LTI system output in time domain

LTI system

A system is linear time-invariant (LTI) if it satisfies:

• Linearity (superposition): if $x_1(t)\to y_1(t)$ and $x_2(t)\to y_2(t)$, then $a x_1(t)+b x_2(t)\to a y_1(t)+b y_2(t)$.
• Time invariance: if $x(t)\to y(t)$ then $x(t-t_0)\to y(t-t_0)$; the system behaves the same at all times.

An LTI system is completely described by its impulse response $h(t)$, the output when the input is $\delta(t)$.

Output in time domain

 x(t) ----->[  LTI system h(t)  ]-----> y(t)

Step 1 – represent the input by impulses. Using the sifting property of the impulse, any input can be written as a sum (integral) of shifted, weighted impulses:

$$x(t)=\int_{-\infty}^{\infty}x(\tau)\,\delta(t-\tau)\,d\tau$$

Step 2 – time invariance. By definition $\delta(t)\to h(t)$, so a shifted impulse gives a shifted response: $\delta(t-\tau)\to h(t-\tau)$.

Step 3 – homogeneity. A weighted impulse gives a weighted response: $x(\tau)\delta(t-\tau)\to x(\tau)h(t-\tau)$, since $x(\tau)$ is a constant for a fixed $\tau$.

Step 4 – superposition. The input is the integral (sum) of such terms, so the output is the integral of the individual responses:

$$y(t)=\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau=x(t)*h(t)$$

This is the convolution integral. By changing the variable ($\lambda=t-\tau$) it can also be written as $y(t)=\int_{-\infty}^{\infty}h(\tau)x(t-\tau)d\tau$.

Special cases:

• Causal system ($h(t)=0$ for $t<0$) with input starting at $t=0$: $y(t)=\int_0^t x(\tau)h(t-\tau)d\tau$.
• Frequency domain: taking the Fourier transform gives $Y(f)=H(f)X(f)$, where $H(f)$ is the transfer function.

Example: for an RC low-pass filter, $h(t)=\frac{1}{RC}e^{-t/RC}u(t)$. With a unit-step input, $y(t)=\int_0^t\frac{1}{RC}e^{-(t-\tau)/RC}d\tau=1-e^{-t/RC}$, the familiar charging curve.