Re: Can a LPF have linear phase when the impuse response is not symmetric?

On 31 Mrz., 21:30, dbd <d...@xxxxxxxx> wrote:
On Mar 31, 10:24 am, "Fred Marshall" <fmarshallx@xxxxxxxxxxxxxxxxxxxx>
wrote:





Greg Berchin's thought experiment has a flaw that explains:

If one cascades two linear systems that each have linear phase then the
resulting system will have linear phase.  One flat delay followed by
another.

But, to add the impulse responses as shown requires that the systems be
paralleled and summed.
It's easy to show that such an arrangement might not have linear phase:

Example:
System 1 is a pure delay of D1 and gain of K1 - clearly linear phase
System 2 is a pure delay of D2 and gain of K2 - clearly linear phase

The resulting unit sample response is a pair of samples spaced D1-D2 for
D1>D2.

We can time shift the response so the two outputs are temporally centered at
t=0.
For D1>D2 as above, the time shift would be D2 + (D1 - D2)/2.
Now the output from System 1 appears at D1 - {D2 + (D1 - D2)/2} =  D1/2 -
D2/2
And the output from System 2 appears at D2 - {D2 + (D1 - D2)/2} = -D1/2 +
D2/2

This is a nicely symmetrically-placed set of samples with generally
different amplitudes.
The Fourier Transform is a sinusoid of amplitude and phase depending on the
system parameters. I don't believe that this one is generally linear phase.

Try this:
Modify K2 going from K2=K1 to K2=-K1.
The beginning situation is even / symmetrical - linear phase.
The ending situation is odd / antisymmetric - linear phase.
But generally the in-between values of K2 don't yield linear phase.

Fred

In my original suggestion about linear combinations in this discussion
I included the constraint of equal delay:

"Note that for weight sets having the same delay, any linear
combination of the weight sets with these symmetries will have linear
phase. "

The delay doesn't matter. Even if the delays are the same, the sum of
an even and odd-symmetric FIR filter does not have linear-phase.
Otherwise, it would be very simple to prove that every FIR filter h
has linear-phase:

set he = 1/2*(h + reverse(h))
set ho = 1/2*(h - reverse(h))

where reverse(h) reverses the coefficients of h. Now he is even, ho is
odd-symmetric, both have the same delay and

h = he + ho,

ie. h can be written as the sum of an even and odd-symmetric FIR
filter. Is h thus linear-phase? Hardly.

Regards,
Andor
.

Relevant Pages

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  • Re: questions raised by reading and thinking with possibly missing background
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