Discrete Convolution
Output of a filter is the convolution of input and impulse response.
Formula
Variables
Example
Every FIR filter computes this weighted sliding sum.
How it works
Discrete convolution tells you how a linear time-invariant system combines a past input sequence with its impulse response to produce the current output. Each output sample is formed by reversing and shifting h[n], multiplying it point-by-point with x[k], and summing the overlap; this works because any input can be decomposed into shifted impulses, and the system’s response to each impulse adds linearly. The formula captures both memory and delay, which is why it is the core operation behind digital filters and many DSP algorithms.
When to use it
Use this whenever you need the output of an LTI discrete-time system from a known input and impulse response, especially for FIR/IIR filter analysis, smoothing, echo simulation, or system modeling. It also appears when combining signals with kernels in image and audio processing.
Worked example, step by step
Let x[n] = {x[0]=2, x[1]=-1, x[2]=3} and h[n] = {h[0]=1, h[1]=4, h[2]=-2}. We compute y[n] = Σ_k x[k] h[n-k] for n = 0 to 4. n = 0: y[0] = x[0]h[0] = 2·1 = 2 n = 1: y[1] = x[0]h[1] + x[1]h[0] = 2·4 + (-1)·1 = 8 - 1 = 7 n = 2: y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = 2·(-2) + (-1)·4 + 3·1 = -4 - 4 + 3 = -5 n = 3: y[3] = x[1]h[2] + x[2]h[1] = (-1)·(-2) + 3·4 = 2 + 12 = 14 n = 4: y[4] = x[2]h[2] = 3·(-2) = -6 Final output sequence: y[n] = {2, 7, -5, 14, -6}.
Common mistakes to avoid
- Forgetting that h[n-k] means the impulse response is time-reversed before shifting
- Using only overlapping indices incorrectly and missing edge terms at the start and end of the output
- Confusing convolution with correlation, which does not reverse the second sequence
Did You Know?
Convolution reverb captures a real space's impulse response, then convolves it with your track to place it 'inside' that room.
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