17.2 Linear time-invariant systems

Linear time-invariant (LTI) systems form the foundation of signal processing analysis. Because they follow strict rules of proportionality and consistency over time, you can fully characterize an LTI system with a single function and predict its output for any input. That predictability is what makes them so central to this course.

Two defining features set LTI systems apart: linearity (the output scales and adds in proportion to the input) and time-invariance (the system behaves the same regardless of when you apply the input).

Properties of Linear Time-Invariant Systems

Fundamental Characteristics of LTI Systems

Linearity means the system obeys the superposition principle. You can break a complicated input into simpler pieces, find each output separately, and add the results together.

Formally: if input $x_1(t)$ produces output $y_1(t)$ and input $x_2(t)$ produces output $y_2(t)$, then the combined input $ax_1(t) + bx_2(t)$ produces the output $ay_1(t) + by_2(t)$ for any constants $a$ and $b$.

This is actually two sub-properties packed into one:

• Additivity: the response to a sum of inputs equals the sum of the individual responses.
• Homogeneity (scaling): if you double the input, the output doubles.

Time-invariance means the system's behavior doesn't depend on when the input arrives.

If input $x(t)$ produces output $y(t)$, then a delayed input $x(t - t_0)$ produces the same delayed output $y(t - t_0)$ for any time shift $t_0$. The system itself isn't changing with time, so shifting the input just shifts the output by the same amount.

Additional Properties of LTI Systems

Not every LTI system is automatically stable or causal. These are additional properties you need to check.

Stability (BIBO) means that every bounded input produces a bounded output ("bounded-input, bounded-output"). For an LTI system, this holds if and only if the impulse response is absolutely integrable:

$\int_{-\infty}^{\infty} |h(t)|\, dt < \infty$

If this integral blows up, the system can produce an output that grows without limit even from a perfectly reasonable input.

Causality means the output at time $t$ depends only on present and past input values, never future ones. For an LTI system, causality requires:

$h(t) = 0 \quad \text{for all } t < 0$

Any physically realizable, real-time system must be causal. You can't build hardware that reacts to an input before it arrives.

System Responses and Characterization

Time-Domain Analysis

The impulse response $h(t)$ is the output you get when the input is a unit impulse $\delta(t)$. This single function completely describes an LTI system. Once you know $h(t)$, you can find the output for any input using the convolution integral:

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

What convolution does, conceptually: it breaks the input into a train of scaled, shifted impulses, runs each one through the system, and sums all the resulting responses. Superposition and time-invariance are exactly what make this work.

The step response $s(t)$ is the output when the input is a unit step $u(t)$. It's useful for characterizing transient behavior (rise time, settling time, overshoot). You can relate it to the impulse response by:

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

Frequency-Domain Analysis

The transfer function $H(s)$ is the Laplace transform of the impulse response $h(t)$. It converts the convolution integral into simple multiplication in the $s$-domain:

$Y(s) = H(s) \cdot X(s)$

This is a huge practical advantage. Instead of evaluating a convolution integral, you multiply two functions, then inverse-transform to get $y(t)$. The transfer function also lets you use algebraic tools like pole-zero analysis to assess stability and system behavior at a glance.

The frequency response $H(j\omega)$ is a special case of the transfer function evaluated along the imaginary axis ($s = j\omega$). It tells you how the system affects sinusoidal inputs at each angular frequency $\omega$:

• The magnitude $|H(j\omega)|$ gives the gain at frequency $\omega$.
• The phase $\angle H(j\omega)$ gives the phase shift at frequency $\omega$.

These are exactly what Bode plots display: magnitude and phase as functions of frequency, giving you a visual picture of how the system filters different frequency components.