Characterizing LTI Systems via the Impulse Response

April 11, 2026 · Luciano Muratore

Building on our understanding of signal transforms, we now examine how the fundamental properties of a system—Linearity and Time-Invariance—combine to provide a complete mathematical characterization through a single sequence: the impulse response.

1. Linearity

A system H\mathcal{H} is linear if the principle of superposition holds for any signals x1[n],x2[n]x_1[n], x_2[n] and scalars a,b∈Ra, b \in \mathbb{R}:

H{ax1[n]+bx2[n]}=aH{x1[n]}+bH{x2[n]}\mathcal{H}\{ax_{1}[n]+bx_{2}[n]\}=a\mathcal{H}\{x_{1}[n]\}+b\mathcal{H}\{x_{2}[n]\}

Why Linearity Matters: It allows us to decompose any complex input signal into a sum of scaled impulses: x[n]=∑k=−∞∞x[k]δ[n−k]x[n]=\sum_{k=-\infty}^{\infty}x[k]\delta[n-k] This guarantees that the system’s total response is simply the sum of its responses to each individual impulse.

2. Time Invariance

A system is time-invariant if a delay in the input signal results in an identical delay in the output:

H{x[n−n0]}=y[n−n0], where y[n]=H{x[n]}\mathcal{H}\{x[n-n_{0}]\}=y[n-n_{0}] \text{, where } y[n]=\mathcal{H}\{x[n]\}

Why Time Invariance Matters: It ensures the system reacts consistently regardless of when an input occurs. Mathematically, this means every shifted impulse generates a shifted version of the same fundamental impulse response h[n]h[n]: H{δ[n−k]}=h[n−k]\mathcal{H}\{\delta[n-k]\}=h[n-k]

3. Stability and Causality

To be physically realizable and useful in practical applications, systems must also satisfy stability and causality:

  • BIBO Stability: A system is stable if every bounded input produces a bounded output. This is equivalent to the impulse response being absolutely summable: ∑n=−∞∞∣h[n]∣<∞\sum_{n=-\infty}^{\infty}|h[n]|<\infty
  • Causality: A system is causal if the output at time nn depends only on present and past inputs. For an LTI system, this requires the impulse response to be zero for all negative time: h[n]=0 for n<0h[n]=0 \text{ for } n<0

4. Impulse Response Characterization

The impulse response is formally defined as the system’s output when the input is a unit impulse: h[n]=H{δ[n]}h[n]=\mathcal{H}\{\delta[n]\}. By combining the properties above, we can derive the response to any input x[n]x[n]:

  1. Decomposition: Represent x[n]x[n] as a sum of impulses.
  2. Linearity: Transform the sum of impulses into a sum of individual responses.
  3. Time Invariance: Replace each individual response with a shifted version of h[n]h[n].

5. Convolution Representation

This process leads to the convolution sum, which is the mathematical foundation of LTI system analysis:

y[n]=x[n]∗h[n]=∑k=−∞∞x[k]h[n−k]y[n]=x[n]*h[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]

Conclusion: Under the conditions of linearity, time-invariance, causality, and stability, the impulse response h[n]h[n] uniquely and completely characterizes the system’s behavior for any possible input.