On Feb 18, 2:03&#4294967295;pm, "mjizhao" <mjiz...@yahoo.com> wrote:
> Thanks for reply.
>
> As you said, if it's a minimum phase system, &#4294967295;it's the same as saying the
> hypothetical system G(f) = log(H(f)) is causal.
>
> But according to minimum phase system definition, it it's a minimum phase
> system, H(f) should also be causal, is it right?

i *may* also be causal, but i am unaware of a theorem that says it
must be.  i dunno.  the effect, on the time-domain behavior, of
logging something in the frequency domain is not clear to me.

there have been, from my experience, two "competing" definitions of a
minimum phase system.  one was restricted to the class of linear
systems that can be built out of adders, scalers, and integrators.
those have H(s) transfer functions (to connect the Fourier and Laplace
notational conventions, the above H(f) is this H(j*2*pi*f)) with all
of the poles and zeros in the left-half plane.  then you can show
that, for these LTI systems, that the Hilbert Transform relationship
exists between the phase and the log magnitude.  the other definition
is the one that begins with the Hilbert Transform relationship.

the first definition above is considering only systems that were
stable and causal to start with.  the latter definition makes no such
assumptions and i am not sure, from that definition, that causality
can be always be assured for every minimum-phase system.

r b-j

Thanks for reply. 

As you said, if it's a minimum phase system,  it's the same as saying the
hypothetical system G(f) =3D log(H(f))
is causal.

But according to minimum phase system definition, it it's a minimum phase
system, H(f) should also be causal, is it right?

Thanks.







>On Feb 18, 1:19=A0pm, "mjizhao" <mjiz...@yahoo.com> wrote:
>> I am a little confused with causal and minimum phase system.
>>
>> If a system is a minimum phase system, then the phase of system could
>> be decided by phase =3D hilbert(log(abs(f))). So the real(f) =3D
>> abs(f)*cos(phase) and imag(f) =3D abs(f)*sin(phase)
>>
>> On the other hand, a minimum phase system is a causal system, so
>> that imag(f) =3D hilbert(real(f)).
>>
>> So, here there are two real/imag number. Which one is the right one?
>> thanks.
>
>your notation leaves a bit to be desired which is, i think, why you're
>confused.  if you express this properly, there is no overt
>contradiction:
>
>
>    H(f) =3D FT{ h(t) }
>
>    H(f) =3D |H(f)| * exp(j*phi(f))  =3D  real{H(f)} + j*imag{H(f)}
>
>
>
>
>if it's causal, h(t) =3D 0  for all t<0 and
>
>    imag{ H(f) } =3D -Hilbert{ real{H(f)} }
>
>
>if it's minimum-phase
>
>    imag{ log(H(f)) } =3D -Hilbert{ real{ log(H(f)) } }
>
>or
>
>    phi(f) =3D -Hlibert{ log|H(f)| }
>
>
>it's the same as saying the hypothetical system
>
>    G(f) =3D log(H(f))
>
>is causal.
>
>r b-j
>

On Feb 18, 1:19&#4294967295;pm, "mjizhao" <mjiz...@yahoo.com> wrote:
> I am a little confused with causal and minimum phase system.
>
> If a system is a minimum phase system, then the phase of system could
> be decided by phase = hilbert(log(abs(f))). So the real(f) =
> abs(f)*cos(phase) and imag(f) = abs(f)*sin(phase)
>
> On the other hand, a minimum phase system is a causal system, so
> that imag(f) = hilbert(real(f)).
>
> So, here there are two real/imag number. Which one is the right one?
> thanks.

your notation leaves a bit to be desired which is, i think, why you're
confused.  if you express this properly, there is no overt
contradiction:


    H(f) = FT{ h(t) }

    H(f) = |H(f)| * exp(j*phi(f))  =  real{H(f)} + j*imag{H(f)}




if it's causal, h(t) = 0  for all t<0 and

    imag{ H(f) } = -Hilbert{ real{H(f)} }


if it's minimum-phase

    imag{ log(H(f)) } = -Hilbert{ real{ log(H(f)) } }

or

    phi(f) = -Hlibert{ log|H(f)| }


it's the same as saying the hypothetical system

    G(f) = log(H(f))

is causal.

r b-j

I am a little confused with causal and minimum phase system.

If a system is a minimum phase system, then the phase of system could
be decided by phase = hilbert(log(abs(f))). So the real(f) =
abs(f)*cos(phase) and imag(f) = abs(f)*sin(phase)

On the other hand, a minimum phase system is a causal system, so
that imag(f) = hilbert(real(f)).

So, here there are two real/imag number. Which one is the right one?
thanks.