Questions: Dirac Delta Function and Impulse Response

Question 1

Why is the Dirac delta function called a 'distribution' rather than a function?

• A. Because it distributes its value evenly across an interval rather than concentrating it at a single point
• B. Because it is defined by its behavior inside integrals (the sifting property) rather than by assigning a value at each point
• C. Because it is used in probability distributions to model rare events
• D. Because Dirac originally derived it from probability theory using the normal distribution as a limit

Answer: (B)

An ordinary function is defined by assigning a value to each point in its domain. The Dirac delta cannot be defined this way — it would need to be zero everywhere except at one point, yet integrate to 1, which is impossible for any standard function. Instead, δ(t) is defined by its sifting property: ∫f(t)δ(t−a)dt = f(a) for any continuous f. It is defined by what it does inside integrals, not by pointwise values. This is the definition of a distribution — a functional that maps test functions to numbers. The 'infinitely tall, infinitely narrow spike' description is a useful intuition, but the formal definition is the integral behavior.

Question 2

A linear system has impulse response h(t) = e^{−2t}u(t). An engineer needs to find the output when the input is f(t) = sin(t)·u(t). What is the most efficient approach?

• A. Re-solve the differential equation from scratch with f(t) = sin(t) on the right-hand side
• B. Approximate sin(t) as a sum of delta functions and solve for each one separately
• C. Compute the convolution (h * f)(t) — since h(t) is the impulse response, convolving it with any input gives the system output
• D. Take the Laplace transform of sin(t) and substitute it into the original ODE in the s-domain without using h(t)

Answer: (C)

The impulse response h(t) is a complete fingerprint of the system's linear dynamics. The convolution theorem guarantees that the output for any input f(t) is simply (h * f)(t) — no need to re-solve the ODE. This is the central payoff of impulse response theory: solve the system once (with δ(t) as input), and you have the tool to handle any input through convolution. The Laplace domain version is equally elegant: Y(s) = H(s)·F(s), where H(s) = L[h(t)] is the transfer function. The impulse response and the ODE are equivalent characterizations of the system.

Question 3

The Dirac delta function δ(t) has the property that ∫_{-∞}^∞ δ(t) dt = 0, because it is zero almost everywhere.

• T. True
• F. False

Answer: False

This is a common and important misconception. The integral of δ(t) over all of ℝ is 1, not 0. This is one of its defining properties. The intuition is that δ(t) concentrates a total 'mass' of 1 at a single point — like an infinitely tall, infinitely narrow spike with unit area. Although it is zero everywhere except at t = 0, the integral captures the total 'mass,' which is 1. This is precisely what makes it the identity element for convolution: f * δ = f. If the integral were 0, the delta would have no effect when convolved with any function.

Question 4

In the Laplace domain, L[δ(t)] = 1, which means that the transfer function H(s) of a system equals the Laplace transform of its impulse response.

• T. True
• F. False

Answer: True

This is correct and fundamental. Since L[δ(t)] = 1, the Laplace transform of the ODE forced by δ(t) has right-hand side 1. Solving gives the transfer function H(s) in the s-domain. Inverting yields h(t), the impulse response. The relationship L[h(t)] = H(s) means the system's frequency-domain behavior and its time-domain response to a unit impulse are Laplace transform pairs. Once you have H(s), the response to any input F(s) is Y(s) = H(s)·F(s) — multiplication in the s-domain corresponds to convolution in the time domain.

Question 5

Explain why the impulse response h(t) is described as a 'fingerprint' of a linear system, and how it is used to find the response to an arbitrary input.

Model answer: The impulse response h(t) completely characterizes a linear time-invariant system because it encodes how the system responds to the most elementary possible input — an instantaneous unit impulse. Any other input f(t) can be thought of as a superposition of scaled, shifted delta functions. By linearity and time-invariance, the system's response to each shifted delta δ(t−τ) is a shifted copy h(t−τ), scaled by f(τ). Integrating over all τ gives the total response: y(t) = ∫f(τ)h(t−τ)dτ = (f * h)(t). So knowing h(t) means never needing to re-solve the ODE for a new input — convolution with h gives the answer directly.

The delta function makes this possible because L[δ(t)] = 1: forcing the system with δ(t) in the Laplace domain gives Y(s) = H(s)·1 = H(s), so the impulse response IS the transfer function (under inverse Laplace). This is why solving the system once with δ(t) as input captures all information about the system's dynamics. The convolution theorem then provides the mechanism: the time-domain convolution (h * f)(t) corresponds to the product H(s)·F(s) in the s-domain, making computation tractable. The impulse response bridges delta-function theory, the convolution theorem, and practical systems analysis.