Subsubsection Dirac Delta Distribution and LTI systems

Definition B.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 B.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.3.2 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 B.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 B.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*}

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