Impulse Response

Introduction

The impulse response is the single most important concept in LTI system theory. It is the output of a system when the input is an impulse function $\delta(t)$ (or $\delta[n]$ in discrete-time). What makes this concept so powerful is the extraordinary fact that for any LTI system, knowing the impulse response tells you EVERYTHING about the system — you can compute the output for any possible input using convolution, determine the frequency response, assess stability, and check causality.

Think of it as a system\'s DNA or fingerprint — a compact description that encodes the system\'s complete behavior. No other single function carries this much information about a system.

Definition

The impulse response $h(t)$ is the output of an LTI system when the input is the unit impulse $\delta(t)$:

$$h(t) = T\{\delta(t)\}$$

For discrete-time: $$h[n] = T\{\delta[n]\}$$

Why It Characterizes the Entire System

The key insight is the combination of two facts:

Fact 1 (Impulse Decomposition): Any signal can be written as a sum of weighted, shifted impulses: $$x(t) = \int_{-\infty}^{\infty} x(\tau)\delta(t-\tau)d\tau$$

Fact 2 (LTI Properties): For an LTI system:

• Linearity: response to a weighted sum = weighted sum of responses
• Time-invariance: response to $\delta(t-\tau)$ is $h(t-\tau)$

Combining these: $y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau = x(t) * h(t)$

The output to ANY input is the convolution of that input with the impulse response.

Relationship to Transfer Function

The transfer function is simply the Fourier Transform (or Laplace/Z-Transform) of the impulse response:

$$H(j\omega) = \mathcal{F}\{h(t)\} = \int_{-\infty}^{\infty}h(t)e^{-j\omega t}dt$$

$$H(s) = \mathcal{L}\{h(t)\} \qquad H(z) = \mathcal{Z}\{h[n]\}$$

The impulse response and transfer function contain identical information — they are time-domain and frequency-domain views of the same system.

System Properties from Impulse Response

Examples

First-Order Continuous System

For the system $\frac{dy}{dt} + ay = x(t)$ with $a > 0$: $$h(t) = e^{-at}u(t)$$

This decays exponentially — stable (integrable) and causal ($h=0$ for $t<0$).

First-Order Discrete System

For $y[n] - 0.8y[n-1] = x[n]$: $$h[n] = (0.8)^n u[n]$$

Decays geometrically — stable ($\sum(0.8)^n = 5 < \infty$) and causal.

Ideal Delay

For $y(t) = x(t - t_0)$: $$h(t) = \delta(t - t_0)$$

The impulse response is a delayed impulse — the simplest dynamic system.

Moving Average Filter

For $y[n] = \frac{1}{M}\sum_{k=0}^{M-1}x[n-k]$: $$h[n] = \frac{1}{M}[\delta[n] + \delta[n-1] + \ldots + \delta[n-M+1]]$$

A rectangular pulse of height $1/M$ and length $M$ — finite impulse response (FIR).

Measuring the Impulse Response

In practice, you cannot apply a true Dirac delta (infinite amplitude, zero width). Instead:

Short pulse approximation: Apply a very short, high-amplitude pulse. If the pulse duration is much shorter than the system\'s response time, the output approximates $h(t)$.

Cross-correlation with white noise: Excite the system with white noise $w(t)$ and cross-correlate the output with the input: $R_{wy}(\tau) \approx h(\tau) \cdot P_w$ (scaled impulse response).

Swept sine (chirp) method: Apply a frequency sweep and deconvolve to obtain $h(t)$. Common in acoustics for measuring room impulse responses.

Maximum length sequences (MLS): Binary pseudo-random sequences with ideal autocorrelation properties. Widely used in audio system measurement.

Physical Interpretation

The impulse response shows how a system "rings" after being excited by a sudden stimulus:

• A fast-decaying $h(t)$ indicates a wideband system (responds quickly)
• A slow-decaying $h(t)$ indicates a narrowband system (rings for a long time)
• Oscillation in $h(t)$ indicates resonant behavior (peaked frequency response)
• Duration of $h(t)$ relates to system memory (how long past inputs affect current output)

Key Takeaways

• Impulse response $h(t) = T\{\delta(t)\}$ completely characterizes any LTI system
• Output for any input is computed via convolution: $y = h * x$
• Transfer function is the Fourier/Laplace/Z-Transform of the impulse response
• Causality requires $h(t) = 0$ for $t < 0$; stability requires absolute integrability
• The impulse response reveals the system\'s memory length, bandwidth, and resonant behavior
• Practical measurement uses short pulses, noise correlation, or swept-sine techniques