>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-]

Reply by robert bristow-johnson●April 29, 20092009-04-29

On Apr 29, 10:57�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 � � � � �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
>

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

Reply by fl●April 29, 20092009-04-29

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.