Summary

This blog post shows how the Goertzel algorithm can be adapted to compute DFT results when the target frequency index is non-integer. It explains the required filter coefficient adjustments, practical implementation options, and the tradeoffs versus FFT-based and interpolation approaches for fractional-bin spectral analysis.

Key Takeaways

• Compute the modified Goertzel coefficients using the fractional index (k fractional) via cos(2π·k/N) and the corresponding complex rotation to target non-integer frequencies.
• Apply windowing or bin-interpolation techniques (e.g., quadratic/parabolic interpolation) to reduce spectral leakage and improve amplitude/phase estimates for fractional bins.
• Implement the fractional-Goertzel using complex mixing (heterodyning) or an equivalent second-order filter to maintain numerical stability and real-time performance.
• Compare computational cost and accuracy versus zero-padding/FFT methods and choose Goertzel when you need sparse, single-tone detection with low overhead.
• Validate amplitude and phase by compensating for the offset from the nearest integer bin and account for finite-sample bias in estimation.

Who Should Read This

DSP engineers and developers working on audio, speech, radar, or communications systems who need efficient single-tone or sparse-frequency detection and want to handle fractional-frequency targets.

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