Convolution is how a linear system responds to any input at all — flip the system's impulse response, slide it across the input, multiply and sum at each position. Slide it yourself and watch the output build one point at a time.
Signals and systems theory studies how signals (functions of time, or discrete sequences) interact with systems (anything that transforms an input signal into an output signal). Convolution is the single mathematical operation that describes exactly how a linear, time-invariant (LTI) system responds to any input signal, once you know its impulse response — the output it produces from an instantaneous spike input.
For any linear, time-invariant system, knowing exactly how it responds to a single instantaneous impulse tells you, via convolution, exactly how it will respond to any arbitrary input signal — because any signal can be thought of as a sum of scaled, shifted impulses, and the system's response to that sum is the sum of its (scaled, shifted) responses to each individual impulse. This is the superposition property that makes LTI systems so mathematically tractable.
Convolution is computed by flipping one signal (typically the impulse response), sliding it across the other, multiplying overlapping samples at each position, and summing the products — that sum is the output value at that position. Slide fully across both signals and you've computed the complete output signal, one point at a time, exactly as shown in the animation above.
Every digital audio filter, image blur/sharpen operation, and vibration transfer function computation is fundamentally a convolution. In structural and mechanical engineering, a building or machine's response to an arbitrary force input (like ground motion during an earthquake) can be computed by convolving that force with the structure's own impulse response — the same operation, applied to a physical system instead of a digital signal.
Flipping accounts for the fact that a system's output at a given time depends on past input values, weighted by how long ago they occurred — the flip correctly aligns "how long ago" with the impulse response's own time axis as the window slides across the input.
Conceptually yes — continuous-time convolution replaces the sum with an integral, but the same flip/slide/multiply/accumulate logic applies. Discrete-time (sampled) convolution, as shown in the animation, is what digital signal processing systems actually compute.
Linear means scaling or summing inputs scales or sums outputs correspondingly (no distortion from combining signals). Time-invariant means the system behaves the same regardless of when an input is applied — a delayed input just produces the same output, delayed. Convolution as the complete input-output relationship depends on both properties holding.
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