Unit Impulse Function

The unit impulse function, also called the Dirac delta function, is arguably the most important singularity function in signals and systems theory. It represents an idealized signal of zero duration but unit area, and its ability to sample a signal at a single point in time, known as the sifting property, makes it indispensable for defining system impulse responses, sampling theory, and Fourier analysis. Understanding it properly removes confusion that many students carry into advanced topics.

Core Concept Explanation

The unit impulse function δ(t) cannot be defined in the ordinary sense as a function that assigns a specific numerical value at every point. At t = 0, it is informally thought of as having infinite amplitude, and it equals zero everywhere else. However, what is precisely defined and physically meaningful is its area, which always equals 1. This is why δ(t) is formally classified as a generalized function or distribution, defined not by its pointwise values but by its behavior inside an integral.

One helpful way to understand δ(t) is through a limiting process. Consider a rectangular pulse of width ε and height 1/ε, centered at t = 0. The area of this rectangle is always exactly 1, regardless of the value of ε. As ε is taken to zero, the width shrinks to zero and the height grows without bound, but the area is preserved at unity. In the limit, this becomes δ(t): a function that is zero everywhere except at t = 0 and has unit area concentrated at that single point.

The discrete time counterpart is the unit impulse sequence δ[n], which equals 1 when n = 0 and equals 0 for all other integer values of n. Unlike the continuous delta function, δ[n] is an ordinary sequence with a perfectly well-defined numerical value at every index. This distinction is important: the continuous δ(t) requires distributional mathematics, while δ[n] is entirely elementary.

Mathematical Expression

The defining property of δ(t) is: it equals zero for all t not equal to 0, and its integral over any interval containing t = 0 equals 1. Formally, the integral from minus infinity to plus infinity of δ(t) dt = 1. The most important derived result is the sifting property: the integral from minus infinity to plus infinity of x(t) δ(t - t₀) dt equals x(t₀), for any continuous function x(t). This says that multiplying any signal by a shifted impulse δ(t - t₀) and integrating picks out the value of that signal precisely at t = t₀.

Other important properties include: δ(at) = δ(t) / |a| for scaling (the impulse narrows but must maintain unit area when the time axis is compressed, requiring amplitude compensation), δ(-t) = δ(t) (even function symmetry), and x(t) δ(t - t₀) = x(t₀) δ(t - t₀) (product property, since δ is nonzero only at t₀, the product collapses x to its value at t₀). The Laplace transform of δ(t) is 1 for all s, and the Fourier transform of δ(t) is also 1 for all frequencies. This flat spectrum interpretation means that an impulse contains all frequencies in equal measure.

Practical Understanding

The impulse response h(t) of a linear time-invariant system is the output when the input is δ(t). This single function completely characterizes the system's behavior because any input x(t) can be written as a superposition of scaled and shifted impulses, and the output is then the corresponding superposition of scaled and shifted impulse responses. This is expressed mathematically as convolution: y(t) = integral of x(τ) h(t - τ) dτ, which is the cornerstone of LTI system analysis.

In sampling theory, the process of ideal sampling is modeled as multiplication of a continuous time signal x(t) by an impulse train, which is an infinite sum of equally spaced impulses separated by the sampling period Ts. Each impulse in the train uses the sifting property to extract the signal value at that sampling instant. The resulting sampled signal is then a weighted impulse train, and the Fourier transform of this train gives the periodic spectrum that leads to the Nyquist sampling theorem.

Given:
Signal x(t) = t² + 3t + 5
Evaluate: I = ∫ x(t) δ(t - 2) dt from -∞ to +∞

Why this formula applies:
The sifting property: ∫ x(t) δ(t - t₀) dt = x(t₀)
Here t₀ = 2, so the impulse at t=2 extracts x(2).

Formula:
∫ x(t) δ(t - t₀) dt = x(t₀)

Substitution:
I = x(t) evaluated at t = 2
I = (2)² + 3(2) + 5

Calculation:
I = 4 + 6 + 5
I = 15

Final Answer:
I = 15 (dimensionless, assuming x(t) is in volts and δ(t) in 1/seconds, result is in volt-seconds)

Exam Tip: For GATE, the sifting property ∫ x(t) δ(t - t₀) dt = x(t₀) is tested directly and indirectly in almost every year. Watch for the scaling property: δ(at) = (1/|a|) δ(t). A common trap is δ(2t - 4) = δ(2(t-2)) = (1/2) δ(t-2). Always factor out the coefficient from inside the argument first, then apply the scaling rule. Also remember that the Fourier transform of δ(t - t₀) = e^(-jωt₀), not just 1.

Quick Revision

• Definition: δ(t) = 0 for t ≠ 0, and ∫δ(t) dt = 1. It is zero everywhere except at the origin, where its area is exactly 1.
• Sifting property: ∫ x(t) δ(t - t₀) dt = x(t₀). The most important property for GATE calculations. Always memorize this.
• Scaling property: δ(at) = (1/|a|) δ(t). For δ(at - b) = (1/|a|) δ(t - b/a). Factor out 'a' first, then identify t₀ = b/a.
• Transforms: L{δ(t)} = 1 (all s), L{δ(t - t₀)} = e^(-st₀). F{δ(t)} = 1 (flat spectrum), F{δ(t - t₀)} = e^(-jωt₀).
• Relation to unit step: d/dt [u(t)] = δ(t) and u(t) = ∫δ(τ) dτ from -∞ to t. Impulse is the derivative of step.
• Impulse response: When input to an LTI system is δ(t), the output is h(t), the impulse response. All outputs are given by convolution y(t) = x(t) * h(t).
• GATE trap: x(t) δ(t - t₀) = x(t₀) δ(t - t₀). The product collapses x(t) to a constant value x(t₀) multiplied by the impulse. Do not leave x(t) inside the product when simplifying.