Section C.2 Signal Analysis

Subsection C.2.1 Linear Time-Invariant Systems

Here, we introduce the fundamental ideas, terminology, and tools used to analyze one-dimensional signals in the time domain. Linear time-invariant (LTI) systems are fully characterized by their impulse response, which connects to the delta function, convolution, and system output in a single conceptual chain.

Subsubsection C.2.1.1 Dirac Delta Distribution and LTI systems

Definition C.2.1. Sifting Property of the Dirac Distribution.

The Dirac delta distribution, denoted \(\delta(t)\text{,}\) is an idealized impulse concentrated at instant in time. Although it is commonly referred to as a function, it is more properly understood as a distribution (generalized function) whose defining property is the sifting property:

\begin{equation*} \int_{-\infty}^{\infty} \delta(t-a) f(a) \, dt = f(a) \, , \end{equation*}

where \(f(t)\) is assumed to be a sufficiently well-behaved function.

Remark C.2.2.

Intuitively, \(\delta(t-a)\) extracts the value of a signal at time \(t=a\text{.}\) The delta distribution as described in Note A.2.1 has a total area,

\begin{equation*} \int_{-\infty}^{\infty} \delta(t-a) \, dt = 1 \, . \end{equation*}

This highlights the fact that the Dirac delta is more appropriately labelled a distribution rather than a function. Specifically, its meaning is determined via its behavior under integration rather than its value at a given point.

Definition C.2.3. Impulse response.

In the context of LTI systems, the Dirac delta plays a fundamental role in the definition of an idealized response of a system to some impulse input. Specifically, if if an LTI system is excited by \(\delta(t)\text{,}\) the resulting output is called the impulse response, \(h(t)\text{.}\)

Remark C.2.4.

Any signal \(x(t)\) can be represented as a continuous superposition of shifted impulses,

\begin{equation*} x(t) = \int_{-\infty}^{\infty} \delta(t-\tau) x(\tau) \, d\tau \, . \end{equation*}

Subsubsection C.2.1.2 Convolution Integral and LTI Systems

Definition C.2.5. Convolution Integral.

The convolution integral,

\begin{equation*} (x * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) \, d\tau \end{equation*}

encapsulates Remark C.2.4 and completely characterizes the input-output behavior of a LTI system.

Subsection C.2.2 Spectral Analysis

This section provides a quick-reference guide to translating mathematical definitions into scientific and engineering terms.

Table C.2.6. Taxonomy of Norms: Mathematics vs. Physical Sciences

Norm Name	Data Type	Notation	Mathematical Formulation	Common Applications
Euclidean norm (equivalent to \(L_2\)-norm)	Vectors	\(\\| \vec{x} \\|_2\)	\(\sqrt{\sum_i^n x_i^2}\)	Distance metrics, linear regression
Frobenius norm	Matrices	\(\\| A \\|_F\)	\(\sqrt{\sum_{i=1}^m \sum_{j=1}^{n}A_{ij}}\)	Matrix magnitude optimization, loss functions, and PCA
Physics Norm	Vectors and Matrices	\mdash	\(\sqrt{A^T W A}\) where \(W\) is a weight matrix	Finite element analysis, fluid dynamics, and control systems

Table C.2.7. Spectral Density and "Energy" Terms

Term	Mathematical Interpretation (Non-Physical Data)	Typical Interpretation (Physical Data in SI Units)
Energy	Total squared variance over time	1 J \(=\) 1 ^kg·m²⁄_s²
Power	Time-averaged variance	1 W \(=\) ^J⁄_s
Density	Quantity distributed over a continuous domain	Quantity per unit of that domain (e.g., ^W⁄_Hz )

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