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

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.

## 1 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

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

$$S_x(\omega)$$ 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).

• thanks, and what if x(t) be a stochastic process? Jul 17 '20 at 9:33
• @m-sh-shokouhi: A stochastic process generally doesn't have a Fourier transform, just a power spectrum. The latter determines whether the process is band-limited or not. Jul 17 '20 at 10:06