Impulse Response & Convolution

The impulse response h[n] of a discrete-time system is the output the system produces when the input is a unit impulse δ[n] — a signal that is 1 at n = 0 and 0 everywhere else. We write this as: if x[n] = δ[n], then y[n] = h[n]. At first glance, feeding a system a single spike and recording its response might seem like a narrow experiment. But for an LTI system, the impulse response is extraordinarily powerful: it completely characterizes the system.

Given h[n] and any input x[n], we can compute the exact output y[n] — no other information about the system's internal structure is needed. This is the central result of LTI systems theory, and it rests on two ideas: any signal can be decomposed into shifted impulses, and linearity plus time-invariance lets us reassemble the response from the pieces.

The unit impulse δ[n] has a remarkable property called sifting: for any signal x[n], we can write x[n] = Σ_k x[k]·δ[n−k]. This identity says that any discrete-time signal is a weighted sum of shifted unit impulses. The weight on the impulse at position k is simply x[k], the signal's value at that index.

Think of it this way: the signal x[n] = {3, 1, 2} at n = 0, 1, 2 is the same as 3·δ[n] + 1·δ[n−1] + 2·δ[n−2]. Each shifted impulse δ[n−k] is a spike at position k, and scaling it by x[k] places the correct amplitude there. This decomposition is exact — it is not an approximation — and it works for any signal, finite or infinite.

Now apply the system T to the decomposed input. The input is a sum of weighted, shifted impulses: x[n] = Σ_k x[k]·δ[n−k]. By linearity, the response to a sum is the sum of the responses, so we can find the system's response to each shifted impulse individually and add them up. By time-invariance, the response to δ[n−k] — a shifted impulse — is just h[n−k], a shifted copy of the impulse response. The weight x[k] scales through by homogeneity.

Putting it all together: the response to x[k]·δ[n−k] is x[k]·h[n−k], and summing over all k gives the convolution formula.

For a linear but time-varying system, the response to δ[n−k] is not h[n−k] but some h[n, k] that depends on both the output time and the shift — a two-variable function, far less convenient. For a time-invariant but nonlinear system, the response to a weighted impulse is not the weighted impulse response, because scaling fails. Convolution, in essence, is the price of admission for LTI analysis: it tells you exactly how to compute the output, and it works because the two properties conspire to make the problem tractable.

In practice, convolution is computed by a sliding overlap-and-add procedure: for each output index n, you flip h, shift it to align with the input at position n, multiply overlapping samples, and sum. The worked example below makes this concrete.