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Minimum Zero-Padding for High-Frequency Peaks


Table: Minimum zero-padding factors $ L_{\hbox {min}}=N_{\hbox {min}}/M$ for keeping peak-frequency bias below approximately $ \Delta $ percent of the sampling-rate divided by the window length [1]. This table is overly conservative for peak-frequencies below 1 kHz. Here, $ N_{\hbox {min}}$ denotes the minimum FFT length, and $ M$ denotes the window length. The zero-padding therefore consists of $ N_{\hbox {min}}-M$ zeros. $ L_{\hbox {min}}$ is calculated using the formulas in [1] and rounded to two significant digits.
Window Type $ \mathbf{\Delta}$ (%) $ \mathbf{L_{\hbox{min}}}$
Rectangular $ 1$ 2.1
Gen. Hamming $ 1$ 1.2
Blackman $ 1$ $ 1.0$
Rectangular $ 0.1$ 4.1
Gen. Hamming $ 0.1$ 2.4
Blackman $ 0.1$ $ 1.8$


Table 5.3 gives zero-padding factors sufficient for keeping the bias below $ 0.01\cdot\Delta\cdot f_s/M$ Hz, where $ f_s$ denotes the sampling rate in Hz, and $ M$ is the window length in samples. For fundamental frequency estimation, $ \Delta $ can be interpreted as the relative frequency error `` $ \Delta f/f$ '' when the window length is one period. In this case, $ f_s/M$ is the fundamental frequency in Hz. More generally, $ f_s/M$ is the bandwidth of each side-lobe in the DTFT of a length $ M$ 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 $ 1$ % accuracy is required in peak-frequency measurement. It should also be understood that a frequency error of $ 0.1$ % is inaudible in most audio applications.6.10


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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