Minimum Zero-Padding for Low-Frequency Peaks

Sharper bounds on the zero-padding factor needed for low-frequency peaks (below roughly 1 kHz) may be obtained based on the measured Just-Noticeable-Difference (JND) in frequency and/or amplitude [276]. In particular, a $ 0.1$ % relative-error spec is good above 1 kHz (being conservative by approximately a factor of 2), but overly conservative at lower frequencies where the JND flattens out. Below 1 kHz, a fixed 1 Hz spec satisfies perceptual requirements and gives smaller minimum zero-padding factors than the $ 0.1$ % relative-error spec.

The following data, extracted from [276, Table I, p. 89] gives frequency JNDs at a presentation level of 60 dB SPL (the most sensitive case measured):

  f =    [    62,    125,    250,    500,   1000,   2000,   4000];
  dfof = [0.0346, 0.0269, 0.0098, 0.0035, 0.0034, 0.0018, 0.0020];
Thus, the frequency JND at 4 kHz was measured to be two tenths of a percent. (These measurements were made by averaging experimental results for five men between the ages of 20 and 30.) Converting relative frequency to absolute frequency in Hz yields (in matlab syntax):
  df = dfof .* f; % = [2.15, 3.36, 2.45, 1.75, 3.40, 3.60, 8.00];
For purposes of computing the minimum zero-padding factor required, we see that the absolute tuning error due to bias can be limited to 1 Hz, based on measurements at 500 Hz (at 60 dB). Doing this for frequencies below 1 kHz yields the results shown in Table 5.4. Note that the Blackman window needs no zero padding below 125 Hz, and the Hamming/Hann window requires no zero padding below 62.5 Hz.


Table: Minimum zero-padding factors $ L_{\hbox {min}}=N_{\hbox {min}}/M$ for keeping peak-frequency bias below approximately 1 Hz (well under 1.75 Hz), assuming the window length $ M$ to span one period of the fundamental frequency.
Window Type $ f$ (Hz) $ \mathbf{L_{\hbox{min}}}$
Rectangular 1000 4.1
  500 3.3
  250 2.6
  125 2.1
  62.5 1.7
Gen. Hamming 1000 2.4
  500 1.9
  250 1.5
  125 1.2
  62.5 1
Blackman 1000 1.8
  500 1.5
  250 1.2
  125 1
  62.5 1



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Matlab for Computing Minimum Zero-Padding Factors
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Minimum Zero-Padding for High-Frequency Peaks