Minimum Zero-Padding for High-Frequency Peaks

Table: Minimum zero-padding factors for keeping peak-frequency bias below approximately percent of the sampling-rate divided by the window length [1]. This table is overly conservative for peak-frequencies below 1 kHz. Here, denotes the minimum FFT length, and denotes the window length. The zero-padding therefore consists of zeros. is calculated using the formulas in [1] and rounded to two significant digits.

Window Type (%) Rectangular 2.1 Gen. Hamming 1.2 Blackman Rectangular 4.1 Gen. Hamming 2.4 Blackman

Table 5.3 gives zero-padding factors sufficient for keeping the bias below Hz, where denotes the sampling rate in Hz, and is the window length in samples. For fundamental frequency estimation, can be interpreted as the relative frequency error `` '' when the window length is one period. In this case, is the fundamental frequency in Hz. More generally, is the bandwidth of each side-lobe in the DTFT of a length rectangular, generalized Hamming, or Blackman window (any member of the Blackman-Harris window family, as elaborated in Chapter 3).

Note from Table 5.3 that the Blackman window requires no zero-padding at all when only % accuracy is required in peak-frequency measurement. It should also be understood that a frequency error of % is inaudible in most audio applications.^6.10

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