What's the relation between frequency band of $X(jomega)$ and $Phi_{xx}(jomega)$?

Question

in which:

$x_{c}(t)$ is a continuous-time signal
$X(jOmega)$ is the Fourier Transform of $x_{c}(t)$
$Phi_{xx}(jOmega)$ is the Power Spectrum Density of $x_{c}(t)$ which defined as Fourier Transform of auto-correlation of $x_{c}(t)$.

in other words I want to know when it is said that a signal is band-limited which is the case? band-limited according to $X(jOmega)$ or $Phi_{xx}(jOmega)$?
or if it's a relationship between two cases what it is.
thanks.

Matt L. · Answer

The autocorrelation of $x(t)$ is
$$r_x(t)=x(t)star x(-t)tag{1}$$
where $star$ denotes convolution. Taking the Fourier transform of $(1)$ gives
$$S_x(omega)=X(omega)X^*(omega)=|X(omega)|^2tag{2}$$
$S_x(omega)$ is the energy density of $x(t)$, and according to $(2)$ it equals the squared magnitude of the Fourier transform of $x(t)$. So if $x(t)$ is band-limited, both $X(omega)$ and $S_x(omega)$ are zero outside the signal's bandwidth.
Note that a deterministic continuous signal which has a Fourier transform (represented by an ordinary function) is usually an energy signal, which doesn't have a power spectrum (only an energy density).

What's the relation between frequency band of $X(jomega)$ and $Phi_{xx}(jomega)$?

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