What is the Goertzel algorithm and when is it used?

Goertzel algorithm efficiently computes single DFT bins without full FFT. Uses recursive filter: s[n] = x[n] + 2*cos(2*pi*k/N)*s[n-1] - s[n-2], then X[k] = s[N] - e^(-j2*pi*k/N)*s[N-1]. Complexity: O(N) per bin vs O(N*log(N)) for full FFT. Efficient when computing fewer than log2(N) DFT bins. Applications: DTMF tone detection (detect 8 specific frequencies), frequency monitoring, and narrow-band analysis. Numerical stability good for large N. Second-order section structure fits DSP hardware well. Preferred for real-time detection of specific frequencies.