Impulse Response

The impulse response h(t) of an LTI system is the output produced when the input is a unit impulse signal delta(t). It completely characterizes the behavior of the system, meaning if you know h(t), you know the response of the system to any arbitrary input.

This concept is foundational in signals and systems because every practical signal can be decomposed into a weighted sum of shifted impulses. The system's response to each of those impulses, superimposed together, gives the total output. This is the physical basis of convolution.

Core Concept Explanation

An LTI system has two key properties: linearity (superposition holds) and time-invariance (a time-shifted input produces an identically time-shifted output). When both properties hold together, knowing the response to one specific input, namely the impulse, is enough to determine the response to any input.

The impulse delta(t) has zero duration but unit area. When passed through a system, it probes every frequency simultaneously. The resulting output h(t) encodes how the system amplifies, delays, or filters each frequency component. If the system is stable and causal, h(t) = 0 for t < 0 and it decays over time.

For a discrete-time LTI system, the equivalent is the impulse response h[n], obtained by exciting the system with the unit sample sequence delta[n]. All discrete LTI analysis follows the same logic as the continuous case.

Mathematical Expression

Mathematically, the output of an LTI system for any input x(t) is expressed as the convolution of x(t) with h(t):

y(t) = x(t) * h(t) = integral from -inf to +inf of x(tau) h(t - tau) d(tau)

This equation is the convolution integral. It says the output at time t is a weighted superposition of all past inputs, weighted by the impulse response. The impulse response acts as the memory of the system.

For a causal LTI system, h(t) = 0 for t < 0, so the integral lower limit effectively becomes 0, and only past inputs contribute to the current output. For a system described by a differential equation, h(t) can be found by solving the equation with x(t) = delta(t).

Practical Understanding

In practice, engineers use h(t) to predict system behavior without solving the differential equation repeatedly for each new input. For example, in filter design, h(t) determines whether the filter is low-pass, high-pass, or band-pass in nature. For a first-order RC circuit, h(t) is an exponential decay of the form (1/RC) e^(-t/RC) u(t).

In GATE problems, impulse responses are used to test stability (BIBO stability requires h(t) to be absolutely integrable), causality (h(t) = 0 for t < 0), and to compute convolution-based output signals.

Numerical Example

Consider an LTI system described by the differential equation dy/dt + 2y(t) = x(t). Find h(t) by applying the impulse input x(t) = delta(t) and solving for y(t) = h(t).

Given:
Differential equation: dy/dt + 2y(t) = x(t)
Input x(t) = delta(t)

Why this formula applies:
For an LTI system, h(t) is the zero-state response to delta(t).

Formula:
For t > 0 (impulse has passed), the system is: dy/dt + 2y = 0
General solution: y(t) = A e^(-2t)

Substitution:
At t = 0+, integrating the equation over [0-, 0+] gives:
y(0+) - y(0-) = 1 (from the delta function)
Since y(0-) = 0 (zero initial conditions):
y(0+) = 1, so A = 1

Calculation:
h(t) = e^(-2t) u(t)

Final Answer with units:
h(t) = e^(-2t) u(t)  [dimensionless, t in seconds]

Exam Tip: For GATE, if h(t) is absolutely integrable (integral of |h(t)|dt from -inf to +inf is finite), the system is BIBO stable. For h(t) = e^(-at)u(t) with a > 0, the integral = 1/a, which is finite, confirming stability.

Quick Revision

• h(t) is the output of an LTI system when input is delta(t).
• It completely characterizes an LTI system.
• Output for any input: y(t) = x(t) * h(t) (convolution integral).
• Causality: h(t) = 0 for t < 0.
• BIBO Stability: integral of |h(t)|dt must be finite.
• For first-order system dy/dt + a*y = x(t), h(t) = e^(-at) u(t).
• GATE trap: A non-causal h(t) does not mean the system is unstable. Causality and stability are independent properties.