cross-correlation
> vector between the input signal vector and the desired response); for
> any system that you would apply a Wiener filter to, these should be
> shift-invariant. Therefore, adding a delay in the Wiener case wouldn't
> have any effect on the statistical characterizations of the system,
> resulting in the same tap-weight vector.
>
> However, the LMS filter uses the desired response sequence as one of
> its inputs, so the delay is important. You need to delay the desired
> response by roughly the same amount as the LMS filter delays the
> input, plus any other systems that might be cascaded (for instance,
> for a zero-forcing equalizer, you need to account for any delay
> imposed by the channel). If you don't delay the desired signal, then
> you'll essentially get garbage out.
>
> Jason

Jason, you are right.  I was thinking more in terms of
"synchronizing" the known system input signal and
the measured, desired signal.  The synchronization is
typically done using correlation with a known sequence,
but the peak of the correlation is not necessarily the
first tap of the channel impulse response.

So I was thinking more in terms of Wiener filtering
if there is no perfect synchronization between the