The Core Operation
The output of any LTI system is the convolution of its input with its impulse response. This single operation is the foundation of every filter, equalizer, and channel model in DSP.
For an LTI system with impulse response h[n] driven by input x[n], the output is the sum of scaled, shifted copies of h[n] — one for each input sample.
The sifting property decomposes any signal into scaled impulses. Linearity passes the sum through T. Time-invariance converts each T{δ[n−k]} into h[n−k]. The convolution sum is the inevitable result.
Multiply, Sum
To compute y[n] at a single output index: flip h to get h[−k], shift by n to get h[n−k], multiply with x[k] element-wise, then sum all products.
- Commutative: x*h = h*x — order doesn't matter
- Associative: (x*h₁)*h₂ = x*(h₁*h₂) — cascade filters merge into one
- Distributive: x*(h₁+h₂) = x*h₁ + x*h₂ — parallel filters add
The unit impulse is the identity element of convolution: x[n] * δ[n] = x[n]. A shifted impulse δ[n−n₀] is a pure delay.
N-sample signal * M-sample kernel → N+M−1 samples output. Direct cost: O(N·M). FFT cost: O((N+M)log(N+M)).
FIR: finite h[n] — always stable, direct convolution sum, linear-phase achievable.
IIR: infinite h[n] — implemented via difference equations, fewer coefficients for sharp selectivity, requires stability check.
- y[n] = Σ x[k]·h[n−k] — the convolution sum for any LTI system
- Derived from sifting + linearity + time-invariance
- Graphical method: flip, shift, multiply, sum
- Commutative, associative, distributive properties
- δ[n] is the convolution identity; δ[n−n₀] is a pure delay
- Output length = N + M − 1; FFT speeds computation
- FIR = finite h[n]; IIR = infinite h[n] via recursion