The output of an LTI system is the input convolved with its impulse response.
y(t)=∫x(τ)h(t-τ)dτ (continuous) or y[n]=Σx[k]h[n-k]. Procedure: flip, shift, multiply, integrate. Properties: commutative, associative, distributive; Fourier transform turns convolution into multiplication. Used for filtering, smoothing and image blur.
Convolving a signal with a 5-point average smooths it.
Convolution gives the output of an LTI system for any input, so it is the central calculation in filtering and system analysis.
The output of an LTI system is the input convolved with its impulse response.
Convolution gives the output of an LTI system for any input, so it is the central calculation in filtering and system analysis.
The output at each time is the sum or integral of the input weighted by the impulse response shifted to that time.
A blur or sharpen filter is a two-dimensional convolution of the image with a small kernel.
Typical applications include computing the output of a digital filter for a long audio signal, predicting the response of a structure to a moving load and smoothing a noisy signal with a moving average filter.
Study "Impulse response" first (topic 11 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
Length of result N+M-1. FFT-based fast convolution. Correlation similar.
Convolving a signal with a 5-point average smooths it.
The next topic is "Differential equation models", which builds directly on this one.
It is classed as Beginner in Signals, Control & Comms (topic 12 of 101). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
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