Properties of Laplace Transform

Time-shifting Property

The time-shifting property states if:

$$x(t)⟺X(s)$$

then for $t_0\ge 0$

$$x(t-t_0)⟺X(s)e^{-s{t}_0}$$

Since $x(t)$ starts at $t=0$ and $x(t-t_0)$ starts at $t=t_0$. To make this more explicit, we can restate this property as:

$$x(t)u(t)⟺X(s)$$

then

$$x(t-t_0)u(t-t_0)⟺X(s)e^{-s{t}_0},t_0\ge 0$$

Frequency Shifting

The frequency shifting property states if

$$x(t)⟺X(s)$$

then

$$x(t)e^{s_{0}t}⟺X(s-s_0)$$

Easy proof:

$$\mathcal{L}[x(t)e^{s_{0}t}]={\int }_{0^{-}}^{\infty }x(t)e^{s_{0}t}{e}^{-st}dt={\int }_{0^{-}}^{\infty }x(t)e^{-(s-s_0)t}dt=X(s-s_0)$$

Scaling Property

If we have:

$$\mathcal{L}[x(t)]=X(s)$$

Then we have:

$$\mathcal{L}[x(at)]=\frac{1}{a}X\left(\frac{s}{a}\right)$$

• If $a>1$, time compression by a factor of $a$ (causes frequency expansion)
• If $a<1$, time expansion by a factor of $a$ (causes frequency compression)

Time Differentiation Property

If $\mathcal{L}[x(t)]=X(s)$, we have:

$$\begin{array}{rl}\mathcal{L}\left[\frac{dx(t)}{dt}\right]& =sX(s)-x(0^{-})\\ \mathcal{L}\left[\frac{dx(t)}{dt}\right]& =s^{2}X[s]-sX(0^{-})-\dot{x}(0)\\ \mathcal{L}\left[\frac{d^{n}x(t)}{dt}\right]& =s^{n}X(s)-s^{n-1}x(0^{-})-s^{n-2}\dot{x}(0^{-})-\cdots -x^{n-1}(0^{-})\end{array}$$

Frequency Differentiation

If $\mathcal{L}[x(t)]=X(s)$, then:

$$\mathcal{L}[tx(t)]=-\frac{d}{ds}X(s)$$

This is different for the discrete Z-transform:

$$\mathcal{Z}[nx[n]]=-z\frac{dX[z]}{dz}$$

Time Integration

Given $\mathcal{L}[x(t)]=X(s)$, then:

$$\mathcal{L}\left[{\int }_0^{t}x(\tau )d\tau \right]=\frac{X(s)}{s}$$

Frequency Domain Integration

Given $\mathcal{L}[x(t)]=X(s)$, then:

$$\mathcal{L}\left[\frac{x(t)}{t}\right]={\int }_s^{\infty }x(\alpha )d\alpha$$

Convolution

Given $\mathcal{L}[x(t)]=X(s)$, then:

$$\mathcal{L}[x_1(t)\ast x_2(t)]=X_1(s)X_2(s)$$

The Z-transform version is:

$$Z[x_1[n]\ast x_2[n]]=x_1[z]x_2[z]$$