🚀 go-pugleaf

In article <wilf.675023233@rigel.sce.carleton.ca> wilf@sce.carleton.ca (Wilf Leblanc) writes:
>
>"... but the zeros are on the unit circle" ??????
>
>O.K. smartass, I'll play your game:
>
>    H(z) = (1+0.5 z^-1)(1+2z^-1)
>         = 1 + 2.5z^-1 + z^-2
>
>Looks symmetric, looks too me like either you are wrong or
>0.5/1 = 2/1 = 1.
>
>
>Nothing like interviewing someone, trying to screw them up, and
>being completely wrong, eh ??
>
>The zeros are not on the unit circle.
>

COMPLETELY wrong?  Half wrong perhaps, (and technically not even that).
Linear phase FIR filters have zeros either on the unit circle in complex
conj. pairs, or in pairs with a reciprocal magnitude (and their complex
conj's).  I ignored the second class (because I forgot about them :( ).
I *think* the remez exchange will generate filters with both kinds of zeros.

Using your filter as an example, changing the second coefficient to anything
less then two will generate complex zeros - which should be on the unit circle.
The time response is still symmetric, so it meets the necessary and sufficent
condition for linear phase, but it does have single zeros on the unit circle.

Example:

	H(z) = 1 + (2/sqrt[2])z^-1 + z^-2 
	H(z) = [(z^-1) - (e^j*pi/4)]*[(z^-1) - (e^-j*pi/4)]

(That's hard to read, should represent a zero an +- 45 deg on the unit circle,
 if I didn't make another mistake)

The point I was trying to make is there are filters which satisfy the
conditions for linear phase, but which place a zero on the unit circle.
I don't know what the significance of this is because I have never used
the linear phase property of FIR filters.  I expect it would give unexpected
results if a filtered signal was summed with one which was just delayed.

As for trying to screw someone up, I think it is important to know why you
know something.  The reason we know symmetric filters have linear phase is
the fourier transform of a symmetric signal is real.  But negative
numbers are real.  So if you plot |gain| and phase, there will be 180 deg
phase shifts.

As an example consider a rectangular pulse, symetric about 0.  It has a sinc
function as a transform - which is real, but bi-phasic.  It's real, so it has
no phase shift.  If it is the impulse response of a filter, does it just
invert some frequencies, or does it have 180 deg phase shift?

Does anyone use the linear phase property of FIR filters in practice?  Is it
necessary avoid zeros on the unit circle?

jt

-- 
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 Jeff Taylor
 Physio Control Corp.
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