9.10 Summary

• A filter is any function that maps an input signal to an output signal. We focused on linear time-invariant (LTI) filters, which sculpt a sound in the frequency domain without adding new frequencies.

• A difference equation defines a filter by writing each output sample as a weighted sum of past input (and possibly output) samples. It is trivial to implement but hard to predict by inspection.

• Convolution, \(y = h * x = \sum_k h[k]\,x[n-k]\), generalizes the (non-recursive) difference equation. It is commutative, associative, and distributive, and produces an output of length \(N + K - 1\).

• The convolution theorem states that convolution in time equals multiplication in frequency, \(Y[k] = H[k] \cdot X[k]\). This is how filters sculpt the spectrum, and (via the FFT) it lets us convolve in \(O(N \log N)\) instead of \(O(N^2)\).

• The impulse response \(h\) is a filter’s output to a unit impulse. It fully characterizes an LTI filter, and every LTI filter is a convolution with its impulse response.

• Recursive filters feed past outputs back into the sum. They are efficient but can have an infinite impulse response and can be unstable.

• Filters are categorized by response shape (low-pass, high-pass, band-pass, band-stop), and a filter’s frequency response can be measured empirically by probing it with sinusoids.

• Subtractive synthesis uses filters as a synthesis tool, carving a timbre out of a harmonically rich source.

• Filtering can be viewed in many equivalent forms: a difference equation, a convolution, an impulse response, a signal-flow diagram, or a multiplication in the frequency domain.