Any one can explain the Hilbert transform related question for me?

Hi,
I read a paragraph of "Theory and application of digital signal
processing" of L. R. Rabiner on page 70. It says:




For v(n), a Hiltert transformed signal, its Fourier transform V(e^
(jw)) has the property

V(e^(jw))=0          pi< w <2*pi        (2.187)

Clearly v(n) is a complex signal since the Fourier transforms of real
signals have the property

V*(e^(-jw))=V(e^(jw))                     (2.188)

which would imply V(e^(jw))=0 if v(n) were real.

I don't understand the above would sentence. (2.188) is a fact for
real signal, right? Why does it get the following:


 V(e^(jw))=0



Thanks.

On Apr 29, 10:57&#4294967295;am, fl <rxjw...@gmail.com> wrote:
> Hi,
> I read a paragraph of "Theory and application of digital signal
> processing" of L. R. Rabiner on page 70. It says:
>
> For v(n), a Hiltert transformed signal, its Fourier transform V(e^
> (jw)) has the property
>
> V(e^(jw))=0 &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295;pi< w <2*pi &#4294967295; &#4294967295; &#4294967295; &#4294967295;(2.187)
>
> Clearly v(n) is a complex signal since the Fourier transforms of real
> signals have the property
>
> V*(e^(-jw))=V(e^(jw)) &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; (2.188)
>
> which would imply V(e^(jw))=0 if v(n) were real.
>
> I don't understand the above would sentence. (2.188) is a fact for
> real signal, right? Why does it get the following:
>
> &#4294967295;V(e^(jw))=0
>

first of all,  V(e^(jw)) is periodic with period 2*pi.  you know why
that is, right?

if v[n] was real, how do you satisfy

    V(e^(jw)) = 0          pi< w <2*pi

*and*

    V*(e^(-jw)) = V(e^(jw))

?

you could have V(e^(j0)) = somthing non-zero.  same with V(e^(j*pi)).
but how could both equations be true for other values of w.

r b-j

On Wed, 29 Apr 2009 07:57:05 -0700 (PDT), fl <rxjwg98@gmail.com>
wrote:

>Hi,
>I read a paragraph of "Theory and application of digital signal
>processing" of L. R. Rabiner on page 70. It says:
>
>
>
>
>For v(n), a Hiltert transformed signal, its Fourier transform V(e^
>(jw)) has the property
>
>V(e^(jw))=0          pi< w <2*pi        (2.187)
>
>Clearly v(n) is a complex signal since the Fourier transforms of real
>signals have the property
>
>V*(e^(-jw))=V(e^(jw))                     (2.188)
>
>which would imply V(e^(jw))=0 if v(n) were real.
>
>I don't understand the above would sentence. (2.188) is a fact for
>real signal, right? Why does it get the following:
>
>
> V(e^(jw))=0
>
>Thanks.

Hello fl,
  I think you're justified in being puzzled 
by those equations. 

First the authors say:

  V(e^(jw))=0      pi< w <2*pi    (2.187)

OK, fine, ...that covers the negative freq range of 
V(e^(jw)), the spectrum of a complex-valued v(n) time 
sequence.  Then they introduce a spectrum that they call 

  V(e^(-jw))

in a conjugated form.  Well, that also looks like an 
expression for a spectrum over a negative freq range.  
I'll bet their explanation confuses many readers. 

Perhaps after the sentence, "... which would imply 
V(e^(jw)) = 0 if v(n) were real." they should have 
added the sentence:

   "Thus, the only way to simultaneously satisfy 
    Equations (2.187) and (2.188) is for v(n) to 
    be complex-valued." 
   
Good Luck,
[-Rick-]