The impulse response is one of the most important concepts in signals and systems. It represents the output of a system when the input is a unit impulse signal. The impulse response completely characterizes a Linear Time-Invariant (LTI) system, meaning that once the impulse response is known, the output of the system for any input can be determined.
In system analysis, the impulse response is usually denoted by \(h(t)\) for continuous-time systems and \(h[n]\) for discrete-time systems.
Table of Contents
Unit Impulse Signal
The unit impulse signal, also called the Dirac delta function, is represented as \(\delta(t)\).
Important properties:
\[
\delta(t) =
\begin{cases}
\infty, & t = 0 \\
0, & t \ne 0
\end{cases}
\]
\[
\int_{-\infty}^{\infty} \delta(t)\,dt = 1
\]
The impulse signal has unit area and occurs at a single instant.
Definition of Impulse Response
If a system is excited with an impulse input \(\delta(t)\), the output obtained is called the impulse response.
\[
\delta(t) \rightarrow h(t)
\]
Thus,
\[
h(t) = \text{output of the system when input is } \delta(t)
\]
Importance in LTI Systems
For a Linear Time-Invariant system, the impulse response completely determines the system behavior.
The output of an LTI system for any input \(x(t)\) is obtained by convolution of the input with the impulse response.
\[
y(t) = x(t) * h(t)
\]
Expanded form:
\[
y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)\,d\tau
\]
Thus, knowing \(h(t)\) allows us to calculate the output for any input signal.
Impulse Response in Discrete-Time Systems
For discrete-time systems, the impulse response is obtained by applying a unit impulse sequence \(\delta[n]\).
\[
\delta[n] =
\begin{cases}
1, & n = 0 \\
0, & n \ne 0
\end{cases}
\]
The output sequence is
\[
h[n]
\]
The convolution expression becomes
\[
y[n] = \sum_{k=-\infty}^{\infty} x[k]\,h[n-k]
\]
Impulse Response and System Properties
The impulse response helps determine several system properties:
Stability
An LTI system is BIBO stable if
\[
\int_{-\infty}^{\infty} |h(t)|\,dt < \infty
\]
Causality
A system is causal if
\[
h(t) = 0 \quad \text{for } t < 0
\]
Memory
If \(h(t)\) extends beyond a single instant, the system has memory.
Impulse Response and Frequency Response
The frequency response of a system is the Fourier transform of the impulse response.
\[
H(\omega) = \mathcal{F}\{h(t)\}
\]
This relationship connects time-domain analysis and frequency-domain analysis.