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

Question

in which:

• $x_{c}(t)$ is a continuous-time signal
• $X(j\Omega)$ is the Fourier Transform of $x_{c}(t)$
• $\Phi_{xx}(j\Omega)$ 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 $X(j\Omega)$ or $\Phi_{xx}(j\Omega)$? or if it's a relationship between two cases what it is.
thanks.

Answer 1

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

$$S_x(\omega)=X(\omega)X^*(\omega)=|X(\omega)|^2\tag{2}$$

$S_x(\omega)$ 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).