Properties

### Linearity

Let $x(t)\longleftrightarrow X(j\omega)$ and $y(t) \longleftrightarrow Y(j\omega)$ then,

(1)
\begin{align} z(t)=ax(t)+by(t) \longleftrightarrow aX(j\omega)+bY(j\omega) \end{align}

### Time Shifting

Let $x(t)\longleftrightarrow X(j\omega)$ then,

(2)
\begin{align} z(t)=x(t-t_0) \longleftrightarrow e^{-j\omega t_0}X(j\omega) \end{align}

Proof:

$Z(j\omega)=\int x(t-t_0)e^{-jwt_0}dt$

Substituting, $\gamma=t-t_0$,we get

$Z(j\omega)=\int x(\gamma)e^{-jw(\gamma+t_0)}dt=e^{-jwt_0}X(j\omega)$

So,magnitude remains unchanged but phase is different.

### Frequency Shifting (Modulation Property)

Let $x(t)\longleftrightarrow X(j\omega)$ then,

(4)
\begin{align} e^{j\gamma t_0}x(t) \longleftrightarrow X(j(\omega-\gamma)) \end{align}

### Time and Frequency Scaling

Let $x(t)\longleftrightarrow X(j\omega)$ then,

(5)
\begin{align} x(at) \longleftrightarrow \frac{1}{|a|}X(j\frac{\omega}{a}) \hspace{0.4cm} \textit{a is a real constant} \end{align}

Special case, $a=-1$

$x(-t) \longleftrightarrow X(-j\omega)$

### Conjugation and Symmetry

Let $x(t)\longleftrightarrow X(j\omega)$ then,

(6)
\begin{align} x^*(t) \longleftrightarrow X^*(-j\omega) \end{align}

Proof:

$x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\omega)e^{j\omega t}d\omega$

$x^*(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X^*(j\omega)e^{-j\omega t}d\omega$

Substitute, $\gamma=-\omega$ and $d\gamma=-d\omega$

(7)
\begin{aligned} x^*(t) &= \frac{-1}{2\pi}\int_{\infty}^{-\infty}X^*(-j\gamma)e^{j\gamma t}d\gamma \\ \nonumber &= \frac{1}{2\pi}\int_{-\infty}^{\infty}X^*(-j\gamma)e^{j\gamma t}d\gamma \\ \nonumber \end{aligned}
• If $x(t)$ is real, i.e $x(t)=x^*(t)$,then we can observe a conjugate symmetry here-

$X(j\omega)=X^*(-j\omega)$

(8)
\begin{align} X(-j\omega)=X^*(j\omega) \end{align}

### Convolution-Multiplication Duality

---------------Put proof here-------------------——

### Differentiation and Integration Properties

• Differentiation:

$\frac{d}{dt}x(t) \longleftrightarrow j\omega X(j\omega)$

(11)
\begin{align} -jtx(t) \longleftrightarrow \frac{d}{d\omega} X(j\omega) \end{align}

Proof:

$x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\omega)e^{j\omega t}d\omega$

$\frac{d}{dt}x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}j \omega X(j\omega)e^{j\omega t}d\omega$

$\frac{d}{dt}x(t)\longleftrightarrow j \omega X(j\omega)$

• Integration:

It is also called Total area under the curve property.

$X(j\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}x(t)e^{-j\omega t}d\omega$

(12)
\begin{align} X(j0)=\frac{1}{2\pi}\int_{-\infty}^{\infty}x(t)dt \end{align}

$x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\omega)e^{j\omega t}d\omega$

(13)
\begin{align} x(0)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\omega)d\omega \end{align}