LTI system

A linear time-invariant (LTI) system is one that is both linear and time-invariant. The most important class of systems in this course, and arguably in all of signal processing.

What LTI buys you: superposition

The superposition principle lets you decompose an input into simple pieces, find the response to each piece, and sum the responses.

Say we can write $x(t)={\sum }_k{\alpha }_{k}p(t-{\tau }_k)$, a sum of shifted and scaled copies of some standard pulse $p$, and we know the system’s response to $p(t)$ is some signal $q(t)$. Then:

• By time-invariance, the response to $p(t-{\tau }_k)$ is $q(t-{\tau }_k)$.
• By homogeneity, the response to ${\alpha }_{k}p(t-{\tau }_k)$ is ${\alpha }_{k}q(t-{\tau }_k)$.
• By additivity, the response to the whole sum is ${\sum }_k{\alpha }_{k}q(t-{\tau }_k)$.

If we know how the system responds to one canonical pulse, we know how it responds to any input we can decompose into shifted, scaled copies of that pulse. Lose linearity, can’t sum responses. Lose time-invariance, can’t shift responses. Both are required.

The canonical pulse: the impulse

The pulse the course picks is the unit impulse $\delta (t)$. The system’s response to $\delta$ is the impulse response $h(t)$. Once we have $h$, the response to any input is given by the convolution:

$y(t)=x(t)\ast h(t)={\int }_{-\infty }^{\infty }x(\tau )h(t-\tau )d\tau .$

A single function $h(t)$ characterizes the entire system, and one operation (convolution) builds the response to any input from it.

Three equivalent ways to analyze

LTI systems can be analyzed in three equivalent ways:

1. Time-domain: differential equation, or convolution with the impulse response.
2. Frequency-domain: multiply input’s spectrum by the frequency response $H(f)=\mathcal{F}\{h\}$.
3. s-domain: multiply input’s Laplace transform by the transfer function $H(s)=\mathcal{L}\{h\}$.

For any given problem, one of the three is usually much easier than the other two, so knowing when to switch between them matters a lot.

Cascaded and parallel LTI systems

Two LTI systems in cascade (output of first feeds input of second) have overall impulse response $h_1\ast h_2$. In the frequency domain that’s multiplication, $H(f)=H_1(f)H_2(f)$.

Two in parallel (same input, outputs summed) have overall impulse response $h_1+h_2$. In frequency, $H_1(f)+H_2(f)$.

Cascade and parallel combine arbitrarily, and at each level the impulse response or frequency response follows from the parts.

What is and isn’t LTI

Real engineering systems are usually approximately LTI in some operating range. Ask first whether the system is LTI; if yes or very nearly yes, apply LTI theory. If not, either linearize (small-signal analysis) or switch to a more general framework.