Linear Systems

Introduction

Linearity is the single most important property in system theory because it enables the powerful principle of superposition: the response to a complex input can be determined by decomposing it into simpler components, analyzing each component separately, and adding the results. This principle transforms intractable problems into manageable ones and is the foundation upon which Fourier analysis, convolution, and all of linear system theory are built.

A linear system satisfies two conditions simultaneously: additivity (the response to a sum of inputs equals the sum of individual responses) and homogeneity (scaling the input scales the output proportionally). Together, these form the superposition property, which is the defining characteristic of linearity.

Formal Definition

A system $T$ is linear if and only if it satisfies the superposition principle:

$$T\{a_1 x_1(t) + a_2 x_2(t)\} = a_1 T\{x_1(t)\} + a_2 T\{x_2(t)\}$$

for all inputs $x_1, x_2$ and all constants $a_1, a_2$.

This can be decomposed into two sub-properties:

Additivity

$$T\{x_1(t) + x_2(t)\} = T\{x_1(t)\} + T\{x_2(t)\}$$

The response to a sum equals the sum of individual responses.

Homogeneity (Scaling)

$$T\{ax(t)\} = aT\{x(t)\}$$

Scaling the input by constant $a$ scales the output by the same constant.

Both must hold simultaneously for the system to be linear. A system can satisfy one without the other (though this is rare in practice).

Testing for Linearity: Systematic Procedure

Step 1: Define two arbitrary inputs $x_1(t)$ and $x_2(t)$ and find their individual outputs $y_1(t) = T\{x_1(t)\}$ and $y_2(t) = T\{x_2(t)\}$.

Step 2: Form the weighted combination $x_3(t) = a_1 x_1(t) + a_2 x_2(t)$ and compute $y_3(t) = T\{x_3(t)\}$.

Step 3: Compare $y_3(t)$ with $a_1 y_1(t) + a_2 y_2(t)$. If equal for ALL choices of $x_1, x_2, a_1, a_2$, the system is linear. If any counterexample exists, it is nonlinear.

Example 1: $y(t) = 3x(t)$ — Linear

Let $x_3 = a_1 x_1 + a_2 x_2$. Then $T\{x_3\} = 3(a_1 x_1 + a_2 x_2) = 3a_1 x_1 + 3a_2 x_2 = a_1(3x_1) + a_2(3x_2) = a_1 y_1 + a_2 y_2$ ✓

Example 2: $y(t) = x^2(t)$ — Nonlinear

$T\{a_1 x_1 + a_2 x_2\} = (a_1 x_1 + a_2 x_2)^2 = a_1^2 x_1^2 + 2a_1 a_2 x_1 x_2 + a_2^2 x_2^2$

But $a_1 T\{x_1\} + a_2 T\{x_2\} = a_1 x_1^2 + a_2 x_2^2$

These are not equal (cross-term exists). Nonlinear. ✗

Example 3: $y(t) = x(t) + 3$ — Nonlinear

Test homogeneity: $T\{ax\} = ax + 3$, but $aT\{x\} = a(x+3) = ax + 3a$. These differ (unless $a=1$). The constant offset $+3$ violates linearity.

Example 4: $y[n] = nx[n]$ — Linear

$T\{a_1 x_1 + a_2 x_2\} = n(a_1 x_1[n] + a_2 x_2[n]) = a_1 n x_1[n] + a_2 n x_2[n] = a_1 y_1 + a_2 y_2$ ✓

Note: This system IS linear but is NOT time-invariant (because the coefficient $n$ changes with time).

Why Linearity Matters

Fourier Analysis Becomes Possible

A signal can be decomposed into sinusoids: $x(t) = \sum c_k e^{jk\omega_0 t}$. For a linear system, the output is the sum of responses to individual sinusoids. This is the entire basis of frequency-domain analysis.

Convolution Describes the System

For linear time-invariant systems, the output is the convolution of input with impulse response: $y = h * x$. This powerful result exists only because linearity allows us to decompose the input into weighted impulses and sum the responses.

Superposition Enables Analysis of Complex Inputs

Instead of analyzing a complicated input directly, decompose it into simple components (impulses, sinusoids, step functions), find each response, and add them. This divide-and-conquer approach is possible only for linear systems.

Common Sources of Nonlinearity

• Saturation/Clipping: Amplifiers become nonlinear when output reaches power supply limits
• Dead zones: Systems that do not respond to small inputs (friction, thresholds)
• Quantization: Rounding to discrete levels violates homogeneity
• Multiplication of signals: $y = x_1 \cdot x_2$ is not linear in either input alone
• Hysteresis: Output depends on the direction of input change (magnetic systems)

Linearization of Nonlinear Systems

Many real systems are nonlinear but operate near a linear region. Small-signal analysis linearizes about an operating point:

$$y \approx y_0 + \frac{\partial f}{\partial x}\bigg|_{x_0} (x - x_0)$$

This Taylor series approximation makes the system approximately linear for small perturbations, enabling linear system tools to be applied within the valid range.

Key Takeaways

• Linear systems satisfy superposition: $T\{a_1 x_1 + a_2 x_2\} = a_1 T\{x_1\} + a_2 T\{x_2\}$
• Superposition = additivity + homogeneity (both must hold simultaneously)
• Any constant offset (like $y = x + 3$) violates linearity
• Linearity enables Fourier analysis, convolution, and all transform-based methods
• To test: apply a weighted combination and check if the output follows superposition
• Most real systems are approximately linear within a limited operating range