Single-frequency detection without an FFT — bin, coefficient, bandwidth and the C routine.

Why Goertzel

When you only care about one or a few frequencies (DTMF digits, a pilot tone, a carrier check), running a full FFT wastes cycles and RAM. The Goertzel algorithm computes a single DFT bin with one multiply and two adds per sample, using one precomputed coefficient:

c = 2 cos (\frac{2 π k}{N}), k = round (\frac{N f _{t}}{f _{s}})

The block length N sets the detection bandwidth $f_{s} / N$ — longer blocks discriminate closer tones but respond more slowly, and the actual detected frequency snaps to the bin center $k f_{s} / N$ , not exactly to your target. The classic DTMF receiver uses $f_{s} = 8 kHz, N = 205$ precisely because all eight tone frequencies then land within ±0.7% of a bin center.

Goertzel Calculator

Why Goertzel