power spectral density of WSS process question

Hi all,

I have a problem which I can't find any way to go:

X is an Wide Sense stationary process. Its power spectral density S(w)
is zero outside [-wmax, wmax]. They ask to prove

R(0)-R(t)< 0.5 *wmax^2 *t^2 *R(0).  where R(t) is autocorrelation of
X.

The question is a hint |sin(x)|<|x|


In this problem I cann't see how to apply the hint to prove the
inequality.

All I can think is it related to something like

R(t)= 1/2pi  * integral_from_-wmax_to w_max (S(w) exp(-jwt)dw.

here I guess sin may be related to the exp(-jwt), but it seems a
blocked way to go.

Can you please give me a hint?

Thanks

Vijay

VijaKhara wrote:
> here I guess sin may be related to the exp(-jwt)

could be ...

On Aug 7, 7:35 pm, VijaKhara <VijaKh...@gmail.com> wrote:
> Hi all,
>
> I have a problem which I can't find any way to go:
>
> X is an Wide Sense stationary process. Its power spectral density S(w)
> is zero outside [-wmax, wmax]. They ask to prove
>
> R(0)-R(t)< 0.5 *wmax^2 *t^2 *R(0).  where R(t) is autocorrelation of
> X.
>
> The question is a hint |sin(x)|<|x|
>
> In this problem I cann't see how to apply the hint to prove the
> inequality.
>
> All I can think is it related to something like
>
> R(t)= 1/2pi  * integral_from_-wmax_to w_max (S(w) exp(-jwt)dw.
>
> here I guess sin may be related to the exp(-jwt), but it seems a
> blocked way to go.
>
> Can you please give me a hint?
>
> Thanks
>
> Vijay

Hint:  S(w) is an even function of w, while the real part
of exp(jwt) is an even function of w and the imaginary
part is an odd function of w.