CTFT linearity and time shifting

Linearity property

If $x_1(t) \leftrightarrow X_1(j\omega)$ and $x_2(t) \leftrightarrow X_2(j\omega)$, then for any constants $a$ and $b$:

$$a\,x_1(t) + b\,x_2(t) \;\longleftrightarrow\; a\,X_1(j\omega) + b\,X_2(j\omega)$$

Proof: $\int[a x_1(t) + b x_2(t)]e^{-j\omega t}dt = a\int x_1e^{-j\omega t}dt + b\int x_2e^{-j\omega t}dt = aX_1 + bX_2$, because integration is linear.

Meaning: The FT of a sum is the sum of the FTs; superposition holds. Example: $\cos\omega_0t = \frac{1}{2}(e^{j\omega_0t} + e^{-j\omega_0t}) \leftrightarrow \pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$.

Time shifting property

$$x(t - t_0) \;\longleftrightarrow\; e^{-j\omega t_0}\,X(j\omega)$$

Proof: Put $\tau = t - t_0$:

$$\int_{-\infty}^{\infty}x(t-t_0)e^{-j\omega t}dt = \int_{-\infty}^{\infty}x(\tau)e^{-j\omega(\tau+t_0)}d\tau = e^{-j\omega t_0}X(j\omega)$$

Meaning: A delay does not change the magnitude spectrum, $|X(j\omega)|$; it only adds a linear phase $-\omega t_0$. Example: $\delta(t) \leftrightarrow 1$, so $\delta(t-2) \leftrightarrow e^{-j2\omega}$ (magnitude 1, phase $-2\omega$).