# pole-zero plot with zeros |a|

Started by March 31, 2004
```A simplified two-path fading radio channel can be modelled as a first-order
FIR filter, with delay T, and the strength of the direct signal being b0 and
the strength of the indirect signal is b1.  The channel can be equalised by
the use of an IIR filter, which has the opposite frequency response.

The frequency response is given by:

G/(1+az^-1)

where G = 1/b0 and a = b1/b0

However, due to the possibility of instability with an IIR filter, an
equivalent FIR filter can be used.  By taking the first 7 IIR impulse
response coefficients, a 7th order FIR filter can be designed.  Seeing as
the FIR impulse response IS its filter coefficients, and the impulse
response has to be the same for the FIR as the IIR to make it an accurate
approximation (!) then you use the IIR impulse response as the FIR
coefficients.

(I hope I have made this clear!!)

The pole zero plot of the FIR equaliser shows that there are 7 poles at the
centre, and 7 zeros, each at magnitude of |a| where a = b1/b0.

Why is this?

My reasoning is this:

This is because of the nature of stability and the Region of Convergence.
For an IIR filter to be stable, the poles need to be on or inside this
circle with radius |a|.  If the poles are outside, then the impulse response
does not converge, and the filter is unstable.  As the IIR coefficients are
being used in the FIR filter, so the rule stays the same.

If you are confused by anything, which I'm sure you are, then please ask
questions!

I hope you can help me!

Phil

```
```"Philip Newman" <nojunkmail@ntlworld.com> writes:
> [...]
> The pole zero plot of the FIR equaliser shows that there are 7 poles at the
> centre, and 7 zeros, each at magnitude of |a| where a = b1/b0.
>
> Why is this?

It depends on what "this" is. Do you mean "Why are there 7 poles at
the centre"?

If so, then that's easy: An FIR is a sum of delayed versions of the signal,
so its z-transform has the form

h[z] = a0 + a1*z^{-1) + ... + aN*z^(-N).

If you rearrange this in terms of positive powers of z instead
of negative powers, you get a rational function with z^N in the
denominator, just by simple algebra.
--
%  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
```
```"Randy Yates" <yates@ieee.org> wrote in message
news:smfoljrc.fsf@ieee.org...
> "Philip Newman" <nojunkmail@ntlworld.com> writes:
> > [...]
> > The pole zero plot of the FIR equaliser shows that there are 7 poles at
the
> > centre, and 7 zeros, each at magnitude of |a| where a = b1/b0.
> >
> > Why is this?
>
> It depends on what "this" is. Do you mean "Why are there 7 poles at
> the centre"?
>

Oops, sorry, no.  I meant why are there 7 zeros each at magnitude |a|.  I
can understand why there are 7 poles and zeros, because it is a 7th order
FIR filter, but I wish to know why they are all at |a|.

Phil

```
```"Philip Newman" <nojunkmail@ntlworld.com> writes:

> "Randy Yates" <yates@ieee.org> wrote in message
> news:smfoljrc.fsf@ieee.org...
>> "Philip Newman" <nojunkmail@ntlworld.com> writes:
>> > [...]
>> > The pole zero plot of the FIR equaliser shows that there are 7 poles at
> the
>> > centre, and 7 zeros, each at magnitude of |a| where a = b1/b0.
>> >
>> > Why is this?
>>
>> It depends on what "this" is. Do you mean "Why are there 7 poles at
>> the centre"?
>>
>
>
> Oops, sorry, no.  I meant why are there 7 zeros each at magnitude |a|.  I
> can understand why there are 7 poles and zeros, because it is a 7th order
> FIR filter, but I wish to know why they are all at |a|.

Who said they were?

I almost just sent the sentence above alone, but that wouldn't be nice. You've
discussed how an equalizer *could* be formed to equalize the channel, but you
haven't said *how* it actually was formed. Is this the output of a simulation and you
are examining the EQ output after it had converged?
--
%  Randy Yates                  % "Midnight, on the water...
%% Fuquay-Varina, NC            %  I saw...  the ocean's daughter."
%%% 919-577-9882                % 'Can't Get It Out Of My Head'
%%%% <yates@ieee.org>           % *El Dorado*, Electric Light Orchestra
```
```>
> Who said they were?

I did :-)  I have simulated this, and I know they are, and I know they
should be!
>
> I almost just sent the sentence above alone, but that wouldn't be nice.
You've
> discussed how an equalizer *could* be formed to equalize the channel, but
you
> haven't said *how* it actually was formed. Is this the output of a
simulation and you
> are examining the EQ output after it had converged?

I am just examining the equaliser output, not the equalised output.

I originally used an IIR filter as an equaliser, but for reasons of
instability under certain conditions, I decided to change to an FIR filter.
I did this by truncating the IIR impulse response at 7T and then using those
numbers printed off by MATLAB as the FIR filter coefficients, the FIR filter
could be used to do a decent job.

I plotted the magnitude response of the FIR and it is a good approximation
of the IIR response, with only a 2dB ripple due to the finite number of
poles.

I then took the impulse response of the FIR equaliser, and observed the
pole-zero plot, using zplane(hFIR) where hFIR is the impulse response of the
FIR equaliser.

The pole zero plot showed 7 poles at the centre, which is obvious seeing as
there is no feedback, and it also showed 4 zeros, each at radius of
magnitude |a| from the centre.

why do they all have radius of mag |a|?

the original channel had FIR coefficients of [b0 b1] and the IIR equaliser
had coefficients [1 a] and [G] for feedback, where a = b1/bo and G = 1/bo

I can send you my report if it will help....

cheers

Phil

```
```>>>>> "Philip" == Philip Newman <nojunkmail@ntlworld.com> writes:

Philip> A simplified two-path fading radio channel can be modelled as a first-order
Philip> FIR filter, with delay T, and the strength of the direct signal being b0 and
Philip> the strength of the indirect signal is b1.  The channel can be equalised by
Philip> the use of an IIR filter, which has the opposite frequency response.

Philip> The frequency response is given by:

Philip> G/(1+az^-1)

Philip> where G = 1/b0 and a = b1/b0

[snip]

Philip> The pole zero plot of the FIR equaliser shows that there are 7 poles at the
Philip> centre, and 7 zeros, each at magnitude of |a| where a = b1/b0.

Philip> Why is this?

Let see if I can do this right.

Let H(z) = (1+a*z^(-1)).  This is basically your frequency
response, but with the G scalar removed.  This is unimportant when
we're looking for the poles and zeroes.n

Divide that out to get:

H(z) = 1 - a/z + (a/z)^2 - (a/z)^3 + ...

You truncated this to 7th order:

H'(z) = 1 - a/z + (a/z)^2 - (a/z)^3 + ... - (a/z)^7.

But this is just a geometric series, whose sum is

H'(z) = (1-(a/z)^8)/(1-a/z)

= (z^8-a^8)/z^7/(z-a)

So we can see the 7 poles at zero, and also the 7 zeroes at a.

Ray, who hopes he got this right.

```
```>>>>> "Raymond" == Raymond Toy <toy@rtp.ericsson.se> writes:

Raymond> H'(z) = (1-(a/z)^8)/(1-a/z)

Raymond>       = (z^8-a^8)/z^7/(z-a)

Raymond> So we can see the 7 poles at zero, and also the 7 zeroes at a.

Raymond> Ray, who hopes he got this right.

Oops, I didn't get it quite right.  There are not 7 zeros at a.  But
there are 7 zeroes with magnitude |a|.

From (z^8-a^8)/(z-a), the numerator has the 8 roots of a^8.  The
denominator removes the real root at a, leaving the 7 other roots.

Ray
```
```"Raymond Toy" <toy@rtp.ericsson.se> wrote in message
news:4ny8pfn3l2.fsf@edgedsp4.rtp.ericsson.se...
> >>>>> "Philip" == Philip Newman <nojunkmail@ntlworld.com> writes:
>
>     Philip> A simplified two-path fading radio channel can be modelled as
a first-order
>     Philip> FIR filter, with delay T, and the strength of the direct
signal being b0 and
>     Philip> the strength of the indirect signal is b1.  The channel can be
equalised by
>     Philip> the use of an IIR filter, which has the opposite frequency
response.
>
>     Philip> The frequency response is given by:
>
>     Philip> G/(1+az^-1)
>
>     Philip> where G = 1/b0 and a = b1/b0
>
> [snip]
>
>     Philip> The pole zero plot of the FIR equaliser shows that there are 7
poles at the
>     Philip> centre, and 7 zeros, each at magnitude of |a| where a = b1/b0.
>
>     Philip> Why is this?
>
> Let see if I can do this right.
>
> Let H(z) = (1+a*z^(-1)).  This is basically your frequency
> response, but with the G scalar removed.  This is unimportant when
> we're looking for the poles and zeroes.n
>
> Divide that out to get:
>
> H(z) = 1 - a/z + (a/z)^2 - (a/z)^3 + ...
>
> You truncated this to 7th order:
>
> H'(z) = 1 - a/z + (a/z)^2 - (a/z)^3 + ... - (a/z)^7.
>
> But this is just a geometric series, whose sum is
>
> H'(z) = (1-(a/z)^8)/(1-a/z)
>
>       = (z^8-a^8)/z^7/(z-a)
>
> So we can see the 7 poles at zero, and also the 7 zeroes at a.
>
> Ray, who hopes he got this right.
>

That looks great Ray, cheers.  I am unsure how you arrived at the
penultimate line though, and which bits divide through by which, could you
go through with brackets so that it is more clear - sorry.

I thought this issue might have something to do with stability, but it seems
not!

Cheers

Phil

```
```>>>>> "Philip" == Philip Newman <nojunkmail@ntlworld.com> writes:

>>
>> H'(z) = (1-(a/z)^8)/(1-a/z)
>>
>> = (z^8-a^8)/z^7/(z-a)
>>
>> So we can see the 7 poles at zero, and also the 7 zeroes at a.
>>
>> Ray, who hopes he got this right.
>>

Philip> That looks great Ray, cheers.  I am unsure how you arrived at the
Philip> penultimate line though, and which bits divide through by which, could you
Philip> go through with brackets so that it is more clear - sorry.

Sure.

(1-(a/z)^8)
----------
(1-(a/z)

Multiply top and bottom by z^8, and we have

z^8 - a^8
-----------
z^8 - a*z^7

So we have (z^8-a^8)/z^7/(z-a) = 1/z^7*(z^8-a^8)/(z-a).  So the
first term gives the 7 poles at 0.

Now look at (z^8 - a^8)/(z-a).  Clearly the numerator has 8 zeroes at
the 8 roots of a^8 (which all have magnitude |a|), one of which is a.
This is cancelled by the 1/(z-a) term, so we have 7 zeroes left, all
with magnitude |a|.

Does that make sense?

Ray
```
```>
> So we have (z^8-a^8)/z^7/(z-a) = 1/z^7*(z^8-a^8)/(z-a).

How did you get this???

Sorry if I am missing something obvious....

Phil

```